Scanners

This is continuation from the previous tutorial - Reflective and catadioptric objectives

 

 

1. INTRODUCTION

This chapter provides an overview of optical scanning techniques in context with their operational requirements. System objectives determine the characteristics of the scanner which, in turn, influence adjacent system elements. For example, the desired resolution, format, and data rate determine the scanner aperture size, scan angle, and speed, which then influence the associated optics.

The purpose of this chapter is to review the diverse options for optical scanning and to provide insight to associated topics , such as scanned resolution and the reduction of spatial errors. This broad perspective is, however, limited to those factors which bear directly on the scanner. Referencing is provided for related system relationships, such as image processing and data display. Topics are introduced with brief expressions of the fundamentals. And, where appropriate, historical and technical origins are referenced.

The subject of scanning is often viewed quite differently in two communities. One is classified as remote sensing and the other, input/output scanning. Associated component nomenclature and jargon are, in many cases, different.

While their characteristics are expanded in subsequent sections, it is useful to introduce some of their distinctions here. Remote sensing detects objects from a distance, as by a space-borne observation platform. An example is infrared imaging of terrain. Sensing is usually passive and the radiation incoherent and often multispectral. Input / output scanning, on the other hand, is local.

A familiar example is document reading (input) or writing (output). Intensive use of the laser makes the scanning active and the radiation coherent. The scanned point is focused via finite-conjugate optics from a local fixed source.

While the scanning components may appear interchangeable, special characteristics and operational modes often preclude this option. This is most apparent for diffractive devices such as acousto-optic and holographic deflectors. It is not so apparent regarding the differently filled scanning apertures, imparting important distinctions in resolution and duty cycle.

The unification of some of the historically separated parameters and nomenclature is considered an opportunity for this writing.

 

System Classifications

The following sections introduce the two principal disciplines of optical scanning, remote sensing and input/output scanning, in preparation for discussion of their characteristics and techniques. 

Remote Sensing. The applications for passive (noninvasive) remote sensing scanners are varied and cover many important aspects of our lives. A signature representative of the target is obtained to form a signal for subsequent recording or display. This process is operationally distinct from active scanning, as expressed further in this chapter. Table 1

 

 

lists typical applications of these techniques. Clearly, remote scanning sensors can be hand-held to satellite-borne. 

A variety of scanning methods has been developed to accomplish the formation of image (or imagelike) data for remote sensing. These methods may be roughly divided into framing, pushbroom, and mechanical. Generally stated, frame scanning requires no physical scan motion and implies that the sensor has a two-dimensional array of detectors which are read out by use of electronic means (e. g., \(\text{CCD}\)), electron beam, or light beam. Such an array requires an optical system that has 2-\(\text{D}\) wide-angle capability. Pushbroom methods typically employ some external means to move the image of a linear array of detectors along the area to be imaged.

Mechanical methods generally include one- and two-dimensional scanning techniques incorporating as few as one detector to multipledetector arrays. As is the case for pushbroom methods, image formation by onedimensional mechanical scanning requires that the platform containing the sensor (or in some cases the object) be moved to create the second dimension of the image. The latter two methods are discussed further in later sections of this tutorial. 

Input/Output Scanning. In contrast to remote sensing, which captures passive radiation, active input/output scanning illuminates an object or medium with a ‘‘flying spot,’’ derived typically from a laser source. Some examples appear in Table 2, divided into two principal

 

TABLE 2 Examples of Input/Output Scanning

functions: input (detecting radiation scattered from the scanning spot) and output (recording or display). Input is modulated by the target to form a signal; output is modulated by a signal.

Some merit clarification. Under input is laser radar—a special case of active remote sensing, using the same coherent and flying-spot scanning disciplines as the balance of those exemplified. Earth resources imaging is the recording of remotely sensed image signals.

Finally, data/image display denotes the general presentation of information, which could include ‘‘hard copy’’ and/or actively projected and displayed images.

Active Scanning is synonymous with flying-spot scanning , the discipline most identified with the ubiquitous cathode-ray tube \(\text{(CRT)}\). While the utilized devices and their performance differ significantly, the distinctions between \(\text{CRT}\) and laser radiation are primarily their degrees of monochromaticity and coherence, as addressed later in this tutorial.

Thus, most high-resolution and high-speed flying-spot scanning are now conducted using the laser as a light source. This work in input/output scanning concentrates on the control of laser radiation and the unique challenges encountered in deflecting photons, devoid as they are of the electric and magnetic fields accompanying the electron beam.  

Scanner Classification. Following the nomenclature introduced in the early ’70s, laser scanners are designated as preobjective, objective, and postobjective. Figure 1 indicates the scan regions within a general conjugate optical transfer of a fixed reference (object) point \(P_o\) to a moving focal (image) point \(P_i\). The component which provides principal focusing of the wavefront identifies the objective lens.

The scanner can perform two functions (see ‘‘Objective , Preobjective, and Postobjective Scanning’’ later in this tutorial): one is translation of the aperture with respect to the information medium.

This includes translation of the lens element(s) or translation of the object, or both, and is identified as an objective scan. The other is angular change of the optical beam with respect to the information medium. Angular scanners are exemplified by plane mirrors on rotating substrates.

Although lenses can be added to an angular scanner, it is seldom so configured. The scanner is either preobjective or postobjective. In holographic scanning, however, the hologram can serve as an objective lens and scanner simultaneously.

Radial Symmetry and Scan Magnification. A basic characteristic of some angular scanners is identified as radial symmetry. When an illuminating beam converges to or diverges from the nodal or rotating axis of an angular scanner, it is said to exhibit radial symmetry.

 

 

 

The collimated beam which is parallel to the rotating axis is a special case of radial symmetry, in which the illuminating beam propagates to or from a very distant point on the axis. Scanners exhibiting radial symmetry provide unity angular optical change for unity mechanical change.

That is \(m=d\Theta/d\Phi=1\) where \(\Theta\) is the optical scan angle and \(\Phi\) is the mechanical change. The parameter \(m\) is called the scan magnification, discussed later under ‘‘Augmented Resolution’’ for Eq. (19).

It ranges typically between 1 and approximately 2, depending on the scanner-illumination configuration, per Table 3. In remote sensing, \(m=\Theta/\Phi=k\). (See ‘‘Compound Mirror Optics Configurations .’’)

The prismatic polygon (see ‘‘Monogon and Polygon Scanners’’) exhibits a variable \(m\), depending on the degree of collimation or focusing of the output beam. When collimated, \(m=2\). When focusing, the value of \(m\) shifts from 2 according to 

\[\tag{1}m'=2+r/f\]

where \(f\) and \(r\) are per Fig. 4 and Eq. (19). This is similar to the ratio of angular velocities of the scanned focal point along the arc of a limaçon,

\[\tag{2}\dot{\Theta}/\dot{\Phi}=2\left(1+\frac{\cos\Phi}{1+f/r}\right)\]

Note that when \(r\rightarrow 0\) or when \(f\rightarrow\infty\), \(\dot\Theta/\dot\Phi\rightarrow 2\).

In holographic scanners which are not radially symmetric, \(m\) depends on the angles of incidence and diffraction of the input and first-order output beam,

\[\tag{3}m=\sin\theta_i+\sin\theta_o=\lambda/d\]

where \(\theta_i\) and \(\theta_o\) are the input and diffracted angles (with respect to the grating normal) and \(d\) is the grating spacing. For example, when \(\theta_i=\theta_o=30^\circ\), \(m=1\), when \(\theta_i=\theta_o=45^\circ\), \(m=\sqrt 2\).

 

 

 

 

 

 

2. SCANNED RESOLUTION 

Remote Sensing Resolution and Data Rates 

Figure 3 illustrates the scanning geometry for an airborne line-scanning system where the aircraft is flying along a ‘‘track’’ at a height \(H\) and velocity \(V\). The total scanned field of view is \(\theta_{max}\) and the cross-track and along-track instantaneous fields of view are \(\Delta\theta\) and \(\Delta\phi\), respectively.\(*\)

The direction directly below the aircraft and normal to the scanned surface is called the nadir. The instantaneous field of view is defined as the geometrical projection of the detector with spatial dimensions \(d_{ct}\) and \(d_{at}\) by the optics having a focal length of \(F\).

Therefore, \(\Delta\theta=d_{ct}/F\) and \(\Delta\theta=d_{at}/F\). Figure 3 shows the ‘‘bow-tie’’ distortion of the scanning geometry which will be discussed further under ‘‘Image Consequences.’’ 

The basic equation relating the aircraft velocity to the angular velocity of the scanning system to produce contiguous scan lines at the nadir is \(V/H=\dot{s}\cdot\Delta\phi\) where \(\dot{s}\) is the scanning system’s scan rate in scans/s. For a system with \(n\) detector elements aligned in the flight direction, \(V/H=n\dot{s}\cdot\Delta\phi\).

The number of resolution elements or pixels in a single scan line is

\[\tag{4a}N=\frac{\theta_{max}}{\Delta\theta}\]

\[\tag{4b}\quad\qquad=\frac{2\pi\theta_{max}}{360^\circ\cdot\Delta\theta}\]

where \(\Delta\theta\) is in radians, \(\theta_{max}\) is the total field of view measured in radians in Eq. (4a) and in degrees in Eq. (4b), for the scanning means employed, taking due regard for the duty cycle given in Eq. (23). The scan rate, in scans per second, may be expressed as a function of the scan mirror speed by 

\[\tag{5}\dot{s}=\frac{R\cdot S}{60}\] 

where \(R\) is the scan mirror rpm and \(S\) is the number of scans produced by the scanning mechanism per revolution of the optics. It follows that the number of resolution elements or pixels per second per scan line is 

\[\tag{6}\dot{N}\frac{2\pi\theta_\text{max}RS}{60\cdot360\cdot\Delta\theta}\]

The angle \(\theta_\text{max}\) (in degrees) is determined by the configuration of the scan mirror and is \(\theta_\text{max}=360\cdot k/S\) where \(k\) is the scanning constant or scan magnification\(^*\) and can have values ranging from 1 to 2.

The specific value is dependent upon the optical arrangement of the scanner as exemplified in Table 3. The pixel rate may now be written as 

\[\tag{7}\dot{N}=\frac{2\pi\text k\text R}{60\cdot\Delta\theta}\]

 

TABLE 3. Typical Features of Pyramidal and Prismatic Polygon Scanners

 

The information retrieval rate of a system is often expressed in terms of the dwell time \(\tau\) or the data bandwidth \(f_e\) as 

\[\tag{8}f_e=\frac{1}{2\tau}=\frac{\dot{N}}{2}\]

By combining the preceding equations , the data bandwidth for a multiple-detector system can be expressed as

\[\tag{9}f_e=\frac{\pi k(V/H)}{nS\Delta\theta\Delta\phi}\]

which illustrates clearly the relationship between important system parameters such as \(f_e\) being inversely proportional to instantaneous field-of-view solid angle \((\Delta\theta/\Delta\phi)\).

 

Input/Output Scanning

Resolution Criteria, Aperture Shape Factor. The resolution of an optical scanner is expressed by the number \(N\) of spots or elements that can be conveyed along a contiguous spatial path. The path is usually (nearly) linear and traversed with uniform velocity.

Although the elements \(\delta\) are analogous to the familiar descriptors pixels or pels (picture elements), such identification is avoided, for pixels often denote spatially digitized scan, where each pixel is uniform in intensity and/or color. Active optical scan, on the other hand, is typically contiguous, except as it may be subjected to modulation.

Normally, therefore, the scanned spots align and convolve to form a continuous spatial function that can be divided into elements by modulation of their intensity. To avoid perturbation of the elemental point spread function \(\text{(PSF)}\) by the modulating (or sampling) process, we assume that the scan function is modulated in intensity with a series of (Dirac) pulses of infinitesimal width, separated by a time \(t\) such that the spatial separation between spot centers is \(w=vt\), where \(v\) is the velocity of the scanned beam.

It is often assumed that the size \(\delta\) of the thus-established elemental spot corresponds to \(w\); that is, the width of the imaged spot is equal to the spacing from its neighbor .

To quantify the number \(N\) of such spots, those which exhibit a gaussian intensity distribution are usually considered overlapping at one of two widths; at their \(1/e^2\) intensity points, or at their 50 percent intensity points (the latter denoted as \(\text{FWHM}\); full width at half maximum). Their relationship is 

\[\tag{10}\delta_\text{FWHM}=0.589\delta_{1/e^2}\]

The resolution \(N\) is identified with its measurement criterion, for the same system will convey dif ferent apparent \(N\), per Eq. (10). That is, it will count approx. \(1.7\times\) as many spots at \(\text{FWHM}\) than at \(1/e^2\) intensity.

These distinctions are accommodated by their aperture shape factors a. For example, the above gaussian aperture distribution is represented by shape factors

\[\tag{11a}a_{1/e^2}=\frac{4}{\pi}=1.27\]

\[\qquad\tag{11b}a_\text{FWHM}=0.589a_{1/e^2}=0.75\]

When adapted to the applicable equation for spot size

\[\tag{12}\delta=aF\lambda\]

in which \(F=f/D\) is the F-number of the cone converging over the distance \(f\) from aperture width \(D\), and \(\lambda\) is the radiation wavelength, the resulting gaussian spot size becomes 

\[\tag{13a}\delta_{1/e^2}=\frac{4}{\pi}\frac{f}{D}\lambda=1.27F\lambda\]

when measured across the \(1/e^2\) intensity points, and 

\[\tag{13b}\delta_\text{FWHM}=0.75F\lambda\]

when measured across \(\text{FWHM}\). 

The factor a further accommodates the resolution changes due to changes in aperture shape, as for apodized and truncated gaussians. Ultimate truncation is that manifest when the illuminating spatial distribution is much larger than the limiting aperture (over-illumination or overfilling), forming the uniformly illuminated aperture.*

Familiar analytic examples are the rectangular and round (or elliptic) apertures , which generate (for the variable \(x\)) the normalized intensity distributions \([\sin x/x]^2\) and \([2J_1(x)/x]^2\) respectively, in which \(J_1(x)\) is the first-order Bessel function of the first kind. 

Figure 2 illustrates\(\dagger\) the \(\text{MTFs}\) of several uniformly illuminated apertures. Their intersections with the 0.5 \(\text{MTF}\) value identifies the spatial frequency at which their modulation is 50 percent. With the rectangular aperture as a reference (its straight line intersects 0.5 \(\text{MTF}\) at 50 percent of the limit frequency, forming \(a=1\)), the intersections of the others with \(\text{MTF}=0.5\) yield corresponding spatial frequencies and relative a-values.

Since the spatial frequency bandpass is proportional to \(D/f=1/F\), the apertures of the others must be widened by their a-values (effectively lowering their \(F\)-numbers) to render equivalent response midrange.

Table 4 summarizes the aperture shape factors (a) for several useful distributions. Truncated, when applied, is two-dimensional. Noteworthy characteristics are:

1. Scanning is in the direction of the aperture width \(D\).

TABLE 4 Aperture Shape Factor a 

 

TABLE 5 Aperture Shape Factor a for One-dimensional Truncation of a Gaussian Intensity Distribution 

 

2. The a-value of 1.25 for the uniformly illuminated round/elliptic aperture corresponds closely to the Rayleigh radius value of 1.22. 

3. The gaussian-illuminated data requires that the width \(D\), measured at the \(1/e^2\) intensity points be centered within the available aperture W. Two conditions are tabulated: untruncated \((W\geq1.7D)\) and truncation at \(W=D\). 

4. The gaussian-illuminated data also provides the a-values for 50 percent \(\text{MTF}\), allowing direct comparison with performance of the uniformly illuminated apertures. 

This data relates to apertures which, if apodized, are truncated two-dimensionally. However, one-dimensional truncation of a gaussian beam by parallel boundaries is not uncommon, typical of that for acousto-optic scanning.

There, the limiting aperture width \(W\) is constant, as determined by the device, while the gaussian width \(D\) is variable. Table 5 tabulates the shape factor a for such conditions. 

To relate to data in Table 4, the case of \(\rho=W/D=0\) represents illumination through a narrow slit. This corresponds to the uniformly illuminated rectangular aperture, whence \(a=1\). When \(\rho=1\), then \(W=D\) and the parallel barriers truncate the gaussian beam at its \(1/e^2\) intensity points.

Compared to symmetric truncation, this allows more of the gaussian skirts to contribute to the aperture width, providing \(a=1.15\) vs. 1.38. When \(\rho=2\), the gaussian beam of half the width of the boundaries is effectively untruncated, halving the resolution \((a=1.75\;\text{vs}\;0.85\), but maximizing radiometric throughput. (See Fig. 24, observing nomenclature, in which \(D=2w_x\) and \(W=2r_o\).)

Fundamental Scanned Resolution. The section on ‘‘Input/Output Scanning’’ introduced the two forms of optical scan: translation and angular deflection. Beam translation is conducted by objective scan, while angular deflection is either pre-objective or post-objective. Examples of each are provided later in this tutorial.

The resolution \(N_s\) of translational scan, by a beam focused to spot size \(\delta\) executing a scanned path distance \(S\), is simply,

\[\tag{14}N_s=\frac{S}{\delta}\]

Extremely high resolutions are practical, but are often limited to moderate speeds and bandwidths . Common implementations provide \(N_s=3000\;\text{to}\;100 ,000\).

 

 

\[\tag{15}N_{\theta}=\frac{\Theta D_0}{a\lambda}\] 

in which \(\Theta\) is the useful deflected optical angle and \(D_o\) is the effective aperture width at its nodal center, discussed in the next section.

Common implementations provide \(N_{\theta}=2,000\) to 30,000. Equation (15) is independent of spot size \(\delta\) and dependent only on the aperture characteristics of \(D_o\) and a, and the wavelength \(\lambda\).

The beam could be converging, collimated, or diverging. When collimated, \(D_o=D\), the actual width of the illuminated portion of the aperture. When converging or diverging, resolution augmentation occurs (see next section).

The numerator of Eq. (9) is a form of the Lagrange invariant, expressed in this nomenclature as

\[\tag{16}n\Theta D=n'\Theta' D'\]

where the primed terms are the refractive index, (small) angular deviation, and aperture width, respectively, in the final image space. For the common condition of \(n=n'\) in air, the \(\Theta D\) product and resolution \(N\) are conserved, invariant with centered optics following the deflector. 

Augmented Resolution, the Displaced Deflector. In general, a scanning system can accumulate resolution \(N\) by adding the two processes described previously, augmentation of angular scan with linear translation, forming 

\[\tag{17}N=N_{\theta}+N_s\]

Augmentation occurs, for example, with conventional multielement scanners (such as polygons) having deflecting elements (facets) which are displaced from the rotating axis by a distance \(r\), and whose output beam is noncollimated.

One active element (of width \(D\)) and its focused output beam is illustrated in Fig. 4. For convenient analysis, the deflecting element appears as overilluminated with an incident beam. The resulting resolution equations and focal spot positions are independent of over- or underillumination (see ‘‘Duty Cycle’’). 

Augmentation for increased resolution is apparent in Fig. 4, in which the output beam is derived effectively from a larger aperture \(D_o\) which is located at \(o\).

By similar triangles, \(D_o=D(1+r/f)\), which yields from Eq. (15), 

\[\tag{18}N=\frac{\Theta D}{a\lambda}\left(1+\frac{r}{f}\right)\]

This corresponds to Eq. (17), for in the \(N_s\) term the aperture \(D\) executes a displacement component \(S\backsimeq r\Theta\), which, with Eq. (12) forms Eq. (14). 

Following are some noteworthy observations regarding the parenthetic augmentation term: 

1. Augmentation goes to zero when \(r=0\) (deflector on nodal axis) or when \(f=\infty\) (output beam collimated). 

2. Augmentation adds when output beam is convergent ( \(f\) positive) and subtracts when output beam is divergent ( \(f\) negative). 

3. Augmentation adds when \(r\) is positive and subtracts when \(r\) is negative (output derived from opposite side of axis \(o\) ). 

The fundamental or nonaugmented portion of Eq. (18), \(N=\Theta D/a\lambda\), has been transformed to a nomograph, Fig. 5, in which the angle \(\Theta\) is represented directly in degrees. \(D/a\lambda\) is plotted as a radius, particularly useful when \(a\lambda=1\;\mu m\), whereupon \(D/a\lambda\) becomes the aperture size \(D\), directly in mm.

The set of almost straight bold lines is the resolution \(N\). Multiples of either scale yield corresponding multiples of resolution. 

 

 

 

Augmenting and Scan Magnification. Equation (18) develops from Fig. 4, assuming that the optimal scan angle \(\Theta\) is equal to the mechanical angle \(\Phi\). This occurs only when the scanner exhibits radial symmetry (see ‘‘Radial Symmetry and Scan Magnification’’). When, however, \(m=d\Theta/d\Phi\neq1\), as for configurations represented in the section on ‘‘Objective, Preobjective, and Postobjective Scanning,’’ account must be taken of scan magnification \(m\). Thus, the more complete resolution equation is represented by  

\[\tag{19}N=\frac{\Theta D}{a\lambda}\left(1+\frac{r}{mf}\right)\]

in which, per Fig. 4,

\(\Theta=\)optical scan angle (active) 
\(D=\) scan aperture width
\(\lambda=\)wavelength (same units as \(D\) )
\(a=\)aperture shape factor
\(m=\)scan magnification \((=d\Theta/d\Phi)\)
\(\Phi=\)mechanical scan angle about \(o\)
\(r=\)distance from \(o\) to \(D\)
\(f=\) 5 distance from \(D\) to \(p\)

 

\((\infty\) for collimated; \(+\) for convergent; \(-\)for divergent). 

Considering \(m=\Theta/\Phi\) as a constant, another useful form is 

\[\tag{20}N=\frac{\Phi D}{a\lambda}\left(m+\frac{r}{f}\right)\] 

whose augmenting term shows a composite magnification

\[\tag{21}m'=m+r/f\] 

which, for the typical prismatic polygon becomes 

\[\tag{21a}m'=2+r/f\]

Duty Cycle. The foregoing resolution equations refer to the active portion of a scan cycle. The full scan period almost always includes a blanking or retrace interval. The ratio of the active portion to the full scan period is termed the duty cycle \(\eta\). The blanking interval can include short overscan portions (straddling the active format), which are used typically for radiometric and timing calibration. The duty cycle is then expressed as 

\[\tag{22}\eta=1-\tau/T\]

in which \(\tau\) is the blanking interval time and \(T\) is the full scan period. A reduced duty cycle increases instantaneous bandwidth for a given average data rate. In terms of the scan angle of polygons, for example, it limits the useful component to 

\[\tag{23}\theta=\eta\;\theta_\text{max}\]

where \(\theta_\text{max}\) is the full available scan angle (see Table 3). 

Over and Underillumination (Over and Underfilling). In overillumination, the light flux encompasses the entire useful aperture. This is usually implemented by illuminating at least two adjacent apertures (e. g., polygon facets) such that the active one is always filled with light flux.

This not only allows unity duty cycle, but provides for resolution to be maximized for two reasons: (1) blanking or retrace may be reduced to zero; and (2) the full available aperture width is operative as \(D\) throughout scan. The tradeoff is the loss of illuminating flux beyond the aperture edges (truncation) and attendant reduction in optical power throughput (see ‘‘Coherent Source’’ under ‘‘Scanning for Input/Output Imaging’’).

An alternative is to prescan the light flux synchronously with the path of the scanning aperture such that it is filled with illumination during its entire transit.

In underillumination, the light flux is incident on a portion of the available aperture, such that this subtense delimits the useful portion \(D\).

A finite and often substantive blanking or retrace interval results, thereby depleting the duty cycle, but maximizing the transfer of incident flux to the scanned output. 

 

 

3. SCANNERS FOR REMOTE SENSING 

Early Single-Mirror Scanners 

Early scanning systems comprised an object-space mirror followed by focusing optics and a detector element (or array). The first scanners were simple rotating mirrors oriented typically at \(45^\circ\) to the axis as illustrated in Fig. 6. The rotational axis of the scan mirror lies parallel to the flight direction. In Fig. \(6a\), the scan efficiency and duty cycle of the oblique or single ax-blade scanner (see monogon under ‘‘Monogon and Polygon Scanners’’) is quite low since only one scan per revolution \((S=1)\) is generated.

The scan efficiency of the wedge or double ax-blade scanner shown in Fig. \(6b\) is twice as great \((S=2)\), although the ef fective optical aperture is less than half that of the oblique scanner for the same mirror diameter. The scanning constant is \(k=1\) for both types (see ‘‘Remote Sensing Resolution and Data Rates’’). 

 

Compound-Mirror-Optics Configurations

The aforementioned scanners suffered from a varying optical aperture as a function of view angle. To overcome this difficulty that causes severe variation in the video resolution during a scan line, several new line scanner configurations were developed.

Most notable among these was the rotating prism scanner invented by Howard Kennedy in the early 1960s and which forms the basis for most of the produced wide-field-of-view line scanners.

 

 

 

 

 

 

 

 

Figures 7 and 8 illustrate two configurations of this scanner. The three-sided scan mirror \(\text{SM}\) shown in Fig. 7 rotates about its longitudinal axis at a high rate and the concomitant folding mirrors \(\text{FM}\) are arranged such that the scanned flux is directed onto suitable focusing optics \(\text{FO}\) which focuses the flux at the detector \(\text{D}\).

As may be seen in the drawing of a four-sided scanner shown in Fig. 8, the effective optical aperture is split into two portions such that their sum is a constant value as a function of view angle. The width of each portion varies as the view angle is changed, with the portions being of equal value at the nadir position. The isometric view in Fig. 8 shows a portion of the scanner comprising the scan mirror, one folding mirror, and the focusing mirror.

For this design, the number of scans per rotation of the scan mirror is equal to the number of faces on the scan mirror, and the scanning constant is \(k=2\), which is also known as optical doubling (see item 3 of the prismatic polygon in Table 3).

Also, two faces of the scan mirror are always used to form the total optical aperture. Another advantage of this scanner configuration is that it produces a compact design for the total scanner system, a major reason for its popularity for over a quarter of a century. 

 

Image Consequences 

In airborne sensing, it is reasonable to assume that the earth is flat beneath the aircraft. When viewing along the nadir, the detector spatial footprint on the ground \(H\Delta\theta\) and \(H\Delta\phi\) in the across- and along-track directions, respectively.

As the view angle \((\theta)\) moves away from the nadir, the geometric resolution on the ground changes as illustrated in Fig. 3, which creates the bow-tie pattern. In the cross-track direction, it is easily shown that the footprint dimension is \(H\Delta\theta\cdot\sec^2\theta\), while in the along-track direction, the footprint dimension is \(H\Delta\phi\cdot\sec\theta\).

The change in footprint as a function of view angle can be significant. For example, if \(\theta_\text{max}=120^\circ\), then the footprint area at the extremes of the scan line are about eight times greater than at the nadir. 

 

 

 

Image Relation and Overlap 

When a linear array of \(n\) detectors is used, it is easily seen that the image of the detector array rotates by exactly the same amount as the view angle if the scanner is pyramidal as shown in Fig. 6. No such rotation occurs for the prismatic polygon, as in the Kennedy scanner, for which each scan comprises \(n\) adjacent detector footprints on the ground that form a segmented bow tie.

The next scan footprint has significant overlap with the preceding scan(s) for \(\theta\neq 0\). A means to compensate for the radiometric difficulties caused by the overlap of scans has been developed. In a single detector system, this artifact is easily compensated by electronic means.

 

Rotating Wedge Scanner 

Figure 9 shows a simple rotating wedge scanner that allows the generation of a wide variety of scan patterns, including a line scan. By controlling the rotational rates and phasing of the wedges, such patterns as included in Fig. 10 can be realized. 

 

Circular Scan 

In some cases, a circular scan pattern has found utility. Typically , the entire optical system is rotated about the nadir with the optical axis inclined at an angle \(\psi\) to the nadir. Figure 11 depicts an object-plane scanner showing how the aircraft or satellite motion creates contiguous scans.

Although the duty cycle is limited, an advantage of such a scanner is that, at a given altitude, the footprint has the same spatial size over the scanned arc. 

 

Pushbroom Scan 

A pushbroom scanner comprises typically an optical system that images onto the ground a linear array of detectors aligned in the cross-track direction or orthogonal to the flight direction.

The entire array of detectors is read out every along-track dwell time which is \(\tau_\text{at}=\Delta\phi/V/H\). Often, when a serial read-out array is employed, the array is rotated slightly such that the read-out time delay between detectors creates an image that is 

 

 

 

properly aligned to the direction of motion. Some state-of-the-art arrays can transfer the image data in parallel to a storage register for further processing.

The principal advantage of the pushbroom scanner is that no moving parts are required other than the moving platform upon which it is located. 

 

 

 

Two-dimensional Scanners 

Two-dimensional scanners have become the workhorses of the infrared community during the past two decades even though line scanners still find many applications, particularly in the area of earth resources. Scanners of this category can be classified into three basic groups, namely, object-space scanner, convergent-beam or image-space scanner, and parallel-beam or intermediate space scanner. Figure 12 depicts the generic form of each group. 

 

Object-space and Image-space Scanners 

The earliest two-dimensional scanners utilized an object-space scan mechanism. The simplest optical configuration is a single flat-mirror (see Fig. 12a ) that is articulated in such a manner as to form a raster scan.

The difficulty with this scan mechanism is that movement of a large mirror with the necessary accuracy is challenging. The size of the mirror aperture when in object space must be greater than that of the focusing optics. By using two mirrors rotating about orthogonal axes, the scan can be generated by using smaller mirrors, although the objective optics must have the capability to cover the entire field of view rather than the \(\text{FOV}\) of the detector. Figure 13 illustrates such a scanner where mirror \(\text{SM1}\) moves the beam in the vertical direction at a slow rate while mirror \(\text{SM2}\) generates the high-speed horizontal scan.

Although the focusing optics \(\text{FO}\) is shown preceding the image-space scan mirrors, the optics could be placed following the mirrors which would then be in object-space .

Although the high-F-number or low-numerical-aperture focusing lens before the mirrors must accommodate the \(\text{FOV}\), it allows the use of smaller mirror facets. The left-hand side of Fig. 13 shows an integral recording mechanism that is automatically synchronized to the infrared receptor side. This feature is one of the more notable aspects of the configuration and sets the stage for other scanner designs incorporating the integrated scene and display scanner.

 

 

 

A disadvantage of this scanner is the large size and weight of the vertical scan mirror, in part, to accommodate both scene and display scan. 

A variation of the two-mirror object-space scanner is known as the discoid scanner, which produces a raster scan at \(\text{TV}\) rates. Figure 14 depicts the scanner configuration which uses a high-speed, multiple-facet scan mirror \(\text{SM1}\) to generate the horizontal scan and a small, oscillating flat mirror \(\text{SM2}\) to produce the vertical scan.

An advantage of this scanner is that only a single detector is needed to cover the \(\text{FOV}\), although a linear array oriented in the scan direction is sometimes used, with time-delay integration, to improve sensitivity.

A feature of the ‘‘paddle’’ mirror scanner is the maintenance of a relatively stable aperture on the second deflector without the use of relay optics (see Figs. 21 and 32 and the section on the ‘‘Parallel Beam Scanner’).

Figure 15 depicts a reflective polygon scanner that generates the high-speed horizontal scan (per facet) by rotation of mirror \(\text{SM}\) about its rotational axis and the vertical movement of the scan pattern by tilting the spinning mirror about pivots \(\text{P1}\) and \(\text{P2}\) using cam \(C\) and its follower \(F\).

The path of the flux from the object reflects from the active facet A of the scan mirror to the folding mirror \(\text{FM}\) to the focusing mirror \(\text{FO}\) back through a hole in mirror \(\text{FM}\) to the detector located in dewar \(D\).

Almost all scanners of this type exhibit scanned-field distortion; i.e., the mapping of object to image space is non-rectilinear (e.g., see the target distortion discussion in the section ‘‘Image Consequences’’). 

 

 

In general, convergent-beam, image-space scanners suffer from severe defocus over the scanned field of view due to the field curvature produced by the rotation of the scan mirror in a convergent beam.

The use of this type scanner is therefore rather limited unless some form of focus correction or curved detector array is employed. A clever invention by Lindberg uses a high-speed refractive prism and a low-speed refractive prism to create the scanned frame. Figure 16 shows the basic configuration for a one-dimensional scanner where the cube \(P\) is rotated about the axis orthogonal to the page.

 

 

 

 

 

where the cube \(P\) is rotated about the axis orthogonal to the page. By proper selection of the refractive index and the geometry of the prism, focus is maintained over a significant and useful field of view. As can be seen from the figure, flux from the object at a given view angle is focused by lens \(\text{L}\) onto surface I which is then directed to the detector \(\text{D}\) by the refraction caused by the rotated prism.

Numerous commercial and military thermographic systems have utilized this principle for passive scanning. Since the field of view, maximum numerical aperture, optical throughput, and scan and frame rates are tightly coupled together, such scanners have a reasonably constrained design region. 

 

 

 

 

 

Other image-space scanners used in a convergent beam are the ‘‘soupbowl’’ and carousel scanners. The soupbowl scanner shown in Fig. 17 uses a rotating array of mirrors to produce a circularly segmented raster scan. The mirror facets may be at the same angle to generate more frames per rotation, given a detector array that has adequate extent to cover the field of view.

The facets could also be tilted appropriately with respect to one another to produce contiguous segments of the field of view if a small detector array is employe. Figure 18 illustrates the configuration of the carousel scanner which uses an array of mirrors arranged such that they create essentially a rectangular scan of the field of view.

Another scanning means that has been used for certain forward-looking infrared systems \(\text{(FLIRs)}\) was to mechanically rotate a detector array of angular extent \(\Phi\) about the optical axis of the focusing optics such that one end of the array was located an angular distance \(\Phi_{os}\) from the optical axis.

The rotating action generated a circular arc scan pattern similar to that of a windshield wiper. The inner radius of the scan pattern is \(\Phi_{os}\) and the outer radius is \(\Phi_{os}+\Phi\).

Clearly, the scan efficiency is rather poor and the necessity to use slip rings or the equivalent to maintain electrical connections to the detector array complicated acceptance of this scanner. The windshield wiper scan can also be generated by rotating only the optics if the optics incorporates anamorphic elements. A pair of

 

 

 

 

cylindrical lenses placed in an afocal arrangement, as illustrated in Fig. 19 at rotational angles \(\theta\) and \(\theta+90^\circ\), will rotate the beam passing through it at twice the rotational rate of the optics. See ‘‘Image Rotation in Derotation’’ in ‘‘Scanner Devices and Techniques.’’ 

 

Multiplexed Image Scanning 

With the advent of detector arrays comprising significant numbers of elements, the use of a single scan mirror became attractive. Figure 20 presents the basic parallel-beam scanner configuration used for the common module \(\text{FLIR}\) and thermal night sights.

The flat scan mirror \(\text{SM}\) is oscillated with either a sawtooth or a triangular waveform such that the detector array \(\text{D}\) (comprising 60, 120, or 180 elements) is scanned over the field of view in the azimuthal direction while the extent of the detector covers the elevation \(\text{FOV}\). Since the detectors are spaced on centers two detector widths apart, the scan mirror is tilted slightly in elevation every other scan to produce a 2:1 interlaced scan of the field of view.

As shown in Fig. 25, the back side of the scan mirror is used to produce a display of the scanned scene by coupling the outputs of the detectors to a corresponding array of \(\text{LEDs}\) which are projected to the user’s eye by lenses \(L1\), \(L2\), \(L3\), and \(L4\). 

 

Parallel-Beam Scanner 

A more complex two-dimensional, parallel-beam scanner configuration of the type shown in Fig. 12c has been developed by Barr & Stroud and is illustrated in Fig. 21 which incorporates an oscillating mirror \(\text{SM1}\), a high-speed polygon mirror \(\text{SM2}\) driven by motor M2, and relay optics \(L1\). (See discussion at end of section on ‘‘Scanning for Input/Optical Imaging’’.)

An afocal telescope is located before the scanner to change the \(\text{FOV}\), as is typical of parallel-beam scanners. Another innovative and compact two-dimensional scanner design by Kollmorgen is depicted in Fig. 22 and features diamond-turned

 

 

 

 

 

 

scanner design by Kollmorgen is depicted in Fig. 22 and features diamond-turned fabrication technology for ease of manufacture and alignment of the mirrors and mounts. Another parallel-beam scanner that uses a simple scan mirror has been developed.

The scan mirror is multifaceted with each facet tilted at an angle that positions the detector array in a contiguous manner in elevation. By having the nominal tilt angle of the facets be \(45^\circ\) to the rotation axis, minimal scanned-field distortion is realized. 

 

 

4. SCANNING FOR INPUT / OUTPUT IMAGING 

Power Density and Power Transfer 

Incoherent Source. This topic merits introduction as the predecessor to laser scanning—cathode-ray tube \(\text{(CRT)}\), flying-spot scanning and recording. 

Adaptation to other forms of incoherent sources, such as light-emitting diodes \(\text{(LEDs)}\) will be apparent. Similarities and contrasts with the handling of coherent sources are expressed. 

In a \(\text{CRT}\), the electron beam power \(P\) (accelerating voltage \(\cdot\) beam current) excites a phosphor of conversion efficiency \(\eta\) and utilization factor \(\gamma\). The resulting radiant power is transferred through an imaging system of optical transmission efficiency \(\text{T}\) and spectral power transfer a to a photosensitive medium of area a during a time \(\text{t}\). The resulting actinic energy density is given by

\[\tag{24}E=\frac{\eta aT\gamma Pt}{A}\quad\text{joules/cm}^2\]

(1 joule \(=1\) watt-sec \(=10^7\) ergs) . 

The first four terms are transfer factors \((\leq1)\)relating to the \(\text{CRT}\), but adaptable to other radiant sources. They are determined for a principal group of \(\text{CRT}\) recording phosphors having varying processes of deposition and aluminizing, and for two typical (silver halide) photosensitive spectral responses: noncolor sensitized and orthochromatic.

The spectral transfer term a is determined from the relatively broad and nonanalytic spectral characteristics of the \(\text{CRT}\) phosphors and the photosensors,

\[\tag{25}a\cong\frac{\displaystyle\sum_{i=1}^n\frac{P_i}{P_\text{max}}\cdot\frac{S_i}{S_\text{max}}\Delta\lambda_i}{\displaystyle\sum^m_{j=1}\frac{P_j}{P_\text{max}}\Delta\lambda_j}\quad(j\geq i)\]

where the \(P_s\) and the \(S_s\) are the radiant power and medium sensitivity respectively, taken at significant equal wavelength increments \(\Delta\lambda\). 

The optical transfer term \(T\) is composed of three principal factors, \(T=T_rT_fT_v\) in which \(T_r\) is the fixed transmission which survives losses due to, for example, reflection and scatter, \(T_f\) is the fixed geometric transfer, and \(T_y\) is the spectrally variable transmission of, for example, different glass types. The fixed geometric transfer is given by 

\[\tag{26}T_f=\frac{\cos^4\Phi V_{\Phi}}{1+4F^2(M+1)^2}\]

The numerator \((\leq1)\) is a transfer factor due to field angle \(\Phi\) and vignetting losses, \(F\) is the lens \(F\)-number, and \(M\) is the magnification, image/object.

The variable component \(T_v\) requires evaluation in a manner similar to that conducted for the \(a\). The resulting available energy density \(E\) is determined from Eq. (24) and compared to that required for satisfactory exposure of the selected storage material. 

Coherent Source. Determination of power transfer is much simplified by utilization of a monochromatic (single-line laser) source. Even if it radiates several useful lines (as a multispectral source), power transfer is established with a finite number of relatively simple determinations.

Laser lines are sufficiently narrow, compared to the spectral characteristics of most transmission and detection media, so that single point evaluations at the wavelengths of interest are usually adequate.

The complexity due to spectral and spatial distributions per Eqs. (25) and (26) are effectively eliminated.

In contrast to the incoherent imaging system described above, which suffers a significant geometric power loss represented by \(T_f\) of Eq.(26), essentially all the radiant power from the laser (under controlled conditions discussed subsequently) can be transferred to the focal spot.

Further, in contrast to the typical increase in radiating spot size with increased electron beam power of a \(\text{CRT}\), the radiating source size of the laser remains essentially constant with power variation. The focused spot size is determined (per the previous section on ‘‘Resolution Criteria, Aperture Shape Factor’’) by the converging beam angle or corresponding numerical aperture or \(F\)-number, allowing for extremely high power densities.

Thus, a more useful form of Eq. (24) expresses directly the laser power required to irradiate a photosensitive material as

\[\tag{27}P=\frac{sR}{T}\left(\frac{A}{t}\right)\text{watts}\]

in which

\(s=\) material sensitivity,  \(J/\text{cm}^2\)
\(R=\)reciprocity failure factor, \(\geq 1\)
\(T=\)optical throughput efficiency,  \(\leq 1\)
\(A=\)exposed area, \(\text{cm}^2\)
\(t=\)time over area \(A\), sec.

 

The reciprocity failure factor \(R\) appears here, since the exposure interval \(t\) (by laser) can be sufficiently short to elicit a loss in sensitivity of the photosensitive medium (usually registered by silver halide media). If the \(A/t\) value is taken as, for example, an entire frame of assumed uniform exposure interval (including blanking),

 

 

then the two-dimensional values of \(\eta\) must appear in the denominator, for they could represent a significant combined loss of exposure time.

The optical throughput efficiency \(T\) is a result of loss factors including those due to absorption, reflection, scatter, diffraction, polarization, diffraction inefficiency in acousto-optic and holographic elements, and beam truncation or vignetting.

Each requires disciplined attention. While the radiation from (fundamental mode) laser sources is essentially conserved in traversing sufficiently large apertures, practical implementation can be burdensome in scanners.

To evaluate the aperture size consistent with throughput power transfer, Figs. 23 and 24 are useful. The data is generalized to elliptic, accommodating the irradiance of typical laser diodes.

Figure 23 shows an irradiance distribution having ellipticity \(\epsilon=w_x/w_y\) (\(w\;@1/e^2\) intensity) apertured by a circle of radius \(r_o\). Figure 24 plots the encircled power (percent) vs. the ellipticity, with the ratio \(r_o/w_x\) as a

 

 

parameter. When \(\epsilon=1\), it represents the circular gaussian beam. Another parameter closely related to this efficiency is the aperture shape factor (discussed previously) affecting scanned resolution. (Note Glossary: \(D=2w_x\) and \(w=2r_o\).) 

 

Objective, Preobjective, and Postobjective Scanning 

Classification Characteristics. The scanner classifications designated as preobjective, objective, and postobjective were introduced previously and represented in Fig. 1 as a general conjugate optical transfer. This section expresses their characteristics. 

Objective Scan (Transverse Translational). Translation of an objective lens transverse to its axis translates the imaged focal point on the information surface. (Axial lens translation which optimizes focus is not normally considered scanning.)

Translation of the information medium (or object) with respect to the objective lens forms the same effect, both termed objective scan.  The two forms of objective scan appear in Fig. 25, the configuration of a drum scanner.

Preobjective Scan (Angular). Preobjective scan can provide a flat image field. This is exemplified by angularly scanning a laser beam into a flat-field or \(f-\theta\) lens, as illustrated in Fig. 26, an important technique discussed further under ‘‘Pyramidal and Prismatic Facets’’ and ‘‘Flat Field Objective Optics.’’ 

Postobjective Scan (Angular). Postobjective scan which is radially symmetric per Fig. 27 generates a perfectly circular scan locus. Departure from radial symmetry (e. g., focal point not on the axis of Fig. 27) generates noncircular (e.g., limaçon) scan, except for the A postobjective mirror with its surface on its axis generates a perfectly circular scan A postobjective mirror with its surface on its axis generates a perfectly circular scan locus, illustrated in Fig. 28. The input beam is focused beyond the axis at point \(o\). Scan magnification \(m=2\).

 

 

 

 

 

Objective Optics 

The objective lens converges a scanned laser beam to a moving focal point. The deflector can appear before, at, or after the lens, forming preobjective, objective, and postobjective scanning, respectively (see previous discussion).

On-axis Objective Optics. The simplest objective lens arrangement is one which appears before the deflector, as in Fig. 27, where it is required only to focus a monochromatic beam on-axis. The (postobjective) deflector intercepts the converging beam to scan its focal point. Ideally, the process is conducted almost aberrationlessly, approaching 

 

 

 

 

diffraction-limited performance. Since the lens operates on-axis only (accommodates no field angle) if the F-number of the converging cone is suf ficiently high (see ‘‘Resolution Criteria, Aperture Shape Factor’’), it can be composed of a single lens element. This simple arrangement can scan a perfectly circular arc [see ‘‘Postobjective Scan (Angular)]’’, the basis for the elegance of the internal drum scanner and the requirement for adapting the information medium to a curved surface. 

Flat-field Objective Optics. Almost all other lens configurations are required to form a flat field by transforming the angular scan to a straight line.  The deflector appears before the lens—preobjective. The most common configuration is similar to that of Fig. 26, as detailed further in ‘‘Design Considerations’’ under ‘‘Monogon and Polygon Scanners,’’ in which the scanned beam is shown collimated. Application is not limited to polygon scanners.

Similar lenses adapt to a variety of techniques, including galvanometer, acousto-optic, electro-optic, and holographic scanners. The lens must accept the scanned angle q from the aperture \(D\) and converge the beam throughout the scanned field to a best-focus along a straight-line locus. Depending on the magnitudes of \(\theta\) and \(D\), the \(\text{F}\)-number of the converging cone and the desired perfection of straight-line focus and linearity, the lens assembly can be composed of from 2 to 7 (or more) elements, with an equal number of choices of index of refraction, 4 to 14 (or more) surfaces and 3 to 8 (or more) lens spacings, all representing the degrees of freedom for the lens designer to accommodate performance. A typical arrangement of three elements is illustrated in Fig. 34. 

Telecentricity. A more demanding arrangement is illustrated in Fig. 29, showing six elements forming a high-performance scan lens in a telecentric configuration. Telecentricity is represented schematically in Fig. 30, in which an ideal thin-lens element depicts the actual arrangement of Fig. 29. Interposed one focal length \(f\) between the scanning aperture \(D\) (entrance pupil) and the flat image surface, the ideal lens transforms the angular change at the input to a translation of the output cone.

The chief ray of the ideal output beam lands normal to the image surface.

 

 

The degree of telecentricity is expressed by the angular departure from normal landing. Telecentricity is applied typically to restrict the spread of the landing beam and/or to retroreflect the probing beam efficiently for internal system calibration. This facility comes dearly, however, for the final lens elements must be at least as wide as the desired scan format.

A further requirement is the need to correct the nonlinearity of the simple system of Fig. 30, in which the spot displacement is proportional to the tangent of the scan angle, rather than to the angle directly. As in all scan lenses, compensation to make displacement proportional to scan angle is termed the \(f\)-\(\theta\) correction. 

Double-pass and Beam Expansion. Another variation of the objective lens is its

 

 

 

 

 

adaptation to double-pass, as depicted in Fig. 31. The lens assembly serves two functions: first, as the collimating portion of a lenticular beam expander and second, upon reflection by the scanner, as a conventional flat-field lens. This not only provides compaction, but since the illuminating beam is normal to the undeflected facet, the beam and facet undergo minimum enlargement, conserving the size of the deflector.

A slight skew of the input and output planes, per Fig. 31, avoids obstruction of the input and scanned beams at the folding mirror. An alternate input method is to make the lens array wide enough to include injection of the input beam (via a small mirror) from the side; at an angle sufficiently off-axis to avoid obstruction of the reflected scanned beam.

This method imposes an of f-axis angle and consequential facet enlargement and beam aberration, but allows all beams to remain in the same plane normal to the axis, avoiding the (typically) minor scanned bow which develops in the aforementioned center-skewed method. Other factors relating to increased surface scatter and reflection need be considered. 

The requirement for beam expansion noted here is fundamental to the formation of the aperture width \(D\) which provides a desired scanned resolution. Since most gas lasers radiate a collimated beam which is narrower than that required, the beam is broadened by propagating it through an inverted telescope beam expander, that is, an afocal lens group having the shorter focal length followed by the longer focal length. Operation may be reversed, forming beam compression, as required. In the previously described double-pass system,

 

 

the objective lens provides the collimating portion (long-focal-length group) of a beam expander. 

Conservation of Resolution. A most significant role of objective optics following the scanner is its determination of the integrity of scanned format, not of scanned resolution, as discussed under ‘‘Input/Output Scanning’’. Denoting \(N\) as the total number of scanned elements of resolution to be conveyed over a full format width, in first analysis, \(N\) is invariant with intervening ideal optics. In reasonably stigmatic systems, the lens determines the size of the spots, not their total number.

The number of spots is determined at the deflector, whether it be galvanometer, acousto-optic, electro-optic, polygonal, holographic, phased array, or any other angular scanner. This invariance is expressed as

\[\tag{28}I=\theta D=\theta'D'\]

an adaptation of the Lagrange invariant [see ‘‘Fundamental Scanned Resolution,’’ Eq. (16)], which is illustrated effectively with telescopic operation. If the scanned beam is directed through a telescope (beam compression), as in Fig. 32, the demagnification of \(f_2/f_1\) reduces \(D\) to \(D'\), but also expands \(\theta\) to \(\theta'\) by the same paraxial factor, sustaining resolution invariance.

If \(D\) were the deflecting aperture width and \(L_1\) were its objective lens (telecentric in this case), then the image along surface \(S\) would exhibit the same number of \(N\) spots as would appear if the output beam from \(D'\) were focused by another objective lens to another image plane. This schematic represents, effectively, a pupil-transferring optical relay.

 

 

5. SCANNER DEVICES AND TECHNIQUES

Many of the techniques addressed here for input/output imaging apply equally to remote sensing. Their reciprocal characteristic can be applied effectively by reversing the positions (and ray directions) of the light source(s) and detector(s). Preobjective and postobjective scanning have their counterparts in object-space and image-space scanning.

A notable distinction, however, is in the option of underillumination or overillumination of the deflecting aperture in input/output imaging, while the aperture is most often fully subtended in collecting flux from a remote source. This leads to the required attention to aperture shape factor in input/output imaging, which is less of an issue in remote sensing.

Another is the need to accommodate a relatively broad spectral range in remote sensing, while input/output operation can be monochromatic, or at least polychromatic. The frequent use of reflective optics in both disciplines tends to normalize this distinction.

 

Monogon and Polygon Scanners

The rotating mirrored polygon is noted for its capacity to render high data rate at high resolution. It is characterized by a multiplicity of facets which are usually plane and disposed in a regular array on a shaft which is rotatable about an axis. When the number of facets reduces to one, it is identified as a monogon scanner. 

Pyramidal and Prismatic Facets. Principal arrangements of facets are termed prismatic (Fig. 33) or pyramidal (Fig. 34). Figure 27 is a single-facet pyramidal equivalent, while Fig. 28 is a single-facet prismatic equivalent (common galvanometer mount). 

The prismatic polygon of Fig. 33 is oriented typically with respect to its objective optics in a manner shown in Fig. 26, while Fig. 34 shows the relationship of the pyramidal polygon to its flat-field lens. The pyramidal arrangement allows the lens to be oriented close to the polygon, while, as in Fig. 26, the prismatic configuration requires space for clear passage of the input beam.  Design consideration for this most popular arrangement is provided later in this chapter. 

Table 3 lists significant features and distinctions of typical polygon scanners. Consequences of Item 3, for example, are that the scan angle of the prismatic polygon is twice that of the pyramidal one for a given rotation. To obtain equal scan angles \(\theta\) of equal 

 

 

 

 

beam width \(D\) (equal resolutions \(N\) ) and to provide equal duty cycle (see ‘‘Augmenting and Scan Magnification’’) at equal scan rates, the prismatic polygon requires twice the number of facets, is almost twice the diameter, and rotates at half the speed of the pyramidal polygon.

The actual diameter is determined with regard for the aperture shape factor (previously discussed) and the propagation of the beam cross section (pupil) across the facet during its rotation (see ‘‘Design Considerations’’). 

Image Rotation and Derotation. When a beam having a round cross section is focused to an isotropic point spread function (psf), the rotation of this distribution about its axis is typically undetectable. If, however, the psf is nonisotropic (asymmetric or polarized), or if an array of 2 or more points is scanned to provide beam multiplexing, certain scanning techniques can cause an undesired rotation of the point and the array of points on the image surface. 

Consider a monogon scanner, per Fig. 27. As shown, the input beam o y er illuminates the rotating mirror. Thus, the mirror delimits the beam, establishing a rectangular cross section which maintains its relative orientation with respect to the image surface.

Thus, if uniformly illuminated , the focal point on the image surface (having in this case a \(\text{sinc}^2x\cdot\text{sinc}^2y\) psf, \(x=\) along-scan and \(y=\) cross-scan) maintains the same orientation along its scanned line. If, however, the input beam is polarized, the axis of polarization of the imaged point will rotate directly with mirror rotation within the rectangular psf. Similarly will be rotation for any radial asymmetry (e. g., intensity or ellipticity) within the aperture, resulting in rotation of the psf. 

Consider, therefore, the same scanner under illuminated with, for example, an elliptical gaussian beam (with major axis horizontal). The axis of the imaged elliptic spot (intended major axis vertical) will rotate directly with the mirror. Similarly, if the scanner is illuminated with multiple beams displaced slightly angularly in a plane (to generate an in-line array of spots), the axis of the imaged array will rotate directly with mirror rotation.

This effect is transferrable directly to the pyramidal polygon which is illuminated per Fig. 34. It may be considered as an array of mirrors, each exhibiting the same rotation characteristics as the monogon of Fig. 27. The mirrors of Fig. 34 are also overilluminated, maintaining a stationary geometric psf during scan (if uniformly illuminated), but subject to rotation of, for example, polarization within the psf. Similarly, it is subject to rotation of an elliptical beam within the aperture, or of a multiple-beam array. 

Not so, however, for the mirror mounted per Fig. 28 (galvanometer mount), or for the prismatic polygon of Figs. 26 & 33, which may be considered a multifacet extension of Fig. 28. When the illuminating beam and the scanned beam form a plane which is normal to the axis of mirror rotation, execution of scan does not alter the characteristics of the psf, except for the possible vignetting of the optical aperture and possible alteration of reflection characteristics (e. g., polarization) with variation in incident angle.

It is noteworthy that in the prior examples, the angles of incidence remained constant, while the image is subject to rotation; and here, the angles of incidence change, while the image develops no rotation.

The distinction is in the symmetry of the scanning system with respect to the illumination. The prior examples (maintaining constant incidence while exhibiting image rotation) are radially symmetric. The latter examples (which vary incidence but execute no image rotation) represent the limit of radial asymmetry.

While mirrored optical scanners seldom operate in regions between these two extremes, holographic scanners can, creating possible complications with, for example, polarization rotation. This is manifest in the variation in diffraction efficiency of gratings for variation in \(p\) and \(s\) polarizations during rotation. (See ‘‘Operation in the Bragg Regime.’’) 

Image Derotation. Image derotation can be implemented by interposing another image rotating component in the optical path to cancel that caused by the scanner. The characteristic of an image rotator is that it inverts an image. 

Thus, with continuous rotation, it rotates the image, developing two complete rotations per rotation of the component. It must, therefore, be rotated at half the angular velocity of the scanner. While the Dove prism  is one of the most familiar components used for image rotation, other coaxial image inverters include 

•  Three-mirror arrangement, which simulates the optical path of the Dove prism
•  Cylindrical/spherical lens optical relay
• Pechan prism, which allows operation in converging or diverging beams 

Design Considerations. A commonly encountered scanner configuration is the prismatic polygon feeding a flat-field lens in preobjective scan, illustrated schematically in Fig. 26. The size and cost of the flat field lens (given resolution and accuracy constraints) is determined primarily by its proximity to the scanner and the demand on its field angle.

A larger distance from the scanner (pupil relief distance) imposes a wider acceptance aperture for a given scan angle, and a wider scan angle imposes more complex correction for of f-axis aberration and field flattening. The pupil relief distance is determined primarily by the need for the input beam (Fig. 26) to clear the edge of the flaT-field lens.

Also, a wider scan angle reduces the accuracy requirement for pixel placement. Since the scan angle q subtends the desired number \(N\) of resolution elements, a wider angle provides a larger angular subtense per element and correspondingly larger allowed error in angle \(\Delta\theta\) for a desired elemental placement accuracy \(\Delta N\). This applies in both along-scan and cross-scan directions, \(\Delta\theta_x\) and \(\Delta\theta_y\), respectively (see ‘‘Scan Error Reduction’’). 

Subsequent consideration of the scanner-lens relationships requires a preliminary estimate of the polygon facet count, in light of its diameter and speed. Its speed is determined by the desired data rates and entails considerations which transcend the optogeometric ones developed here.

Diffraction-limited relationships are used throughout, requiring adjustment for anticipated aberration in real systems. The wavelength \(\lambda\) is a convenient parameter for buffering the design to accommodate aberration.

For example, an anticipated fractional spot growth of 15 percent due to systematic aberration is accommodated by using \(\lambda_+=1.15\lambda\).

Performance characteristics which are usually predisposed are the resolution \(N\) (elements per scan), the optical scan angle \(\theta\), and the duty cycle \(\eta\). Their interrelationships are presented under ‘‘Input/Output Scanning,’’ notably by Eqs. (15) and (23). The values of \(N\) and \(\theta\) for a desired image format width must be deemed practical for the flat-field lens.

Following these preliminary judgments , the collimated input beam width \(D\) is determined from [see ‘‘Fundamental Scanned Resolution’’ Eq.(15)]

\[\tag{29}D=Na\lambda/\theta\]

in which \(a\) is the aperture shape factor and \(\lambda\) is the wavelength. For noncollimated beams, see ‘‘Augmented Resolution, the Displaced Deflector,’’ notably Eq. (19). The number of facets is determined from Table 3 and Eq. (23),

\[\tag{30}n=4\pi\eta/\theta\]

whereupon it is adjusted to an integer.

Scanner-Lens Relationships. The polygon size and related scan geometry into the flat-field lens may now be determined. Figure 35 illustrates a typical prismatic polygon

 

 

and its input and output beams, all in the same plane. One of n facets of width W is shown in three positions: undeflected and in its limit-rotated positions. The optical beams are shown in corresponding undeflected and limit positions, deflected by \(\pm\theta/2\).

A lens housing edge denotes the input surface of a flat-field lens. Angle \(\gamma\) provides clear separation between the input beam and the down-deflected beam or lens housing.

The pupil relief distance \(P\) (distance \(ac\)) and its slant distance \(P_e\) (distance \(bc\)) are system parameters which establish angle \(a\) such that \(\cos a =P/P_e\). Angle \(a\) represents the off-axis illumination on the polygon which broadens the input beam on the facet.

The beam width \(D_m\) on the facet is widened due to a and due to an additional safety factor \(t\) \((1\leq t\leq1.4)\) which limits one-sided truncation of the beam by the edge of the facet at the end of scan. Applying these factors, the beam width becomes 

\[\tag{31}D_m\frac{Dt}{\cos a}\]

Following Eq. (15), the duty cycle is represented by \(\eta=1-D_m/W\), yielding the facet width

\[\tag{32}W=D_m/(1-\eta)\]

from which the outer (circumscribed) polygon diameter is developed; expressed by

\[\tag{33}D_p=\frac{Dt}{(1-\eta)\sin\pi/n\cos a}\]

Solution of Eq. (33) or expressions of similar form entails determination of a, the angle of of f-axis illumination on the facet . This usually requires a detailed layout, similar to that of Fig. 35. Series approximation of \(\cos a\) allows transformation of Eq. (33) to replace a with more direct dependence on the important lens parameter \(P\) (pupil relief distance), yielding,

\[\tag{34}D_p\frac{Dt}{(1-\eta)\sin\pi/n}\cdot\frac{1+\theta Ds/2P}{1-\theta^2/8}\]

in which, per Fig. 35, \(s\approx2\) is a safety multiplier on \(D\) for secure input/output beam separation and clearance.

Orientation of the scanner and lens also requires the height \(h\), the normal distance from the lens axis to the polygon center. This is developed as

\[\tag{35}h=R_c\sin(\gamma/2+\theta/4)\]

in which \(R_c\) is the radial distance \(o_c\), slightly shorter than the outer radius \(R\), approximated to be

\[\tag{36}R_c=R\left[1-\frac{1}{4}(\pi/n)^2\right]\]

 

Holographic Scanners 

General Characteristics. Almost all holographic scanners comprise a substrate which is rotated about an axis, and utilize many of the concepts representative of polygons. An array of holographic elements disposed about the substrate serves as facets, to transfer a fixed incident beam to one which scans.

As with polygons, the number of facets is determined by the optical scan angle and duty cycle (see ‘‘Duty Cycle’’) , and the elemental resolution is determined by the incident beam width and the scan angle (see Eq. 15).

In radially symmetric systems, scan functions can be identical to those of the pyramidal polygon. While there are many similarities to polygons, there are significant advantages and limitations. The most attractive features of holographic scanners are:

1. Reduced aerodynamic loading and windage with elimination of radial discontinuities of the substrate. 

2. Reduced inertial deformation with elimination of radial variations.

3. Reduced optical-beam wobble when operated near the Bragg transmission angle.

Additional favorable factors are: 

1. Operation in transmission, allowing efficient beam transfer and lens-size accommodation. 

2. Provision for disk-scanner configuration, with facets disposed on the periphery of a flat surface, designable for replication. 

3. No physical contact during exposure. Precision shaft indexing between exposures allows for high accuracy in facet orientation. 

4. Filtering in retrocollection, allowing spatial and spectral selection by rediffraction.  

5. Adjustability of focus, size, and orientation of individual facets.

Some limiting factors are: 

1. Need for stringent design and fabrication procedures, with special expertise and facilities in diffractive optics, instrumentation, metrology, and processing chemistry. 

2. Accommodation of wavelength shift: exposure at one wavelength (of high photosensitivity) and reconstruction at another (for system operation). Per the grating equation for first-order diffraction, 

\[\tag{37}\sin\theta_i+\sin\theta_o=\lambda/d\] 

where \(\theta_i\) and \(\theta_o\) are the input and diffracted output angles with respect to the grating normal and d is the grating spacing, a plane linear grating reconstructs a collimated beam of a shifted wavelength at a shifted angle.

Since wavefront purity is maintained, it is commonly employed,  although it requires separate focusing optics (as does a polygon). When optical power is added to the hologram (to provide self-focusing), its wavelength shift requires compensation for aberration.

Further complications arise when intended for multicolor operation, even if plane linear gratings. Further, even small wavelength shifts, as from laser diodes, can cause unacceptable beam misplacements, requiring corrective action.

3. Departure from radial symmetry develops complex interactions which require critical balancing to achieve good scan linearity, scan-angle range, wobble correction, radio-metric uniformity, and insensitivity to input beam polarization.

 

 

 

This is especially demanding in systems having optical power in the holograms. 

4. Systems which retain radial symmetry to maintain scan uniformity may be limited in Bragg angle wobble reduction, and can require auxiliary compensation, such as anamorphic error correction.

Holographic Scanner Configurations. A scanner which embodies some of the characteristics expressed above is represented in Fig. 36. A cylindrical glass substrate supports an array of equally spaced hololenses which image the input beam incident at o to the output point at \(P\).

Since point \(o\) intersects the axis, the scanner is radially symmetric, whereupon \(P\) executes a circular (arced) scan concentric with the axis, maintaining magnification \(m=\theta/\Phi=1\).

A portion of the radiation incident on the image surface is backscattered and intercepted by the hololens, and reflected to a detector which is located at the mirror image \(o'\) of point \(o\). The resolution of this configuration is shown to be analogous to that of the pyramidal polygon.

An even closer analogy is provided by an earlier reflective form illustrated in Fig. 37, emulating the pyramidal polygon, Fig. 34. It scans a collimated beam which is transformed by a conventional flat-field lens to a scanned focused line.

This is one of a family of holofacet scanners,  the most prominent of which tested to the highest performance yet achieved in combined resolution and speed—\(20,000\) elements per scan at 200 Mpixels/s. This apparatus is now in the permanent collection of the Smithsonian Institution.

Operation in the Bragg Regime. The aforementioned systems are radially symmetric and utilize substrates which allow derivation of the output beam normal to the rotating axis.

 

 

While operation with radial asymmetry was anticipated in 1967, it was not until operation in the Bragg regime was introduced that major progress developed in disk configurations. Referring to Fig. 38, the input and output beams \(I\) and \(O\) appear as principal rays directed to and diffracted from the holographic sector \(\text{HS}\), forming angles \(\theta_i\) and \(\theta_o\) with respect to the grating surface normal. 

For the tilt-error reduction in the vicinity of Bragg operation, the differential in output angle \(d\theta_o\) for a differential in hologram tilt angle \(da\) during tilt error \(\Delta_a\) is given by

\[\tag{38}d\theta_o=\left[1-\frac{\cos(\theta_i+\Delta_a)}{\cos(\theta_o-\Delta_a)}\right]da\]

whence, when \(\theta_i=\theta_0\), a small \(\Delta a\) is effectively nulled. While the \(\theta_i\) and \(\theta_o\) depart from perfect Bragg symmetry during hologram rotation and scan, the reduction in error

 

 

remains significant. An analogy of this important property is developed for the tilting of a refractive wedge operating at minimum deviation. When \(\theta_i=\theta_o\simeq45^\circ\), another property develops in the unbowing of the output scanned beam: the locus of the output beam resides (almost) in a plane normal to that of the paper over a limited but useful range.

Further, the incremental angular scan for incremental disk rotation becomes almost uniform: their ratio m at small scan angles is shown to be equal to the ratio \(\lambda/d\) of the grating equation (see the section on ‘‘Radial Symmetry and Scan Magnification’’ and Eq. 3).

At \(\theta_i=\theta_o=45^\circ\), \(m=\lambda/d=\sqrt 2\). This results in the output-scan angle to be \(\sqrt 2\) larger (in its plane) than the disk-rotation angle. While such operation provides the above attributes, it imposes two practical restrictions. 

1.  For high diffraction efficiency from relief gratings (e. g., photoresist), the depth-to-spacing ratio of the gratings must be extremely high, while the spacing \(d=\lambda/\sqrt 2\) must be extremely narrow. This is difficult to achieve and maintain, and difficult to replicate gratings which provide efficient diffraction. 

2.  Such gratings exhibit a high polarization selectivity, imposing a significant variation in dif fraction efficiency with grating rotation (see ‘‘Image Rotation and Derotation’’).

Accommodation of these limitations is provided by reducing the Bragg angle and introducing a bow correction element to straighten the scan line. This is represented in Fig. 39; a high-performance scanner intended for application to the graphic arts.

The Bragg angle is reduced to \(30^\circ\). This reduces the magnification to \(m=1=\lambda/d\) (as in radially symmetric systems), increases \(d\) to equal \(\lambda\) for more realizable deep-groove gratings, and reduces significantly the angular polarization sensitivity of the grating. 

 

 

 

 

 

The elegance of the \(45^\circ\) Bragg configuration has been adapted to achieve self-focusing in less demanding tasks (e. g., laser printing). This is exemplified in Fig. 40, which includes a holographic lens to balance the wavelength shift of the laser diode,  to shape the laser output for proper illumination of the scanner and to accommodate wavelength shift reconstruction.

However, such multifunction systems are compounded by more critical centration requirements and balancing of characteristics for achievement of a discrete set of objectives. 

 

Galvanometer and Resonant Scanners 

To avoid the scan nonuniformities which can arise from facet variations (see ‘‘Scan Error Reduction’’) of polygons or holographic deflectors, one might avoid multifacets. Reducing the number to one, the polygon becomes a monogon.

This adapts well to the internal drum scanner (Fig. 27), which achieves a high duty cycle, executing a very large angular scan within a cylindrical image surface. Flat-field scanning, however, as projected through a flat-field lens, allows limited optical scan angle, resulting in a limited duty cycle from a rotating monogon. If the mirror is vibrated rather than rotated completely, the wasted scan interval may be reduced.

Such components must, however, satisfy system speed, resolution, and linearity. Vibrational scanners include the familiar galvanometer and resonant devices and the less commonly encountered piezoelectrically driven mirror transducer. 

The Galvanometer. Referring to Fig. 41a, a typical galvanometer driver is similar to a torque motor. Permanent magnets provide a fixed field which is augmented \((\pm)\) by the variable field developed from an adjustable current through the stator coils.

Seeking a new balanced field, the rotor executes a limited angular excursion \((\pm\Phi/2)\). With the mirror and principal ray per Fig. 28, the reflected beam scans through \(\pm\theta/2\), twice that of the rotor. 

The galvanometer is a broadband device , damped sufficiently to scan within a wide

 

 

 

range of frequencies, from zero to an upper value close to its mechanical resonance. Thus, it can provide the sawtooth waveform with a longer active linearized portion and shorter retrace time\(=\tau\). This is represented in Fig. 42 (solid lines) showing rotation angle \(\Phi\) vs. time.

As a broadband device, it can also serve for random access, positioning to an arbitrary location within its access-time limitations. For this feature of waveform shaping, the galvanometer was categorized as a low inertia scanner. 

The Resonant Scanner. When damping is removed almost completely, large vibrations can be sustained only very near the resonant frequency of the oscillating system. The 

 

 

 

 

 

resonant scanner is thus characterized by larger angular excursions at a fixed and usually higher frequency, executing near-perfect sinusoidal oscillations. A typical driver configuration is illustrated in Fig. 41b. Figure 42 (dashed lines) shows a sinusoid with the same zero-crossings as those of the sawtooth waveform. Contrary to its popular designation as ‘‘low-inertia ,’’ the resonant scanner provides rigid time increments, as though it exhibits a high inertia.

While the rotary inertia of the suspension system is low to allow high repetition rates, it permits no random access and no scan waveform shaping, as do the galvanometer, acousto-optic, electro-optic, and other wideband scanners designated as low-inertia devices 

Suspension Systems. In the vibrational scanners, the bearings and suspension systems are the principal determinants of scan uniformity. The galvanometer shaft must be sufficiently stiff and long to inhibit cross-scan wobble.

However, to maximize the oscillating frequency, the armature is restricted in size and mass. Fortunately, its reciprocating motion tends to retrace its path (and its perturbations) faithfully over many cycles, making adjacent scans more uniform than if the same shaft rotated completely within the same bearings, as in a motor.

Some bearings are flexure, torsion, or taut-band devices which insert almost no along-scan perturbations. Because of their low damping, these suspensions are most often applied to the resonant scanner.

When damped, they can serve for the galvanometer, suffering a small sacrifice in bandwidth and maximum excursion, but gaining more uniform scan with very low noise and almost unlimited life. Some considerations are their low radial stiffness and possible coupling perturbation from torsion, shift of the axis of rotation with scan angle, and possible appearance of spurious modes when lightly damped. Most of these factors can be well-controlled in commercial instrument designs. 

Adaptations and Comparisons. Because the resonant scanner oscillates sinusoidally, and we seek typically a linearized scan, some significant adaptations are often required. As illustrated in Fig. 42 (dashed lines), we must select from the sine function a central portion which is sufficiently linear to be linearized further by timing the pixels or extracting them out of memory at a corresponding rate.

To limit the variation in pixel rate to 2:1 (i.e., velocity at zero crossover will be twice that at the same limit), then the useful excursion must be restricted to \(60^\circ/90^\circ\) or \(66. 7\) percent of its peak angle. When scanning with only one slope of the sinusoid (as for generation of a uniformly spaced raster), this represents a duty cycle of only 33.3 percent.

To raise the duty cycle, one must accommodate a greater variation in data rate. If, for example, the useful scan is 80 percent of its full excursion (40 percent when using one slope) , then the velocity variation rises to 3.24\(\times\). That is, the data rate or bandwidth at crossover is 3.24 times that at the scan limit.

Also, its bandwidth at crossover is approximately \(2\frac{1}{2}\) times that of the galvanometer, as represented by their relative slopes in Fig. 42 

There is a corresponding variation in the dwell-time of the pixels, resulting in predictable but significant variation in image exposure or detectivity: 2:1 for 33.3 percent duty cycle and \(3\frac{1}{4}\):1 for 40 percent duty cycle. This may require compensation over the full scan interval, using position sensing and control. In contrast, the broadband galvanometer with feedback can provide a highly linearized scan at a duty cycle of approximately 70 percent.

 

Acousto-optic Scanners 

Acousto-optic diffraction serves effectively for high-speed low-inertia optical deflection. It can provide random beam positioning within extremely short access times, or generate repetitive linear scans at very high rates, or divide a single beam into multiple beams for multiplexing applications.

The tradeoff is, however, relatively low resolution, seldom providing more than \(N=1000\) elements per scan. 

The principles of acousto-optics were formulated in 1932 and its attributes were applied only five years later to the Scophony TV projection system. Its potential for laser scanning was explored in the mid-60s. While various acousto-optic interactions exist, laser scanning is dominated by operation in the Bragg regime.

Fundamental Characteristics. Diffraction from a structure having a periodic spacing \(\Lambda\) is expressed as \(\sin\theta_i+\sin\theta_0=n\lambda/\Lambda\), in which \(\theta_i\) and \(\theta_o\) are the input and output beam angles respectively, \(n\) is the diffractive order, and \(\lambda\) is the wavelength.

Bragg operation requires that \(\theta_i=\theta_o=\theta_B\). In a ‘‘thick’’ diffractor, length \(L\geqq\Lambda^2/\lambda\), wherein all the orders are transferred efficiently to the first, and the Bragg angle reduces to 

\[\tag{39}\theta_B=\frac{1}{2}\frac{\lambda}{\Lambda}\]

Per Fig. 43, the grating spacing \(\Lambda\) is synthesized by the wavefront spacing formed by an acoustic wave traveling through an elastic medium. An acoustic transducer at one end converts an electrical drive signal to a corresponding pressure wave which traverses the medium at the velocity \(v_s\), whereupon it is absorbed at the far end to suppress standing waves.

The varying pressure wave in the medium forms a corresponding variation in its refractive index. An incident light beam of width \(D\) is introduced at the Bragg angle (angle shown exaggerated). An electrical drive signal at the center frequency \(f_o\) develops a variable index grating of spacing \(\Lambda\) which diffracts the output beam at \(\theta_B\) into position \(b\).

The drive signal magnitude is adjusted to maximize beam intensity at position \(b\), minimizing intensity of the zero order beam at position \(a\). When \(f_o\) is increased to \(f_s=f_o+\Delta f\), the grating spacing is decreased, diffracting the output beam through a larger angle, to position \(c\). The small scan angle \(\theta\) is effectively proportional to the change in frequency \(\Delta f\). 

The scan angle is \(\theta=\lambda/\Delta\Lambda=(\lambda/v_s)\Delta f\). The beam width, traversed by the acoustic wave over the transit time \(\tau\) is \(D=v_s\tau\).

Substituting into Eq.(15) and accounting for duty cycle per Eq. (22), the resolution of the acousto-optic scanner (total \(N\) elements for total \(\Delta f)\) is 

\[\tag{40}N=\frac{\tau\Delta f}{a}(1-\tau/T)\]

The \(\tau\Delta f\) component represents the familiar time-bandwidth product, a measure of information-handling capacity. 

Deflection Techniques.  Because the clear aperture width \(W\) of the device is fixed, anamorphic optics is often used to illuminate \(W\) with an adjusted beam width \(D —\) encountering selective truncation by the parallel boundaries of \(W\). The beam height (in quadrature to \(D\) ) can be arbitrarily narrow to avoid apodization by the aperture. This one-dimensional truncation of the gaussian beam requires assignment of an appropriate aperture shape factor a, summarized in Table 5. 

Additional topics in acousto-optic deflection are cylindrical lensing due to linearly swept \(f_s\) correction for decollimation in random access operation, Scophony operation, traveling lens or chirp operation, correction for color dispersion, polarization effects, and multibeam operation.

 

Electro-optic (Gradient) Scanners

The gradient deflector is a generalized form of beam scanner in which the propagating wavefronts undergo increasing retardation transverse to the beam, thereby changing the wavefront spacing (wavelength) transverse to the beam.

To maintain wavefront continuity, the rays (orthogonal trajectories of the wavefronts) bend in the direction of the shorter wavelength. Referring to Fig. 44a, this bend angle \(\theta\) through such a deflection cell may be expressed as 

\[\tag{41}\theta=k_o(dn/dy)l\]

where \(n\) is taken as the number of wavelengths per unit axial length \(l,\text{y}\) is the transverse distance, and \(k_o\) is a cell system constant. For the refractive material form in which the wavefront traverses a change \(\Delta n\) in index of refraction and the light rays traverse the change in index over the full beam aperture \(D\) in a cell of length \(L\), then the relatively small deflection angle becomes 

\[\tag{42}\theta(\Delta n/n_f)L/D\]

 

 

where \(n_f\) is the refractive index of the final medium (applying Snell’s law and assuming \(\sin\theta=\theta\)). Following Eq. (15), the corresponding resolution in elements per scan angle is expressed as 

\[\tag{43}N=(\Delta n/n_f)L/a\lambda\]

The \(\Delta n\) is given by 

\[\tag{44a}\text{for}\;\text{class}\;\text{I}\;\text{materials}\;\Delta n=n^3_0r_{ij}E_z\] 

\[\tag{44b}\text{for}\;\text{class}\;\text{II}\;\text{materials}\;\Delta n=n^3_er_{ij}E_z\]

where \(n_{o,e}\) is the (ordinary, extraordinary) index of refraction, \(r_{ij}\) is the electro-optic coefficient, and \(E_z=V/Z\) is the electric field in the \(z\) direction (see Fig. 45). 

Methods of Implementation. An electroacoustic method of developing a time-dependent index gradient was proposed in 1963 utilizing the (harmonic) pressure variations in an acoustically driven cell (of a transparent elastic material).

Although this appears similar to acousto-optic deflection (see ‘‘Acousto-optic Scanners’’) , it differs fundamentally in that the cell is terminated reflectively rather than absorptively (to support a standing wave).

Also, the acoustic wavelength is much longer than the beam width, rather than much shorter for Bragg operation. A method of approaching a linearly varying index gradient utilizes a quadrupolar array of electrodes bounding an electro-optic material; and is available commercially.

A continuous index gradient can be simulated by the use of alternating electro-optic prisms. A single stage biprism is illustrated in Fig. 44b and an iterated array for practical implementation appears in Fig. 45.

Each interface imparts a cumulative differential in retardation across the beam. The direction and speed of retardation is controlled by the index changes in the electro-optic material. While resolution is limited primarily by available materials, significant experiment and test is reported for this form of deflector.

Drive Power Considerations. The electrical power dissipated within the electro-optic material is given by 

\[\tag{45}P=\frac{1}{4}\pi V^2Cf/Q\]

where \(V\) is the applied (p-p sinusoidal) voltage in volts, \(C\) is the deflector capacitance in farads, \(f\) is the drive frequency in \(\text{Hz}\), and \(Q\) is the material \(Q\)-factor \([Q=1/\text{loss}\;\text{tangent}(\tan\delta)\simeq1/\text{power}\;\text{factor}\), \((Q>5)]\).

The capacitance \(C\) for transverse electroded deflectors is approximately that for a parallel-plate capacitor of (rectangular) length \(L\), width \(Y\), and dielectric thickness \(Z\) (per Fig. 45)

\[\tag{46}C=0.09_KLY/Z\quad \text{pF}\]

where \(k\) is the dielectric constant of the material \((L,Y,Z\) in cm).

The loss characteristics of materials which determine their operating \(Q\) are often a strong function of frequency beyond \(10^5\;\text{Hz}\). The dissipation characteristics of some electro-optic materials are provided, and a resolution-speed-power figure of merit has been proposed. 

Unique Characteristics. Most electro-optic coefficients are extremely low, requiring high drive voltages to achieve even moderate resolutions (to \(N\simeq 100)\). However, these devices can scan to very high speeds (to \(10^5/s)\) and suffer effectively no time delay (as do acousto-optic devices), allowing use of broadband feedback for position control. 

 

 

 

 

6. SCAN - ERROR REDUCTION 

High-resolution scanners often depend on precise rotation of a shaft about its axis, said shaft supporting a multiplicity of deflecting elements (facets, mirrors, holograms). The control of angular uniformity of these multielements with respect to the axis, and of the axis with respect to its frame, can create an imposing demand on fabrication procedures and consequential cost.

Since uniformity of beam position in the cross-scan direction may not be approached by phasing and timing of the data (as can be the along-scan errors), several noteworthy techniques have been developed to alleviate this burden. 

 

Available Methods 

The general field of cross-scan error reduction is represented in Table 6. Fabrication accuracy may be selected as the only discipline, or it may be augmented by any of the auxiliary methods. The active ones utilize high-speed low-inertia \((A-O\) or \(E-O)\) or piezoelectric deflectors or lower-speed (galvanometer) deflectors which are programmed to rectify the beam-position errors.

While open-loop programming is straightforward (while accounting for angular magnification/demagnification as a function of the accessed beam size), elegant closed-loop methods may be required to rectify pseudorandom perturbations. This must, however, be cost-effective when compared to the alternatives of increased fabrication accuracy and of the passive techniques. 

 

Passive Methods

Passive techniques require no programming. They incorporate optical principles in novel configurations to reduce beam misplacement due to angular error in reflection or diffraction. Bragg-angle error reduction of tilted holographic deflectors is discussed in the section, ‘‘Operation in the Bragg Regime.’’ 

Anamorphic Error Control. Anamorphic control, the most prominent treatment, may be applied to any deflector. The basics and operational characteristics are summarized here. Separating the nonaugmented portion of the resolution equation [Eq. (19)] into quadrature components and denoting the cross-scan direction as \(y\), then the error, expressed in the number of resolvable elements, is 

\[\tag{47}N_\text{y}=\frac{\theta\text{y}D\text{y}}{a\lambda}\]

 

TABLE 6 Techniques for Cross-Scan Error Reduction 

 

in which \(a\lambda\) is assumed constant, \(\theta_y\) is the angular error of the output beam direction, and \(D_\text{y}\) is the height of the beam illuminating the deflector. The objective is to make \(N_\text{y}\rightarrow 0\).

Mechanical accuracies determine \(\theta_\text{y}\), while anamorphics are introduced to reduce \(D_\text{y}\); usually accomplished with a cylindrical lens focusing the illuminating beam in the \(\text{y}\) direction upon the deflector.

[The quadrature (along-scan) resolution is retained by the unmodified \(D_\text{x}\) and scan angle \(\theta_x\).] As \(D_\text{y}\) is reduced, the \(\text{y}\) displacement error is reduced. Following deflection, the \(\text{y}\) direction scanned spot distribution is restored by additional anamorphics—restoring the nominal converging beam angle (via \(F_\text{y}\), the \(F\)-number forming the scanning spot in the \(\text{y}\) direction). 

The error reduction ratio is

\[\tag{48}R=D'_\text{y}/D_\text{y}\]

where \(D'_\text{y}\) is the compressed beam height and \(D_\text{y}\) is the original beam height on the deflector. 

A variety of anamorphic configurations has been instituted, with principal variations in the output region, in consort with the objective lens, to reestablish the nominal \(F_\text{y}\) while maintaining focused spot quality and uniformity. 

Double-Reflection Error Control.  In double-reflection (Table 6), the deflector which creates a cross-scan error is reilluminated by the scanned beam in such phase as to tend to null the error. This can be conducted in two forms: internal and external.

An internal double-reflection scanner is exemplified by the pentaprism monogon in Fig. 46a; a (glass) substrate having two of its five surfaces mirrored. This is an optically stabilized alternate to the \(45^\circ\) monogon of Fig. 27, operating preobjective in collimated light.

Tipping the pentaprism cross-scan (in the plane of the paper) leaves the \(90^\circ\) output beam unaffected. A minor translation of the beam is nulled when focused by the objective lens. The pentamirror per Fig. 46b, requires, however, significant balancing and support of the mirrors, since any shift in the nominal \(45^\circ\) included angle causes twice the error in the output beam.

A stable double-reflector is the open mirror monogon of Fig. 46c. Its nominal \(135^\circ\) angle serves identically to maintain the output beam angle at \(90^\circ\) from the axis, independently of cross-scan wobble. With a rigid included angle and simple balancing, it can provide high speed operation.

Two variations which double the duty cycle, as would a two-faceted pyramidal polygon or ax-blade scanner (see ‘‘Early Single-Mirror Scanners’’) appear in Fig. 47. Figure 47a is effectively two pentamirrors forming a butterfly scanner and Fig. 47b is effectively a pair of open mirrors.

The absolute angles of each half-section must maintain equality to

 

 

 

 

 

 

within half of the allowed error in the output beam. Also, the center section of (a) must be angularly stable to within one-quarter of the allowed error, because an increased included angle on one side forms a corresponding decrease on the other.

Other dynamic considerations involve inertial deformation, and the beam displacements and mirror widths (not shown) to accommodate the distance of the input beam from the axis during rotation. 

The need for near-perfect symmetry of the multiple double-reflectors can be avoided by transferring the accuracy requirement to an external element that redirects recurrent beam scans.

One such form is illustrated in Fig. 48. A prismatic polygon illuminated with a collimated beam of required width (only principal rays shown) deflects the beam first to a roof mirror, which returns the beam to the same facet for a second deflection toward the flat-field lens.

 

 

The roof mirror phases the returned beam such as to null the cross-scan error upon the second reflection. Several characteristics are noteworthy: 

1. The along-scan angle is doubled. That is, scan magnification \(m=4\) rather than 2.
2. This normally requires increasing the number of facets to provide the same angle with the same duty cycle.
3. However, during polygon rotation, the point of second reflection shifts significantly along the facet and sacrifices duty cycle.
4. The pupil distance from the flat-field lens is effectively extended by the extra reflections, requiring a larger lens to avoid vignetting.
5. The roof mirror and flat-field lens must be sized and positioned to minimize obstruction of the input and scanned beams. Allow for finite beam widths (see ‘‘Scanner-Lens Relationships’’).