# Introduction to Binary Optics

This is a continuation from the previous tutorial - ** spectrally efficient multiplexing - OFDM**.

## 1. Introduction

Binary optics is a surface-relief optics technology based on VLSI fabrication techniques (primarily photolithography and etching), with the ‘‘binary’’ in the name referring to the binary coding scheme used in creating the photolithographic masks. The technology allows the creation of new, unconventional optical elements and provides greater design freedom and new materials choices for conventional elements.

This capability allows designers to create innovative components that can solve problems in optical sensors, optical communications, and optical processors. Over the past decade, the technology has advanced sufficiently to allow the production of diffractive elements, hybrid refractive-diffractive elements, and refractive micro-optics which are satisfactory for use in cameras, military systems, medical applications, and other demanding areas.

The boundaries of the binary optics field are not clearly defined, so in this section, the concentration will be on the core of the technology: passive optical elements which are fabricated using VLSI technology.

As so defined, binary optics technology can be broadly divided into the areas of optical design and VLSI-based fabrication. Optical design can be further categorized according to the optical theory used to model the element: geometrical optics, scalar diffraction theory, or vector diffraction theory; while fabrication is composed of two parts: translation of the optical design into the mask layout and the actual micromachining of the element.

The following sections discuss each of these topics in some detail, with the emphasis on optical design.

Directly related areas which are discussed in other sections but not in this section include micro-optics and diffractive optics fabricated by other means (e. g., diamond turning, conventional manufacturing, or optical production), display holography (especially computer-generated holography), mass replication technologies (e. g., embossing, injection molding, or epoxy casting), integrated optics, and other micromachining technologies.

## 2. Design - Geometrical Optics

In many applications, binary optics elements are designed by ray tracing and ‘‘classical’’ lens design principles. These designs can be divided into two classes: broadband and monochromatic.

In broadband applications, the binary optics structure has little optical power in order to reduce the chromatic aberrations and its primary purpose is aberration correction. The device can be viewed as an aspheric aberration, corrector, similar to a Schmidt corrector, when used to correct the monochromatic aberrations and it can be viewed as a material with dispersion an order of magnitude greater than and opposite in sign to conventional materials when used to correct chromatic aberrations.

In monochromatic applications, binary optics components can have significant optical power and can be viewed as replacements for refractive optics.

In both classes of designs, binary optics typically offers the following key advantages:

- Reduction in system size, weight, and/or number of elements
- Elimination of exotic materials
- Increased design freedom in correcting aberrations, resulting in better system performance
- The generation of arbitrary lens shapes (including micro-optics) and phase profiles

### Analytical Models

#### Representation of a Binary Optics Element

As with any diffractive element, a binary optics structure is defined by its phase profile \(\phi(x,y)\) (\(z\) is taken as the optical axis), design wavelength \(\lambda_0\), and the surface on which the element lies.

For simplicity, this surface is assumed to be planar for the remainder of this section, although this is commonly not the case. For example, in many refractive/diffractive systems, the binary optics structure is placed on a refractive lens which may be curved. The phase function is commonly represented by either explicit analytical expression or decomposition into polynomials in \(x\) and \(y\) (e. g., the HOE option in CODE V).

Explicit analytic expressions are used in simple designs, the two most common being lenses and gratings. A lens used to image point \((x_o,y_o,z_o)\) to point \((x_i, y_i,z_i)\) at wavelength \(\lambda_0\) has a phase profile

\[\tag{1}\begin{align}\phi(x,y)&=\frac{2\pi}{\lambda_0}\left[z_o\left(\sqrt{(x-x_o)^2/z_o^2+(y-y_o)^2/z_o^2+1}-1\right)\right.\\&\left.\quad-z_i\left(\sqrt{(x-x_i)^2/z_i^2+(y-y_i)^2/z_i^2+1}-1\right)\right]\end{align}\]

where \(z_o\) and \(z_i\) are both taken as positive to the right of the lens. The focal length is given by the Gaussian lens formula:

\[\tag{2}\frac{1}{f_0}=\frac{1}{z_i}-\frac{1}{z_o}\]

with the subscript indicating that \(f_0\) is the focal length at \(\lambda_0\). A grating which deflects a normally incident ray of wavelength \(\lambda_0\) to the direction with direction cosines \((L,M)\) is described by

\[\tag{3}\phi(x,y)=\frac{2\pi}{\lambda_0}(xL+yM)\]

Axicons are circular gratings and are described by

\[\tag{4}\phi(x,y)=\frac{2\pi}{\lambda_0}(\sqrt{x^2+y^2}L)\]

where \(L\) now describes the radial deflection.

For historical reasons, the polynomial decomposition of the phase profile of the element commonly consists of a spheric term and an aspheric term:

\[\tag{5}\phi(x,y)=\phi_S(x,y)+\phi_A(x,y)\]

where

\[\phi_A(x,y)=\frac{2\pi}{\lambda_0}\sum_{k}\sum_{l}\alpha_{kl}x^ky^l\]

and the spheric term \(\phi_S(x,y)\) takes the form of Eq. (1).

Since the phase profiles produced by binary optics technology are not constrained to be spheric, \(\phi_S(x,y)\) is often set to zero by using the same object and image locations and the aspheric term alone is used to describe the profile. The binary optics element is then optimized by optimizing the polynomial coefficients \(\alpha_{kl}\). If necessary, the aspheric term can be forced to be radially symmetric by constraining the appropriate polynomial coefficients.

It is possible to describe the phase profile of a binary optics element in other ways. For example, \(\phi(x,y)\) could be described by Zernicke polynomials or could be interpolated from a two-dimensional look-up table. However, these methods are not widely used since lens design software currently does not support these alternatives.

#### Ray Tracing by the Grating Equation

A binary optics element with phase \(\phi(x,y)\) can be ray traced using the grating equation by modeling the element as a grating, the period of which varies with position. This yields

\[\tag{6}L'=L+\frac{m\lambda}{2\pi}\frac{\partial\phi}{\partial{x}}\]

\[\tag{7}M'=M+\frac{m\lambda}{2\pi}\frac{\partial\phi}{\partial{y}}\]

where \(m\) is the diffracted order, \(L,M\) are the direction cosines of the incident ray, and \(L'\), \(M'\) are the direction cosines of the diffracted ray. In geometrical designs, the element is usually blazed for the first order (\(m=1\)).

Note that it is the phase gradient \(\nabla\phi(x,y)\) (a vector quantity proportional to the local spatial frequency) and not the phase \(\phi(x,y)\) which appears in the grating equation.

The magnitude of the local period is inversely proportional to the local spatial frequency and given by

\[\tag{8}D(x,y)=2\pi/|\nabla\phi|\]

where \(| |\) denotes the vector magnitude. The minimum local period determines the minimum feature size of the binary optics structure, a concern in device fabrication (see ‘‘Fabrication’’ later in this tutorial).

#### Ray Tracing by the Sweatt Model

The Sweatt model, which is an approximation to the grating equation, is another method for ray tracing. The Sweatt approach models a binary optics element as an equivalent refractive element and is important since it allows results derived for refractive optics to be applied to binary optics.

In the Sweatt model, a binary optics element with phase \(\phi(x,y)\) at wavelength \(\lambda_0\) is replaced by a refractive equivalent with thickness and refractive index given by

\[\tag{9}t(x,y)=\frac{\lambda_0}{n_0-1}\frac{\phi(x,y)}{2\pi}+t_0\]

\[\tag{10}n(\lambda)-1=\frac{\lambda}{\lambda_0}(n_0-1)\]

Here, \(t_0\) is a constant chosen to make \(t(x,y)\) always positive and \(n_0\) is the index of the material at wavelength \(\lambda_0\). The index \(n_0\) is chosen by the designer and as \(n_0\rightarrow\infty\), the Sweatt model approaches the grating equation. In practice, values of \(n_0=10,000\) are sufficiently high for accurate results.

In the special case of a binary optics lens described by Eq. (1), the more accurate Sweatt lens can be used. In this case, the element is modeled by two surfaces of curvature

\[\tag{11}c_o=\frac{1}{(1-n_0)z_o}\]

\[\tag{12}c_i=\frac{1}{(1-n_0)z_i}\]

and conic constant \(-n_0^2\), with the axis of each surface passing through the respective point source. The refractive index is still modeled by Eq. (10).

### Aberration Correction

#### Aberrations of a Binary Optics Singlet

As a simple example of a monochromatic imaging system, consider a binary optics singlet which is designed to image the point \((0, 0, z_o)\) to the point \((0, 0, z_i)\) at wavelength \(\lambda_0\). The phase profile of this lens can be derived from Eq. (1) and the focal length \(f_0\) from Eq. (2).

Now consider an object point of wavelength \(\lambda\) located at \((0, \xi_o, l_o)\). The lens will form an image at \((0, \xi_i, l_i)\) (see Fig. 1), with the paraxial image position \(l_i\) and height \(\xi_i\) given by

\[\tag{13}\frac{1}{l_i}=\frac{\lambda}{f_0\lambda_0}+\frac{1}{l_o}\]

\[\tag{14}\frac{\xi_i}{l_i}=\frac{\xi_o}{l_o}\]

Note that the first equation is just the Gaussian lens law but using a wavelength-dependent focal length of

\[\tag{15}f(\lambda)=f_0\frac{\lambda_0}{\lambda}\]

The focal length being inversely proportional to the wavelength is a fundamental property of diffractive lenses. In addition, due to the wavelength shift and position change of the object point, the lens will form a wavefront with a primary aberration of

\[\tag{16}\begin{align}W(x,y)&=\frac{1}{8}\left[\left(\frac{1}{l_i^3}-\frac{1}{l_o^3}\right)-\frac{\lambda}{\lambda_0}\left(\frac{1}{z_i^3}-\frac{1}{z_o^3}\right)\right](x^2+y^2)^2\\&\quad-\frac{1}{2l_i}\left(\frac{1}{l_i^2}-\frac{1}{l_o^2}\right)\xi_iy(x^2+y^2)\\&\quad+\frac{3}{4l_i^2}\left(\frac{1}{l_i}-\frac{1}{l_o}\right)\xi_i^2y^2+\frac{1}{4l_i^2}\left(\frac{1}{l_i}-\frac{1}{l_o}\right)\xi_i^2x^2\end{align}\]

where the ray strikes the lens at \((x, y)\).

The first term is spherical aberration, the second is coma, and the last two are tangential and sagittal field curvature. As noted by Welford, all the off-axis aberrations can be eliminated if and only if \(l_i=l_o\), a useless configuration. In most systems of interest, the limiting aberration is coma.

The performance of the binary optics singlet can be improved by introducing more degrees of freedom: varying the stop position, allowing the binary optics lens to be placed on a curved surface, using additional elements, etc.

#### Chromatic Aberration Correction

Binary optics lenses inherently suffer from large chromatic aberrations, the wavelength-dependent focal length [Eq. (15)] being a prime example. By themselves, they are unsuitable for broadband imaging and it has been shown that an achromatic system consisting only of diffractive lenses cannot produce a real image.

However, these lenses can be combined successfully with refractive lenses to achieve chromatic correction. The chromatic behavior can be understood by using the Sweatt model, which states that a binary optics lens behaves like an ultrahigh index refractive lens with an index which varies linearly with wavelength [let \(n_0\rightarrow\infty\) in Eq. (10)].

Accordingly, they can be used to correct the primary chromatic aberration of conventional refractive lenses but cannot correct the secondary spectrum. For the design of achromats and apochromats, an effective Abbe number and partial dispersion can also be calculated. For example, using the \(C\), \(d\), and \(F\) lines, the Abbe number is defined as \(V_d=[n(\lambda_d)-1]/[n(\lambda_F)-n(\lambda_C)]\).

Substituting Eq. (10) and letting \(n_0\rightarrow\infty\) yields

\[\tag{17}V_d=\lambda_d/(\lambda_F-\lambda_C)=-3.45\]

In a similar fashion, the effective partial dispersion using the \(g\) and \(F\) lines is

\[\tag{18}P_{gF}=(\lambda_g-\lambda_F)/(\lambda_F-\lambda_C)=0.296\]

By using these effective values, the conventional procedure for designing achromats and apochromats can be extended to designs in which one element is a binary optics lens.

Figure 2 plots the partial dispersion \(P_{gF}\) versus Abbe number \(V_d\) for various glasses. Unlike all other materials, a binary optics lens has a negative Abbe number. Thus, an achromatic doublet can be formed by combining a refractive lens and a binary optics lens, both with positive power.

This significantly reduces the lens curvatures required, allowing for larger apertures. In addition, the binary optics lens has a position in Fig. 2 which is not collinear with the other glasses, thus also allowing the design of apochromats with reduced lens curvatures and larger apertures.

#### Monochromatic Aberration Correction

As a simple example, consider a refractive system which suffers from third-order spherical aberration and has a residual phase given by

\[\tag{19}\phi_r(x,y)=\frac{2\pi}{\lambda}C(x^2+y^2)^2\]

where \(C\) describes the spherical aberration. Then, a binary optics corrector with phase

\[\tag{20}\phi_b(x,y)=-\frac{2\pi}{\lambda_0}C(x^2+y^2)^2\]

will completely correct the aberration at wavelength \(\lambda_0\) and will reduce the aberration at other wavelengths to

\[\tag{21}\phi_r+\phi_b=\frac{2\pi}{\lambda}C(1-\lambda/\lambda_0)(x^2+y^2)^2\]

The residual aberration is spherochromatism.

### Micro-optics

Binary optics technology is especially suited for the fabrication of micro-optics and micro-optics arrays, as shown in Fig. 3. The advantages of binary optics technology include the following:

- Uniformity and coherence. If desired, all micro-optics in an array can be made identical to optical tolerances. This results in coherence over the entire array (see Fig. 4).
- Refractive optics. Binary optics is usually associated with diffractive optics. This is not a fundamental limit but results primarily from fabrication constraints on the maximum achievable depth (typically, 3 μm with ease and up to 20 μm with effort). However, for many micro-optics, this is sufficient to allow the etching of refractive elements. For example, a lens or radius \(R_0\) which is corrected for spherical aberration and focuses collimated light at a distance \(z_0\) (see Fig. 5) has a thickness of \[\tag{22}t_\text{max}=n\left[\sqrt{R_0^2+z_0^2}-z_0\right]/(n-1)\] where \(n\) is the index of the material.
- Arbitrary phase profiles. Binary optics can produce arbitrary phase profiles in micro-optics just as easily as in macro-optics. Fabricating arrays of anamorphic lenses to correct the astigmatism of semiconductor lasers, for example, is no more difficult than fabricating arrays of conventional spherical lenses.
- 100 percent fill factor. While many technologies are limited in fill factor (e. g., round lenses on a square grid yield a 79 percent fill factor), binary optics can achieve 100 per cent fill factor on any shape grid.
- Spatial multiplexing. Each micro-optic in an array can be different from its neighbors and the array itself can compose an arbitrary mosaic rather than a regular grid. For example, a binary optics array of individually designed micro-optics can be used to optimally mode match one-dimensional laser arrays to laser cavities or optical fibers.

### Optical Performance

#### Wavefront Quality

The wavefront quality of binary optics components is determined by the accuracy with which the lateral features of the element are reproduced. Since the local period (typically several μm) is usually much larger than the resolution with which it can be reproduced (of order 0.1 μm), wavefront quality is excellent. In fact, wavefront errors are typically limited by the optical quality of the substrate rather than the quality of the fabrication.

#### Diffraction Efficiency

The diffraction efficiency of a device is determined by how closely the binary optics stepped-phase profile approximates a true blaze. The theoretical efficiency at wavelength \(\lambda\) of an element with \(I\) steps designed for use at \(\lambda_0\) is:

\[\tag{23}\eta(\lambda,I)=\left|\text{sinc}(1/I)\frac{\sin(I\pi\alpha)}{I\sin\pi\alpha}\right|^2\]

where \(\text{sinc}(x)=\sin(\pi{x})/(\pi{x})\), \(\alpha=(\lambda_0/\lambda-1)/I\).

This result is based on scalar theory, assumes perfect fabrication, and neglects any material dispersion. Figure 6 plots the efficiency \(\eta(\lambda,I)\) for different numbers of steps \(I\); while Table 1 gives the average efficiency over the bandwidth \(\Delta\lambda\) for a perfectly blazed element (\(I\rightarrow\infty\)). The efficiency equation is asymmetric in \(\lambda\) but symmetric in \(1/\lambda\).

The use of scalar theory in the previous equation assumes that the local period \(D(x,y)\) [see Eq. (8)] is large compared to the wavelength. As a rule of thumb, this assumption begins to lose validity when the period dips below 10 wavelengths (e. g., a grating with period less than \(10\lambda_0\) or a lens faster than F/5) and lower efficiencies can be expected in these cases.

The efficiency discussed here is the diffraction efficiency of an element. Light lost in this context is primarily diffracted into other diffraction orders, which can also be traced through a system to determine their effect. As with conventional elements, binary optics elements will also suffer reflection losses which can be minimized in the usual manner.

## 3. Design - Scalar Diffraction Theory

Designs based on scalar diffraction theory are based on the direct manipulation of the phase of a wavefront. The incident wavefront is generally from a coherent source and the binary optics element manipulates the phase of each point of the wavefront such that the points interfere constructively or destructively, as desired, at points downstream of the element. In this regime, binary optics can perform some unique functions, two major applications being wavefront multiplexing and beam shaping.

### Analytical Models

In the scalar regime, the binary optics component with phase profile \(\phi(x, y)\) is modeled as a thin-phase screen with a complex transmittance of

\[\tag{24}c(x,y)=\exp[j\phi(x,y)]\]

The phase screen retards the incident wavefront and propagation of the new wavefront is modeled by the appropriate scalar formulation (e. g., angular spectrum, Fresnel diffraction, Fraunhofer diffraction) for nonperiodic cases, or by Fourier series decomposition for periodic cases.

The design of linear gratings is an important problem in the scalar regime since other problems can be solved by analogy. A grating with complex transmittance \(c(x)\) and period \(D\) can be decomposed into its Fourier coefficients \(C_m\), where

\[\tag{25}C_m=\frac{1}{D}\int_{0}^Dc(x)\exp(-j2\pi{mx/D})dx\]

\[\tag{26}c(x)=\sum_{m=-\infty}^{\infty}C_m\exp(j2\pi{mx/D})\]

The relative intensity or efficiency of the \(m\)th diffracted order of the grating is

\[\tag{27}\eta_m=|C_m|^2\]

Due to the fabrication process, binary optics gratings are piecewise flat. The grating transmission in this special case can be expressed as \(c(x)=c_i,\) for \(x_i\lt{x}\lt{x}_{i+1}\), where \(c_i\) is the complex transmission of step \(i\) of \(I\) total steps, \(x_0=0\), and \(x_I=D\). The Fourier coefficients then take the form

\[\tag{28}C_m=\sum_{i=0}^{I-1}c_i\delta_i\exp(-j2\pi{m}\Delta_i)\text{sinc}(m\delta_i)\]

where \(\delta_i=(x_{i+1}-x_i)/D\), \(\Delta_i=(x_{i+1}+x_i)/(2D)\).

The sinc term is due to the piecewise flat nature of the grating. If, in addition to the above, the grating transition points are equally spaced, then \(x_i=iD/I\) and Eq. (28) reduces to

\[\tag{29}C_m=\exp(-j\pi{m}/I)\text{sinc}(m/I)\left[\frac{1}{I}\sum_{i=0}^{I-1}c_i\exp(-j2\pi{mi/I})\right]\]

The bracketed term is the FFT of \(c_i\), which makes this case attractive for numerical optimizations. If the complex transmittance is also stepped in phase by increments of \(\phi_0\), then \(c_i=\exp(ji\phi_0)\) and Eq. (29) further reduces to

\[\tag{30}C_m=\exp[j\pi((I-1)\alpha-m/I)]\text{sinc}(m/I)\frac{\sin(I\pi\alpha)}{I\sin\pi\alpha}\]

where \(\alpha=\phi_0/(2\pi)-m/I\).

This important case occurs whenever a true blaze is approximated by a stepped-phase profile. The efficiency equation [Eq. (23)] is a further specialization of this case.

### Wavefront Multiplexers

#### Grating Designs

Grating multiplexers (also known as beam-splitter gratings) split one beam into many diffracted beams which may be of equal intensity or weighted in intensity.

Table 2 shows some common designs. In general, the designs can be divided into two categories: continuous phase and binary. Continuous phase multiplexers generally have better performance, as measured by the total efficiency and intensity uniformity of the diffracted beams, while binary multiplexers are easier to fabricate (with the exception of several naturally occurring continuous phase profiles).

If the phase is allowed to be continuous or nearly continuous (8 or 16 phase levels), then the grating design problem is analogous to the phase retrieval problem and iterative techniques are commonly used. A generic problem is the design of a multiplexer to split one beam into \(K\) equal intensity beams. Fanouts up to 1:50 with perfect uniformity and efficiencies of 90 – 100 percent are typical.

The complex transmittance of a binary grating has only two possible values [typically \(+1\) and \(-1\), or \(\exp(j\phi_0)\) and \(\exp(-j\phi_0)\)], with the value changing at the transition points of the grating. By nature, the response of these gratings have the following properties:

- The intensity response is symmetric; that is, \(\boldsymbol{\eta}_m=\boldsymbol{\eta}_{-m}\).
- The relative intensities of the nonzero orders are determined strictly by the transition points. That is, if the transition points are held constant, then the ratios \(\boldsymbol{\eta}_m/\boldsymbol{\eta}_n\) for all \(m, n\ne0\) will be constant, regardless of the actual complex transmittance values.
- The complex transmittance values only affect the balance of energy between the zero and nonzero orders.

Binary gratings are usually designed via the Dammann approach or search methods and tables of binary designs have been compiled. Efficiencies of 60 to 90 percent are typical for the \(1:K\) beam-splitter problem.

#### Multifocal Lenses

The concepts used to design gratings with multiple orders can be directly extended to lenses and axicons to design elements with multiple focal lengths by taking advantage of the fact that while gratings are periodic in \(x\), paraxial lenses are periodic in \((x^2+y^2)\), nonparaxial lenses in \(\sqrt{x^2+y^2+f_0^2}\), and axicons in \(\sqrt{x^2+y^2}\).

For gratings, different diffraction orders correspond to plane waves traveling in different directions, but for a lens of focal length \(f_0\), the \(m\)th diffraction order corresponds to a lens of focal length \(f_0/m\). By splitting the light into different diffraction orders, a lens with multiple focal lengths (even of opposite sign if desired) can be designed.

As an example, consider the paraxial design of a bifocal lens, as is used in intraocular implants. Half the light should see a lens of focal length \(f_0\), while the other half should see no lens. This is a lens of focal length \(f_0\), but with the light split evenly between the \(0\) and \(+1\) orders.

The phase profile of a single focus lens is given by \(\phi(r)=-2\pi{r^2}/(2\lambda_0f_0)\), where \(r^2=x^2+y^2\). This phase, with the \(2\pi\) ambiguity removed, is plotted in Fig. 7a as a function of \(r\) and in Fig. 7b as a function of \(r^2\), where the periodicity in \(r^2\) is evident. To split the light between the \(0\) and \(+1\) orders, the blaze of Fig. 7b is replaced by the 1:2 continuous splitter of Table 2, resulting in Fig. 7c. This is the final design and the phase profile is displayed in Fig. 7d as a function of \(r\).

### Beam Shapers and Diffusers

In many cases, the reshaping of a laser beam can be achieved by introducing the appropriate phase shifts via a binary optics element and then letting diffraction reshape the beam as it propagates.

If the incoming beam is well characterized, then it is possible to deterministically design the binary optics element. For example, Fig. 8a shows the focal spot of a Gaussian beam without any beam-forming optics. In Fig. 8b, a binary optics element flattens and widens the focal spot. In this case, the element could be designed using phase-retrieval techniques, the simplest design being a clear aperture with a \(\pi\) phase shift over a central region.

If the beam is not well-behaved, then a statistical design may be more appropriate. For example, in Fig. 8c, the aperture is subdivided into randomly phased subapertures. The envelope of the resulting intensity profile is determined by the subaperture but is modulated by the speckle pattern from the random phasing. If there is some randomness in the system (e. g., changing laser wavefront), then the speckle pattern will average out and the result will be a design which reshapes the beam and is robust to variations in beam shape.

### Other Devices

Other Fourier optics-based applications which benefit from binary optics include the coupling of laser arrays via filtering in the Fourier plane or other means, the fabrication of phase-only components for optical correlators, and the implementation of coordinate transformations. In all these applications, binary optics is used to directly manipulate the phase of a wavefront.

## 4. Design - Vector Diffraction Theory

Binary optics designs based on vector diffraction theory fall into two categories: grating-based designs and artificial index designs.

Grating-based designs rely on solving Maxwell’s equations for diffraction from the element. At present, this is practical only for periodic structures. Two major methods for this analysis are the expansion in terms of space harmonics (coupled wave theory) and the expansion in terms of modes (modal theory). In this category, optical design is difficult since it can be both nonintuitive and computationally intensive.

Artificial index designs are based on the following premise. When features on the component are small compared to the wavelength, then the binary optics element will behave as a material of some average index. Two common applications are shown in Fig. 9.

In Fig. 9a, the device behaves as an antireflection coating (analogous to anechoic chambers) since, at different depths, the structure has a different average index, continuously increasing from \(n_1\) to \(n_2\).

In Fig. 9b, the regular, subwavelength structure exhibits form birefringence. For light polarized with the electric vector perpendicular to the grooves, the effective index is

\[\tag{31}\frac{1}{n_\text{eff}^2}=p\frac{1}{n_1^2}+(1-p)\frac{1}{n_2^2}\]

where \(p\) is the fraction of total volume filled by material 1. However, for light polarized with the electric vector parallel to the grooves,

\[\tag{32}n_\text{eff}^2=pn_1^2+(1-p)n_2^2\]

In both these cases, the period of the structure must be much less than the wavelength in either medium so that only the zero order is propagating.

## 5. Fabrication

### Mask Layout

At the end of the optical design stage, the binary optics element is described by a phase profile \(\phi(x, y)\). In the mask layout process, this profile is transformed into a geometrical layout and then converted to a set of data files in a format suitable for electron-beam pattern generation. From these files, a mask maker generates the set of photomasks which are used to fabricate the element.

The first step is to convert the phase profile \(\phi(x, y)\) into a thickness profile (see Fig. 10a,b ) by the relation

\[\tag{33}t(x,y)=\frac{\lambda_0}{2\pi(n_0-1)}(\phi\text{ mod }2\pi)\]

where \(\lambda_0\) is the design wavelength and \(n_0\) is the index of the substrate at \(\lambda_0\). The thickness profile is the surface relief required to introduce a phase shift of \(\phi(x, y)\). The thickness varies continuously from 0 to \(t_0\), where

\[\tag{34}t_0=\lambda_0/(n_0-1)\]

is the thickness required to introduce one wave of optical path difference.

To facilitate fabrication, \(t(x, y)\) is approximated by a multilevel profile \(t'(x, y)\) (Fig. 10c), which normally would require one processing cycle (photolithography plus etching) to produce each thickness level. However, in binary optics, a binary coding scheme is used so that only \(N\) processing cycles are required to produce

\[\tag{35}I=2^N\]

thickness levels (hence the name binary optics).

The photomasks and etch depths required for each processing cycle are determined from contours of the thickness \(t(x, y)\) or equivalently the phase \(\phi(x, y)\), as shown in Table 3.

The contours can be generated in several ways. For simple phase profiles, the contours are determined analytically. Otherwise, the contours are determined either by calculating the thickness at every point on a grid and then interpolating between points or by using a numerical contouring method, analogous to tracing fringes on an interferogram.

To generate the photomasks, the geometrical areas bounded by the contours must be described in a graphics format compatible with the mask vendor (see Fig. 11a,b). Common formats are GDSII and CIF, both of which are high-level graphics descriptions which use the multisided polygon (often limited to 200 sides) as the basic building block. Hierarchical constructions (defining structures in terms of previously defined structures) and arraying of structures are also allowed.

The photomasks are usually written by electron-beam generators using the MEBES (Moving Electron Beam Exposure System) format as input. Most common high-level graphics descriptions can be translated or ‘‘fractured’’ to MEBES with negligible loss in fidelity via existing translation routines.

Currently, commercial mask makers can achieve a minimum feature size or ‘‘critical dimension’’ (CD) of 0.8 μm with ease, 0.5 μm with effort, and 0.3 μm in special cases. The CD of a binary optics element is determined by the minimum local period [see Eq. (8)] divided by the number of steps, \(D_\text{min}/I\). For lenses,

\[\tag{36}D_\text{min}\dot{=}2\lambda_0F\]

where \(F\) is the F-number of the lens; while, for gratings, \(D_\text{min}\) is the period of the grating.

In MEBES, all geometrical shapes are subdivided into trapezoids whose vertices lie on a fixed rectangular grid determined by the resolution of the electron-beam machine (see Fig. 11c). The resolution (typically 0.05 μm) should not be confused with the CD achievable by the mask maker.

In summary, the description of the photomask begins as a mathematical description based on contours of the thickness profile and ends as a set of trapezoids whose vertices fall on a regular grid (see Fig. 11).

This series of translations results in the following artifacts. First, curves are approximated by straight lines. The error introduced by this approximation (see Fig. 12) is

\[\tag{37}\delta=R(1-\cos\theta/2)\dot{=}R\theta^2/8\]

Normally, the maximum allowable error is matched to the electron-beam resolution. Second, all coordinates are digitized to a regular grid. This results in pixelization artifacts (which are usually negligible), analogous to the ziggurat pattern produced on video monitors when plotting gently sloped lines. Finally, the MEBES writing process itself has a preferred direction since it uses electrostatic beam deflection in one direction and mechanical translation in the other.

In addition to the digitized thickness profile, photomasks normally include the following features which aid in the fabrication process. Alignment marks are used to align successive photomasks, control features such as witness boxes allow the measurement of etch depths and feature sizes without probing the actual device, and labels allow the fabricator to easily determine the mask name, orientation, layer, etc.

### Micromachining Techniques

Binary optics uses the same fabrication technologies as integrated circuit manufacturing. Specifically, the micromachining of binary optics consists of two steps: replication of the photomasks pattern into photoresist (photolithography) and the subsequent transfer of the pattern into the substrate material to a precise depth (etching or deposition).

The replication of the photomasks onto a photoresist-covered substrate is achieved primarily via contact, proximity, or projection optical lithography. Contact and proximity printing offer lower equipment costs and more flexibility in handling different substrate sizes and substrate materials.

In contact printing, the photomask is in direct contact with the photoresist during exposure. Vacuum-contact photolithography, which pulls a vacuum between the mask and photoresist, results in the highest resolution (submicron features) and linewidth fidelity.

Proximity printing, which separates the mask and photoresist by 5 to 50 μm, results in lower resolution due to diffraction. Both contact and proximity printing require 1:1 masks.

In projection printing, the mask is imaged onto the photoresist with a demagnification from 1x to 20x. Projection printers are suitable for volume manufacturing and can take advantage of magnified masks. However, they also require expensive optics, strict environmental controls, and can only expose limited areas (typically 2 cm x 2 cm).

Following exposure, either the exposed photoresist is removed (positive resist) or the unexposed photoresist is removed (negative resist) in a developer solution. The remaining resist serves as a protective mask during the subsequent etching step.

The most pertinent etching methods are reactive ion etching (RIE) and ion milling. In RIE, a plasma containing reactive neutral species, ions, and electrons is formed at the substrate surface. Etching of the surface is achieved through both chemical reaction and mechanical bombardment by particles.

The resulting etch is primarily in the vertical direction with little lateral etching (an anisotropic etch) and the chemistry makes the etch attack some materials much more vigorously than others (a selective etch). Because of the chemistry, RIE is material-dependent. For example, RIE can be used to smoothly etch quartz and silicon, but RIE of borosilicate glasses results in micropatterned surfaces due to the impurities in the glass.

In ion milling, a stream of inert gas ions (usually Ar) is directed at the substrate surface and removes material by physical sputtering. While ion milling is applicable to any material, it is usually slower than RIE.

For binary optics designed to be blazed for a single order (i. e., designs based on geometrical optics), the major effect of fabrication errors is to decrease the efficiency of the blaze. There is little or no degradation in the wavefront quality. Fabrication errors can be classified as lithographic errors, which include alignment errors and over/underexposure of photoresist, and etching errors, which include depth errors and nonuniform etching of the substrate.

As a rule of thumb, lithographic errors should be held to less than 5 per cent of the minimum feature size (\(\lt0.05D_\text{min}/I\)), which can be quite challenging; while etching errors should be held to less than 5 percent of \(t_0\), which is usually not too difficult.

For binary optics designed via scalar or vector diffraction theory, manufacturing tolerances are estimated on a case-by-case basis through computer simulations.

The next tutorial introduces ** silica nanofibers and subwavelength-diameter fibers**.