Synthetic diffractive optics in the resonance domain
Eero Noponen and Antti Vasara
Department of Technical Physics, Helsinki University of Technology, SF-02150 Espoo, Finland
Jari Turunen, J. Michael Miller, and Mohammad R. Taghizadeh
Department of Physics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK
Received September 30, 1991; revised manuscript received November 18, 1991; accepted November 19, 1991
Resonance-domain diffractive optics covers the region where the characteristic feature sizes in the surface-relief modulation structure of the diffractive optical element are comparable to the wavelength of light; it maybe viewed as a bridge between synthetic holography and the electromagnetic theory of diffraction gratings andrough surfaces. We consider the problem of synthesizing, in the framework of the electromagnetic theory,various types of periodic resonance-domain diffractive optical elements that utilize several diffraction orders.Parametric optimization is used to design one-to-N fan-out elements, N-to-N star couplers, and polarization-controlled optical beam splitters and switches with close to 100% efficiencies and no undesired diffractionorders in the image half-space. Reflection-type fan-out gratings with six and seven output beams are demon-strated experimentally at A = 10.6 m.
1. INTRODUCTIONConsider a diffractive optical element with the index-modulated region occupying the volume -H < z < 0. Incomputer-generated holography, the effect of this elementon a wave front U1 (x, y, z) incident from the negative z di-rection is customarily described by the complex-amplitudetransmission function' t(x, y) = UT(x, y, 0)/U,(x, y, 0),where UT(X,y,0) is the diffracted field across the planez = 0 and U(x, y, 0) is the (incident) field that would beobserved across the same plane in the absence of the dif-fractive element (hologram). Analogously, a reflectionfunction r(x,y) may be defined as the ratio of the dif-fracted and the incident wave fronts across the plane z =-H: r(x, y) = UR(X, -H)/UI(x, y, -H).
On some occasions both U and UT (or UR) are known,and to solve the wave-front transformation problem it suf-fices to find an index-modulation structure n(x, y, z) thatrealizes t(x, y) with the accuracy permitted by the fabrica-tion method. This is the case, e.g., in the design of dif-fractive lenses and aspheric holographic elements foroptical testing.2 On other occasions, e.g., in image dis-play by computer-generated holograms and in array gen-eration by synthetic Fourier-domain gratings,3 UT needsto satisfy only some constraints such as the possession of aspecified optical power spectrum within a certain spatialfrequency window W. Here, degrees of freedom such asthe Fourier-space phase distribution within W and theamplitude distribution outside W are available for the op-timization of the wave-front transformation.4
The design of computer-generated holograms has so farbeen based almost exclusively on Fourier optics, 5 i.e., onthe approximate Fresnel and Fraunhofer diffraction theo-ries involving the paraxial approximation. In this domainthere is a direct correspondence between the index-modulation profile and the transmission function:
t(x, y) = exp ik J n(x, y, z)dz], (1)
where k = 2r/A is the wave number in vacuum. Thetransmission function is, however, independent of theform of the incident field only within the paraxial approxi-mation. Nonparaxial diffraction theories based onKirchhoff boundary conditions have been developed,6 buttheir validity for surface-relief gratings with H A is en-sured only if the smallest transverse features are severaltimes larger than the wavelength.
In the resonance domain, where, by definition, the char-acteristic transverse features are comparable to A, the vol-ume (multiple-scattering) effects inside the modulatedregion as well as the state of polarization of the incidentlight become important. Here, rigorous electromagneticdiffraction theory must be used: numerical methods de-veloped for the analysis of gratings and rough surfaces areavailable.`~
With the development of the microlithographic fabrica-tion of surface-relief profiles containing submicrometerfeatures, it has become possible to scale the dimensions ofcomputer-generated hologram structures down to a regionwhere the Fourier optical designs may no longer be valid.The need to analyze diffractive optical elements in or nearthe resonance domain by using the electromagnetic theoryhas been widely recognized, and some investigations onphase Fresnel lenses'0"' and optical array illuminators'2 "'have been performed, although in the studies reported inRefs. 10 and 11 the geometry of the lens was not takeninto account in a fully rigorous manner. For diffractivelenses, the primary effect is the decrease of efficiency intothe desired wave front, but the designs of array generators(and other diffractive elements whose function is based onthe simultaneous use of several diffraction orders) failcompletely near the resonance region.
The effects that appear in the resonance domain havethus far been regarded as limitations in computer-generated holography. We aim to demonstrate that thisneed not be the case: the problem of synthesizing peri-odic resonance-domain diffractive optical elements that
When illuminated by an obliquely incident, linearly po-larized plane wave [the y component of the electric (TEpolarization) or the magnetic (TM polarization) field]
U1(x, z) = exp[ikno(x sin 0 + z cos 6)], (3)
the periodic scatterer of Fig. 1 produces a reflected and atransmitted wave field, UR(X, z -H) and UT(X, z 2 0),respectively, which contain both homogeneous (propagat-ing) and inhomogeneous (evanescent) waves. These fieldscan be expressed in the form of Rayleigh expansions (see,e.g., Refs. 7 and 8) as
UR(x, z) = R exp[i(ymx - rz)],mn= _
-H 0 ZFig. 1. Diffraction of a plane wave by a binary, periodic diffrac-tive optical element.
utilize several diffraction orders is addressed by usingparametric optimization and the electromagnetic diffrac-tion theory. By exploiting the new degrees of freedomavailable as the result of strong volume and polarizationeffects, we can enhance the performance of many well-understood diffractive optical components. The examplesgiven include almost lossless fan-out elements (connectingone source to N detectors) and star couplers (connecting Nmutually uncorrelated sources to N detectors). It is alsopossible to synthesize elements that have no counterpartsin the framework of Fourier optics: as an example, wedesign a polarization node that either passes through orswaps two beams with different angles of incidence, de-pending on their states of polarization.
The fabrication of these elements for visible light ap-pears to be somewhat beyond the present-day lithographictechnology, but two of the fan-out designs are demon-strated at A = 10.6 /im.
2. SYNTHESIS PROBLEM IN THERESONANCE DOMAIN
We restrict the discussion here to two-dimensional refrac-tive-index-modulation structures n(x, y, z) = n(x, z).This greatly simplifies the mathematical formalism andthe numerical implementation of the synthesis problem,since cross-polarization coupling is excluded and the dif-fraction problem reduces to the two fundamental cases ofpolarization (TE and TM) that may be treated separately. 7
Furthermore, we concentrate on periodic (period d),lamellar structures, as illustrated in Fig. 1, with J groovesin one period:
(nj when x E [aj,bj) (2)n2 otherwise
with -H z s , j = 1,...,J, and 0 al < b < ... <aj < b < ... < a < b • d. The assumption of a lamel-lar profile is by no means essential, but such structuresare the easiest to fabricate with microlithography. More-over, in the resonance domain these binary gratings canbe designed to exhibit properties that serve well to illus-trate the new design prospects. This restriction is actu-ally not explicit in much of the discussion to follow.
that describe the response of the grating to illuminationby an incident plane wave. Expressions (9) and (10) areformally similar to those appearing in Fourier optics, butthe amplitudes Rm and Tm depend in the resonance domainstrongly on the angle of incidence, the state of polarizationof the incident wave, and the scale of the groove structurein relation to A. There is no simple connection [such asEq. (1)] between n(x, z) and t(x) [or r(x)], and the ampli-tudes Rm and Tm must be determined by solving the fullelectromagnetic boundary-value problem. Therefore r(x)and t(x) cannot be called reflection and transmissionfunctions of the grating in the same sense as in Fourieroptics (we use the term response function).
The infinite sets {Rm} and {Tn} each divide into two sub-sets containing the amplitudes of the propagating and theevanescent diffraction orders, respectively. The two setsof propagating orders, Rt = {m I rm real} and = {m I tmreal}, are of primary interest since only these orders areobservable in the far field. However, in the resonancedomain the evanescent waves cannot be neglected, sincethey contribute significantly to the amplitudes of thepropagating orders. The number of evanescent wavesthat is needed for achievement of reliable results must, inpractice, be determined by numerical experiments on theconvergence of the amplitudes Rm and Tm toward theirasymptotic values (obtained when an infinite number ofevanescent waves is included).
X
d
.................................;
X no n3
Noponen et al.
1208 J. Opt. Soc. Am. A/Vol. 9, No. 7/July 1992
The relative intensities of the reflected propagating dif-fraction orders are
I. = (rm/ro)JR. 1j2, (11)
and for transmitted orders they are
J = C(tm/ro)ITm12 , (12)
where C = 1 for TE polarization and C = (no/n3)2 for TMpolarization. We define the image window W C R U 5- asthe spatial frequency range inside which we wish to obtaina specified diffraction pattern. When electromagnetictheory is used to solve the diffraction problem, even W =R U is meaningful. In the examples given in Section 3,we choose W = R for reflection gratings (no = n arereal; n2 = n 3 are complex) and W = 9- for transmissiongratings (no = n, n2 = n3 are all real), i.e., we try to con-trol the diffracted field in a full half-space: this is insharp contrast with the scalar theoretic methods that arevalid only if a relatively small subset of either a or 9 ischosen. Naturally, narrower windows can be used also inthe resonance domain.
For simplicity, we restrict the attention here to binaryintensity signals by introducing a goal power spectrum
(13)= fIm Im E {0, 1}, m EE V}
consisting of
N= E Yimew
signal orders with equal intensities. To characterize thefidelity of the diffractive element, we introduce the dif-fraction efficiency
r7= I. .gn (15)mEW
as the ratio of the power diffracted into the signal beamsinside W and the power of the incident beam. Further,we define the reconstruction error
E = max I1 - NIn/lq (16)
as a maximum norm that gives the distance between thepower spectrum synthesized so far and the goal 9?. Fi-nally, the functions
rw(x) = _ Rm'exp(i27rmx/d), (17)mew
tw(x) = TM exp(i27rmx/d) (18)mEW
can be defined by extending the summations in Eqs. (9)and (10) over the image window only. These functionsrepresent the wave-front transformations stored in thediffractive element.
We are now in a position to state the synthesis problemof a resonance-domain diffractive optical element illumi-nated by a single plane wave, Eq. (3):
Find an index-modulation structure n(x, z) that mini-mizes the reconstruction error, Eq. (16), and simulta-neously maximizes the diffraction efficiency, Eq. (15).
(14)
As was stated above, the synthesis problem is analogous tothat often used in the conventional (scalar) theory of peri-odic computer-generated holograms. In the resonancedomain, however, meaningful extensions exist: one can,e.g., define different goal distributions for the two funda-mental cases of polarization or for two or more differentangles of incidence and/or wavelengths. Consider Q mu-tually uncorrelated " unit-amplitude input waves
U,q(X, z) = exp[ikqno(x sin 0+ z cos 0,)] (19)
where kq = 2
/Aq and q = 1,...,Q For these waves wedefine in an obvious manner the image windows W,, thegoal spectra
9'q = {Im,q I Im,q = {0,1}, m E wq} (20)
with
(21)Nq = 2 IMq)mEd
the diffraction efficiencies
71q = Im,q Im,qME I Cq
the reconstruction errors
Eq = maxImq11 - NqImq/'1iqj,mE=-W
and the functions rlwrq(x) and trq(x) for q =
generalized synthesis problem can nowfollows:
Find n(x, z) that maximizes q=1..Q{7q}
neously minimizes a,. {Ej.
(22)
(23)
1,...,Q Thebe stated as
and simulta-
We note that alternative definitions could be given andthat the synthesis problem could be further generalized,e.g., to permit partially (or fully) correlated input waveswith arbitrary relative intensities.
In the framework of Fourier optics the expanded syn-thesis problem defined above is meaningful for differentwavelengths only (see the design of so-called color separa-tion gratings 5 ). In nonparaxial Kirchhoff theory, it ispossible to employ different angles of incidence, althoughthe design prospects are limited. 6 For different states ofpolarization, however, this extended definition is mean-ingful only in the context of electromagnetic theory.
The synthesis problem described above bears a degreeof similarity to the problem of electromagnetic inversescattering by diffraction gratings. 7 In the inverse (re-construction) problem the aim is to determine the indexmodulation or the surface-relief profile from insufficientdata such as the efficiency curves of one or more diffrac-tion orders measured as a function of the angle of inci-dence. Here uniqueness of the solution is not ensured.In the synthesis problem, we also have partial informationon the diffraction pattern, i.e., the specified goal spec-trum QP. Uniqueness is irrelevant as long as the solutionmeets the design goals and the fabrication constraints,but the existence of a good solution is not guaranteed.The techniques of inverse diffraction have been applied tograting-profile optimization, the goal being to maximize
Noponen et al.
Vol. 9, No. 7/July 1992/J. Opt. Soc. Am. A 1209
the efficiency into one order (to blaze the grating), as isrequired in spectroscopy.'7 "8 However, continuous-surface-relief structures that are difficult to fabricate bymicrolithography are obtained. We use a different ap-proach, i.e., parametric optimization, that allows us tospecify explicitly the type of the grating structure.
3. DESIGN METHOD AND RESULTS
To solve the (generalized) synthesis problem, we use anapproach that is rather similar in nature to the methodapplied in previous studies in the Fourier domain6 92 0 andthat is therefore described only briefly here. We aim tominimize the quadratic merit function
QMf = E (, - Imq/qlNq)2, (24)
q=l mE-Wq
where i&q represents the goal efficiencies assigned to thediffraction patterns resulting from different input waves:we often use = 1 for all q = 1, ... , Q The parametersof the grating profile (the relief depth H and the transi-tion points {aj, bj} in the case of a lamellar profile of Fig. 1)represent the degrees of freedom that we employ tominimize Mf For this purpose, we apply a conventionalgradient algorithm, in which each of the N parameters isperturbed, to evaluate the direction of steepest descent inthe N-dimensional parameter space. In general, thismethod stagnates in a local minimum, which means thatwe may have to repeat the optimization a number of times(starting from different random configurations) to find anacceptable solution. Therefore it is difficult to give esti-mates on the computational complexity, which is in gen-eral considerable since the full diffraction problem has tobe solved each time a trial change on one of the transitionpoints or a move along the gradient direction is made.
To solve the electromagnetic boundary-value problemfor Rm and T,,,, we employ two different methods: fordielectric and metallic gratings (with finite conductivity)we use an implementation 2 "2 0 of the space-harmonicexpansion method developed in Refs. 21-23, whereas amultiple-groove generalization2 4 of a modal method forlamellar gratings 2 ' is employed for the (idealized) case ofperfectly conducting gratings. Although the formermethod is capable of solving the diffraction problem for allrealistic (mutlilevel) lamellar gratings, it requires a largenumber of space harmonics if the conductivity is high; inthese circumstances the computationally efficient modalmethod for perfectly conducting materials gives realisticresults. In the final determination of the coefficients Tmand Rm, as many as 500 evanescent orders were included,but a much smaller number was sufficient during the opti-mization: typically, the inclusion of approximately 20-40 evanescent waves gave satisfactory convergence fordielectric and perfectly conducting gratings. When thespace-harmonic method was used to determine the validityof the perfectly conducting solutions for finitely conduct-ing metallic gratings, good convergence required as manyas 300-500 Rayleigh orders for very-high-conductivitygratings (e.g., n = 25 + 89i for aluminum at A = 9.92 gim).
We now proceed to give some design examples to illus-trate the prospects of resonance-domain diffractive opticsin areas other than finding the optimal blazing of a dif-
fraction grating. Unless stated otherwise, the examplesare for TE-polarized light.
Consider first the problem of designing an optimal dif-fractive beam splitter that divides a linearly polarizedinput beam into two equal-intensity orders. According tothe coding theory of Fourier-domain computer-generatedholography,2 6 27 the necessary restriction t(x)l 1 impliesan upper bound -qu < 100% for the efficiency inside anyfinite spatial frequency window W. For the present caseof a fan-out to two, this upper bound can be found explic-itly: 71u = 8 2 = 81%.2' Extraordinarily, a hologramstructure that reaches the upper bound exactly can also befound: this is the binary-phase grating with a phasedelay of 'r rad and a groove width of half a period. Out-side the domain of Fourier optics, the values of t(x) are notnecessarily restricted by It(x)l c 1. This is due to volume(multiple-scattering) effects: it is well known that anequal-intensity fan-out to two with close to 100% effi-ciency is possible with a thick (H >> A) sinusoidally strati-fied index-modulation grating if use is made of the zerothand the first Bragg orders.29 To investigate the problemof a fan-out to two in the framework of the electromag-netic theory, we consider two geometries, which are illus-trated in Fig. 2.
For normal incidence [Fig. 2(a)], we use a goal P ={1,0,1} for orders W = {-1,0,1}. The choice 1 < n3d/A <2 ensures that only the three central orders can propa-gate, i.e., W = 9-. We optimized the period d, the reliefdepth H, and the groove width c = b - al of a simplelamellar grating, assuming (as is done throughout thissection for dielectric gratings) that n = n = 1.5, n2 =n3 = 1. The solution is {d/A, H/A, c/A} = {1.769,1.396,0.322}, with I = 47.3% and Io = 0.2%. The remaining5.2% of incident energy is distributed among the reflectedorders. The amplitudes and the phases of the responsefunction t(x) and the stored wave-front transformationtw(x) are plotted in Figs. 3(a) and 3(b), respectively.Clearly, tw(x) approximates well the ideal wave-fronttransformation:
t(x) = exp(ia)cos(2rrx/d - a) (25)
(here a is an arbitrary, real number originating from thefreedom to choose the relative phases inside ). Thesmall differences between t(x) and tv(x) are due to re-flections and the remaining weak undiffracted beam.The functions t(x) and tw(x) differ significantly, however,
(a)
(b)
1< s
Fig. 2. Fan-out to two with resonance-domain transmissiongratings: geometries. (a) Normal incidence: orders m = +1are used. (b) Bragg incidence: orders m = 0 and m = -1 areused.
Noponen et al.
1210 J. Opt. Soc. Am. A/Vol. 9, No. 7/July 1992
(a)2.5
2.0
01.\ \
0.00.0 0.2 0.4 0.6 0.8 1.0
x/d(b)
1.0
,0.6 -
2.0.2 0 I
-0.6
-1.0 I . . . . .0.0 0.2 0.4 0.6 0.8 1.0
X/d
(C)
1.6
X 1.2
- 0.8
0.4
0.0C
(d)1.0
! 0.6
X 0.2
2.0.2co
'.0 0.2 0.4 0.6x/d
-0.6
-1.00.0 0.2 0.4 0.6 0
x/d
Fig. 3. Fan-out to two with resonance-domain transngratings: results. (a) Amplitude and (b) phase of the re(transmission) function t(x) for the geometry of Fig. 2(a).curves: the summation extends over a large number of ecent waves. Dashed curves: the summation extends overgating waves only. In (c) and (d) the corresponding resugiven for the geometry of Fig. 2(b).
indicating the important role of the evanescent wathe resonance domain. Note also that both t(x) andhaving amplitudes that exceed unity, are complete]ferent from the binary-phase Fourier optical transmfunction:
=exp(8.77i) when x/A E [0,0.322)t(x) =
1 otherwise
determined by using Eq. (1).For Bragg incidence sin 0 = A/2nod [Fig. 2(1
specify the goal 9 = {1,1} for = {-1,0} and c1/2 < d/A < 3/2 to ensure that = Sf. A sol{d/A, H/A, c/A} = {1.000, 0.784, 0.468} was obtained,L = 48.8% and 1 = 48.5%. Now the reflected po'
only 2.7%, which means that the synthetic grating acreflects less light than a simple boundary between itn = 1.5 and n = 1 (see Ref. 30, where the electromaltheory is used to design antireflection gratings).amplitudes and the phases of t(x) and t(x) are plotFigs. 3(c) and 3(d). The ideal wave-front transformris now
tv(x) = exp(ia)exp(-iirx/d)cos(rrx/d - a)
(a arbitrary, real), which formally indicates that theing should first collimate the incident beam and therit in two. These functions are indeed performedhigh accuracy. The differences between t(x) and tvnot appear to be quite so dramatic as they are at nincidence.
Higher fan-out is also possible with resonance-dcdiffractive optics: the design concepts outlinedneed to be extended only slightly. As in Fourier cmore than one groove is needed for a fan-out higherthree. For an odd-numbered fan-out to N = 2Aequal-intensity beams, we choose Im = 1 forO= {-IM} and impose the restriction that M < d/A < M ensure that = ff. The simplest way to obtain anfan-out to N = 2M beams is to use, analogous
Fig. 2(b), (first-order) Bragg incidence sin 0 = A/2nod, tochoose Im = 1 for = {-M,...,M - 1}, and to requirethat d satisfy the constraint M - 1/2 < d/A < M + 1/2.In general, we prefer periods d near the upper end of thepermissible range to avoid diffraction angles close to 7r/2.
For example, a dielectric grating with d = 4.86A,. .A H = 1.26A, and three grooves defined by the boundaries
08 1.0 {(aj/A,bj/A)} = {(0.00,0.69), (1.41,1.87), (2.59,3.28)} gener-ates nine beams with 'i = 93.3% and E = 3.0%. Thisprofile is symmetric; rigorously, the Fourier-domain sym-metry I-n = Im of the diffraction patterns of binary grat-ings holds for symmetric profiles only. On the other
I hand, a reflection grating (no = n = 1, n2 = n3 = ia,l a~ co ) with d = 4.47A, H = 0.30A, and three grooves
A~ with boundaries {(aj/A,bj/A)} = {(0.00,0.55),(1.07,1.63),(2.41, 3.69)} generates, at an angle of incidence 0 = 6.42°,
.8 1.0 eight beams with 7 = 1 and E = 3.5%. The efficiency of100% is achieved because perfect conductivity has been as-spsone sumed and all the propagating orders have been equalized.
Solid To compare the resonance-domain designs with thosevanes- achievable in Fourier optics, we note that the upperpropa- bounds r7u cannot, in general, be predicted precisely forlts are intensity signals, nor can one be certain whether the best
Fourier-domain configuration has been found. It is, how-ves in ever, widely agreed that the maximum efficiencies of one-t-WW, dimensional binary designs (often called Dammannly dif- gratings') are in the range 77-86.5%, while for uncon-ission strained phase gratings they are in the range 92-99%,
depending on N. In all these figures, reflections at theboundary of the modulated region have been neglected
26) (antireflection coating of this boundary is extremely diffi-(26) cult), amounting to an additional loss of approximately 4%.
We concentrate next on the generalized synthesis prob-lem involving more than one input wave. As an example,
hwe we consider the so-called star coupler illustrated in Fig. 4.hoose The function of this diffractive element is to connect eachution of the N (mutually uncorrelated) sources A, B, C, etc. to allwith the N receivers A', B', C', etc. The resonance-domain
wer is devices for odd and even N, illustrated in Figs. 4(a) andtually 4(b), respectively, are extensions of the one-to-N fan-outidices elements in the sense that only the N orders propagatinggnetic toward the N receivers exist in the image space, but now
The several inputs are required for generation of the sameted in power spectrum. Note, however, that the response for
nation each input q consists of a.different set 0W of diffraction
(27)
grat-n splitwith
(x) doormal
)mainaboveIptics,r thanI + 1
F 1 to
l even;ly to
A < a < 'C.
j- B'
(a)L JIA '
A
B C'
B'(b)
Fig. 4. Operating principle of a resonance-domain star couplerconnecting N sources to N detectors: (a) odd fan-out, (b) evenfan-out with incident waves at Bragg angles.
21.
Noponen et al.
Vol. 9, No. 7/July 1992/J. Opt. Soc. Am. A 1211
(a)2.5 -
2.0 -
3Z 1.5-
_ 1.0 H0.5
0.0 _0.0
(b)
X
2CO
2.0
1.6
_1.2
0.8
0.4
0.00.0 0.2
(c)
0.2 0.4 0.6 0.8
x/d1.0
2.5
2.0
X 1.5
1.0
0.5
0.00.0
(d)
0.4 0.6 0.8 1.0x/d
0.2 0.4 0.6x/d
0.8 1.0
0.8 1.0
Fig. 5. Response functions r(x) and the wave-front transforma-tions r(x) for the 4-to-4 reflective star coupler at the two anglesof incidence.
orders, which in general complicates the relief structure.The synthesis problem may be simplified by assuming asymmetric lamellar profile, which implies that the re-sponse at an angle of incidence -0 is the mirror image ofthe response at 0. Then, for odd N = 2M + 1, we haveQ = M + 1 terms in Eq. (24), while for even N = 2M,Q = M inputs need to be treated.
Consider first a 3-to-3 dielectric star coupler, as illus-trated in Fig. 4(a). The restriction 1 < d/A < 2 removesall the undesired transmitted beams; the image windowsare OWl = {-1,0,1} for normal incidence (source B) andW2 = {-2, -1,0} for incidence from the direction ofsource A. We found a symmetric solution with d/A =1.773, H/A = 0.957, and two grooves {(aj/A,bj/A)} ={(0.104,0.397),(1.376,1.669)} characterized by ql = 96.9%,_72 = 96.0%o, El = 0.8%, and E2 = 3.0%. Here again, theefficiencies are higher than or equal to the transmissioncoefficient of the incident beam through a dielectricboundary from n = 1.5 to n = 1.
As another example, we synthesized a 4-to-4 perfectlyconducting star coupler, i.e., the reflective equivalent ofthe geometry of Fig. 4(b). Now we impose the restriction1.5 < d/A < 2.5 and choose OWl = {-2,-1,0,1} for inci-dence at the first Bragg angle [source B in Fig. 4(b)] andW2 = {-3, -2, -1,0} for the second Bragg angle [source Ain Fig. 4(b)]. A symmetric solution with d/A = 2.351,H/A = 0.43, and three grooves {(aj/A,bj/A)} = {(0.272,0.554), (0.608,1.743), (1.797,2.079)} gives El = 0.9% andE2 = 0.9%. The efficiencies are equal to unity. The re-sponse functions r(x) and the optimized wave-front trans-formations r(x) of this solution are shown in Fig. 5 forthe two angles of incidence. Again, there are large differ-ences between the desired wave-front transformationsrw(x), the response functions r(x) needed to generatethem, and the (binary-phase) Fourier optical reflectionfunctions.
The star couplers considered here have their counter-parts in the domain of Fourier optics: a conventionalfan-out element such as a Dammann grating can beused.3 2 Because of the space invariance of the Fourieroptical gratings, an N-to-N star coupler requires a grating
with a fan-out to 2N - 1 for odd N and a fan-out to 2Nfor even N. Therefore the upper bound for the efficiency(apart from reflection losses) is 7uN/(2N - 1) for odd Nand -q u!2 for even N, where 7 u is the upper bound for theappropriate fan-out element.
Our final example concerns the utilization of polariza-tion effects in resonance-domain diffractive optics. Wewish to design a four-port polarization-sensitive device,illustrated in Fig. 6, that accepts two mutually uncorre-lated input beams at angles ±0, letting them through un-diffracted if they are TM polarized and swapping them ifthey are TE polarized. If we again choose a symmetriclamellar profile, the response at -0 is for both polariza-tions the mirror image of the response at 0. Then theproblem of synthesizing this polarization-controlled nodereduces to the problem of designing the diffractive polar-ization beam splitter illustrated in Fig. 7. The dielectricsolution with d = 0.724A and H = 1.720A requires onlyone groove of width 0.140A: at 0 = 27.4 (inside n = 1.5),the efficiencies into the zeroth and the (minus) first ordersare 98.5% and 0.02% for TM polarization and 0.05% and95.9% for TE polarization. Hence both the efficienciesand the signal-to-noise ratios are remarkably good, therest of the incident energy being reflected. A correspond-ing reflection-mode node gives a virtually ideal perfor-mance with grating parameters d/A = 1.368, H/A = 0.391,groove width 1.061A, and angle of incidence 21.45.
There is no counterpart for these diffractive optical ele-ments in the framework of Fourier optics (with isotropicmaterials). Active nodes that can time sequentially passor swap two input beams can be constructed, e.g., by uti-lization of acousto-optic or electro-optic Bragg diffraction(Ref. 9, p. 929). The present node is fundamentally dif-ferent, being a passive polarization-sensitive component;however, one can easily make it active by controlling thepolarization of the input beams, e.g., with twisted nematicliquid-crystal devices.
Fig. 6. Operating principle of a polarization-sensitive node thateither lets through or swaps two mutually uncorrelated inputs at±0, depending on their (common) state of polarization.
] -iI7 TM
,P TEFig. 7. Operating principle of a diffractive polarization beamsplitter: TM-polarized light is passed straight through (zerothorder), while TE-polarized light is diffracted into the first order.
-i
51
� . I . . . .
Noponen et al.
; _
1212 J. Opt. Soc. Am. A/Vol. 9, No. 7/July 1992
Fig. 8. Scanning electron microscope photograph of a one-to-sixresonance-domain fan-out grating.
Fig. 9. Optical reconstruction of the resonance-domain one-to-six fan-out grating.
4. EXPERIMENTAL ILLUSTRATIONNumerical simulations in which the effects of random per-turbations of the optimized parameters on the figures ofmerit 7 and E were determined have revealed that an ac-curacy of the transition points in the relief profile shouldbe better than -A/20 for meaningful demonstrations: inparticular, the reconstruction error is a sensitive functionof fabrication errors. With the present-day status of mi-crolithographic fabrication technology, demonstrationsare therefore restricted to infrared wavelengths. We re-alized some of the reflection-type resonance-domain fan-out gratings for A = 10.6 m by using the followingtechnique.
A chrome-on-glass mask containing the grating struc-ture was first written by using an electron-beam patterngenerator. To produce a surface-relief structure withcorrect depth H, a layer of SiO2 with thickness H wasgrown on a silicon substrate and subsequently coated witha layer of photoresist (thickness 1 gim). The mask was
then contact copied on photoresist; an optical shearing mi-croscope was used to observe and control the linewidth.The SiO2 layer was reactive ion etched for 2 h in CHF3 andHe; the silicon substrate acted as an etch stop. Finally,after the remaining photoresist was removed, a 50-nm Allayer was sputtered onto the grating to achieve almost per-fect reflectance at A = 10.6 Aum: the reflectivity of sucha layer is close to 99%. Figure 8 is a scanning electronmicroscope photograph of the groove structure of a one-to-six fan-out grating: the grating period is d = 36.6 Am,and the width of the narrower of the two grooves is 6.1 ,m.
A conical diffraction mount with an angle of 100 be-tween the incident wave and its projection onto the xzplane was used to reconstruct the hologram; from the de-sign point of view, this requires only a simple scale changein the grating parameters (Ref. 8, pp. 56-57). The opticalreconstruction of the one-to-six fan-out grating is shownin Fig. 9. The input CO2 laser beam passes through therectangular hole in the heat-sensitive liquid-crystal paper,where the diffracted orders are recorded after reflectionby the grating (third from the top in the array visible onthe wafer): owing to Bragg incidence at 120, the order -1appears next to the hole.
By scanning the diffraction pattern with a large-areadetector we determined that the uniformity of the spotarray is E - 11% (the design value is 2.3%), which isroughly in agreement with the numerical analysis of themanufacturing errors, assuming that the fabrication ac-curacy of the transverse features is -0.2 Am. A one-to-seven fan-out grating was also reconstructed, with auniformity of E 15%. This was due mainly to a relief-depth error of -30 nm (H = 2.68 m instead of the de-sign value 2.65 gim): at normal incidence, thezeroth-order intensity is a sensitive function of the reliefdepth. Excluding the zeroth order, which was weakerthan the others, we measured E 8%.
Obviously, with resonance-domain diffractive optics,the ultimate degree of miniaturization of diffractive opti-cal systems can be achieved. However, factors such asthe large diffraction angles and the nonconstant angularseparation between the diffraction orders clearly visiblein Fig. 9 are evidently introduced. The conventional re-construction geometries of computer-generated hologramsmust therefore be replaced by novel schemes. A feasibil-ity study of the resonance-domain components in variousapplications is beyond the scope of this paper, but we notethat, because of the large diffraction angles, multifacetcollimation and focusing elements can be used.
5. CONCLUSIONS
In this paper we have considered the synthesis problem ofoptical elements in a field that we call resonance-domaindiffractive optics. Some of the new design prospects thatare available when the vectorial nature of light is fullytaken into account have been demonstrated.
We view synthetic resonance-domain diffractive opticsas an extension of computer-generated holography intothe region where Kirchhoff diffraction theory has to bereplaced by the electromagnetic theory of diffraction grat-ings. Whether the resonance-domain diffractive opticalelements synthesized here can (or should) be called holo-grams is a matter of opinion; we have, however, retained
Noponen et al.
Vol. 9, No. 7/July 1992/J. Opt. Soc. Am. A 1213
and extended the basic objective of synthetic holography,i.e., the storage of wave-front transformations t(x) thatgive rise to specified diffraction patterns 9P covering afinite spatial frequency window W (whereas the tradi-tional motivation of electromagnetic grating theory is theanalysis and the synthesis of grating profiles with optimalblazing for maximum efficiency into a single order).
As in conventional computer-generated holography withintensity signals, the freedom of phase distribution insideW and the complex-amplitude distribution outside W (seeRef. 4) are used (although somewhat implicitly with ourparticular algorithm) to ensure that a certain type of re-lief profile, binary in our case, generates the desired in-tensity pattern inside OW. An important difference is thatthe restriction t(x) < 1, which limits the diffraction effi-ciency, can be avoided. This is because the complex am-plitudes outside W that we use as a design freedom areassociated with evanescent waves that carry no energy.
ACKNOWLEDGMENTS
The authors thank F Wyrowski and A. T. Friberg forstimulating discussions and the Electron Beam Lithogra-phy Facility group at Rutherford Appleton Laboratory andthe staff of Edinburgh Microfabrication Facility for fabri-cating the resonance-domain demonstration gratings.This work was financed by the Academy of Finland andthe Science and Engineering Research Council, UK.
REFERENCES AND NOTES
1. M. Born and E. Wolf, Principles of Optics (Pergamon, Ox-ford, 1986).
2. W-H. Lee, "Computer-generated holograms: techniques andapplications," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. XVI, pp. 121-231.
3. N. Streibl, "Beam shaping with optical array generators,"J. Mod. Opt. 36, 1559-1573 (1989).
4. 0. Bryngdahl and F. Wyrowski, "Digital holography-computer-generated holograms," in Progress in Optics,E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. XXVIII,pp. 1-86.
5. J. W Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
6. P. Beckmann, "Scattering of light by rough surfaces," in Prog-ress in Optics, E. Wolf, ed. (North-Holland, Amsterdam,1967), Vol. VI, pp. 53-69.
7. R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 ofTopics in Current Physics (Springer-Verlag, Berlin, 1980).
8. D. Maystre, "Rigorous vector theories of diffraction grat-ings," in Progress in Optics, E. Wolf, ed. (North-Holland,Amsterdam, 1984), Vol. XXI, pp. 1-67.
9. T. K. Gaylord and M.G. Moharam, 'Analysis and applicationsof optical diffraction by gratings," Proc. IEEE 73, 894-937(1985).
10. N. C. Gallagher, Jr., and S. S. Naqvi, "Diffractive optics:scalar and non-scalar design analysis," in Holographic Op-tics: Optically and Computer Generated, I. N. Cindrich andS. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1052,32-40 (1989).
11. J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, and J.
Bergstrom, "Diffraction efficiency of binary optical ele-ments," in Computer and Optically Formed Holographic Op-tics, I. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116-124 (1990).
12. A. Vasara, E. Noponen, J. Turunen, J. M. Miller, and M. R.Taghizadeh, "Rigorous diffraction analysis of Dammanngratings," Opt. Commun. 81, 337-342 (1991).
13. A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R.Taghizadeh, and J. Tuovinen, "Rigorous diffraction theory ofbinary optical interconnects," in Holographic Optics III:Principles and Applications, G. M. Morris, ed., Proc. Soc.Photo-Opt. Instrum. Eng. 1507, 224-238 (1991).
14. Two monochromatic waves with the same wavelength and thesame state of polarization but incident at different anglesare, strictly speaking, mutually coherent. However, mutualnoncorrelation is assumed here, meaning, e.g., that indepen-dent (quasi-monochromatic) sources are used.
15. H. Dammann, "Color separation gratings," Appl. Opt. 17,2273-2279 (1978).
16. J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Wester-holm, M. R. Taghizadeh, and J. M. Miler, "Storage of multipleimages in a thin synthetic Fourier hologram," Opt. Commun.84, 383-392 (1991).
17. A. Roger and D. Maystre, "Inverse scattering method in elec-tromagnetic optics: application to diffraction gratings,"J. Opt. Soc. Am. 70, 1483-1495 (1980).
18. A. Roger, "Grating profile optimizations by inverse scatter-ing methods," Opt. Commun. 32, 11-13 (1980).
19. J. Turunen, A. Vasara, J. Westerholm, G. Jin, and A. Salin,"Optimization and fabrication of grating beamsplitter,"J. Phys. D 21, S102-S105 (1988).
20. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E.Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, and S.Kuisma, "Binary surface-relief gratings for array illumina-tion in digital optics," Appl. Opt. 31, 3320-3336 (1992).
21. C. B. Burckhardt, "Diffraction of a plane wave at a sinu-soidally stratified dielectric grating," J. Opt. Soc. Am. 56,1502-1509 (1966).
22. F. G. Kaspar, "Diffraction by thick, periodically stratifiedgratings with complex dielectric constant," J. Opt. Soc. Am.63, 37-45 (1973).
23. K. Knop, "Rigorous diffraction theory for transmission phasegratings with deep rectangular grooves," J. Opt. Soc. Am. 68,1206-1210 (1978).
24. J. M. Miller, J. Turunen, M. R. Taghizadeh, A. Vasara, and E.Noponen, "Rigorous modal theory for perfectly conductinglamellar gratings," IEEE Conf. Publ. 342, 99-102 (1991).
25. D. Maystre and R. Petit, "Diffraction par un reseau lamel-laire infinement conducteur," Opt. Commun. 5, 90-93 (1972).
26. F. Wyrowski and 0. Bryngdahl, "Digital holography as part ofdiffractive optics," Rep. Prog. Phys. 54, 1481-1571 (1991).
27. F. Wyrowski, "Theory of digital holography," IEEE Conf.Publ. 342, 93-97 (1991).
28. F Wyrowski, "Characteristics of diffractive optical elements/digital holograms," in Computer and Optically Formed Holo-graphic Optics, I. N. Cindrich, ed., Proc. Soc. Photo-Opt.Instrum. Eng. 1211, 2-10 (1990).
29. H. Kogelnik, "Coupled wave theory for thick hologram grat-ings," Bell Syst. Tech. J. 48, 2909-2947 (1967).
30. M. G. Moharam and T. K. Gaylord, "Diffraction analysis ofdielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1392 (1982).
31. H. Dammann and K. Grtler, "High-efficiency in-line mul-tiple imaging by means of multiple phase holograms," Opt.Commun. 3, 312-315 (1971).
32. U. Killat, G. Rabe, and W Rave, "Binary phase gratings forstar couplers with high splitting ratio," Fiber Integr. Opt. 4,159-167 (1982).
