Introduction to Miniature and Micro-Optics

This is a continuation from the previous tutorial - spectrally efficient multiplexing - Nyquist-WDM.

 

1. Introduction

Optical components come in many sizes and shapes. A class of optical components that has become very useful in many applications is called micro-optics. We define micro-optics very broadly as optical components ranging in size from several millimeters to several hundred microns. In many cases, micro-optic components are designed to be manufactured in volume, thereby reducing cost to the customer.

The following paragraphs describe micro-optic components that are potentially useful for large-volume applications. The discussion includes several uses of micro-optics, design considerations for micro-optic components, molded glass and plastic lenses, distributed-index planar lenses, Corning’s SMILE^TM lenses, microFresnel lenses, and, finally, a few other technologies that could become useful in the near future.

Micro-optics are becoming an important part of many optical systems. This is especially true in systems that demand compact design and form factor.

Some optical fiber-based applications include fiber-to-fiber coupling, laser-diode-to-fiber connections, LED-to-fiber coupling, and fiber-to-detector coupling.

Microlens arrays are useful for improving radiometric efficiency in focal-plane arrays, where relatively high numerical aperture (NA) microlenslets focus light onto individual detector elements.

Microlens arrays can also be used for wavefront sensors, where relatively low-NA lenslets are required. Each lenslet is designed to sample the input wavefront and provide a deviation on the detector plane that is proportional to the slope of the wavefront over the lenslet area.

Micro-optics are also used for coupling laser diodes to waveguides and collimating arrays of laser diodes. An example of a large-volume application of micro-optics is data storage, where the objective and collimating lenses are only a few millimeters in diameter.

 

2. Micro-Optics Design Considerations

Conventional lenses made with bulk elements can exploit numerous design parameters, such as the number of surfaces, element spacings, and index/dispersion combinations, to achieve performance requirements for NA, operating wavelength, and field of view.

However, fabricators of micro-optic lenses seek to explore molded or planar technologies, and thus the design parameters tend to be more constrained. For example, refractive microlenses made by molding, ion exchange, mass transport, or the SMILE^TM process resemble single-element optics.

Performance of these lenses is optimized by manipulating one or possibly two radii, the thickness, and the index or index distribution. Index choices are limited by the available materials. Distributed-index and graded-index lenses have a limited range of index profiles that can be achieved.

Additional performance correction is possible by aspherizing one or both surfaces of the element. This is most efficiently done with the molding process, but molded optics are difficult to produce when the diameter of the lens is less than 1.0 mm. In general, one or two aberrations may be corrected with one or two aspheres, respectively.

Due to the single-element nature of microlenses, insight into their performance may be gained by studying the well-known third-order aberrations of a thin lens in various configurations.

Lens bending and stop shift are the two parameters used to control aberrations for a lens of a given power and index. Bending refers to distribution of power between the two surfaces, i. e., the shape of the lens.

The shape is described by the shape factor \(X\) which is

\[\tag{1}X=\frac{C_1+C_2}{C_1-C_2}\]

where \(C_1\) and \(C_2\) are the curvatures of the surfaces.

The third-order aberrations as a function of \(X\) are shown in Fig. 1. 

 

These curves are for a lens with a focal length of 10.0 mm , an entrance pupil diameter of 1.0 mm, field angle \(\bar{u}=20^\circ\), an optical index of refraction of 1.5, \(\lambda\) = 0.6328 μm, and the object at infinity.

For any given bending of the lens, there is a corresponding stop position that eliminates coma, and this is the stop position plotted in the figure. The stop position for which coma is zero is referred to as the natural stop shift, and it also produces the least curved tangential field for the given bending. Because the coma is zero, these configurations of the thin lens necessarily satisfy the Abbe sine condition.

When the stop is at the lens (zero stop shift), the optimum shape to eliminate coma is approximately convex-plano (\(X=+1\)) with the convex side toward the object. The optimum shape is a function of the index, and the higher the index, the more the lens must be bent into a meniscus.

Spherical aberration is minimized with the stop at the lens, but astigmatism is near its maximum. It is interesting to note that biaspheric objectives for data storage tend toward the convex-plano shape.

Astigmatism can be eliminated for two different lens-shape/stop-shift combinations, as shown in Fig. 1. The penalty is an increase in spherical aberration. Note that there is no lens shape for which spherical, coma, and astigmatism are simultaneously zero in Fig. 1, that is, there is no aplanatic solution when the object is at infinity. The aplanatic condition for a thin lens is only satisfied at finite conjugates.

The plano-convex shape (\(X=-1\)) that eliminates astigmatism is particularly interesting because the stop location is in front of the lens at the optical center of curvature of the second surface. All chief rays are normally incident at the second surface. Thus, the design is monocentric. (Obviously, the first surface is not monocentric with respect to the center of the stop, but it has zero power and only contributes distortion.)

 

Two very common configurations of micro-optic lenses are \(X=+1\) and \(X=-1\) with the stop at the lens. Typically, the object is at infinity. In Fig. 2, we display contours of normalized rms wavefront deviation, \(\bar{\sigma}=\sigma\lambda_\text{rms}/2y\), versus \(\bar{u}\) and NA, where \(2y\) = diameter of the stop.

Aberration components in \(\sigma_\text{rms}\) include third-order spherical, astigmatism, and coma. The focus is adjusted to give minimum rms deviation of the wavefront, so effects of Petzval curvature are not included. Tilt is also subtracted.

As NA or field angle is increased, rms wavefront aberration increases substantially. The usable field of view of the optical system is commonly defined in terms of Marechal’s criterion as field angles less than those that produce \(2y\bar{\sigma}/1000\lambda\le0.07\) wave. For example, if the optical system operates at \(2y=1.0\) mm, \(\lambda\) = 0.6328 μm, NA =0.1, \(X=+1\), \(n=1.5\), and \(\bar{u}=2^\circ\), the wavefront aberration due to third-order contributions is

\[\tag{2}\sigma_\text{rms}=\frac{2y\bar{\sigma}}{1000\lambda}\approx\frac{(1.0\times10^{-3}\text{m})(0.015)}{(10^3)(0.6328\times10^{-6}\text{m/wave})}=0.024\text{ wave}\]

which is acceptable for most situations.

Note that the configuration for \(X=-1\) yields \(\sigma_\text{rms}\approx0.079\) wave, which is beyond the acceptable limit. When large values of \(\sigma_\text{rms}\) are derived from Fig. 2, care must be taken in interpretation of the result because higher-order aberrations are not included in the calculation. Also, if field curvature is included in the calculation, the usable field of view is significantly reduced.

Coma and astigmatism are only significant if the image field contains off-axis locations. In many laser applications, like laser diode collimators, the micro-optic lens is designed to operate on axis with only a very small field of view. In this case, spherical aberration is very significant.

A common technique that is used to minimize spherical aberration is to aspherize a surface of the lens. Third-, fifth-, and higher orders of spherical aberration may be corrected by choosing the proper surface shape. In some lens design codes, the shape is specified by

\[\tag{3}Z=\frac{ch^2}{1+\sqrt{1-(1+k)c^2h^2}}+Ah^4+Bh^6+Ch^8+Dh^{10}\]

where \(Z\) is the sag of the surface, \(c\) is the base curvature of the surface, \(k\) is the conic constant (\(k=0\) is a sphere, \(k=-1\) is a paraboloid, etc.), and \(h=\sqrt{x^2+y^2}\) is the radial distance from the vertex. The \(A\), \(B\), \(C\), and \(D\) coefficients specify the amount of aspheric departure in terms of a polynomial expansion in \(h\).

When a plane-parallel plate is inserted in a diverging or converging beam, such as the window glass of a laser diode or an optical disk, spherical aberration is introduced. The amount of aberration depends on the thickness of the plate, the NA of the beam, and to a lesser extent the refractive index of the plate, as shown in Fig. 3.

The magnitude of all orders of spherical aberration is linearly proportional to the thickness of the plate. The sign is opposite that of the spherical aberration introduced by an \(X=+1\) singlet that could be used to focus the beam through the plate.

Therefore, the aspheric correction on the singlet compensates for the difference of the spherical aberration of the singlet and the plate. This observation follows the fact that minimum spherical aberration without aspheric correction is achieved with the smallest possible air gap between the lens and the plate. For high-NA singlet objectives, one or two aspheric surfaces are added to correct the residual spherical aberration.

 

3. Molded Microlenses

Molded micro-optic components have found applications in several commercial products, which include compact disk players, bar-code scanners, and diode-to-fiber couplers.

Molded lenses become especially attractive when one is designing an application that requires aspheric surfaces. Conventional techniques for polishing and grinding lenses tend to be time-expensive and do not yield good piece-to-piece uniformity. Direct molding, on the other hand, eliminates the need for any grinding or polishing.

Another advantage of direct molding is that useful reference surfaces can be designed directly into the mold. The reference surfaces can take the form of flats. The reference flats are used to aid in aligning the lens element during assembly into the optical device. Therefore, in volume applications that require aspheric surfaces, molding becomes a cost-effective and practical solution.

The molding process utilizes a master mold, which is commonly made by single-point diamond turning and post polishing to remove tooling marks and thus minimize scatter from the surface. The master can be tested with conventional null techniques, computer-generated null holograms, or null Ronchi screens. Two types of molding technology are described in the following paragraphs. The first is molded glass technology. The second is molded plastic technology.

 

Molded Glass

One of the reasons glass is specified as the material of choice is thermal stability. Other factors include low birefringence, high transmission over a broad wavelength band, and resistance to harsh environments.

Several considerations must be made when molding glass optics. Special attention must be made to the glass softening point and refractive index. The softening point of the glass used in molded optics is lower than that of conventional components. This enables the lenses to be formed at lower temperatures, thereby increasing options for cost-effective tooling and molding. The refractive index of the glass material can influence the design of the surface. For example, a higher refractive index will reduce the surface curvature. Smaller curvatures are generally easier to fabricate and are thus desirable.

An illustration is Corning’s glass molding process. The molds that are used for aspheric glass surfaces are constructed with a single-point diamond turning machine under strict temperature and humidity control. The finished molds are assembled into a precision-bored alignment sleeve to control centration and tilt of the molds. A ring member forms the outside diameter of the lens, as shown in Fig. 4.

The glass material, which is called a preform, is inserted between the molds. Two keys to accurate replication of the aspheric surfaces are forming the material at high glass viscosity and maintaining an isothermal environment. After the mold and preform are heated to the molding temperature, a load is applied to one of the molds to press the preform into shape. After molding, the assembly is cooled to below the glass transformation point before the lens is removed. Optical performance characteristics of the finished lens are determined by the quality of the mold surfaces, the glass material, and the preform volume, which also determines the thickness of the lens when pressed. 

 

An alternative process is used at Kodak, Inc., where molded optics are injection molded and mounted into precision lens cells. In this process, a tuned production mold can reproduce intricate mounting datum features and extremely well-aligned optics. It can also form a stop, baffle, or a film-plane reference in the system. Table 1 lists preferred and possible tolerances for molded glass components. The Kodak process has been tested with over 50 optical glasses, which include both crowns and flints. This provides a wide index-of-refraction range, \(1.51\lt{n}\lt1.85\), to choose from.

 

Most of the molded glass microlenses manufactured to date have been designed to operate with infrared laser diodes at \(\lambda\) =  780 – 830 nm. The glass used to make the lenses is transparent over a much broader range, so the operating wavelength is not a significant factor if designing in the visible or near infrared.

Figure 5 displays a chart of the external transmission of several optical materials versus wavelength. LaK09 (curve B) is representative of the type of glass used in molded optics. The external transmission from 300 nm to over 2200 nm is limited primarily by Fresnel losses due to the relatively high index of refraction (\(n\) = 1.73). The transmission can be improved dramatically with antireflection coatings. 

 

Figure 6 displays the on-axis operating characteristics of a Corning 350110 lens, which is used for collimating laser diodes. The rms wavefront variation and effective focal length (EFL) are shown versus wavelength. The highest aberration is observed at shorter wavelengths. As the wavelength increases, the EFL increases, which decreases the NA slightly. 

 

Table 2 lists several optical properties of molded optical materials. The trend in molded glass lenses is to make smaller, lighter, and higher NA components. Reduction in mass and size allows for shorter access times in optical data storage devices, and higher NA improves storage density in such devices.

 

Molded Plastic

Molded plastic lenses are an inexpensive alternative to molded glass. In addition, plastic components are lighter than glass components. However, plastic lenses are more sensitive to temperatures and environmental factors. The most common use of molded plastic lenses is in compact disk (CD) players.

Precision plastic microlenses are commonly manufactured with injection molding equipment in high-volume applications. However, the classical injection molding process typically leaves some inhomogeneities in the material due to shear and cooling stresses. Improved molding techniques can significantly reduce variations, as can compression molding and casting.

The current state of the art in optical molding permits master surfaces to be replicated to an accuracy of roughly one fringe per 25 mm diameter, or perhaps a bit better. Detail as small as 5 nm may be transferred if the material properties and processing are optimum and the shapes are modest. Table 3 lists tolerances of injection-molded lenses. 

 

The tooling costs associated with molded plastics are typically less than those associated with molded glass because of the lower transition temperature of the plastics. Also, the material cost is lower for polymers than for glass. Consequently, the costs associated with manufacture of molded plastic microlenses are much less than those for molded glass microlenses. The index of refraction for the plastics is less than that for the glass lenses, so the curvature of the surfaces must be greater, and therefore harder to manufacture, for comparable NA.

The glass map for molded plastic materials is shown in Fig. 7. The few polymers that have been characterized lie mainly outside the region containing the optical glasses and particularly far from the flint materials. Data on index of refraction and Abbe number are particularly difficult to obtain for molded plastic.

The material is supplied in pelletized form, so it must first be molded into a form suitable for measurement. The molding process subjects the material to a heating and annealing cycle that potentially affects the optical properties. Typically, the effect of the additional thermal history is to shift the dispersion curve upward or downward, leaving the shape unchanged.

 

Changes in dimension or refractive index due to thermal variations occur in both molded glass and molded plastic lenses. However, the effect is more pronounced in polymer optical systems because the thermal coefficients of refractive index and expansion are ten times greater than for optical glasses, as shown in Table 2.

When these changes are modeled in a computer, a majority of the optical systems exhibit a simple defocus and a change of effective focal length and corresponding first-order parameters. An experimental study was made on an acrylic lens designed for a focal length of 6.171 mm at \(\lambda\) = 780 nm and 20\(^\circ\)C. At 16\(^\circ\)C, the focal length changed to 6.133 mm. At 60\(^\circ\)C, the focal length changed to 6.221 mm. Thermal gradients, which can introduce complex aberrations, are a more serious problem. Therefore, more care must be exercised in the design of athermalized mounts for polymer optical systems.

The transmission of two common optical plastics, polystyrene and polycarbonate, are shown in Fig. 5. The useful transmittance range is from 380 to 1000 nm. The transmission curve is severely degraded above 1000 nm due to C-H vibrational overtone and recombination bands, except for windows around 1300 nm and 1500 nm. Sometimes, a blue dye is added to the resins to make the manufactured part appear ‘‘water clear,’’ instead of slightly yellowish in color. It is recommended that resins be specified with no blue toner for the best and most predictable optical results.

The shape of the lens element influences how easily it can be manufactured. Reasonable edge thickness is preferred in order to allow easier filling. Weak surfaces are to be avoided because surface-tension forces on weak surfaces will tend to be very indeterminate. Consequently, more strongly curved surfaces tend to have better shape retention due to surface-tension forces.

However, strongly curved surfaces are a problem because it is difficult to produce the mold. Avoid clear apertures that are too large of a percentage of the physical surface diameter. Avoid sharp angles on flange surfaces. Use a center/edge thickness ratio less than 3 for positive lenses (or 1/3 for negative lenses). Avoid cemented interfaces. Figure 8 displays a few lens forms. 

 

The examples that mold well are C, E, F, and H. Form A should be avoided due to a small edge thickness. Forms A and B should be avoided due to weak rear surfaces. Form D will mold poorly due to bad edge/center thickness ratio. Form G uses a cemented interface, which could develop considerable stress due to the fact that thermal differences may deform the pair, or possibly even destroy the bond.

Since polymers are generally softer than glass, there is concern about damage from ordinary cleaning procedures. Surface treatments, such as diamond films, can be applied that greatly reduce the damage susceptibility of polymer optical surfaces.

A final consideration is the centration tolerance associated with aspheric surfaces. With spherical optics, the lens manufacturer is usually free to trade off tilt and decentration tolerances. With aspheric surfaces, this tradeoff is no longer possible. The centration tolerance for molded aspherics is determined by the alignment of the mold halves. A common specification is 4 to 6 μm, although 3 to 4 μm is possible.

 

4. Monolithic Lenslet Modules

Monolithic lenslet modules (MLMs) are micro-optic lenslets configured into close-packed arrays. Lenslets can be circular, square, rectangular, or hexagonal. Aperture sizes range from as small as 25 μm to 1.0 mm. Overall array sizes can be fabricated up to 68 x 68 mm.

These elements, like those described in the previous section, are fabricated from molds. Unlike molded glass and plastic lenses, MLMs are typically fabricated on only one surface of a substrate, as shown in the wavefront sensing arrangement of Fig. 9.

 

An advantage of MLMs over other microlens array techniques is that the fill factor, which is the fraction of usable area in the array, can be as high as 95 to 99 percent. Applications for MLMs include Hartman testing, spatial light modulators, optical computing, video projection systems, detector fill-factor improvement, and image processing.

There are three processes that have been made used to construct MLMs. All three techniques depend on using a master made of high-purity annealed and polished material. After the master is formed, a small amount of release agent is applied to the surface. In the most common fabrication process, a small amount of epoxy is placed on the surface of the master. A thin glass substrate is placed on top. The lenslet material is a single-part polymer epoxy. A slow-curing epoxy can be used if alignment is necessary during the curing process. The second process is injection molding of plastics for high-volume applications. The third process for fabrication of MLMs is to grow infrared materials, like zinc selenide, on the master by chemical vapor deposition. Also , transparent elastomers can be used to produce flexible arrays.

MLMs are advertised to be diffraction-limited for lenslets with NA < 0.10. Since the lens material is only a very thin layer on top of the glass substrate, MLMs do not have the same concerns that molded plastic lenses have with respect to birefringence and transmission of the substrate.

For most low-NA applications, individual lenslets can be analyzed as plano-convex lenses. Aspheres can be fabricated to improve imaging performance for higher NAs. Aspheres as fast as NA = 0.5 have been fabricated with spot sizes about twice what would be expected from a diffraction-limited system. The residual error is probably due to fabrication imperfections observed near the edges and corners of the lenslets.

 

5. Distributed-Index Planar Microlenses

A distributed-index planar microlens, which is also called a Luneberg lens, is formed with a radially symmetric index distribution. The index begins at a high value at the center and decreases to the index value of the substrate at the edge of the lens. The function that describes axial and radial variation of the index is given by

\[\tag{4}n(r,z)\approx{n}(0,0)\sqrt{1-g^2r^2-\frac{2g\Delta{n^2}(0,0)}{d}z^2}\]

where \(r\) is the radial distance from the optical axis, \(z\) is the axial distance, \(n(0, 0)\) is the maximum index at the surface of the lens, \(g\) is a constant that expresses the index gradient, \(d\) is the diffusion depth, and \(\Delta=(n(0,0)-n_2)/n(0,0)\), where \(n_2\) is the substrate index.

Typical values are \(\Delta=0.05\), \(d=0.4\) mm, \(r_\text{max}=0.5\) mm, \(n_2=1.5\), and \(g=\sqrt{2\Delta}/r_\text{max}=0.63\text{ mm}^{-1}\).

These lenses are typically fabricated on flat substrates and yield hemispherical index profiles, as shown in Fig. 10. Two substrates placed together will produce a spherical lens. Several applications of light coupling with distributed-index microlenses have recently been demonstrated. These include coupling laser diodes to fibers, LEDs to fibers, fibers to fibers, and fibers to detectors. In the future, arrays of lenslets might aid in parallel communication systems.

 

One way to introduce the index gradient is through ion exchange. As shown in Fig. 10, a glass substrate is first coated with a metallic film. The film is then patterned with a mask that allows ions to diffuse from a molten salt bath through open areas of the mask.

Ions in the glass substrate are exchanged for other ions in the molten salt at high temperatures. The diffused ions change the refractive index of the substrate by an amount that is proportional to their electric polarizability and concentration.

To increase the index, diffusing ions from the salt bath must have a larger electronic polarizability than that of the ions involved in the glass substrate. Since ions that have larger electron polarizability also have larger ionic radius, the selective ion exchange changes the index distribution and creates local swelling where the diffusing ion concentration is high. The swelling can be removed with polishing for a smooth surface. Alternatively, the swelling can be left to aid in the lensing action of the device.

To obtain the proper index distribution, the mask radius and diffusion time must be chosen carefully. If the mask radius, \(r_\text{mask}\), is small compared to the diffusion depth, the derivative of the index distribution with respect to radial distance \(r\) monotonically decreases.

Since the curvature of a light ray passing through the medium is proportional to the gradient of the logarithm of the refractive index, the rays tend not to focus. A suitable combination of diffusion time \(t\) and mask radius is given by \(Dt/r_\text{mask}^2\approx0.4\), where \(D\) is the diffusion constant of the dopant in the substrate. Table 4 displays the diffusion time necessary for making a planar microlens with a radius of 0.5 mm. Typically, the paraxial focal length in the substrate is \(l_0\approx20r_\text{mask}\), and the numerical aperture is \(\text{NA}\approx{n_2}/20\).

 

 

Other fabrication techniques can also be used. Planar lens arrays in plastics are fabricated with monomer-exchange diffusion. Plastics are suitable for making larger-diameter lenses because they have large diffusion constants at relatively low temperatures (100\(^\circ\)C).

The electromigration technique is more effective for creating devices with short focal length. For example, by applying an electric field of 7 V/mm for 8 h, it is possible to obtain a planar microlens with radius of 0.6 mm and focal length of 6.8 mm.

A distributed-index microlens array using a plasma chemical vapor deposition (CVD) method has also been reported. In this process, hemispherical holes are etched into a planar glass substrate. The holes are filled with thin layers of a combination of SiO₂ and Si₂N₄. These materials have different indices of refraction, and the composition is varied from the hemispherical outside shell to the center to provide a Luneburg index distribution.

Shearing interferometry can be used to measure the index distribution from thinly sliced samples of lenslets. Samplets are acquired laterally or longitudinally, as shown in Fig. 11.

 

 

Results of the measurement on a lateral section are shown in Fig. 12 for the ion-exchange technique. The solid line is the theoretical prediction, and the dotted line corresponds to measured data. The large discrepancy between measured and theoretical results is probably due to concentration-dependent diffusion or the interaction of the dopants. 

 

Figure 13 shows the two-dimensional index profile resulting from a deep electromigration technique. These data correspond much more closely to the theoretical values in Fig. 12.

 

 

The ray aberration of a distributed-index lens is commonly determined by observing the longitudinal aberration at infinite conjugates, as shown in Fig. 14. The paraxial focusing length, \(l_0\), is given by

\[\tag{5}l_0=d+\frac{\sqrt{1-2\Delta}}{g}\cot\left[\frac{gd\sin^{-1}\sqrt{2\Delta}}{\sqrt{2\Delta}}\right]\]

The amount of longitudinal aberration is defined by \(\text{LA}=(l-l_0)/l_0\), where \(l\) is the distance at which a ray crosses the optical axis. LA increases with the radius \(r\) of the ray. In order to display the effects of different \(D\) and \(n_2\) parameters, we define a normalized numerical aperture that is given by

\[\tag{6}\overline{\text{NA}}=\frac{\text{NA}}{n_2\sqrt{2\Delta}}\]

and is plotted in Fig. 15 versus diffusion depth for several values of LA. Notice that, for small values of LA, the maximum NA occurs at a diffusion depth of \(d\approx0.9/g\). 

 

Wave aberration of a planar distributed index microlens is shown in Fig. 16. The large departure at the maximum radius indicates severe aberration if used at full aperture. Swelled-structure lenses can exhibit much improved performance. It has been determined that the index distribution contributes very little to the power of the swelled-surface element. Most of the focusing power comes from the swelled surface-air interface. A few characteristics of ion-exchanged distributed-index microlenses are shown in Table 5. 

 

6. SMILE^TM MICROLENSES

Spherical Micro Integrated Lenses (SMILE^TM) are also micro-optic lenslets configured into arrays. Unlike MLMs, SMILE^TM lenses are formed from a photolytic technique in photosensitive glass.

Applications for SMILE^TM lenses include facsimile machines and photocopiers, LED/LCD enhancers, autofocus devices in video and SLR cameras, optical waveguide connectors, and others.

The process used for construction of SMILE^TM lenses involves an exposure of the glass to ultraviolet light through a chrome mask. The mask is patterned so that circular areas, which correspond to the lens diameter, are opaque on a clear background. The arrangement of the opaque circles on the mask determines the layout of the lenses in the final device.

The exposure is followed by a thermal development schedule that initiates the formation of noble metal particles, which in turn serve as nuclei for the growth of a lithium metasilicate microcrystalline phase from the homogeneous glass.

The thermo-optically developed crystallized region is slightly more dense than the unexposed homogeneous glass. The exposed region contracts as the crystalline phase develops. This squeezes the soft undeveloped glass and forces it beyond the plane of the original surface. Minimization of surface energy determines the spherical nature of the surface. The spherical eruption constitutes the lenslet.

In addition to the lens-forming process, the exposed and developed region surrounding the lens is rendered optically opaque, thus providing optical isolation. Figure 17 displays an electron photomicrograph showing the spherical protrusions in perspective. 

 

Figure 18 shows an optical micrograph of the lenses in a close-packed geometry. 

 

The lens sag, as well as the optical density and color of the exposed region, is a function of the excitation light exposure and the thermal schedule. The lens sag is also determined to some extent from the distance between the lenslets.

The glass can be exposed from both sides simultaneously through a pair of precisely aligned masks. This permits a lens pattern that has symmetric lens curvatures on opposing sides. It is advertised that SMILE^TM lens substrates can be as thin as 0.25 mm with diameters from 75 to 1000 μm. The minimum separation of the lenses must be greater than 15 μm. Effective focal lengths are available between 50 and 200 μm, with \(\text{NA}\le0.35\).

SMILE^TM lenses suffer from aberration near the lenslet edges, as shown in the interferogram of Fig. 19. Over most of the lenslet area, straight and equally spaced fringes are observed. However, near the edges, a strong curve indicates the presence of spherical aberration. 

 

An irradiance spot profile is displayed in Fig. 20 that results from focusing a 0.2-NA lenslet at \(\lambda\) = 0.6238 μm. The full-width-at \(1/e^2\) is 3.7 μm. It has been shown that, for facsimile applications, arrays of SMILE^TM lenses exhibit better irradiance uniformity and have an enhanced depth of focus when compared to a rod-lens array of GRIN lenses. 

 

7. Micro-Fresnel Lenses

The curvature of an optical beam’s wavefront determines whether the beam is converging, diverging, or collimated. A bulk lens changes the wavefront curvature in order to perform a desired function, like focusing on a detector plane. The micro-Fresnel lens (MFL) performs the same function as a bulk lens, that is, it changes the curvature of the wavefront. In a simple example, the MFL converts a plane wavefront into a converging spherical wavefront, \(A(x,y,z)\), as shown in Fig. 21. The difference between an MFL and a bulk lens is that the MFL must change the wavefront over a very thin surface.

 

 

A Fresnel lens is constructed of many divided annular zones, as shown in Fig. 22. Fresnel lenses are closely related to Fresnel zone plates. Both zone patterns are the same. However, unlike a Fresnel zone plate, the Fresnel lens has smooth contours in each zone, which delay the phase of the optical beam by \(2\pi\) radians at the thickest point. In the central zone, the contour is usually smooth enough that it acts as a refractive element.

Toward the edges, zone spacing can become close to the wavelength of light, so the Fresnel lens exhibits diffractive properties. Also, due to the quasi-periodical nature of the zones and the diffractive properties, Fresnel lenses have strong wavelength dependencies.

Advantages of the Fresnel lens are that they can be made small and light compared to bulk optical components. Note that binary optics are stepped approximations to the MFL smooth-zone contour.

To understand the zonal profiles of the MFL, we return to our example problem illustrated in Fig. 21. The converging spherical wavefront is given by

\[\tag{7}A(r,z)=\frac{A_0}{\rho}[-i(k\rho+\omega{t})]=\frac{A_0}{\rho}\exp[i\phi(r,z)]\]

where \(A_0\) is the amplitude of the wave, \(\rho^2=(z-f)^2+r^2\), \(r=\sqrt{x^2+y^2}\), \(f\) is the focal length, and \(k=2\pi/\lambda\). The phase of \(A(x, y, z)\) at \(t=0\) and in a plane just behind the MFL is given by

\[\tag{8}\phi(x,y,0^+)=-k\sqrt{f^2+r^2}\]

We could add a constant to Eq. (8) and not change any optical properties other than a dc phase shift. Let

\[\tag{9}\psi^+(r)=\phi(x,y,0^+)+kf+2\pi=2\pi+k(f-\sqrt{f^2+r^2})\]

Zone radii are found by solving

\[\tag{10}k(f-\sqrt{f^2+r_m^2})=-2\pi{m}\]

where \(m=1, 2, 3, . . .\) is the zone number. The result is

\[\tag{11}r_m=\sqrt{2\lambda{fm}+(\lambda{m})^2}\]

Equation (9) becomes

\[\tag{12}\psi^+_m(r)=2\pi(m+1)+k(f-\sqrt{f^2+r^2})\]

The job of the MFL is to provide a phase change so that the incident wavefront phase, \(\psi^-(r)\), is changed into \(\psi^+(r)\). The phase introduced by the MFL, \(\psi_\text{MFL}(r)\), must be

\[\tag{13}\psi_\text{MFL}(r)=\psi^+(r)-\psi^-(r)\]

A phase change occurs when a wave is passed through a plate of varying thickness, as shown in Fig. 23.

 

 

\(\psi^+(r)\) is given by

\[\tag{14}\begin{align}\psi^+(r)&=\psi^-(r)+kn_\text{MFL}d(r)+kn_i[\Delta-d(r)]\\&=\psi^-(r)+k(n_\text{MFL}-n_i)d(r)+kn_i\Delta\end{align}\]

where \(d(r)\) is the thickness profile, \(n_i\) is the refractive index of the image space, \(n_\text{MFL}\) is the refractive index of the substrate, and \(\Delta\) is the maximum thickness of the MFL pattern. \(d(r)\) is found by substituting Eq. (14) into Eq. (12) . Note that the factor \(\Delta\) is a constant and only adds a constant phase shift to Eq. (14). Therefore, we will ignore \(\Delta\) in the remainder of our development. If \(n_i=1\), the result is

\[\tag{15}d_m(r)=\frac{\lambda(m+1)}{n_\text{MFL}-1}-\frac{\sqrt{f^2+r^2}-f}{n_\text{MFL}-1}\]

where we have arbitrarily set \(\psi^-(r)=0\). The total number of zones \(M\) for a lens of radius \(r_M\) is

\[\tag{16}M=\frac{r_M(1-\sqrt{1-\text{NA}^2})}{\lambda\text{NA}}\]

The minimum zone period, \(\Lambda_\text{min}\), occurs at the outermost part of the lens and is given by

\[\tag{17}\Lambda_\text{min}=r_M-r_{M-1}=r_M\left(1-\sqrt{1-\frac{2\lambda{f}+(2M-1)\lambda^2}{2M\lambda{f}+(M\lambda)^2}}\right)\]

The following approximations may be used without significant error if \(\text{NA}\lt0.2\) and \(M\gg1\):

\[\tag{18}d_m(r)\approx\frac{m\lambda{f}-0.5r^2}{f(n_\text{MFL}-1)}\]

\[\tag{19}r_m\approx\sqrt{2m\lambda{f}}\]

\[\tag{20}M\approx\frac{r_M}{2\lambda}\text{NA}\]

and

\[\tag{21}\Lambda_\text{min}\approx\frac{\lambda}{\text{NA}}\]

The consequence of using Eqs. (18) and (19) for NA > 0.2 is that a small amount of spherical aberration is introduced into the system.

The aberration characteristics of the MFL and the Fresnel zone plate are very similar. For convenience, we describe a zone plate with the stop at the lens that is illuminated with a plane wave at angle \(\alpha\).

For an MFL made according to Eq. (15) and used at the proper conjugates, there will be no spherical aberration or distortion. Coma, astigmatism, and field curvature are given by \(W_{131}=\alpha{r_M^3}/2\lambda{f^2}\), \(W_{222}=\alpha^2r_M^2/2\lambda{f}\), and \(W_{220}=\alpha^2r_M^2/4\lambda{f}\), respectively.

When \(M\gg1\) and \(\alpha\) is small, the dominant aberration is coma, \(W_{131}\). If the substrate of the zone plate is curved with a radius of curvature equal to the focal length, coma can be eliminated. Chromatic variations in the focal length of the MFL are also similar to a Fresnel zone plate. For NA < 0.2,

\[\tag{22}\lambda{f}\approx\frac{r_M^2}{2M}\]

A focal-length-shift versus wavelength comparison of a Fresnel (hologram) lens and some single-element bulk-optic lenses are shown in Fig. 24. Note that the dispersion of the MFL is much greater than the bulk lens, and the dispersion of the MFL is opposite in sign to that of the bulk lenses. These facts have been used to design hybrid achromats by combining bulk lenses and diffractive lenses into the same system.

 

 

The thermal variations in MFLs primarily result in a change of focal length given by

\[\tag{23}\Delta{f}=2f\alpha_g\Delta{T}\]

where \(f\) is the nominal focal length, \(\alpha_g\) is the coefficient of thermal expansion for the substrate material, and \(\Delta{T}\) is a uniform temperature change of the element. For most optical glasses, \(\alpha_g\) ranges from \(5\times10^{-4}{}^\circ\text{C}^{-1}\) to \(10\times10^{-4}{}^\circ\text{C}^{-1}\).

There are several technologies that have been used to fabricate MFLs. These include electron-beam writing in resist, laser writing in resist, diamond turning, and molding.

Electron-beam writing in resist usually involves complicated translation stages under computer control in a high-vacuum environment, as shown in Fig. 25.

 

 

The focused electron beam is scanned over the sample, and the amount of exposure is controlled by varying the electron-beam current, dwell time, or number of repetitive scans. After exposure, the resist is developed. Smooth-zone profiles are obtained by properly varying the exposure. An example of a depth-versus-dose curve for PMMA resist is given in Fig. 26.

 

 

Notice that this exposure differs from that used in an electronic semiconductor device process, where only binary (fully exposed or unexposed) patterns are of interest. For a 0.2-μm-thick resist, this technique has produced 0.1 μm patterning of grating profiles. However, according to Eq. (15), the medium must be at least \(\lambda/(n_\text{MFL}-1)\) deep, which is greater than 1.0 μm for most applications.

If the pattern is to be transferred into a glass substrate with ion milling, the resist thickness must be adjusted according to the differential etching rate between the resist and the glass. It has been argued that, due to the relatively thick resist requirements, the resolution of electron-beam writing is similar to that of optical writing. The minimum blazed zone width that can be fabricated reliably with either technique is 2 to 3 μm. The problem of using thick resists can be avoided to some degree if one uses a reflection MFL rather than a transmission MFL, because less surface-height relief is required.

Optical writing in photoresist requires similar positioning requirements to those used in electron-beam writing. However, it is not necessary to perform the exposure in a high-vacuum environment. Instead of using an electron beam, optical writing uses a focused laser beam, as shown in Fig. 27.

 

 

Usually, some form of autofocus control is implemented to keep the spot size as small as possible. Standard photoresists, like Shipley 1400-27, may be used, so the wavelength of the exposure laser is typically 442 nm (HeCd) or 457 nm (Ar\(^+\)). Exposure is controlled by varying the power of the laser beam or by varying the number of repetitive scans.

The diffraction efficiency, \(\eta\), of an MFL is defined as the ratio of the power in the focused spot to the power in the unfocused beam transmitted through the lens. At best, Fresnel zone plates exhibit \(\eta=40.5\%\). Blazing the grating profile can significantly increase the efficiency of the lens.

Theoretically, \(\eta\) of an MFL can be 100 percent with the proper profile. However, there are several process parameters that limit \(\eta\), as shown in Fig. 28, where a perfect zone profile has width \(T\) and height \(d_\text{max}=\lambda/(n_\text{MFL}-1)\).

 

 

Variation of film thickness, over-etching, and swell all exhibit sinc-squared dependency on the errors. Shoulder imperfection is the most critical parameter, with \(\eta\) proportional to \((s/T)^2\). For \(\eta\) > 90%, \(s/T\ge0.95\), which implies that the falling edge of the zone profile must take no more than 5 percent of the grating period. This is possible with low NA systems, where the zone spacing is large compared to the resolution of the exposure system, but it becomes difficult in high NA systems, where the zone spacing is on the order of several microns.

Analysis of the three remaining parameters indicates fairly loose tolerances are acceptable. For \(\eta\) > 98%, tolerance on individual parameters are: \(|d(n-1)/\lambda-1|\lt0.25\), \(a/T\gt0.5\), and \(\Delta{d}(n-1)/\lambda\lt0.50\).

Due to the increasing difficulty in fabricating correct zone profiles with decreasing zone width, most MFLs exhibit a variation in diffraction efficiency versus radius. In the center of the zone pattern, where the zone spacing is large, the measured diffraction efficiency can be in excess of 90 percent. At the edge of the zone pattern, where the zone spacing can be on the order of a few wavelengths, the measured diffraction efficiency is much lower.

One possible solution to this problem is to use ‘‘superzone’’ construction, in which the zone radii are found from a modified form of Eq. (10), that is

\[\tag{24}k(f-\sqrt{f^2+r_M^2})=2\pi{Nm}\]

where \(N\) is the superzone number. This results in a maximum thickness of \(d_\text{max}=N\lambda/(n_\text{MFL}-1)\). Note that \(N=1\) corresponds to the standard MFL. \(N=2\) implies that zones are spaced at every \(4\pi\) phase transition boundary instead of at every \(2\pi\) phase transition boundary. Although this makes the zones wider apart, the surface relief pattern must be twice as thick.

Molding provides a potentially valuable process for fabricating large quantities of MFLs economically. MFLs can be produced with conventional injection molding, but due to the large thermal expansion coefficient of polymers, the lenses are sensitive to thermal variations.

An alternative MFL molding process is shown in Fig. 29, where a glass substrate is used to avoid large thermal effects.

 

 

First, a master lens is formed with electron-beam or laser writing. A stamper is prepared using conventional nickel electro-forming methods. After potting a UV-curable resin between the stamper and the glass substrate, the replica lenses are molded by the photopolymerization (2P) process. The wavefront aberration versus temperature for a \(\lambda\) = 780 nm, NA = 0.25, diameter = 0.5 mm lens formed with this technique is shown in Fig. 30.

 

 

A variation on this technique is to use the stamper as a substrate in an electron-beam evaporation device. Inorganic materials of various refractive indices can be deposited on the stamper, resulting in a thin lens of high refractive index. The high refractive index of a material like ZnS (\(n\) = 2.35) can be used to lower the maximum thickness requirement of the lens, which makes fabrication of the master with electron-beam writing easier.

 

8. Other Technologies

There are several other technologies that are potentially valuable for micro-optic components. Four particularly interesting technologies are melted-resin arrays, laser-assisted chemical etching, mass transport, and drawn preform cylindrical lenses.

Melted-resin arrays are formed with the process shown in Fig. 31. First, an Al film is deposited on a quartz substrate and patterned with holes that serve as aperture stops for the array. Next, circular pedestals are formed on top of the aperture holes. The pedestals are hardened so that they are insoluble and stable for temperatures in excess of 180\(^\circ\)C. Cylinders of resin are then developed on top of the pedestals. The device is heated to 140\(^\circ\)C to melt the resin. The pedestals serve to confine the melting resin. The lenses form into hemispherical shapes due to surface tension forces. Lens diameters have been demonstrated at 30 μm with good wavefront performance and uniformity.

 

 

Laser-assisted chemical etching (LACE) can be used to make arrays of F/0.7 to F/10 lenslets with spacings of 50 to 300 μm. Microlenses have been fabricated in glass, silicon, CdTe, and sapphire with 95 percent fill factors and figure quality better than 1/10th wave. In the LACE process, a focused laser beam is scanned over a thick layer of photoresist. The irradiance of the laser beam is modulated in order to vary the exposure and thus the thickness of the developed resist. With the proper irradiance mapping, accurate lens profiles can be produced in the developed resist. If lenslet material other than photoresist is required, a pattern can be exposed in the photoresist and transferred into the new material by ion milling.

In the mass-transport process, a multilevel mesa structure is first etched into a semiconductor, as shown in Fig. 32a. The semiconductor must be a binary compound in which the evaporation rate of one element is negligible compared to that of the other. For example, InP has been used successfully.

 

 

The mesa structure is placed in a furnace at an elevated temperature. Since some surface decomposition occurs with InP, a minimum phosphorus vapor pressure must be maintained in the gas ambient to prevent the sample from being transformed into metallic In.

The decomposition produces free In atoms located at the crystal surface, which are in equilibrium with phosphorus in the vapor and InP in the crystal. The concentration of In in the vapor is negligible. The equilibrium concentration of free In atoms increases with increasing positive surface curvature, since the higher surface energy of the high-curvature regions translates into a lower bonding energy in the decomposition process. Consequently, a variation in curvature across the surface will result in diffusion of free In atoms from regions of high positive curvature, where the concentrations are high, to low-curvature regions, where the In-diffused atoms exceed the equilibrium concentration and have to be reincorporated into the crystal by reaction with P to form InP. (The diffusion of P in the vapor phase is presumably much faster than the diffusion of free In atoms on the surface. The latter is therefore assumed to be the rate-limiting process.)

The mass transport of InP, resulting from decomposition in high-curvature regions, will continue until the difference in curvature is completely eliminated. After mass transport, a smooth profile is obtained, as shown in Fig. 32b. The design of the mesa structure can result in very accurate control of the lens profile, as shown in Fig. 33. Mass-transport lens arrays have been used to collimate arrays of laser diodes, with diffraction-limited performance at NA ~ 0.5.

 

 

Very accurate cylindrical lenses can be drawn from preforms. An SEM photo of an elliptical cylindrical lens is shown in Fig. 34. The first step in the process is to make a preform of a suitable glass material. Since the cross-sectional dimensions of the preform are uniformly reduced (typically, 50 – 100X) in the fiber drawing process, small manufacturing errors become optically insignificant. Therefore, standard numerically controlled grinding techniques can be utilized to generate a preform of any desired shape.

Besides maintaining the preform shape, the drawing process also polishes the fiber. Results are presented that demonstrate a 200-μm-wide elliptical cylindrical lens. The SFL6 preform was about 0.75 cm wide. The lens has a nominal focal length of 220 μm at \(\lambda\) = 800 μm. The lens is diffraction-limited over about a 150-μm clear aperture, or NA ~ 0.6. The application is to collimate the fast axis of laser diodes.

 

 

The next tutorial discusses in detail about what are hollow-core fibers?