Construction of one-dimensional multilevel phase gratings is described that, when illuminated by acoherent plane wave, lead to the formation of amplitude binary gratings with an arbitrary value of theopening ratio. The gratings are proposed as array illuminators that can provide a significantly highcompression factor together with a large number of uniformly illuminated points.
1. Introduction
An array illuminator is an optical device that providesillumination to an array of points. Various construc-tions, those that usually convert a uniform field of aplane wave into an array of beams, have been pro-posed.'
In this paper we analyze a Talbot array illumina-tor2 3 based on properties of the diffraction field of aphase grating. This construction has some potentialadvantages over other systems, such as a large num-ber of light spots, uniform illumination, and simplic-ity of construction. Its main drawback is that it re-quires complicated and difficult-to-fabricate phasestructures.
The aim of this work is to design a one-dimensional(1-D) multilevel, phase structure that, when illumi-nated by a coherent plane wave, leads to the forma-tion of a binary amplitude grating with a significantlysmall value for the opening ratio, thus making itusable as an array illuminator. The generalizationof our results for the two-dimensional (2-D) case isobvious.
Theoretical studies of the Fresnel diffraction fieldof phase gratings usually lead to elaborate and te-dious analyses.4 Diffraction gratings, however, be-long to a group of objects whose Fresnel diffractionfield changes periodically along its optical axis.5This property allows one to simplify analysis by
When this study was performed, the authors were with theInstituto Nacional de Astrofisica, Optica y Electronica, ApartadoPostal 216, Puebla 72000, Puebla, Mexico. V. Arrizon is now withthe Departamento de Semiconductores, Universidad Autonoma dePuebla, Puebla, Mexico.
reversing the problem of finding a phase structurethat is capable of producing the desired amplitudepatterns. Yet, it is possible to analyze the necessaryconditions under which pure phase distributions areformed in the diffraction field of an amplitude grating.It follows from the periodicity of the diffraction fieldthat, if we can reproduce this phase distribution (inthe form of a phase grating) and illuminate it with acoherent plane wave, the images of the amplitudestructure will emerge at certain planes behind thephase grating.
In Section 2 we follow the procedure of Winthropand Worthington6 to establish the conditions underwhich an amplitude binary grating produces diffrac-tion patterns of constant irradiance. This result isused to design 1-D multilevel phase gratings that arecapable of producing amplitude, binary structures.As examples, we present calculated profiles of phasegratings that can produce 1-D amplitude binarystructures with opening ratios of 1/4, 1/5, 1/7, and1/9.
2. Theory
We analyze the Fresnel diffraction field of a binaryamplitude grating that is characterized by an arbi-trary value of opening ratio a and is illuminated by acoherent plane wave. The geometry of the setup isshown in Fig. 1.
For the purpose of this work it is possible to limitanalysis of the Fresnel diffraction field to a fractionalTalbot distance:
z = Zt/n, n = 2, 3 1
where Zt = 2d2
/X is the self-imaging distance of thegrating of period d and X is the wavelength of light.
Following the analysis of Winthrop and Worthing-ton,6 we express the transmittance of such a grating
Introducing the condition given by Eq. (1) into Eq. (6)we obtain the function A(x, z = Zt/n) for a fractionalTalbot distance:
1icA(x,z=Zr/n)=- z exp(-i2,m 2/n)exp(i2irmx/d).
(7)
The summation in Eq. (7) can be expressed with thehelp of two new parameters q and L, such that
m = nL + q,
Fig. 1. Geometry of the optical system.
as a convolution of the comb function with transmit-tance of a unitary cell, that is,
(2)t(x) = to(x) X = 8(x - md).m= -.
In our case the function that describes unitary celltransmittance to(x) is defined as
to = f1, for lx < w/20, for Ix > w/2(
where w < d (see Fig. 2). The opening ratio of thegrating is given by a = w/d.
The diffraction field f(x, z) at a distance z behindthe grating can be expressed (under paraxial approxi-mation) as a convolution of a transmittance functiont(x) with the function h(x, z) = exp(ikx2/2z):
f(X ) iz t(x) h(x, z)
= exp(ikz) tox g A(x, z),.i t(x xx )
(8)
where L is an integer number and q = 0, 1, .... n - 1.Thus, Eq. (7) can be rewritten as
In Eq. (9) it is possible to replace the summation overindex L by the comb function:
z exp(i27rnLx/d) = - z B(x - Ld/n).L=-c n L=-cc
(10)
Thus, Eq. (9) can be transformed into the form
A(x, z = Zt/n) = z C(L, n)B(x - Ld/n), (11)L= -s
where the coefficient C(L, n) is given by(4)
where k = 2r/X and A(x, z) is defined asC(L, n) = - , exp i q(L - q)] .
A(x, z) = z 8(x - md) 0 h(x, z). (5)
After some manipulation this function can be pre-sented in the following form6:
1 XA(x, z) = - exp[-irz(m/d) 2 ]exp(i2rrmx/d).
m= -c
-d/2 -w/2
t0 (x)
0
w/2 d/2Fig. 2. Transmittance function to(x) of a unitary cell.
(6)
Finally, we can introduce Eq. (11) into Eq. (5) andobtain the expression for the Fresnel diffraction fielddistribution at the plane z = Zt/n behind the binaryamplitude grating:
f (x, z = Zt/n) = C(L, n)to(x - Ld/n), (13)L=-cc
One can observe that f (x, z = Zt/n) is constructedby means of the summation of a set of functionsC(L, n)to(x) that are shifted in the x direction by thefactor Ld/n. From this formulation and definition ofto(x) it follows that all future diffraction images arefully described by coefficients C(L, n). Thus, let usinvestigate the properties of C(L, n) that will be
x helpful in determining locations of planes of constantirradiance.
As it follows directly from Eq. (12), C(L, n) isperiodic in index L with period n, that is, for any
The term exp[(i27r/n)q(L - q)] is also periodic in qwith a period n. This allows us to express Eq. (12) inthe equivalent form
1 n '" [i2=rrQLn) _ exp[-q(L
nq=-p n
wherep is any integer number.In Appendix A we show that, for any n 2 1 and for
any integer number K, we have
C(L = 2K, n)j = JC(L = 0, n)j, (16)
IC(L = 2K+ 1,n)l = C(L = 1,n)I. (17)
It is possible to show, however (see Appendix B), that,for any odd n, we have
C(1, n)I = C(0, n)I • 0. (18)
Equation (18) together with Eqs. (16) and (17) impliesthat in this case all the coefficients C(L, n) retain thesame, constant value of the modulus.
In Appendix C we show that, when n is an evennumber, the following relationships are fulfilled forany integer numbers K and M:
C(L = 2K, n = 4M + 2)1 = 0, (19a)
C(L = 2K, n = 4M)l = IC(L = 0, n = 4M)l • 0,
obtain uniform intensity distribution in the observa-tion plane. Coefficients C(L, n) are characterized bydifferent phase factors, and we can conclude that theobtained distribution forms a multilevel, binary phasestructure of period d. Each unitary cell of such astructure is divided into n subcells characterized bydifferent values of phase coefficients [see Fig. 3(a)].
B. Even Number n
When n is an even number we obtain a result that issimilar to the one described above provided that theopening ratio a = 2/n. In this case there are twoslightly different situations that depend on whethern/2 is even or odd and both are illustrated in Figs. 3(b)and 3(c). The most significant difference between nodd and the present situation exists because now thecoefficients C(L, n) are equal to zero for all even orodd values of L [see Eqs. (19) and (20)]. This causesevery second component in the series describing thefield distribution [see Eq. (13)] to vanish. As aresult, function f(x, z = Zt/n) is constructed by plac-ing rect(xw) functions that are spaced by a distance2Zt/n. Thus, it is necessary to increase the value ofthe opening ratio of the grating from a = 1/n to (x =2/n to assure formation of uniform intensity distribu-tion in the observation plane. All these results aresummarized in Table 1.
3. Multilevel Phase Gratings
From the properties of self-imaging it follows that, ifwe can reproduce the phase profile given explicitly byEq. (13) and illuminate it by a coherent plane wave,then at the distance Zt - (Zt/n) an amplitude gratingwith opening ratio a will occur. It is evident that the
C(L = 2K + 1, n = 4M)l = 0,
C(L = 2K + 1, n = 4M + 2)1
= C(L = 1, n = 4M +2)1 • 0.
(20a)1
(20b)
Equations (18)-(20) show that coefficients C(L, n)exhibit slightly different properties for even and oddvalues of parameter n. This implies that there aretwo different cases that depend on value n for whichthe possibility of obtaining a uniform intensity distri-bution should be investigated separately.
A. Odd Number n
It follows from Eq. (18) together with Eq. (14) thatwhen n is an odd number I C(L, n) I = const for anyvalue of L. Thus, for a = w/d = 1/n a constantirradiance distribution is obtained in the plane ofobservation. This occurs because, according to Eq.(18), the modulus of coefficients C(L, n) in Eq. (13)does not depend on value L. Moreover, because ofthe definition of function to(x), we can determine thatf (x, z = Zt/n) is constructed by placing a series ofrect(xw) functions of a width w = d/n that are spacedby a distance d/n. Thus, they fill space in the x direc-tion tightly and without overlapping. Since the mod-ulus of coefficients C(L, n) is independent of L, we
Fig. 3. Intensity distribution at the distance z = Zt/n behind thebinary amplitude grating for (a) n odd and a = 1/n; (b) for both nand n/2 even and a = 2/n; (c) for n even but n/2 odd and a = 2/n.
IC(-3,,,)I' IC(-2 '�,)12 IC(_,,,,)12 JC(40',,)j1 IC(ln )12 JC(2 ,,)12 1 C (.3, n I'
- q) ,
x
Table 1. Basic Properties of the Coefficients C(L, n) and Values ofOpening Ratio Required for Uniform Intensity Distribution at the
Observation Plane z = Zt/n
RequiredL Value of
Openingn Odd Even Ratio a
Odd I C(L, n) I = const ; 0 C(L, n) =const. Of 0 1/nEven C(L, n) = 0 C(L, n) I const. 0 2/n
(n/2 even)Even I C(L, n) I = const • 0 C(L, n) = 0 2/n
(n/2 odd)
phase grating will also have a period d and its unitarycell will consist of n (for n odd) or n/2 (for n even)subcells, each characterized by a different (but con-stant) value of the phase factor. The phase valuescan be explicitly calculated by extracting imaginaryparts of C(L, n) coefficients for L = 0, 1, ... n = 1.
One can note that the opening ratio of the ampli-tude grating as well as the phase structure of thephase grating can be determined if the number n isgiven. Thus, we can treat value n as a parameterthat fully characterizes both the amplitude and phasegratings. Because we obtain a = 1 for n = 2 and thiscase is not physically realizable, we can limit therange of variation for this parameter to n > 2.
4. Examples
As far as we know, only two kinds of phase grating(n = 4 and n = 3) have been analyzed and proposed asarray illuminators.2' 3 Thus, we have designed a fewmultilevel phase gratings for different values of param-eter n as examples. The phase profiles were calcu-lated for n = 5, 7, 8, and 9 by extracting the imaginarypart from the expression that describes C(L, n) [seeEq. (12)]. The calculations were made for eachsubcell of the phase structure, i.e., for L = 0, 1,....n - 1, by applying numerical methods. Each calcu-lated value was adjusted to fit into the interval (0, 27r)by subtracting the integer multiple of 2'r. The phasevalues that were calculated this way are shown inTable 2, and appropriate phase profiles are plotted inFigs. 4(a)-4(d).
It is interesting to note that the number of differentphase levels is at most (n + 1)/2 for the phasestructures associated with odd n and n/2 when n iseven. This property shows that the only two-levelphase structures that can be used as Talbot arrayilluminators (they can be called binary phase grat-
Table 2. Phase Profiles of Difraction Gratings Calculated for n = 5, 7, 8,and 9 by Extracting the Imaginary Part of C(L, n)
Fig. 4. Profiles of the phase gratings plotted for the values ofparameters (a) n = 5; (b) n = 7; (c) n = 8; (d) n = 9.
ings) can produce amplitude binary gratings withopening ratios of a = 0.5 (n = 4) and a = 0.33 (n = 3).
5. Conclusions
We have investigated conditions under which an am-plitude binary grating, illuminated by a coherent planewave, produces a constant irradiance distribution inthe Fresnel diffraction region. It has been shownthat, for the grating with an opening ratio of a = 1/n(for n odd) or a = 2/n (for n even), a pure phase distri-bution can be observed at a distance z = Zt/n. Apply-ing the property of longitudinal periodicity of diffrac-tion fields of periodic objects, we conclude that, whenwe replace the phase distribution by the phase diffrac-tion grating and illuminate it by a coherent planewave, exact replicas of amplitude gratings emerge atthe distances z = Z,(N - 1/n); N = 1, 2....
The results obtained allow us to calculate profilesof multilevel phase gratings that can be used toproduce binary amplitude gratings with a signifi-cantly low value of the opening ratio. We propose touse these kinds of grating as array illuminators.However, the possibility of applying them in practiceis strongly limited by the difficulty of fabricating suchphase structures. It seems that simple methods donot exist that allow us to manufacture multilevelphase profiles with the required precision, except fortwo-level phase gratings. Thus, some technologicaldevelopment necessary before is the proposed methodcan be used efficiently.
Assuming that L is an even number we use Eq. (12) toobtain
i n- -i2rr 1C(L = 2K, n) = exp- q(2K - q)J
i n-1 - i2 7 _= - Iexp j [(q -K)2 - K2]}
=-exp( K2) z exp n (q - K)2]-exp( K2) -exp( q2)fl \ f l / = -K /
i27r = exp n K2 C(L = 0, n), (Al)
which leads directly to Eq. (16). In order to proveEq. (17) we assume that L is odd and use Eq. (12) toget
C(L = 2K + 1, n)
1 n-l i2Tr= - exp - q(2K + 1
1 irr= -exp-K(1 +K)
n-1 i2rX E exp - (q - K)(1
q=O n
1 i2Kr= -exp - K(1 + K)
n-1-K i2r1K exP - q(l
- q)]
- q + K)]
- q)]
i2,Tr= exp n K(1 + K) C(L = 1, n). (A2)
Also Eq. (A2) leads directly to Eq. (17).
Appendix B
Assuming that n is an odd natural number, we provethat C(1, n = 2p + 1) I = C(0, n = 2p + 1) . UsingEq. (12) we obtain
C(L = 1, n)
1 2 exPp + 1 q(l - q)]n = ep2p +
= I, 2p xpn¶ 2 [-q + q2 + (2p + 1)q +p 2 ~n q= p
I i2 rr 2p -i2r= exp 2 +lP 2 1 exp 2 + (q
i 2,r -i2,rr2= exp(2>+ 1 P2 ) 2 I exp(2 + q)
= exp(2p + l p2)C(L = 0, n = 2p + 1). (B1)
From the definition of C(0, n) it follows that thiscoefficient is different from 0 for any odd n. Fromthis we determine that Eq. (18) holds for any oddnatural n and integer number L.
Appendix C
In order to prove Eqs. (19) and (20) first we analyzethe case of n = 4M + 2 and L = 0, where M is anatural number. Using Eq. (12) we obtain
C(L 0, n = 4M + 2)
4M+1 i/T
I exp TM-1 qnq=O q= LV.+1L/
[ exP2M +l q2) + 7 exp q2
i2M [-i~r- 71 ex1p1)q]
n (q=O (2M + )
+ exP 2M (q + 2M + 1)2
n I: exp(2M + l q ) eP(2 M q2) =0.
Thus, the desired property is established for a partic-ular case of L = 0. Using Eq. (16) we see, however,that it is also valid for any even L. Proceeding in thesame way as in Eq. (Cl) we can also prove thatC(L = 0, n = 4M) 0.
Now, we analyze the case of n = 4M and L =1. Assuming this, we can express Eq. (12) as
Equation (C2) together with Eq. (17) leads directly toEq. (20a). Also in this case, by repeating the analysisfrom Eq. (C2) for L = 1 and n = 4M + 2 we can easilydetermine that C(L = 1, n = 4M + 2) • 0.
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3. A. W. Lohmann and J. A. Thomas, "Making an array illumina-tor based on the Talbot effect," Appl. Opt. 29, 4337-4340(1990).
4. K. Patorski and P. Szwaykowski, "Light intensity distributionin the Fresnel diffraction region of a non-sinusoidal phasegrating," Opt. Appl. 11, 627-631 (1981).
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