CHAPTER 8 BINARY OPTICS - Photonics Research...

CHAPTER 8 BINARY OPTICS

Michael W . Farn and Wilfrid B . Veldkamp MIT / Lincoln Laboratory Lexington , Massachusetts

8 . 1 GLOSSARY

A aspheric

C describes spherical aberration

C m Fourier coef ficients

c curvature

c ( x , y ) complex transmittance

D local period

f focal length

k , l running indices

l i paraxial image position

L , M direction cosines

m dif fraction order

P partial dispersion

s spheric

t thickness

V d Abbe number

x , y , z Cartesian coordinates

l wavelength

h dif fraction ef ficiency

j i paraxial image height

f ( x , y ) phase

0 , i iterative points

9 dif fracted

8 .1

8 .2 OPTICAL ELEMENTS

8 . 2 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 suf ficiently to allow the production of dif fractive elements , hybrid refractive- dif fractive 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 dif fraction theory , or vector dif fraction 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 . For a more general overview , the reader is referred to Refs . 1 for many of the original papers , 2 and 3 for a sampling of applications and research , and 4 6 for a lay overview .

Directly related areas which are discussed in other sections but not in this section include micro-optics and dif fractive 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 .

8 . 3 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 of fers 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

BINARY OPTICS 8 .3

Analytical Models

Representation of a Binary Optics Element . As with any dif fractive element , a binary optics structure is defined by its phase profile f ( x , y ) ( z is taken as the optical axis) , design wavelength l 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 / dif fractive 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 l 0 has a phase profile

f ( x , y ) 5 2 l 0

[ z o ( 4 ( x 2 x o ) 2 / z 2 o 1 ( y 2 y o ) 2 / z 2 o 1 1 2 1)

2 z i ( 4 ( x 2 x i ) 2 / z 2 i 1 ( y 2 y i ) 2 / z 2 i 1 1 2 1)] (1)

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 :

1 / f 0 5 1 / z i 2 1 / z o (2)

with the subscript indicating that f 0 is the focal length at l 0 . A grating which deflects a normally incident ray of wavelength l 0 to the direction with direction cosines ( L , M ) is described by

f ( x , y ) 5 2 l 0

( xL 1 yM ) (3)

Axicons are circular gratings and are described by

f ( x , y ) 5 2 l 0

( 4 x 2 1 y 2 L ) (4)

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 :

f ( x , y ) 5 f S ( x , y ) 1 f A ( x , y ) (5)

where

f A ( x , y ) 5 2 l 0 O

k O

l a k l x

k y l

and the spheric term f S ( x , y ) takes the form of Eq . (1) . Since the phase profiles produced by binary optics technology are not constrained to be spheric , f 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 coef ficients a k l . If necessary , the aspheric term can be forced to be radially symmetric by constraining the appropriate polynomial coef ficients .

It is possible to describe the phase profile of a binary optics element in other ways . For example , f ( 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 f ( 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

L 9 5 L 1 m l 2

f

x (6)

M 9 5 M 1 m l 2

f

y (7)

where m is the dif fracted order , L , M are the direction cosines of the incident ray , and L 9 , M 9 are the direction cosines of the dif fracted ray . 7 In geometrical designs , the element is usually blazed for the first order ( m 5 1) . Note that it is the phase gradient = f ( x , y ) (a vector quantity proportional to the local spatial frequency) and not the phase f ( 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

D ( x , y ) 5 2 / u = f u (8)

where u u 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 chapter) .

Ray Tracing by the Sweatt Model . The Sweatt model , 8 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 f ( x , y ) at wavelength l 0 is replaced by a refractive equivalent with thickness and refractive index given by

t ( x , y ) 5 l 0

n 0 2 1 f ( x , y )

2 1 t 0 (9)

n ( l ) 2 1 5 l

l 0 ( n 0 2 1) (10)

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 l 0 . The index n 0 is chosen by the designer and as n 0 5 , the Sweatt model approaches the grating equation . In practice , values of n 0 5 10 , 000 are suf ficiently high for accurate results . 9

In the special case of a binary optics lens described by Eq . (1) , the more accurate Sweatt lens 1 0 can be used . In this case , the element is modeled by two surfaces of curvature

c o 5 1 / [(1 2 n 0 ) z o ] (11)

c i 5 1 / [(1 2 n 0 ) z i ] (12)

BINARY OPTICS 8 .5

FIGURE 1 Primary aberrations of a binary optics lens . 7

and conic constant 2 n 2 0 , 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 l 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 l located at (0 , j o , l o ) . The lens will form an image at (0 , j i , l i ) (see Fig . 1) , with the paraxial image position l i and height j i given by

7

1 l i

5 l

f 0 l 0 1

1 l 0

(13)

j i / l i 5 j o / l o (14)

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

f ( l ) 5 f 0 l 0

l (15)

The focal length being inversely proportional to the wavelength is a fundamental property of dif fractive 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 7

W ( x , y ) 5 1 8 F S 1

l 3 i 2

1 l 3 o D 2 l

l 0 S 1

z 3 i 2

1 z 3 o D G ( x 2 1 y 2 ) 2

2 1 2 l i

S 1 l 2 i

2 1 l 2 o D j i y ( x 2 1 y 2 )

1 3

4 l 2 i S 1

l i 2

1 l o D j 2 i y 2 1 1 4 l 2 i S

1 l i

2 1 l o D j 2 i x 2 (16)

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 of f-axis aberrations can be eliminated if and only if l i 5 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 . For a more detailed discussion , see Refs . 1 , 7 , and 11 .

Chromatic Aberration Correction . Binary optics lenses inherently suf fer 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 dif fractive lenses cannot produce a real image . 1 2

However , these lenses can be combined successfully with refractive lenses to achieve chromatic correction (for a more detailed discussion than what follows , see Refs . 4 , 13 , and 14) . 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 5 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 ef fective 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 5 [ n ( l d ) 2 1] / [ n ( l F ) 2 n ( l C )] . Substituting Eq . (10) and letting n 0 5 yields

V d 5 l d / ( l F 2 l C ) 5 2 3 . 45 (17)

In a similar fashion , the ef fective partial dispersion using the g and F lines is

P g F 5 ( l g 2 l F ) / ( l F 2 l C ) 5 0 . 296 (18)

By using these ef fective values , the conventional procedure for designing achromats and apochromats 1 5 can be extended to designs in which one element is a binary optics lens .

Figure 2 plots the partial dispersion P g F 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 .

FIGURE 2 Partial dispersion vs . Abbe number . 1 4

BINARY OPTICS 8 .7

Monochromatic Aberration Correction . For a detailed discussion , the reader is referred to Refs . 1 and 11 . As a simple example , 4 consider a refractive system which suf fers from third-order spherical aberration and has a residual phase given by

f r ( x , y ) 5 2 l

C ( x 2 1 y 2 ) 2 (19)

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

f b ( x , y ) 5 2 2 l 0

C ( x 2 1 y 2 ) 2 (20)

will completely correct the aberration at wavelength l 0 and will reduce the aberration at other wavelengths to

f r 1 f b 5 2 l

C (1 2 l / l 0 )( x 2 1 y 2 ) 2 (21)

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 :

FIGURE 3 96 3 64 Array of 51 3 61 m m CdTe microlenses .


FIGURE 4 Micro-optic telescope using ( a ) coherent arrays ; ( b ) incoherent arrays .

$ 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) .

$ Refracti y e optics . Binary optics is usually associated with dif fractive optics . This is not a fundamental limit but results primarily from fabrication constraints on the maximum achievable depth (typically , 3 m m with ease and up to 20 m m with ef fort) . However , for many micro-optics , this is suf ficient to allow the etching of refractive elements . For example , a lens or radius R 0 which is corrected for spherical aberration 1 5 and focuses collimated light at a distance z 0 (see Fig . 5) has a thickness of

t m a x 5 n [ 4 R 2 0 1 z

2 0 2 z 0 ] / ( n 2 1) (22)

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 dif ficult than fabricating arrays of conventional spherical lenses .

$ 1 0 0 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 dif ferent 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 . 1 6

FIGURE 5 Thickness of a refractive lens . 1 5

BINARY OPTICS 8 .9

Optical Performance

Wa y efront 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 m m) is usually much larger than the resolution with which it can be reproduced (of order 0 . 1 m m) , 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 .

Dif fraction Ef ficiency . The dif fraction ef ficiency of a device is determined by how closely the binary optics stepped-phase profile approximates a true blaze . The theoretical ef ficiency at wavelength l of an element with I steps designed for use at l 0 is :

4

h ( l , I ) 5 U sinc (1 / I ) sin ( I a ) I sin a U

2

(23)

where sinc ( x ) 5 sin ( x ) / ( x )

a 5 ( l 0 / l 2 1) / I

This result is based on scalar theory , assumes perfect fabrication , and neglects any material dispersion . Figure 6 plots the ef ficiency h ( l , I ) for dif ferent numbers of steps I ;

FIGURE 6 Dif fraction ef ficiency of binary optics . 4


TABLE 1 Average Dif fraction Ef ficiency for Various Bandwidths 4

D l / l 0 h #

0 . 00 0 . 10 0 . 20 0 . 30 0 . 40 0 . 50 0 . 60

1 . 00 1 . 00 0 . 99 0 . 98 0 . 96 0 . 93 0 . 90

while Table 1 gives the average ef ficiency over the bandwidth D l for a perfectly blazed element ( I 5 ) . 4 The ef ficiency equation is asymmetric in l but symmetric in 1 / l .

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 l 0 or a lens faster than F / 5) and lower ef ficiencies can be expected in these cases . For a more detailed discussion , see Ref . 17 .

The ef ficiency discussed here is the dif fraction ef ficiency of an element . Light lost in this context is primarily dif fracted into other dif fraction orders , which can also be traced through a system to determine their ef fect . As with conventional elements , binary optics elements will also suf fer reflection losses which can be minimized in the usual manner .

8 . 4 DESIGN SCALAR DIFFRACTION THEORY

Designs based on scalar dif fraction 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 f ( x , y ) is modeled as a thin-phase screen with a complex transmittance of

c ( x , y ) 5 exp [ j f ( x , y )] (24)

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 dif fraction , Fraunhofer dif fraction) 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 coef ficients C m , where

C m 5 1 D E D

0 c ( x ) exp ( 2 j 2 mx / D ) dx (25)

c ( x ) 5 O m 52

C m exp ( j 2 mx / D ) (26)

BINARY OPTICS 8 .11

The relative intensity or ef ficiency of the m th dif fracted order of the grating is

h m 5 u C m u 2 (27)

Due to the fabrication process , binary optics gratings are piecewise flat . The grating transmission in this special case can be expressed as c ( x ) 5 c i for x i , x , x i 1 1 , where c i is the complex transmission of step i of I total steps , x 0 5 0 , and x I 5 D . The Fourier coef ficients then take the form

C m 5 O I 2 1 i 5 0

c i d i exp ( 2 j 2 m D i ) sinc ( m d i ) (28)

where d i 5 ( x i 1 1 2 x i ) / D

D i 5 ( x i 1 1 1 x i ) / (2 D )

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 5 iD / I and Eq . (28) reduces to

C m 5 exp ( 2 j m / I ) sinc ( m / I ) F 1 I O I 2 1

i 5 0 c i exp ( 2 j 2 mi / I ) G (29)

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 f 0 , then c i 5 exp ( ji f 0 ) and Eq . (29) further reduces to 1 8

C m 5 exp [ j (( I 2 1) a 2 m / I )] sinc ( m / I ) sin ( I a ) I sin a

(30)

where a 5 f 0 / (2 ) 2 m / I

This important case occurs whenever a true blaze is approximated by a stepped-phase profile . The ef ficiency 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 dif fracted beams which may be of equal intensity or weighted in intensity . 1 9 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 ef ficiency and intensity uniformity of the dif fracted beams , while binary multiplexers are easier to fabricate (with the exception

TABLE 2 Grating Multiplexers of Period D , 0 , x , D

Phase profile h 2 1 h 0 h 1 Remarks

f ( x , y ) 5 H 0 x , D / 2 D / 2 , x 0 . 41 0 0 . 41 Binary 1 : 2 splitter f ( x , y ) 5 H 0 2 . 01 x , D / 2 D / 2 , x 0 . 29 0 . 29 0 . 29 Binary 1 : 3 splitter f ( x , y ) 5 x / D

f ( x , y ) 5 arctan [2 . 657 cos (2 x / D )] 3

0 . 31 0 . 41 0 . 31

0 . 41 0 . 31

Continuous 1 : 2 splitter Continuous 1 : 3 splitter


of several naturally occurring continuous phase profiles) . Upper bounds for the ef ficiency of both continuous and binary types are derived in Ref . 20 .

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 . 2 1 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 ef ficiencies of 90 100 percent are typical .

The complex transmittance of a binary grating has only two possible values [typically 1 1 and 2 1 , or exp ( j f 0 ) and exp ( 2 j f 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 , h m 5 h 2 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 h m / h n for all m , n ? 0 will be constant , regardless of the actual complex transmittance values .

$ The complex transmittance values only af fect 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 . 2 2 , 2 3 Ef ficiencies 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 1 y 2 ) , nonparaxial lenses in 4 x 2 1 y 2 1 f 2 0 , and axicons in 4 x 2 1 y 2 . For gratings , dif ferent dif fraction orders correspond to plane waves traveling in dif ferent directions , but for a lens of focal length f 0 , the m th dif fraction order corresponds to a lens of focal length f 0 / m . By splitting the light into dif ferent dif fraction 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 1 orders . The phase profile of a single focus lens is given by f ( r ) 5 2 2 r 2 / (2 l 0 f 0 ) , where r 2 5 x 2 1 y 2 . This phase , with the 2 ambiguity removed , is plotted in Fig . 7 a as a function of r and in Fig . 7 b as a function of r 2 , where the periodicity in r 2 is evident . To split the light between the 0 and 1 1 orders , the blaze of Fig . 7 b is replaced by the 1 : 2 continuous splitter of Table 2 , resulting in Fig . 7 c . This is the final design and the phase profile is displayed in Fig . 7 d as a function of r .

Beam Shapers and Dif fusers

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 dif fraction reshape the beam as it propagates . If the incoming beam is well characterized , then it is possible to deterministically design the binary optics element . 2 4 For example , Fig . 8 a shows the focal spot of a Gaussian beam without any beam-forming optics . In Fig . 8 b , 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 phase shift over a central region . If the beam is not well-behaved , then a statistical design may be more appropriate . 2 5 For example , in Fig . 8 c , 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

BINARY OPTICS 8 .13

FIGURE 7 Designing a bifocal lens : ( a ) lens with a single focus ; ( b ) same as ( a ) , but showing periodicity in r 2 ; ( c ) substitution of a beam-splitting design ; ( d ) same as ( c ) , but as a function of r .

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 , 2 6 the fabrication of phase-only components for optical correlators , 2 7 and the implementation of coordinate transformations . 1 6 , 2 8 In all these applications , binary optics is used to directly manipulate the phase of a wavefront .

FIGURE 8 Reshaping a focused beam : ( a ) Gaussian focus ; ( b ) deterministic beam-shaper ; ( c ) statistical dif fuser .


FIGURE 9 Artificial index designs : ( a ) antireflection layer ; ( b ) form birefringence .

8 . 5 DESIGN VECTOR DIFFRACTION THEORY

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

Grating-based designs rely on solving Maxwells equations for dif fraction 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) . 2 9 In this category , optical design is dif ficult 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 . 9 a , the device behaves as an antireflection coating (analogous to anechoic chambers) since , at dif ferent depths , the structure has a dif ferent average index , continuously increasing from n 1 to n 2 . In Fig . 9 b , the regular , subwavelength structure exhibits form birefringence . 3 0 For light polarized with the electric vector perpendicular to the grooves , the ef fective index is

1 n 2 ef f

5 p 1 n 2 1

1 (1 2 p ) 1 n 2 2

(31)

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

n 2 ef f 5 pn 2 1 1 (1 2 p ) n

2 2 (32)

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 .

8 . 6 FABRICATION

Mask Layout

At the end of the optical design stage , the binary optics element is described by a phase profile f ( 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 .

BINARY OPTICS 8 .15

FIGURE 10 Translation from f ( x , y ) to micromachined surface : ( a ) phase f ( x , y ) ; ( b ) thickness t ( x , y ) ; ( c ) binary optics profile t 9 ( x , y ) .

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

t ( x , y ) 5 l 0

2 ( n 0 2 1) ( f mod 2 ) (33)

where l 0 is the design wavelength and n 0 is the index of the substrate at l 0 . The thickness profile is the surface relief required to introduce a phase shift of f ( x , y ) . The thickness varies continuously from 0 to t 0 , where

t 0 5 l 0 / ( n 0 2 1) (34)

is the thickness required to introduce one wave of optical path dif ference . To facilitate fabrication , t ( x , y ) is approximated by a multilevel profile t 9 ( x , y ) (Fig .

10 c ) , 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

I 5 2 N (35)

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 f ( 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 3 1 or by using a numerical contouring method , 3 2 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 . 11 a ,b ) . Common formats are GDSII and CIF , 3 3 both of which are high-level graphics descriptions which


TABLE 3 Processing Steps for Binary Optics

Layer Etch region , defined by t ( x , y ) Etch region , defined by f ( x , y ) Etch depth

1 2 3 4

0 , t mod ( t 0 ) , t 0 / 2 0 , t mod ( t 0 / 2) , t 0 / 4 0 , t mod ( t 0 / 4) , t 0 / 8 0 , t mod ( t 0 / 8) , t 0 / 16

0 , f mod 2 , 0 , f mod , / 2 0 , f mod / 2 , / 4 0 , f mod / 4 , / 8

t 0 / 2 t 0 / 4 t 0 / 8 t 0 / 16

FIGURE 11 Mask layout descriptions : ( a ) mathematical description based on thickness contours ; ( b ) high-level graphics description ; ( c ) MEBES .

FIGURE 12 Quantization angle .

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 m m with ease , 0 . 5 m m with ef fort , and 0 . 3 m m 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 m i n / I . For lenses ,

D m i n 8 2 l 0 F (36)

where F is the F-number of the lens ; while , for gratings , D m i n 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 . 11 c ) . The resolution (typically 0 . 05 m m) 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

d 5 R (1 2 cos / 2) 8 R 2 / 8 (37)

BINARY OPTICS 8 .17

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 3 4 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 . 3 4 3 5 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 of fer lower equipment costs and more flexibility in handling dif ferent 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 m m , results in lower resolution due to dif fraction . Both contact and proximity printing require 1 : 1 masks . In projection printing , the mask is imaged onto the photoresist with a demagnification from 1 3 to 20 3 . 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 3 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 ef fect of fabrication errors is to decrease the ef ficiency 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 ( , 0 . 05 D m i n / 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 dif ficult . For


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

8 . 7 REFERENCES

1 . Holographic and Dif fractive Lenses and Mirrors , Proc . Soc . Photo - Opt . Instrum . Eng . , Milestone Series 34 , 1991 .

2 . Computer and Optically Generated Holographic Optics series , Proc . Soc . Photo - Opt . Instrum . Eng . , 1052 , 1989 ; 1211 ; 1990 ; 1555 , 1991 .

3 . Miniature and Microoptics series , Proc . Soc . Photo - Opt . Instrum . Eng . , 1544 , 1991 and 1751 , 1992 .

4 . G . J . Swanson , Binary Optics Technology : The Theory and Design of Multi-level Dif fractive Optical Elements , M . I . T . Lincoln Laboratory Technical Report 854 , NTIS Publ . AD-A213-404 , 1989 .

5 . M . W . Farn and W . B . Veldkamp , Binary Optics : Trends and Limitations , Conference on Binary Optics , NASA Conference Publication 3227 , 1993 , pp . 19 30 .

6 . S . H . Lee , Recent Advances in Computer Generated Hologram Applications , Opt . and Phot . News , 16 : 7 , 1990 , pp . 18 23 .

7 . W . T . Welford , Aberrations of Optical Systems , Adam Hilber , Ltd ., Boston , 1986 , pp . 75 78 , 217 225 .

8 . W . C . Sweatt , Mathematical Equivalence between a Holographic Optical Element and an Ultra-high Index Lens , J . Opt . Soc . Am . , 69 , 1979 , pp . 486 487 .

9 . M . W . Farn , Quantitative Comparison of the General Sweatt Model and the Grating Equation , Appl . Opt . , 1992 , pp . 5312 5316 .

10 . W . C . Sweatt , Describing Holographic Optical Elements as Lenses , J . Opt . Soc . Am . , 67 , 1977 , pp . 803 808 .

11 . D . A . Buralli and G . M . Morris , Design of Dif fractive Singlets for Monochromatic Imaging , Appl . Opt . , 30 , 1991 , pp . 2151 2158 .

12 . D . A . Buralli and J . R . Rogers , Some Fundamental Limitations of Achromatic Holographic Systems , J . Opt . Soc . Am . , A6 , 1989 , pp . 1863 1868 .

13 . C . W . Chen , Application of Dif fractive Optical Elements in Visible and Infrared Optical Systems , Proc . Soc . Photo - Opt . Instrum . Eng . CR41 , 1992 , pp . 157 172 .

14 . T . Stone and N . George , Hybrid Dif fractive-refractive Lenses and Achromats , Appl . Opt . , 27 , 1988 , 2960 2971 .

15 . R . Kingslake , Lens Design Fundamentals , Academic Press , Inc ., New York , 1978 , pp . 77 78 , 112 114 .

16 . J . R . Leger and W . C . Goltsos , Geometrical Transformation of Linear Diode-laser Arrays for Longitudinal Pumping of Solid-state Lasers , IEEE J . of Quant . Elec . , 28 , 1992 , pp . 1088 1100 .

17 . G . J . Swanson , Binary Optics Technology : Theoretical Limits on the Dif fraction Ef ficiency of Multilevel Dif fractive Optical Elements , M . I . T . Lincoln Laboratory Technical Report 914 , 1991 .

18 . H . Dammann , Spectral Characteristics of Stepped-phase Gratings , Optik , 53 , 1979 , pp . 409 417 .

19 . A . Vasara , et al ., Binary Surface-Relief Gratings for Array Illumination in Digital Optics , Appl . Opt . 31 , 1992 , pp . 3320 3336 .

20 . U . Krackhardt , et al ., Upper Bound on the Dif fraction Ef ficiency of Phase-only Farnout Elements , Appl . Opt . , 31 , 1992 , pp . 27 37 .

21 . D . Prongue , et al ., Optimized Kinoform Structures for Highly Ef ficient Fan-Out Elements , Appl . Opt . 31 , 1992 , pp . 5706 5711 .

BINARY OPTICS 8 .19

22 . U . Killat , G . Rabe , and W . Rave , Binary Phase Gratings for Star Couplers with High Splitting Ratios Fiber and Integrated Optics , 4 , 1982 , pp . 159 167 .

23 . U . Krackhardt , Binaere Phasengitter als Vielfach-Strahlteiler , Diplomarbeit , Uni y ersitaet Erlangen - Nuernberg , Erlangen , Germany , 1989 .

24 . J . Hossfeld , et al ., Rectangular Focus Spots with Uniform Intensity Profile Formed by Computer Generated Holograms , Proc . Soc . Photo - Opt . Instrum . Eng . , 1574 , 1991 , pp . 159 166 .

25 . C . N . Kurtz , Transmittance Characteristics of Surface Dif fusers and the Design of Nearly Band-Limited Binary Dif fusers , J . Opt . Soc . Am . , 62 , 1972 , pp . 982 989 .

26 . J . R . Leger , et al ., Coherent Laser Beam Addition : An Application of Binary-optics Technology , The Lincoln Lab Journal , 1 , 1988 , pp . 225 246 .

27 . M . A . Flavin and J . L . Horner , Amplitude Encoded Phase-only Filters , Appl . Opt . 28 , 1989 , pp . 1692 1696 .

28 . O . Bryngdahl , Geometrical Transforms in Optics , J . Opt . Soc . Am . , 64 , 1974 , pp . 1092 1099 . 29 . T . K . Gaylord , et al ., Analysis and Applications of Optical Dif fraction by Gratings , Proc . IEEE

73 , 1985 , pp . 894 937 . 30 . D . H . Raguin and G . M . Morris , Antireflection Structured Surfaces for the Infrared Spectral

Region , Appl . Opt . 32 , 1993 , pp . 1154 1167 . 31 . J . Logue and M . L . Chisholm , General Approaches to Mask Design for Binary Optics , Proc .

Soc . Photo - Opt . Instrum . Eng . , 1052 , 1989 , pp . 19 24 . 32 . A . D . Kathman , Ef ficient Algorithm for Encoding and Data Fracture of Electron Beam Written

Holograms , Proc . Soc . Photo - Opt . Instrum . Eng . , 1052 , 1989 , pp . 47 51 . 33 . S . M . Rubin , Computer Aids for VLSI Design , Addison-Wesley Publishing Co ., Reading , MA ,

1987 . 34 . N . G . Einspruch and R . K . Watts (eds . ) , Lithography for VLSI , VLSI Electronics Series 16 ,

Academic Press , Inc ., Boston , MA , 1987 . 35 . N . G . Einspruch and D . M . Brown (ed . ) , Plasma Processing for VLSI , VLSI Electronics Series 8 ,

Academic Press , Inc ., Boston , MA , 1984 .

Volume IIntroductionPart 1 - Geometric OpticsChapter 1 -General Principles of Geometric Optics

Part 2 - Physical OpticsChapter 2 - InterferenceChapter 3 - DiffractionChapter 4 - Coherence TheoryChapter 5 - PolarizationChapter 6 - Scattering by ParticlesChapter 7 - Surface Scattering

Part 3 - Quantum OpticsChapter 8 - Optical Spectroscopy and Spectroscopic LineshapesChapter 9 - Fundamental Optical Properties of Solids

Part 4 - Optical SourcesChapter 10 - Artificial SourcesChapter 11 - LasersChapter 12 - Light Emitting DiodesChapter 13 - Semiconductor LasersChapter 14 - Ultrashort Laser Sources

Part 5 - Optical DetectorsChapter 15 - PhotodetectorsChapter 16 - PhotodetectionChapter 17 - High-Speed PhotodetectorsChapter 18 - Signal Detection and AnalysisChapter 19 - Thermal Detectors

Part 6 - Imaging DetectorsChapter 20 - Photographic FilmsChapter 21 - Image Tube Intensified Electronic ImagingChapter 22 - Visible Array DetectorsChapter 23 - Infrared Detector Arrays

Part 7 - VisionChapter 24 - Optics of the EyeChapter 25 - Visual PerformanceChapter 26 - ColorimetryChapter 27 - Displays for Vision ResearchChapter 28 - Optical Generation of the Visual StimulusChapter 29 - Psychophysical Methods

Part 8 - Optical Information and Image ProcessingChapter 30 - Analog Optical Signal and Image ProcessingChapter 31 - Principles of Optical Disk Data Storage

Part 9 - Optical Design TechniquesChapter 32 - Techniques of First-Order LayoutChapter 33 - Aberration Curves in Lens DesignChapter 34 - Optical Design SoftwareChapter 35 - Optical SpecificationsChapter 36 - Tolerancing TechniquesChapter 37 - Mounting Optical ComponentsChapter 38 - Control of Stray LightChapter 39 - Thermal Compensation Techniques

Part 10 - Optical FabricationChapter 40 - Optical FabricationChapter 41 - Fabrication of Optics by Diamond Turning

Part 11 - Optical Properties of Films and CoatingsChapter 42 - Optical Properties of Films and Coatings

Part 12 - Terrestrial OpticsChapter 43 - Optical Properties of WaterChapter 44 - Atmospheric Optics

Volume IIIntroductionPart 1 - Optical ElementsChapter 1 - LensesChapter 2 - Afocal SystemsChapter 3 - PolarizersChapter 4 - Nondispersive PrismsChapter 5 - Dispersive Prisms and GratingsChapter 6 - Integrated OpticsChapter 7 - Miniature and Micro-OpticsChapter 8 - Binary Optics8 . 1 GLOSSARY8 . 2 INTRODUCTION8 . 3 DESIGN GEOMETRICAL OPTICS8 . 4 DESIGN SCALAR DIFFRACTION THEORY8 . 5 DESIGN VECTOR DIFFRACTION THEORY8 . 6 FABRICATION8 . 7 REFERENCES

Chapter 9 - Gradient Index OpticsChapter 10 - Optical Fibers and Fiber-Optic CommunicationsChapter 11 - X-Ray OpticsChapter 12 - Acousto-Optic Devices and ApplicationsChapter 13 - Electro-Optic ModulatorsChapter 14 - Liquid Crystals

Part 2 - Optical InstrumentsChapter 15 - CamerasChapter 16 - Camera LensesChapter 17 - MicroscopesChapter 18 - Reflective and Catadioptric ObjectivesChapter 19 - ScannersChapter 20 - Optical SpectrometersChapter 21 - InterferometersChapter 22 - PolarimetryChapter 23 - Infrared Detector Arrays

Part 3 - Optical MeasurementsChapter 24 - Radiometry and PhotometryChapter 25 - The Measurement of Transmission, Absorption, Emission, and ReflectionChapter 26 - ScatterometersChapter 27 - EllipsometryChapter 28 - Spectroscopic MeasurementsChapter 29 - Optical MetrologyChapter 30 - Optical testingChapter 31 - Use of Computer-Generated Holograms in Optical TestingChapter 32 - Transfer Function Techniques

Part 4 - Optical and Physical Properties of MaterialsChapter 33 - Properties of Crystals and GlassesChapter 34 - Polymetric OpticsChapter 35 - Properties of MetalsChapter 36 - Optical Properties of SemiconductorsChapter 37 - Black Surfaces for Optical Systems

Part 5 - Nonlinear and Photorefractive OpticsChapter 38 - Nonlinear OpticsChapter 39 - Photorefractive Materials and Devices

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