Today, many types of diffractive optical element�DOE� are known. Prominent examples are diffrac-tion gratings, computer-generated holograms, kino-forms, diffractive phase structures, and volumeholograms. A DOE transforms an illuminating lightbeam into a desired output light distribution. Thereis a wide range of applications, such as in opticalcomputing, data storage, security, beam shaping, andidentification.
DOEs are roughly categorized as thin or thick, andas computer generated or optically produced. Thickor volumetric gratings and holograms have prefera-ble properties for many applications.1 They are ableto exhibit only one diffraction order and achieve highdiffraction efficiencies. Their angular, wavelength,and phase selectivity make it possible to store mul-tiple pages of information in a single hologram simul-taneously �multiplexing�, which can be read outseparately. With the multiplexing technique a largeamount of information can be stored in a small vol-
S. Borgsmuller and S. Noehte are with European Media Labo-ratory, Schloss-Wolfsbrunnenweg 31c, 69118 Heidelberg, Ger-many. C. Dietrich is with tesa scribos GmbH, Sickingenstraße65, 69126 Heidelberg, Germany. T. Kresse is with tesa AG,Quickbornstraße 24, 20253 Hamburg, Germany. R. Manner iswith the Institute of Computer Science V, University of Mann-heim, Quadrat B6 23-29C, 68131 Mannheim, Germany.
Received 14 November 2002; revised manuscript received 14April 2003.
ume. Because the computational effort and the pro-duction capabilities forbid the calculation andexposure of volume holograms in a point-by-pointmanner, they have to be exposed holographically,which brings disadvantages such as a need for highexperimental precision. Synthetic objects cannot bestored directly, because the light field that has to berecorded must be generated optically.
On the other hand, for thin DOEs many calculationand optimization routines are known.2 But thinDOEs do not exhibit the above mentioned advantagesof the volume holograms. Only a few examples forwavelength and angular multiplexing can be found inthe literature, which need a high computational ef-fort and precise production techniques.3,4
A hybrid type of diffractive element is stratifiedDOEs �SDOEs�. They consist of multiple layers ofthin or thick DOEs, which are interleaved with opti-cally homogenous buffer layers and are exposed ho-lographically. Experiments and the theoreticaltreatment of such devices5–7 have shown that theyhave the properties of wavelength and angular selec-tivity, even in the case of thin SDOEs. Also higher-order suppression and a high diffraction efficiencyhave been observed. But, to our knowledge, thereexists no calculation scheme to store arbitrary data inthese elements.
In this paper we introduce computer-generatedSDOEs consisting of multiple layers of computer-generated thin DOEs interleaved with optically ho-mogeneous buffer layers, analogical to the above-mentioned SDOEs. Synthetic objects can be codedinto the SDOEs, and coding schemes known from
computer-generated DOEs can be adapted for thecalculation. Computer-generated SDOEs showsome properties of volumetric DOEs, i.e., the appear-ance of only one diffraction order and a high angularand wavelength selectivity, which can be utilized formultiplexing. The properties of Fourier-type DOEs,e.g., ease of the readout setup and the shift invari-ance of the reconstruction on the lateral position ofthe DOE can be mixed with those of the SDOEs.Thus computer-generated SDOEs promise a wide ap-plicability.
In Section 2 we present a model for SDOEs anddescribe several calculation algorithms, including al-gorithms for multiplexing. We go into detail for aspecific SDOE, the two-layer binary DOE. Further,fabrication tolerances are discussed. Simulated re-constructions of binary two-layer DOEs follow in Sec-tion 3, and experimental results are presented inSection 4.
2. Theory
An SDOE consists of two or more layers of thin DOEs�Fig. 1�. These layers are parallel and so close to-gether that the Fresnel approximation is not valid todescribe the propagation of light between the layers.If the SDOE is illuminated by a monochromatic laserbeam the reconstruction arises at a certain distancebehind the element in the reconstruction plane,where the Fresnel or Fraunhofer approximation isvalid.
A. Mathematical Description
If K is the total number of single DOEs in a stack, thekth DOE hk attenuates the amplitude and shifts thephase of an illuminating monochromatic light beamin its plane z � zk due to its complex transmittancefunction hk�x, y�:
Denoting the spatial distribution of the complex lightfield as
A� x, y, z� � � A� x, y, z��exp�i�A� x, y, z��, (2)
one can write down the effect of a single DOE on thefield:
A� x, y, zk�� � hk� x, y� A� x, y, zk
��
� �hk� x, y�� A� x, y, zk���expi��k� x, y�
� �A� x, y, zk���, (3)
where z � zk��z � zk
�� is the plane immediate behind�before� the DOE hk. The amplitude of the light fieldis therefore attenuated by the factor �hk�x, y��, and thephase is shifted by the addend �k�x, y�. Because theDOEs are suggested as thin, they change the lightfield discontinuously.
The light field just before the following DOE hk�1 isconnected to the light field emerging from hk by thenear-field transform �NFT� �Fig. 2�:
A� x, y, zk�1� � � NFT� zk�1�zk��A� x, y, zk
���, (4)
Fig. 1. Schematic illustration of the setup of a SDOE. The SDOE consists of two or more layers of separate DOEs. If the SDOE isilluminated by a laser beam, a reconstruction image arises in the reconstruction plane.
Fig. 2. Schematic illustration of the propagation of light betweentwo single DOEs, represented by a near-field transform.
with the angular spectrum of plane waves for thepropagation in free space:8
A� x, y, z� � NFT�z�A� x, y, z � �z��
� FT�1�FT�A� x, y, z � �z��exp��ikz�z��
� �0
�0
FT�A� x, y, z � �z����x, �y�
� exp��i2���x x � �y y��
� exp��ikz�z�d�xd�y, (5)
where FT denotes the Fourier transform, �x and �yare spatial frequencies, kz is the z component of thewave vector k, kz � �k2 � kx
2 � ky2�1�2 � 2��1��2 � �x
2
� �y2�1�2, and �z is the propagation distance.
The SDOE can be viewed as an input-output devicethat transforms an incoming light distribution Ain�x,y� � A�x, y, z1
�� into an output light distributionAout�x, y� � A�x, y, zK
��. The incoming light is atten-uated and phase-shifted by the first DOE h1, thenpropagates to the second layer, is attenuated andphase shifted by h2, etc. Thus there exists a clearrelationship between the input and the output of theSDOE that can be represented by a series of complexmultiplications of the light field with the hk�x, y�followed by free-space propagations:
for clarity the �x, y� dependence of the hk is omittedhere and in the following equations. The action of asingle DOE is mathematically reversible if the abso-lute values of the hk are nonzero for all �x, y, k� �thishas to be considered in the design process�. TheNFT is reversible in all cases, so one can evaluate anoutput-input relation for the SDOE:
Ain� x, y� �1h1
NFT� z2�z1��1 � · · ·
1hK�2
� NFT� zK�1�zK�2��1 � 1
hK�1
� NFT� zK�zK�1��1 � 1
hKAout� x, y��� · · ·� .
(7)
The reconstruction R � �R�x, y��exp�i�R�x, y�� of theSDOE arises in the reconstruction plane z � zR,which is connected to the output plane z � zK
� by anoptical transform T:
R� x, y� � T�Aout� x, y��. (8)
Depending on the distance between the output planeand the reconstruction plane one can use the Fresnelor the FT. The latter can also be used if a Fourierlens is placed between the two planes and the recon-struction plane coincides with the focal plane of thelens. In all cases T is reversible.
B. Calculation of the SDOE
Usually, a DOE is designed to transform a specificinput field Ain�x, y�, e.g., a Gaussian wave or a planewave, into a desired field R�x, y� which is also calledthe desired reconstruction. The desired reconstruc-tion determines the output Aout�x, y� that has to beachieved by the element:
Aout� x, y� � T�1�R� x, y��. (9)
The tilde denotes desired functions. In most casesan approximate solution or an exact solution only fora set of constraints is required. It is clear, that formost desirable reconstructions R�x, y� there is a mul-titude of sets hk that meets the requirements, i.e.,R�x, y� � R�x, y�.
1. Basic CodingTo pick a solution out of the solution space one setsK � 1 single DOEs arbitrarily �e.g., randomly� andcalculates the remaining DOE hr. Therefore weneed the field just in front of the selected DOE andthe field that should emerge behind it to yield thedesired reconstruction:
The task for hr is now to transform A�x, y, zr�� into
A�x, y, zr��. So hr should match this function:
hr� x, y� �A� x, y, zr
��
A� x, y, zr��
. (12)
To avoid division by zero hr�x, y� must be set arbi-trarily if A�x, y, zr
�� � 0. The difficulty is now to codethe desired complex transmittance function hr�x, y�into the DOE function hr�x, y�, which has to be afunction that can be realized in the production pro-cess. Usually, a DOE cannot amplify light, so �hr�x,y�� � 1. There are other constraints for hr�x, y� thatmainly mirror the production capabilities for the sin-gle DOEs: sampling, dynamic ranges, phase-onlycoding, amplitude-only coding, amplitude-phase cor-relations, etc. This depends on the material usedand the writing device. Therefore
hr� x, y� � C�hr� x, y��, (13)
where C is a coding operator. Many coding opera-tors are known from one-layer diffractive elements2
and are applicable here, so one can choose an appro-priate one from these.
2. Iterative CodingIn most applications the coding will cause a largeerror in the reconstruction. To improve the qualityof the reconstruction, iterative coding schemes areapplicable here, i.e., generalized Gerchberg–Saxtonalgorithms.9,10 In these algorithms the coding oper-ator C represents the constraints in the domain of thediffractive element. In most applications the recon-struction has to meet only a small number of proper-ties �e.g., a certain intensity in a signal window�,represented by the constraints operator CR in thereconstruction domain, leaving degrees of freedom forthe reconstruction and thus for the DOE. These de-grees of freedom can be used to optimize the DOE byiteratively going back and forth between the plane ofthe diffractive element and the reconstruction plane,applying the constraints operators in both domains.Usually, the reconstruction error decreases with thenumber of iterations and the algorithm can bestopped if a certain error condition is reached. Inthe case of SDOEs the backward operation is given byEqs. �9� and �11�, and the forward operation is givenby
R� x, y� � T�hKNFT� zK�zK�1��· · ·hr�1
� NFT� zr�1�zr��hr A� x, y, zr
���· · ·��. (14)
The constraints operators in the DOE domain and inthe reconstruction domain can be defined, thus gen-eralized Gerchberg–Saxton algorithms are applicablehere to optimize the calculated DOE.
3. Distribution-on-Layers AlgorithmWith the above calculation schemes the relevant in-formation is coded only in one element hr. The otherelements have been set arbitrarily and are notadapted to the problem. It is preferable to spreadthe information over the whole SDOE, so each singleDOE carries the same amount of information. Thiscan be done by an iterative algorithm that we calledthe distribution-on-layers algorithm: you keep hrfrom Eq. �13� constant and pick out another DOE forcalculation, e.g., h1. Then h1 is calculated by use ofEq. �13�, and h1 can be further improved by a gener-alized Gerchberg–Saxton algorithm and set constant.After that the next element, e.g., h2, is calculated andso on, until a tolerable error in the reconstruction isreached or the error cannot be minimized further.This algorithm is assumed to reduce the coding errorsignificantly especially for a large information load inthe reconstruction, because the information is dis-tributed on many layers.
C. Selectivity and Multiplexing
Because the input-output relation of SDOEs is rathercomplex �Eq. �6��, small derivations in the input fieldAin lead to a large error in the output field Aout.Thus we have a strong selectivity on the amplitude,the phase, and on the angle of the input beam, whichmeans that small derivations in these variables candestroy the reconstruction. Because the NFT is de-
pendent on the wavelength �kz in Eq. �5�� we expectalso a wavelength selectivity of the device. Theseselectivities can be used to store more than one re-construction image �page� in the SDOE �multiplex-ing�. Each of the stored pages can be read out underthe appropriate conditions. Angular and wave-length multiplexing are the most common multiplex-ing schemes, because the readout conditions can beeasily established experimentally. Here we intro-duce two algorithms for calculating multiplexed ho-lograms, namely, the complex addition algorithm andan algorithm that uses an iterative approach. Thesealgorithms have in common that they code all pagesinto a single layer. Owing to the �usually expected�restrictions of the DOEs in resolution and dynamicrange, a high error for the individual pages can beexpected, because a lot of information has to be storedin a restricted medium. To reduce the error theabove described iterative methods can be used. Es-pecially with the distribution-on-layers algorithm,the information gets scattered across all the layers ofthe SDOE, so the storage capacity can be increased.How many pages can be coded in the entire SDOEmust be decided individually by considering the re-construction quality.
1. Multiplexed Coding by Complex AdditionFor the calculation of a multiplexed SDOE with Jpages you need J starting conditions Ain
j �x, y� and Jdesired outputs Aout
j �x, y�. Again K � 1 DOEs areset arbitrarily, and the desired function hr�x, y� forthe remaining DOE is calculated by use of Eq. �12� forevery pair �Ain
j �x, y�, Aoutj �x, y�� yielding a set of hr
j�x,y�. To store the information in the DOE, the singledesired functions hr
j�x, y� will be added before coding:
hr� x, y� � C�1J �
j�1
J
hrj� x, y�� . (15)
Thus all pages are coded into the DOE hr by a super-position of the individual signals. A similar codingscheme has been used successfully for coding three-dimensional objects �composed out of two-dimensional �2D� planes� into a single hologram.11
2. Multiplexed Coding with an Iterative AlgorithmThe iterative approach described here is again a gen-eralized Gerchberg–Saxton algorithm. It is as-sumed that every desired reconstruction Rj�x, y� canbe represented by a constraints operator CR
j . Firstthe DOE hr is calculated �Eq. �13�� for the first pageAout
1 �x, y� and the first initial condition Ain1 �x, y�.
Then the reconstruction is calculated by use of Eq.�14�. Now the constraints operator for the secondpage CR
2 is applied on the calculated reconstruction.If the operator is not too restrictive, parts of the oldinformation are conserved. The back transformyields the second desired output Aout
2 �x, y�, and hr iscalculated for the pair �Ain
2 �x, y�, Aout2 �x, y��. This is
repeated until all pages are coded into hr. Thisscheme is expected to be successful only for a fewpages, because the information from a page decays
with every further iteration. The pages are notequally represented in the diffractive element such asin the complex addition scheme, but there is a hier-archy of page quality connected to the order of calcu-lation.
D. Two-Layer Binary Phase Elements
Binary diffractive phase elements have advantagescompared with multilevel phase elements. They areeasy to produce, because only one phase level has tobe written and have a high diffraction efficiency.For multilayer elements, pure phase elements arepreferable to avoid absorption losses. Thus two-layer binary phase elements are simple SDOEs,where the basic properties of such devices can bestudied.
If a two-layer binary phase element is illuminatedby a plane wave Ain�x, y� � 1 and the reconstructionlies in the Fourier domain of the device, we have thefollowing basic equations for the system:
h1� x, y� � exp�i��1� x, y�� (16)
h2� x, y� � exp�i��2� x, y�� (17)
R� x, y� � FT�h2� x, y�NFT�z�h1� x, y���, (18)
where �z � z2 � z1. Because the element is binary,�1�x, y� and �2�x, y� can only have the values 0 or 1.The phase-shift difference between exposed and non-exposed points is given by �, which is usually in theinterval �0, ��. There is a relationship between abinary phase element and a binary amplitude ele-ment with the same function:
h1� x, y� � exp�i��1� x, y��
� 1 � �ei� � 1��1� x, y�;
�1� x, y� � 0 � 1. (19)
Thus we can rewrite the NFT �Eq. �5��:
NFT�z�h1� x, y�� � NFT�z�1 � �ei� � 1��1� x, y��
� FT�1�FT�1 � �ei� � 1��1� x, y��exp��ikz�z��
� exp��ik�z� � �ei� � 1�NFT�z��1� x, y��,(20)
and the reconstruction �Eq. �18��:
R� x, y� � FT�h2NFT�z�h1��
� FT��1 � �ei� � 1��2� x, y��
� exp��ik�z�
� �ei� � 1�NFT�z��1� x, y���
� �� x, y�exp��ik�z� � �ei� � 1�
� (FT��2� x, y��exp��ik�z�
� FT�NFT�z��1� x, y���) � �ei� � 1�2
� FT��2� x, y�NFT�z��1� x, y���, (21)
where ��x, y� is the Dirac delta function and repre-sents the zeroth order. Thus the reconstruction of atwo-layer binary phase element is a superposition ofthe Fourier transforms of the individual layers and across-talk term �last term in Eq. �21��.
If you would place the individual layers next toeach other you would get the superposition of theindividual Fourier transforms in the reconstructionplane only. Thus the cross-talk term is the maindifference provided by the stratified arrangement.It is preferable to code the information into the cross-talk term, if we are looking for properties that areunknown to one-layer diffractive elements. There-fore we have to evaluate the coding process to seeunder which conditions the information is not codedin the individual layers but in the cross-talk term.
The information is coded using one of the followingequations �see Eqs. �10�–�13��:
h1� x, y� � C�NFT�z�1�FT�1�R� x, y��
h2� x, y� �� (22)
h2� x, y� � C� FT�1�R� x, y��
NFT�z�h1� x, y��� . (23)
In the coding of Eq. �22� we start in the reconstructionplane with the desired reconstruction R, go back intothe plane of the second binary phase element h2,subtract the arbitrary chosen phase values of thiselement from the phase �A�x, y, z2
�� of the light wave,propagate into the plane of the first element h1, andcode the resulting light field into the first element.The subtraction of the phase values of the elementfrom the phase of the light field changes the informa-tion contained in the light field. If the second ele-ment has a phase shift � of � in the exposed points,this operation is equivalent to an exclusive OR �XOR�operation of the highest bit, if you define the highestbit of a phase value as 0, if the phase is in the interval�0, �� and as 1, if the phase is in the interval ��, 2��.If the second element is chosen at random, you get areal XOR encryption of the highest bits of the phaseof the light field and thus a resulting phase that isalmost random �information is still contained in thelower bits�.
It is known from kinoforms that if the desired re-construction has a random phase, the informationcontained in the amplitude of the reconstruction canbe found to a bigger part in the phase of its Fouriertransform. Setting the amplitude of the Fouriertransform constant and transforming back yields ex-actly the desired reconstruction with a diffractionefficiency of 78%.12 Owing to the encryption of thehighest bit of the phase, the main part of the infor-mation of the desired reconstruction cannot be foundin FT�1�R�x, y���h2�x, y� and thus not in the firstdiffractive element h1. So the information is neithercoded only in the first nor only in the second element�which is set arbitrarily�. The information is there-fore situated in the cross-talk term only, if the phasevalues of the second element are 0 and �, otherwiseno XOR encryption takes place, and the information
is additionally coded into the first element. Consid-erations about the coding after Eq. �23� yield analog-ical results.
E. Fabrication Tolerances
In principle, one can use a classical method for thefabrication of the single DOEs of a computer-generated SDOE and arrange the single DOEs inparallel afterward. A more preferable method is toarrange the layers in parallel before exposing thesingle DOEs. In this case the writing process can bedone with a laser lithograph, and the depth of focus ofthe writing beam can be controlled to write into thedifferent layers. In all cases one needs to satisfycertain design tolerances.
Consider a two-layer DOE, with a sampling dis-tance of ds and a distance between the layers of �z �z2 � z1. A lateral displacement xerr of the first DOEleads to a lateral shift of the diffraction image of thefirst DOE in the second layer. Apparently, the lat-eral displacement must not exceed the sampling dis-tance ds: xerr � ds �in the case of two-layer DOEsone can correct a lateral displacement with a tiltedreadout beam�. An error in the distance betweenthe two layers leads to a change of the diffractionimage of the first DOE in the second layer. Thesmallest controllable grating in the first DOE has aperiod of 2ds. A distance error of zerr leads to alateral shift of the first order of this grating in thesecond layer of �zerr�2ds. This shift should besmaller than the sampling distance, so we can definezerr � �2ds���ds as distance tolerance.
3. Simulation
For the simulations we choose two-layer binary phaseelements, because they are very simple SDOEs andshow properties that are unknown to one-layerDOEs. The two DOEs are equally sized and aresampled with N � 512 � 512 sampling points of asampling distance of ds � 1 �m in both x and ydirection. The two layers are arranged in a distance�z � z2 � z1 � 58 �m. The two-layer DOE is illu-minated by a plane wave of wavelength � � 632.8 nm,and the reconstruction is in the Fourier domain of theelement, i.e., a lens is placed between the last ele-ment and the reconstruction plane. As a reconstruc-tion image we choose the capital letter A in a signalwindow that is placed outside the center of the recon-struction �Fig. 3�. Outside the signal window wechoose R � 0, and for the phase inside the signalwindow we choose a random phase. We imple-mented the near-field transform with two fast Fou-rier transforms and the Fourier transform with asingle fast Fourier transform, ignoring any shape ofthe single sampling points. The coding operator inthe DOE domain is a phase hard clipping operator,setting all amplitude values to 1 and the phase valuesin the interval �0, �� to 0, and the remaining phasevalues to �.
As a measure for the reconstruction quality we
choose the normalized simple mean squared error�MSE�:
MSE �1
� x1 � x0�� y1 � y0� �x0
x1
�y0
y1
��R� x, y��
� a�R� x, y���2 dxdy, (24)
where �x0, x1, y0, y1� are the coordinates of the signalwindow and the factor a is chosen such that the MSEbecomes independent of a scaling factor of the recon-struction:
a �
�x0
x1
�y0
y1
�R� x, y��R� x, y�dxdy
�x0
x1
�y0
y1
�R� x, y��2 dxdy
. (25)
A second important quality for diffractive elements isthe diffraction efficiency �, that we define as the ratioof the intensity in the signal window and the overallintensity:
� �
�x0
x1
�y0
y1
�R� x, y��2 dxdy
�� �R� x, y��2 dxdy
. (26)
First we examine the consequences of Eq. �21�.We set the second element of the SDOE random withphase levels 0 and ��2. Then we calculate the firstelement using Eq. �22�. The reconstruction of thewhole element �Fig. 4�a�� consists of the coded infor-mation and a weaker twin image, resulting from theFourier transform of the first element �Fig. 4�b��. Ifwe set the second element random with phase values0 and � and calculate the first DOE, the twin imagecannot be found in the reconstruction �Fig. 4�c��, i.e.,only the first reconstruction order appears. TheFourier transform of only the first element yieldsnoise �Fig. 4�d��, which means that the encryptionwas successful. The effect of the appearance of only
Fig. 3. Capital letter A in a signal window off center.
one diffraction order is known from kinoforms andother multilevel phase diffractive elements, volumeholograms, and blazed gratings, but not from one-layer binary DOEs �except from sub-wavelength grat-ings�. In our case the �1 diffraction order doesn’teven appear when the SDOE is reconstructed withother phase values than those used in the calculation,as it does in the case of multilevel phase diffractiveelements. Thus this property results from the cross-talk term in Eq. �21� and is a property of the stratifiedstructure.
If we put a binary Fourier-type kinoform of anothersignal �capital letter F� into the second element with
phase values 0 and � � � instead of a binary randomdistribution, and calculate the first layer using Eq.�22�, the reconstruction is similar to that of a randomsecond element �Fig. 4�c��, thus the Fourier-transform terms in Eq. �21� get compensated. If thesame element �calculated with phase values 0 and� � �� is reconstructed with phase values 0 and � ���2, the reconstruction of the kinoform appears inaddition to the coded image �Fig. 5�, so the Fourier-transform terms in Eq. �21� become visible. This isdue to the different dependency on � of the Fouriertransforms and of the cross-talk term in Eq. �21�.
Next we examine the different calculation schemesfor the element. One diffractive element is chosenrandomly with phase values 0 and � � �, and thesecond is calculated with Eq. �22� or Eq. �23� with thephase hard clipping coding operator. The values forthe MSE and the diffraction efficiency can be found inTable 1. With the first element set at random andthe second element calculated, the MSE has an ac-ceptable value for image information of 3.2%. If thesecond element is chosen randomly and the first ele-ment is calculated, the MSE is significantly smaller�1.5%�. The diffraction efficiency in both cases issufficient with a value of approximately 9.5%. If thecalculated DOE gets iteratively optimized by a gen-eralized Gerchberg–Saxton algorithm, utilizing thephase freedom in the Fourier domain, i.e., the con-straints operator in the reconstruction plane is reset-ting the amplitude to the initial value and keepingthe phase, we get an improvement in the MSE �2.3%versus 1.3%�, while the diffraction efficiency does not
Fig. 4. �a� Simulated reconstruction of a two-layer binary diffractive optical element with a binary random distribution in the secondlayer, calculated with the phase levels 0 and ��2. �b� Fourier transform of the first layer of Fig. 6�a�. �c� Same as Fig. 6�a� calculatedwith the phase levels 0 and �. �d� Fourier transform of the first layer of Fig. 6�c�.
Fig. 5. Simulated reconstruction of a two-layer binary diffractiveoptical element calculated with the phase levels 0 and �, with abinary kinoform �F� in the second layer, reconstructed with thephase levels 0 and ��2.
Table 1. Mean Squared Error �MSE� and Diffraction Efficiency �a
First element Second element MSE �%� � �%�
Random Calculated 3.2 9.5Calculated Random 1.5 9.4Random Optimized �10 iterations� 2.3 9.4Optimized �10 iterations� Random 1.3 9.5Distribution-on-layers Distribution-on-layers 0.8 14.9
change significantly. Essential improvements canbe obtained with the distribution-on-layers algo-rithm, using the same constraints operator. We getan MSE of 0.8% and a diffraction efficiency of 14.9%.
In Table 1 the values for a binary kinoform and anoptimized kinoform, improved with a generalizedGerchberg–Saxton algorithm, are given for compari-son. The MSE in the case of the distribution-on-layers algorithm is much better than in the case ofthe optimized kinoform �1.6%�, while the diffractionefficiency is slightly smaller.
Two-layer binary phase elements exhibit a strongangular selectivity. While at an angle of incidenceof 0.5 degree the reconstruction is still visible, it hasvanished at an angle of 1 deg �Fig. 6�a��. The dif-fraction efficiency decreases from 9.5% at 0 degree tothe half at 0.71 deg and goes into the noise level of3.3% at approximately 1 degree. The MSE growsfrom 1.48% to the noise level of 18.5%. Beyond 1degree a sinc-like oscillation of the MSE and the dif-fraction efficiency can be seen. A variation of theangle of incidence results in a shift of the diffractionimage of the first DOE in the plane of the secondDOE. So the sinc-like oscillations are explained bysampling theory, where sampled and non band-limited signals have the exact value in their samplingpoints, are zero in all other sampling points, and havea sinc-like behavior in between. If the diffractionimage of the first DOE is shifted in the plane of thesecond DOE, the sampling points of the second DOE
see a sinc variation of the diffraction-image functionplus noise from other sampling points.
Figure 6�b� shows the wavelength selectivity of thedevice. Here the diffraction efficiency and the MSEapproach the noise level at a wavelength of approxi-mately 520 nm �green� for a device calculated for632.8 nm �red�.
Another property is the phase-shift insensitivity ofthe 2-layer binary phase element. Although the bi-nary two-layer phase element is calculated with aphase shift of 0 and � � � it shows acceptable resultsfor lower phase-shift values � during the reconstruc-tion. This can be understood with Eq. �21�, because� appears here as a factor. Figure 8�c� shows thedependency of the diffraction efficiency and of theMSE of the reconstruction on the phase shift of thesingle DOEs.
It is remarkable that only one order appears in thereconstruction for all variations of angle, wavelength,phase shift, and alignment. Thus the whole recon-struction plane can be used for the reconstruction,because the first order is not disturbed by the minusfirst order.
The various selectivities of the device can be utilizedto code more than one reconstruction image �page� intothe device �multiplexing�. Common modes are angu-lar and wavelength multiplexing. In the simulationwe choose two pages, the capital letter A and the cap-ital letter F in different signal windows. The secondDOE is set randomly with phase values 0 and � � �,
Fig. 6. �a� Angular selectivity of a two-layer binary phase element: diffraction efficiency � �solid curve� and MSE �dashed curve�. �b�Wavelength selectivity of a two-layer binary phase element: diffraction efficiency � �solid curve� and MSE �dashed curve�. �c� Phaseshiftinsensitivity of a two-layer binary phase element: diffraction efficiency � �solid curve� and MSE �dashed curve�.
Fig. 7. �a� Angular multiplexed two-layer binary phase element: diffraction efficiency � A �solid curve�, MSE A �dashed curve�,diffraction efficiency � F �dashed and dotted curve�, MSE F �dotted curve�. �b� Wavelength multiplexed two-layer binary phase element:diffraction efficiency � A �solid curve�, MSE A �dashed curve�, diffraction efficiency � F �dashed and dotted curve�, MSE F �dotted curve�.
and the first DOE is coded with the complex additionalgorithm �Eq. �15��. Figures 7�a� and 7�b� show theresults for these devices. The diffraction efficienciesfor the individual pages are lower than for a singlepage ��6.8%� and the MSE values are higher ��3.1%�.These values still are acceptable, so at least two pagescan be coded into a binary two-layer diffractive opticalelement. Further improvements can be made withthe above described optimization algorithms.
4. Experimental Results
In this section we present some experimental resultswith two-layer binary DOEs. We use the same pa-rameters as in the simulations, except for the numberof sampling points, where we choose 1024 � 1024.Even though the shape of the exposed points is ig-nored during the calculation, acceptable experimen-tal results still are obtained. Some suggestions havebeen made to include the point shape in the NFT,13
which have to be considered in the future. As re-cording material we used a stack of two polymerfilms. The element is written in a point by pointmanner with a focused laser beam. Owing to the
angle of divergence of the laser of �17 degrees, writ-ing into one layer does not disturb the other. Be-cause some parameters, like the exact phase shift andthe influence of the point shape, are not known ex-actly, the results presented here have only a qualita-tive character, but deliver a good proof of principle.
Figure 8�a� shows the reconstruction of a two-layerbinary diffractive phase element, calculated with adistribution-on-layers algorithm. Only one diffrac-tion order appears in the reconstruction, thus thewhole reconstruction plane can be used without dis-turbance from negative orders �Fig. 8�b��. The in-tensity is sufficient above the noise level. Figure9�a� shows the reconstruction of a device with a bi-nary kinoform in one layer. The reconstruction ofthe binary kinoform is the 2D barcode and its twinimage, and the cross-talk term of Eq. �21� delivers theletters O.K. Figure 9�b� shows the reconstruction ofthe same device illuminated under a different angleof incidence. The 2D barcode is still visible, whilethe letters O.K. disappeared owing to the angularselectivity of the device.
Figure 10 shows the multiplexing capability of the
Fig. 8. Experimental reconstructions of two-layer binary diffractive optical elements with a binary random distribution in the secondlayer. �a� Only one diffraction order appears. �b� The whole reconstruction plane can be used.
Fig. 9. Experimental reconstructions of two-layer binary diffractive optical elements with a kinoform in the second layer. �a� Whenilluminated under 0 degree the reconstruction consists of the reconstruction of the kinoform �2D barcode� and the coded signal �lettersO.K.�. �b� When illuminated under a slightly different angle of incidence the letters O.K. disappear.
device. Here we used angular multiplexing with 0and 1 degree. The two pages �b and 6� have beencoded with the iterative approach. At 0 degree thereconstruction of b appears, at 0.5 degrees both re-constructions appear with lower diffraction effi-ciency, and at 1 degree the reconstruction of the page6 appears.
5. Conclusion
We have presented a model and various calculationschemes for computer-generated SDOEs. The prop-erties of computer-generated SDOEs, such as the ap-pearance of only one diffraction order and thepossibility of angular and wavelength multiplexing,have been demonstrated in simulations and experi-mentally. Computer-generated SDOEs are promis-ing for various applications, e.g., in opticalcomputing, data storage, and security. In the fu-ture, improvements must be made in the calculationof the SDOEs, e.g., to include the point shape in thenear-field transform or to find coding operators thatare more suitable. Investigations must be madeabout the storage capacity and the reconstructionquality as a function of the number of layers andother parameters, such as the number of phase levelsor the layer distance. A classification of computer-generated SDOEs should be given by a detailed com-parison of them with computer-generated one-layerdiffractive optical elements and with volume holo-grams.
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Fig. 10. Experimental reconstruction of a angular multiplexed two-layer binary diffractive optical element with page b for 0 degree andpage 6 for 1 degree: �a� 0 degree, �b� 0.5 degrees, �c� 1 degree.
