Design of fan-out kinoforms in the entire scalar
`diffraction regime with an optimal-rotation-angle method
`
`Jo¨ rgen Bengtsson
`
`An algorithm for the design of diffractive optical phase elements 共kinoforms兲 that give rise to fan-out 共i.e.,
`spot兲 patterns was developed and tested. The algorithm is based on the Helmholtz–Kirchhoff rigorous
`scalar diffraction integral for the evaluation of the electric field behind the kinoform. The optimization
`of the kinoform phase modulation is performed with an efficient optimal-rotation-angle method. The
`algorithm permits any spatial configuration of the locations of the desired spots. For example, the spots
`共all or some兲 can be located at large angles to the optical axis 共nonparaxial case兲 or they can be located
`in the near near field of the kinoform, i.e., where the Fresnel approximation is no longer valid. Two
`examples of fabricated kinoforms designed with this algorithm are presented. © 1997 Optical Society of
`America
`Key words: Diffractive optics design, kinoforms, fan-out diffraction pattern, rigorous scalar diffrac-
`tion, optimal rotation angle.
`
`Introduction
`1.
`A kinoform 共or diffractive optical element兲 is a thin
`component with a depth relief on one surface. The
`relief modulates the phase of a laser beam that
`passes through the kinoform. The kinoform relief
`can be designed so that the distribution of light be-
`hind the kinoform closely resembles some desired
`one.
`There are two principal types of desired light dis-
`tributions. One is the even distribution of light over
`some area in a plane behind the kinoform. This is
`often called beam shaping after its most common
`application.1–3 At least as important is the distribu-
`tion of light to small 共diffraction-limited兲 spots in
`space. This is sometimes called beam splitting, as it
`is often used for dividing the energy of a laser beam
`into many beams, or, to put it differently, for sending
`a signal from one channel 共optical fiber or semicon-
`ductor laser emitter兲 into many. This paper treats
`the design of kinoforms performing this latter task.
`They are often called fan-out kinoforms.
`The design algorithm described in this paper is
`
`The author is with the Optics Group, Department of Microwave
`Technology, Chalmers University of Technology, S-412 96 Go¨te-
`borg, Sweden.
`Received 14 March 1997; revised manuscript received 28 May
`1997.
`0003-6935兾97兾328435-10$10.00兾0
`© 1997 Optical Society of America
`
`based on scalar diffraction theory. For this theory to
`be valid, the feature size of the kinoform must not be
`too small.
`It is still debated as to what “too small”
`should mean.4 – 6 Fan-out diffractive elements de-
`signed with scalar diffraction theory with a smallest
`feature size of only approximately two wavelengths
`have been fabricated, and they work fairly well.7
`In
`the scalar diffraction regime, several methods for de-
`signing kinoforms exist. The methods are often
`used when the diffraction pattern is in the Fresnel or
`Fraunhofer region of the kinoform. Then the field in
`the diffraction plane is given as the Fourier trans-
`form of the product of the field in the kinoform plane
`and the Fresnel 共near-field兲 phase factor. Since the
`Fourier transform directly relates a spatial field dis-
`tribution 共in the kinoform plane兲 to another 共in the
`diffraction plane兲, this will be referred to as a spatial
`共This relation is as opposed to a
`Fourier transform.
`different way of evaluating the diffracted field, the
`so-called propagation of the angular spectrum, which
`also utilizes Fourier transforms but in which the
`transforms relate fields to angular spectra.兲
`The Fourier transform is often evaluated with the
`efficient fast Fourier transform 共FFT兲 algorithm.
`The so-called direct-search methods, like direct bi-
`nary search and simulated annealing,8 –11 are gener-
`ally demanding in terms of computer capacity. Less
`demanding are the iterative Fourier transform algo-
`rithms 共IFTA’s兲.12–14 One iterates with the FFT be-
`tween the kinoform plane and the diffraction plane.
`In both planes manipulations of the phase, ampli-
`
`10 November 1997 兾 Vol. 36, No. 32 兾 APPLIED OPTICS
`FNC 1033
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`8435
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`

`tude, or both, of the fields are performed to force the
`algorithm to give a good solution.
`There are also methods based on the spatial Fou-
`rier transform that use ways of optimizing the phase
`other than that of the IFTA’s.15,16 The use of the
`efficient FFT algorithm makes the IFTA’s the fastest
`of the design methods. But they also have serious
`disadvantages: First, they cannot readily be used to
`calculate a kinoform whose spot pattern is three di-
`mensional; instead they require that all spots be con-
`fined to a single plane at a specified distance behind
`the kinoform. Second, the position of the spots in
`this plane 共the diffraction plane兲 cannot be chosen
`fully arbitrarily. This is because the FFT works
`with matrices, or grids, and the grid spacing in the
`diffraction plane is given after the grid size in the
`kinoform plane is specified. A spot in the diffraction
`plane will then always be in a grid position, and the
`distance along the axes of the grid between two spots
`will always be an integer times the grid size. Third,
`spatial Fourier transform methods require that the
`Fresnel approximation 共of which the Fraunhofer ap-
`proximation is a special case兲 be valid. This approx-
`imation is not valid if light is diffracted at large
`angles to the optical axis 共nonparaxial case兲, nor is it
`valid if the desired spots are to be produced in the
`near near field of the kinoform 共i.e., so close behind
`the kinoform that the Fresnel phase factor is no
`longer appropriate兲.
`The design method described in this paper has
`none of these disadvantages.
`It can be used for a
`general configuration of spots. The accuracy in the
`evaluation of the electric field is in principle the same
`as that for the method of angular-spectrum propaga-
`tion since, in both cases, the full scalar wave equation
`is used. The accuracy should be good as long as
`scalar theory can be used. Thus it cannot be used for
`extremely nonparaxial situations in which the longi-
`tudinal component of the field cannot be neglected.
`The use of an optimal-rotation-angle 共ORA兲 opti-
`mization, a technique used earlier for a spatial Fou-
`rier transform method,17 also makes the method
`reasonably efficient.
`It has been tested successfully
`on an ordinary workstation for a kinoform with
`250,000 elements, each allowed to impose a continu-
`ous phase modulation in the interval 关0, 2␲兴 and with
`60 spots in the diffraction pattern. Of course, it still
`requires more computer time than do the IFTA’s.
`The algorithm gives the phase modulation in the
`kinoform plane as output. Normally, this can be
`interpreted directly as the relief depth; the depth and
`phase modulation of a pixel are simply proportional
`to each other. Only when the light is obliquely in-
`cident at a large angle does the relation between
`phase modulation and relief depth get more compli-
`cated. However, for these extreme cases the validity
`of the scalar theory itself is often questionable.
`The design method in this paper has two major
`parts.
`In the first part, described in Section 2, the
`contribution to the electric field at some point in
`space 共which will be a point where a spot should be兲
`from a small portion of the kinoform 共called a pixel兲 is
`
`8436
`
`APPLIED OPTICS 兾 Vol. 36, No. 32 兾 10 November 1997
`
`Fig. 1. Kinoform plane with pixel k indicated and one of the
`locations m in space.
`
`determined. The Helmholtz–Kirchhoff integral and
`a minimum of approximations are used to make the
`calculations as accurate as possible but still efficient.
`The contribution to the electric field from a pixel
`depends on, for example, the obliquity of the incident
`field, the distance between the pixel and the point in
`space, and the amount of phase modulation of the
`pixel. The amount of phase modulation is the vari-
`able that is to be optimized. This optimization is the
`second major part of the method and is described in
`Section 3. The optimization is performed with an
`ORA method. With this method, the amount of
`phase modulation of the pixel is chosen so that the
`total field amplitude in the desired spot locations in
`space is maximized under the constraint that the
`relative intensities in the spots are the desired ones.
`
`2. Contribution from One Pixel to the Field at a
`Spot Location
`The kinoform is made up of small, rectangular seg-
`ments called pixels.
`In this section we find the con-
`tribution to the electric field at some point m in space
`from pixel number k 共see Fig. 1兲. This contribution
`is called Ukm, and we want to write it as
`
`Ukm ⫽ Akm exp共 j␸km兲exp关 j共␸inc ⫹ ␸k兲兴,
`
`(1)
`
`where ␸inc is the known phase value of the field in the
`center of the pixel without phase modulation, i.e., as
`if the kinoform had no surface relief. The value of
`␸inc depends on only the incident wave; for a plane
`wave it can be set to zero for all pixels. The term ␸k
`is the amount of phase modulation imposed by the
`surface relief.
`It is the variable that is optimized in
`the design algorithm. The phase modulation ␸k is
`assumed to be constant in the whole pixel, i.e., we
`assume that every pixel is a plateau at a constant
`depth.
`It should also be a good approximation for a
`continuous relief, provided the depth does not vary
`too much within one pixel.
`We must now find the complex transfer function
`Akm exp共 j␸km兲 from pixel k to spot m.
`In Goodman’s
`book,18 the integral theorem developed by Helmholtz
`
`

`

`and Kirchhoff with the Kirchhoff boundary condi-
`tions is stated as
`
`Ukm ⫽
`
`1
`
`4␲兰 兰
`
`⳵Uk
`⳵n
`
`G ⫺ Uk
`
`dxdy,
`
`(2)
`
`⳵G
`⳵n
`
`pixel k
`where G is the scalar free-space Green’s function 共or,
`to put it more simply, the scalar field from a point
`source兲 from point m to a position in the pixel:
`
`G共x, y, z, u, v, L兲兩z⫽0 ⫽
`
`,
`
`(3)
`
`exp共 jkr01兲
`r01
`where r01 is the distance from point m with coordi-
`nates 共u, v, L兲 to a position in the pixel with coordi-
`nates 共x, y, z兲:
`r01 ⫽ 关共u ⫺ x兲2 ⫹ 共v ⫺ y兲2 ⫹ 共L ⫺ z兲2兴1兾2兩z⫽0,
`and k is the magnitude of the free-space wave vector
`共the k-vector兲
`
`(4)
`
`k ⫽
`
`.
`
`(5)
`
`2␲
`␭0
`Further, Uk is the complex electric field in pixel k.
`We assume that this field within each pixel can be
`approximated with a plane wave with a constant am-
`plitude Ak and a direction of propagation 共i.e., the
`direction of the k-vector兲 that is the direction of prop-
`agation in the center of that pixel.
`If we use coordi-
`nates ␰, ␩, and ␨ with the origin in the pixel center 共cf.
`Fig. 1兲 and also use our definition of ␸inc as the value
`of the phase in the pixel center without modulation
`共i.e., if ␸k ⫽ 0兲, then we have
`Uk共␰, ␩, ␨兲兩␨⫽0 ⫽ Ak exp关 j共␸inc ⫹ ␸k兲兴
`⫻ exp关 j共kx␰ ⫹ ky␩ ⫹ kz␨兲兴兩␨⫽0.
`We start by calculating the normal derivative in
`the kinoform plane of Uk in Eq. 共6兲. From the con-
`struction of Eq. 共2兲 from the Green’s theorems the
`normal vector of the kinoform plane points in the
`negative z direction 共or ␨ direction兲, as indicated in
`Fig. 1. Then
`
`(6)
`
`⳵Uk
`⳵n
`
`⫽ ⫺
`
`⳵Uk
`⳵␨
`
`⫽ ⫺jkzUk.
`
`(7)
`
`Before taking the normal derivative of G, let us cal-
`culate the components of the k-vector for the plane
`wave in the pixel 共cf. Fig. 2兲 expressed in terms of the
`radius of the outgoing wave R, in the absence of a
`phase relief, and the pixel-center positions xc and yc.
`From Fig. 2 it can be seen that
`k៮
`R៮
`xcxˆ ⫹ ycyˆ ⫹ Rzˆ
`2 ⫹ R2兲1兾2 .
`k兩 ⫽
`兩R៮
`共xc
`2 ⫹ yc
`k
`But the k-vector components are defined as
`k៮ ⫽ kxxˆ ⫹ kyyˆ ⫹ kzzˆ,
`
`(8)
`
`(9)
`
`⫽
`
`k
`
`Fig. 2. Construction of the components of the k-vector in the
`center of pixel k.
`
`and so the components kx, ky, and kz can be identified
`directly from Eq. 共8兲.
`If R happens to be negative,
`i.e., if the incident wave converges, all components
`should be minus the value obtained with Eqs. 共8兲 and
`共9兲.
`Now we calculate the normal derivative of G in the
`kinoform plane:
`
`⳵G
`⳵n
`
`⫽ ⫺
`
`⳵G
`⳵z
`
`⫽
`
`L
`r01
`
`再 jk ⫺
`
`冎G,
`
`1
`r01
`
`and so we have, from Eq. 共2兲,
`
`1
`
`Ukm ⫽
`
`4␲兰 兰
`4␲(⫺jkz ⫺
`⬇ 1
`
`pixel k
`
`L
`r01
`
`⫺jkzUkG ⫺ Uk
`
`c再 jk ⫺
`
`L
`r01
`
`1
`r01
`
`pixel k
`
`再 jk ⫺
`c冎)兰 兰
`
`(10)
`
`冎Gd␰d␩
`
`1
`r01
`
`UkGd␰d␩.
`
`(11)
`
`In expression 共11兲 we approximated the two explicitly
`written r01 values by their value in the pixel center:
`
`r01
`
`c ⫽ 关共u ⫺ xc兲2 ⫹ 共v ⫺ yc兲2 ⫹ L2兴1兾2.
`
`(12)
`
`We will be a little more careful with the r01 value in
`the phase factor of G.
`The integral in expression 共11兲 is evaluated by the
`insertion of Eqs. 共6兲 and 共3兲:
`
`兰 兰
`
`pixel k
`
`UkGd␰d␩ ⫽
`
`Ak exp关 j共␸inc ⫹ ␸k兲兴
`c
`r01
`
`⫻兰 兰
`
`exp关 j共kx␰ ⫹ ky␩兲兴
`
`pixel k
`⫻ exp共 jkr01兲d␰d␩,
`
`(13)
`
`where r01 in the denominator of G was approximated
`by its value in the pixel center. To obtain a simple
`analytical solution of the integral, we expand the
`
`10 November 1997 兾 Vol. 36, No. 32 兾 APPLIED OPTICS
`
`8437
`
`

`

`time is optimized. The information needed to opti-
`mize a pixel is the current value of the phase modu-
`lation ␸k, the values Akm and ␸km, representing the
`transfer function between that pixel and all the spot
`locations, and the phase of the total field ␸m from all
`pixels in these spots. The influence of the other pix-
`els enters through only the values of ␸m.
`In practice,
`to save time all the pixels in the kinoform are opti-
`mized before calculation of the fields in the spots, and
`thus ␸m, again. Then a new optimization of all the
`pixels is performed. The procedure is repeated until
`the kinoform performance is the desired one. The
`ORA method is described in the following.
`Figure 3 shows the complex-number plane where
`the complex amplitude Um 共absolute value and
`phase兲 of one spot is shown. Also shown is the con-
`It can be seen
`tribution to Um from pixel k, Ukm.
`that, if the phase modulation of that pixel is changed
`by ⌬␸k, thus rotating the contribution by the same
`angle, the length of the Um vector 共i.e., the absolute
`value of the field at point m兲 is changed by ⌬l. With
`␾km defined as
`
`␾km ⫽ ␸m ⫺ 兵␸km ⫹ ␸inc ⫹ ␸k其,
`
`(18)
`
`we can see from Fig. 3 that
`
`⌬l ⫽ Akm cos共␾km ⫺ ⌬␸k兲 ⫺ Akm cos ␾km.
`
`(19)
`
`For the particular choice of ⌬␸k shown in Fig. 3, ⌬l
`is positive, which means that the light intensity in-
`creases in spot m. What about in the other spot
`locations? Will the intensity increase there as well?
`No, not in general, of course. However, there does
`exist a value of ⌬␸k, the ORA, for which the sum of
`the changes ⌬l for all spot locations is maximized.
`Changing the phase modulation of pixel k by the ORA
`thus increases the kinoform efficiency.
`
`Fig. 3. Complex-number plane showing the amplitude change ⌬l
`of the field in spot m from changing the phase modulation of pixel
`k by ⌬␸k. Re, real; Im, imaginary.
`
`remaining r01 in a Maclaurin series about the pixel-
`center coordinates:
`r01 ⫽ {(u ⫺ 关xc ⫹ ␰兴)2 ⫹ (v ⫺ 关yc ⫹ ␩兴)2 ⫹ L2}1兾2
`xc␰ ⫺ u␰ ⫹ yc␩ ⫺ v␩
`c
`r01
`where only the terms linear in ␰ and ␩ are retained.
`The integral is now solved readily. With the sub-
`stitutions
`
`,
`
`(14)
`
`⬇ r01
`
`c ⫹
`
`k˜ x ⫽ kx ⫹
`
`k˜ y ⫽ ky ⫹
`
`k共xc ⫺ u兲
`c
`r01
`k共yc ⫺ v兲
`c
`r01
`
`,
`
`,
`
`the integral becomes
`
`兰 兰
`
`pixel k
`
`UkGd␰d␩ ⫽
`
`4Ak exp关 j共␸inc ⫹ ␸k兲兴
`c
`r01
`
`sin冉k˜ x
`
`⫻ exp共 jkr01
`
`c兲
`
`k˜ x
`
`(15)
`
`.
`
`冊
`
`b 2
`
`sin冉k˜ y
`
`k˜ y
`
`冊
`
`a 2
`
`(16)
`Finally, we insert the factor from expression 共11兲 be-
`fore the integral to obtain the expression for the con-
`tribution from pixel k to the field in point m, Ukm.
`The transfer function in Eq. 共1兲 has now been found:
`
`Akm exp共 j␸km兲 ⫽
`
`1
`
`4␲( ⴚ jkz ⫺
`
`L
`r01
`
`c再 jk ⫺
`sin冉k˜ x
`
`1
`r01
`
`a 2
`
`c冎)
`冊
`sin冉k˜ y
`
`,
`
`冊
`
`b 2
`
`k˜ y
`
`⫻
`
`4Ak
`c exp共 jkr01
`r01
`
`c兲
`
`k˜ x
`
`(17)
`where Akm and ␸km are the absolute value and the
`argument, respectively, of this complex number.
`They do not change with the modulation ␸k and can
`thus be calculated for all 共k, m兲 at the start of the
`design algorithm and stored in the computer mem-
`ory.
`One might worry about the approximations we
`have made to arrive at the desired result, Eq. 共17兲.
`However, the small size of a pixel, typically a factor of
`a hundred 共length scale兲 or more smaller than the
`whole kinoform, makes these approximations per-
`fectly justified for virtually any situation; if they are
`not, one can simply increase the number of pixels.
`
`3. Optimization of the Pixel Phase Modulation
`The optimization of the phase modulation of a pixel is
`performed with an ORA17 method.
`It gives an ana-
`lytical expression for how much the phase modula-
`tion of a pixel should change, ⌬␸k. One pixel at a
`
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`
`

`

`The phase modulation ␸k is now changed by this
`value of ⌬␸k, called the ORA. One then proceeds to
`the next pixel and performs a similar optimization.
`To obtain exactly the desired intensity in each spot,
`we use modified forms of the sums S1 and S2:
`S1 ⫽ 兺
`S2 ⫽ 兺
`
`wmAkm cos ␾km,
`
`wmAkm sin ␾km.
`
`(23)
`
`m
`
`m
`
`The only difference between Eqs. 共23兲 and the sums in
`Eqs. 共21兲 is the introduction of the real, positive num-
`bers wm that are weights for each of the spot loca-
`tions.
`If the intensity in spot m is found to be too
`low, wm is increased, and so the corresponding term
`becomes more important in the sum and vice versa if
`the intensity is too high. The weights are changed
`according to
`
`old冉Im
`
`desired
`
`冊0.35
`
`wm
`
`new ⫽ wm
`
`.
`
`(24)
`
`Im
`Here Im is the intensity 共the square of the absolute
`value of the field兲 in spot location m. The exact
`value of the exponent, here 0.35, is not important but
`should be low enough to avoid unstable behavior.
`The procedure described above assumes that the
`phase modulation is continuous, i.e., ␸k can take any
`value in the interval 关0, 2␲兴.
`It is not difficult, how-
`ever, to modify the method for any kind of phase
`restriction. For instance, if the phase modulation is
`quantized, one simply changes ␸k by that integer
`multiple of the phase-quantization step that is closest
`to the value of ⌬␸k obtained with the ORA method.
`A flowchart describing the whole algorithm is
`shown in Fig. 4.
`In the boxes with a dashed border
`are the two main procedures described in this section
`and in Section 2. The algorithm starts with the cal-
`culation of the transfer functions 共Akm and ␸km兲 from
`all pixels to the desired spot locations.
`If the num-
`
`Fig. 4. Flowchart of the complete design algorithm.
`
`To find the ORA we calculate the sum of the ⌬l
`values for all spots for a given pixel k:
`⌬l ⫽ 兺
`
`关Akm cos共␾km ⫺ ⌬␸k兲 ⫺ Akm cos ␾km兴
`
`m
`⫽ S1 cos ⌬␸k ⫹ S2 sin ⌬␸k ⫺ S1
`
`⫽再S3 cos共⌬␸k ⫺ ␣k兲 ⫺ S1
`
`⫺S3 cos共⌬␸k ⫺ ␣k兲 ⫺ S1
`
`if S1 ⬎ 0
`if S1 ⬍ 0,
`
`(20)
`
`兺m
`
`where
`
`Akm cos ␾km,
`
`Akm sin ␾km,
`
`S1 ⫽ 兺
`S2 ⫽ 兺
`
`m
`
`m
`S3 ⫽ 共S1
`
`␣k ⫽ arctan冉S2
`
`2 ⫹ S2
`
`2兲1兾2,
`
`冊 .
`
`(21)
`
`S1
`It is directly seen from Eq. 共20兲 that this sum is
`maximized if the phase modulation is changed by
`⌬␸k ⫽ ␣k,
`⌬␸k ⫽ ␣k ⫹ ␲,
`⌬␸k ⫽ ␲兾2,
`⌬␸k ⫽ ⫺␲兾2,
`
`if S1 ⬎ 0,
`if S1 ⬍ 0,
`if S1 ⫽ 0 and S2 ⬎ 0,
`if S1 ⫽ 0 and S2 ⬍ 0.
`
`(22)
`
`Fig. 5. Uniformity error versus the number of iterations for the
`design of the near near-field kinoform.
`
`10 November 1997 兾 Vol. 36, No. 32 兾 APPLIED OPTICS
`
`8439
`
`

`

`Now the weights wm for each spot are changed up-
`ward or downward, depending on whether the inten-
`sity is too low or too high 关Eq. 共24兲兴, respectively.
`Then the values of the phase modulation ␸k of the
`pixels are changed with the ORA method. When all
`pixels have been changed the field in the spot loca-
`tions is again calculated with Eq. 共25兲 but with the
`new values of ␸k. As a quality measure, the unifor-
`mity error
`
`unif. err. ⫽
`
`共Im兾Im
`共Im兾Im
`
`desired兲max ⫺ 共Im兾Im
`desired兲max ⫹ 共Im兾Im
`
`desired兲min
`desired兲min
`
`(26)
`
`is often used. The iteration stops when the unifor-
`mity error becomes less than some preset value.
`
`4. Examples
`Two examples of kinoforms designed with the algo-
`rithm outlined in Fig. 4 are presented. They have
`been tested experimentally.
`
`A. Example 1: Fan-Out Kinoform in the Near Near Field
`The first example is a kinoform producing an array of
`3 ⫻ 3 spots in the near near field of the kinoform, i.e.,
`where the Fresnel approximation is not valid. The
`kinoform consists of 512 ⫻ 512 pixels, each with a
`size of a ⫽ b ⫽ 10 ␮m.
`共The only reason for choosing
`such an FFT-like number of pixels is that the soft-
`ware for the electron-beam lithograph used in the
`fabrication currently requires it.兲 The whole kino-
`form is thus 5.12 mm ⫻ 5.12 mm. The kinoform is
`illuminated by a plane wave 共␭ ⫽ 633 nm, from a
`He–Ne laser兲 with a constant intensity over the
`whole kinoform. The spot array is produced at a
`
`Fig. 6. Designed phase modulation of the near near-field kino-
`form.
`
`ber of pixels and spots is large, it may be necessary to
`store the values on the computer’s hard drive. The
`modulation of each pixel is set initially to a random
`value. Then the field in the spot locations Um is
`calculated.
`It is given by the sum of the contribution
`from all pixels:
`
`Um ⫽ 兺
`
`k
`
`Ukm ⫽ 兺
`
`k
`
`Akm exp共 j␸km兲exp关 j共␸inc ⫹ ␸k兲兴.
`
`(25)
`
`Fig. 7. Measured intensity distribution from the fabricated kinoform with the designed phase modulation of Fig. 6. The two arrows
`indicate line-scan endpoints for Figs. 8 and 9.
`
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`Fig. 8. Calculated intensity along the line whose endpoints are
`indicated by the white arrows in Fig. 7.
`
`distance of 20 mm behind the kinoform. The spot
`separation was chosen to be 5.12兾3 mm ⫽ 1.71 mm.
`The reason for this particular choice is that then
`there exists a very simple intuitive solution to the
`problem: The kinoform is divided into 3 ⫻ 3 equal
`quadratic segments, each segment being an ordinary
`symmetric Fresnel phase lens that focuses the light
`to a point on its optical axis 20 mm from the lens.
`Figure 5 shows the progress of
`the design
`algorithm—the uniformity error versus the number
`of iterations. An iteration is one change of all the
`pixel phase-modulation values ␸k 共and a calculation
`of the electric field in the spots兲. The time for one
`iteration increases both with the number of pixels
`and with the number of desired spot locations. For
`this example an iteration took approximately 200 s
`on a Hewlett-Packard Model 715兾33 workstation.
`The computer code for the design algorithm was writ-
`ten in MATLAB.
`The final result is a kinoform giving a uniformity
`error of less than 0.1%. The efficiency 共the total
`power in the desired spots divided by the total
`power in the kinoform plane兲 was calculated to be
`⬃40%. The low efficiency occurs because we chose
`to have extremely large pixels, a ⫽ b ⫽ 10 ␮m, for
`easy fabrication.
`It is no problem for the design
`algorithm to handle this situation because all the
`approximations made in Section 2 are still perfectly
`valid. However, a structure with such large pixels
`is inherently bad at performing the strong focusing
`from the kinoform plane to the nearby 3 ⫻ 3 spot
`locations. The algorithm makes the best of the
`situation, but inevitably there will be much light
`going to positions other than the desired ones,
`hence the low efficiency.
`The kinoform designed with this method is shown
`in Fig. 6. The gray scale represents phase-
`modulation values from 0 to 2␲.
`It is certainly in-
`teresting to notice that the solution shown in Fig. 6
`closely resembles the intuitive solution mentioned
`above, which was an array of 3 ⫻ 3 Fresnel lenses 共at
`
`Fig. 9. Measured intensity reconstructed along the same line as
`for Fig. 8. The slight defocusing leads to the spots being broader
`than those shown in Fig. 8.
`
`least near the center of each so-called lens the struc-
`ture is very much like a Fresnel lens兲. No structure
`like this could be expected to result from design meth-
`ods based on the spatial Fourier transform.
`A kinoform with the above phase modulation was
`fabricated. The phase modulation was converted to
`a relief depth according to
`
`depthk ⫽
`
`␸k
`2␲
`
`,
`
`(27)
`
`␭0
`nresist ⫺ 1
`where nresist is the refractive index of the resist ma-
`terial 关poly共methylglutarimide兲兴. The surface relief
`was exposed in resist spun on a quartz substrate.
`The exposure was made with electron-beam lithogra-
`phy with an acceleration voltage of 25 kV. Because
`of the large pixels, no compensation for the proximity
`effect was made. However, the proximity effect will
`have some influence, especially in the peripheral
`parts of each Fresnel lens, so the performance of
`
`Fig. 10. Uniformity error versus the number of iterations for the
`three-dimensional fan-out kinoform.
`
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`

`more than 20 mm behind the kinoform兲. This was
`necessary because the spot size in focus is smaller
`than the size of the pixels in the CCD array, which
`leads to peculiar effects, for instance, if a spot falls in
`the dead area between two neighboring CCD cells.
`As can be seen from Fig. 7, the desired 3 ⫻ 3 spots
`共measured uniformity error of 8%兲 are the most in-
`tense, but indeed other faint light spots are also
`present, as would be expected from the discussion
`earlier in this subsection.
`To compare the actual results with those predicted,
`one calculates the intensity along the line whose end
`points are indicated by the white arrows in Fig. 7.
`For each calculation point along the line, values of
`Akm and ␸km were calculated with m now represent-
`ing the calculation point. The contributions Ukm
`from all pixels were then summed to obtain the total
`field in the point, as in Eq. 共25兲. The result is shown
`in Fig. 8.
`It can be qualitatively compared with the
`measured results shown in Fig. 9. Figure 9 actually
`shows, in every position, the sum of the intensities in
`a few CCD cells that have the same horizontal posi-
`tion and are nearest to the calculation line in the
`vertical direction. The measured spots are broader
`because, as was mentioned, the camera had to be
`slightly out of focus.
`
`B. Example 2: Kinoform with a Three-Dimensional
`Fan-Out Pattern
`In this example the spots were to be produced at
`different distances from the kinoform plane. For
`the design it means simply that the coordinate L, as
`well as coordinates u and v, can be different for
`different spots. A kinoform was designed for an
`incoming plane wave with a constant intensity.
`
`Fig. 11. Designed phase modulation of the kinoform giving a
`three-dimensional fan-out.
`
`the actual kinoform will be somewhat degraded com-
`pared with the calculated one. The resist was de-
`veloped
`until
`a maximum relief
`depth of
`approximately 1.2 ␮m was reached, corresponding to
`a phase modulation of 2␲. The kinoform was illu-
`minated by an expanded beam from a He–Ne laser,
`and the fan-out pattern was captured by a CCD cam-
`era.
`The measured fan-out at a distance of 20 mm is
`shown in Fig. 7.
`In fact, the camera had to be posi-
`tioned slightly out of focus 共at a distance slightly
`
`Fig. 12. Measured intensity distribution in a plane 5 cm behind the kinoform.
`
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`

`Fig. 13. Measured intensity distribution in a plane 10 cm behind the kinoform.
`
`The kinoform should produce spot patterns at three
`different distances:
`the figure 5 at 5 cm, the figure
`10 at 10 cm, and the figure 20 at a distance of 20 cm
`from the kinoform. Of course, then we must be in
`the near field 共Fresnel region兲 of the kinoform, be-
`cause in the far field the diffraction pattern does not
`change its character with the distance— only the
`size of the diffraction pattern scales with the dis-
`tance.
`The spot-pattern figures were all designed to have
`
`a height of 3.5 mm 共to fit the CCD array兲.
`In all, the
`number of desired spot locations was 51. The de-
`sired intensity in the spots was proportional to the
`inverse square of the distance L so that the power in
`every spot would be approximately the same 共because
`the area of a spot increases as the square of L兲. The
`kinoform consisted of 256 ⫻ 256 pixels 共again, only
`because of fabrication requirements兲, each with a size
`of a ⫽ b ⫽ 4 ␮m, giving a total kinoform size of 1.02
`mm ⫻ 1.02 mm.
`
`Fig. 14. Measured intensity distribution in a plane 20 cm behind the kinoform.
`
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`

`

`The evolution of the uniformity error is shown in
`Fig. 10. The efficiency was calculated to be ⬃75%.
`The designed phase modulation of the kinoform is
`shown in Fig. 11. A kinoform with this design was
`fabricated following the procedures mentioned in the
`previous example 共Subsection 4.A兲. The measured
`spot patterns at the three distances are shown in Figs.
`12–14. The measured uniformity error within each
`figure pattern 共i.e., 5, 10, and 20兲 was 10%, 8%, and
`13%, respectively.
`In the figures one can see some
`faint defocused spots belonging to a different figure
`pattern, e.g., the defocused number 20 in Fig. 13.
`
`5. Conclusion
`A design algorithm for fan-out kinoforms has been
`described and tested experimentally.
`It is based on
`rigorous scalar wave-propagation theory. No seri-
`ously limiting approximations are made, so the
`method is general and accurate as long as the as-
`sumption of scalar diffraction is valid. The optimi-
`zation uses an ORA method with an analytical
`expression for the change of the phase modulation of
`a pixel, which makes the optimization efficient. The
`algorithm was tested successfully on an ordinary
`workstation for kinoforms with up to 250,000 pixels
`and 60 desired spot locations. The proposed algo-
`rithm requires more computer time than do IFTA’s
`especially when there is a large number of desired
`spot locations, since the run time, unlike the case for
`the IFTA, increases with this number.
`Two kinoforms designed with this algorithm were
`fabricated. One produced an array of spots in the
`near near field of the kinoform where the Fresnel
`approximation is not valid. The other kinoform pro-
`duced a three-dimensional spot array with different
`fan-out patterns at three different distances from the
`kinoform. The experimental results agree well with
`those predicted.
`It can be concluded that use of the
`algorithm is advantageous when full freedom of the
`configuration of spots is desired. The spots may be
`located at any distance relative to each other; they
`may be at large angles to the optical axis; they may be
`close to the kinoform; they may form a three-
`dimensional pattern.
`
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`