1-s2.0-003040189400650j-main.pdf

1 March 1995

OPTICS COMMUNICATIONS Optics Communications 115(1995) 40-44

HSEVIER

Simple equations for the calculation of a multilevel phase grating for Talbot array illumination Changhe Zhou, Liren Liu Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800-216, Shanghai 201800, China

Received 30 June 1994; revised version received 3 October 1994

Abstract Compared with Szwaykowski’s equation [Appl. Optics 32 ( 1993 ) 11091, simple equations for the calculation of a multilevel phase grating for array illumination based on the fractional Talbot effect are given. The characteristics of the multilevel phase gratings can also be easily analyzed with these simple equations.

put forward [ 1 ] and demonstrated [2] the array illuminator based on the fractional Talbot effect, many researchers published their works on this subject recently. Da [ 31 and later Arriz6n [ 41 discussed in how many cases binary phase gratings can be used for array illuminations based on the fractional Talbot effect. Their conclusions are that only six cases are possible for a single binary phase grating to be used for Talbot array illuminations. Although Arriz6n [ 5 ] also gave an architecture in which binary phase gratings are cascaded in the specific serial distances to realize a required (not arbitrary) high compression array illumination, the cost of making such binary phase gratings is still high and light efficiency will be reduced due to a longer distance between the first binary phase plane and the required array illumination plane. Generally, a multilevel phase grating is required to realize a arbitrary high compression array illumination at the fractional Talbot distance. Leger et al. [ 61 have given a simple equation to calculate the multilevel phase grating for some even-number compression array illumination and they also gave a experimental example to support his equation. Szwaykowski Since

Lohmann

al. [ 7 ] gave their equation to design the multilevel phase grating for arbitrary-number compression array illumination, but Szwaykowski’s equation [ 7 ] is very complex. In this paper, simple equations for the calculation of a multilevel phase grating for Talbot array illumination are given. For the cases of N multiples of 4, Leger et al. [ 61 have given the following equation to calculate the multilevel phase

et

(1) where k= 1, .... N/2; N is a coefficient corresponding to the fractional Talbot distance ( 1/N)Z,; 2, is the Talbot distance Z, =2d2/A,

(2)

where d is the periodic distance of the one-dimensional grating and ;i is the illuminated light wavelength. For arbitrary-number compression ratio array illumination, Szwaykowski et al. [ 71 gave the following equation:

0030-4018/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved .SSDI0030-4018(94)00650-4

C. Zhou, L. Liu /Optics Communications N-l

N)] ,

=Aexp[$(k.

1

= (27rIN)P(k-P)

(4)

and N-l S(kW=

,&

sin[&kp)l,

(5)

N-l

pmmol

C(kW=

2

then @(k, N) can be calculated

(6)

(7)

where 6(k, N) is the adjusting phase (0, 7cor 27r), according to the signs of S( k, N) and C( k, N), making $( k, N) be in the area (0,27r). In fact, the fractional Talbot effect is the very well known phenomenon which has been studied by many researchers, Liu [ 8 ] has also given the following equations to describe this phenomenon. It is assumed that the object grating E, (x) is f

.

Rect(x-kd)

E*(x) =A ,=E

m

distance

Rk Rect(x-kd/P)

(1 /N)Zr,

(12)

,

,

where p= N/2, k= 1, 2, .... N/2. To calculate the multilevel phases from Eqs. ( 1O)(12), the procedure in Eqs. (4)-(7) should also be used. So the computation complexity in Liu’s method is a little larger than that in Szwaykowski’s method, but both the methods are in the same complexity degree. In this paper, simplified equations for the calculation of the multilevel phases, compared with the above two methods, are given. Based on Guigay’s deduction result [ 9 1, one knows =A

E2 x,~Z~ (

>

2 E,(x+md/N) m=l

Xexp(ilrm2/2N)[1+iN(-I)“],

(13)

where A is a factor we don’t concern. In the case N=4q, q= 1, 2, .... the terms with oddnumber FYIin Eq. ( 13) is zero, so the phase factor in the above equation ( 13 ) is 2

@(k, N)= g

nk? = (N,2)

,

(14)

If N

@(k,N)=

(10)

where/?=N, k= 1, 2, .... N. If N is even and (N/2) is also even, i.e., N=4q, q= 1,2, .... the phase factor will be ~~,exp[-i(2Rkml~+nm’iP)l,

f exp[-i(n(2k-1)m//I+nm2/p)], m=l

(9)

the re-

where A is a amplitude factor we don’t concern. is odd, the phase factor will be

&=

Rk=

where m= 2k, k= 1,2, .... (N/2). This equation is that which Leger et al. [ 6 ] have given [ Eq. ( 1) 1. This expression is simple to be calculated. In other cases of N, similar simple equations can also be obtained from Eq. ( 13). In the case N=4q+ 2, the terms with even-number nz in Eq. ( 13 ) is zero, so the phase factorintheEq. (13)is

(8)

k=-co

In the fractional Talbot sponse E2 (x) should be

q= 1,

as

@(k,N)=arctan[S(k,N)/C(k,N)]+6(k,N),

E,(x)=

whereP=N/2, k= 1,2, .... (N/2). IfNiseven but (N/2) isodd, i.e., N=4q+2, 2, .... the phase factor will be (3)

where A is a amplitude factor we don’t concern and @(k, N) is its relative phase. The step to calculate the multilevel phase $(k, N) is rather complex as follows. If one defines d(kP)

41

115 (I 995) 40-44

(11)

n(2;;1)‘,

(15)

where k= 1, 2, .... (N/2). In the case N= 4q+ 1, the phase factor in Eq. ( 13) is $(k,N)=n[k2/2N++(-1)k],

(16)

where k= 1, 2, .... N. In case N= 4q+ 3, the phase factor becomes c$(k,N)=n

[k2/2N-$(-l)k],

(17)

42

C. Zhou. L. Liu /Optics Communications 1 IS (I 995) 40-44

where k= 1,2, .... N. To verify these equations ( 14)-( 17), computer simulations have been used. Computer simulations show that the simplified equations given in Eqs. (14)-( 17) accord well with Szwaykowski’s equation and Liu’s equations. One advantage of the equations given in Eqs. (14)-( 17) is that they are easy to be calculated. Some values of multilevel phases for Talbot array illumination are given in Table 1. In addition, another advantage of the equations given in Eqs. ( 14)- ( 17 ) is that the characteristics of the designed multilevel phase grating can be easily analyzed. In the case N=4q, it is easy to prove that Eq. ( 14) has the following characteristic

where k= 1, 2, .... 2q+ 1. From this equation ( 19), we know that the structure of the multilevel phases in one period in this case is also symmetric about the center phase with k=q+ 1. In the case N= 4q+ 1, from Eq. ( 16), we can prove

exp(irr&)

exp[$(N-k,

=exp(in

(N~~k)2),

(18)

where k= 1, 2, .... (q- 1). From this equation, we know that the structure of the multilevel phases in

one period is symmetric about the para-center phase with k= q except the last one phase (k= 2q), which is shown in Fig. 1. In the case N=4q+2, it is easy to prove that Eq. ( 15 ) has the following characteristic: exp(i,

(2til)2)=exp(irc

]N-(~~-1)12),

N)]=exp[i@(k,

N)] ,

(20)

where k= 1, 2, .... 2q. In the case N= 4qS 3, from Eq. ( 17 ), we can prove

exp[i~(N-k,N)l=exp[i~(k,N)l, Table I Some values of the multilevel phases in one period at ( 1/N) Zr distance which can be used for the fractional Talbot array illumination. L is the number of the different multilevel phases N

Multilevel phases

3 4 5 6 7 8 9 10 11

o,o, 4n/3 R/2,0 0,4n/5,4n/5,0,2n/5 0, 2R/3, 0 0, 12R/7,4n/7,4n/l, 12R/l,O, Ton/7 R/4, R, R/4,0 0,2n/3,4R/9,4a/3,4a/3,4R/9,2R/3,0,4R/9 0,2n/5,6R/5, 2n/5,0, 0, 18R/11,4R/11,2R/l1, 12R/ll, 12R/ll, 2R/11,4R/11,18n/11,0, 16rr/ll n/6,2R/3,3~/2,2R/3, n/6,0 0,8R/13,4n/13, 14x/13, 12R/l3,24R/l3, 24n/l3,12R/l3, 14~/13,4R/13,8n/13,0,6n/13 0,2n/7,6n/7, 12n/7,6n/l, 2n/l, 0 0,8R/5,4n/15,0.4n/5,2R/3,8n/5,8n/5, 2R/3,4n/5,0,4n/15,8RJ5,0,22n/l5 x/8, R/2,9R/8,0,9R/8, n/2, R/8,0 0,10R/17,4n/17, 16R/tl, 12R/17,26R/l7, 24n/17,6R/17,6x/17,24R/17,26n/17, 12R/ll, 16~/17,4n/17,10n/17,0,8R/17 0.2R/9,2n/3,4R/3,2R/9,4R/3,2~/3,2R/9,0 0,30R/19,4n/19,36R/l9,12R19,8R/19, 24n/19,22R/l9,2R/19,2n/l9,22n/l9,24n/l9, 8~119, 12n/l9,36R/l9,4n/l9,30R/19,0,28n/19 n/l0,2x/5,9n/lO, 8x/5, n/2,8R/5,9n/10, 2R/5, R/IO, 0

12 13 14 15 16 17

18 19

20

-

L -

2 2 3 2 4 3 4 3 6 4 7

(19)

(21)

where k= 1,2, .... 2q+ 1. The symmetrical characteristics in all the above cases are shown in Fig. 1. From Fig. 1, we know that corresponding to N=4q, 4q+ 1, 4q+2, 4q+ 3, the maximum numbers of the different multilevel phases 2q+ 1, q+ 1, 2q+2, respectively. are q+l, Szwaykowski’s conclusion [ 7 ] is “the number of different phase level is at most (N+ 1) /2 for the phase structure associated with odd N and N/2 for N is symmetric

symmetric

k=l.Z,

...

q,

...

2q

k=

qtl,

1, 2,

Zqtl

a symmetric

I& k=l.Z.

4 10

6

‘.’

ICI

4qtl

k=l.

2.

4qt3

.‘.

14

Fig. 1. The structure of the multilevel phase gratings in one period formed by a binary amplitude grating with a appropriate opening ratio in the different fractional Talbot distances ( 1/N)&, (a) N=49, (b) N=4q+2, (c) N=49+ I, (d) N=49+3.

C. Zhou, L. Liu /Optics Communications 115 (1995) 40-44

even”, corresponding to N= 4q, 4q+ 1,4q+ 2,4q+ 3, the numbers of different phase levels are 2q, 2q+ 1, 2q+ 1, 2q+2, respectively. So the phase structure shown in Fig. 1 is more accurate than Szwaykowski’s conclusion. The numbers of the different multilevel phases in all the above cases as indicated by the symmetric structures in Fig. 1, however, are not the practical numbers of the different multilevel phases; e.g., to consider the case of N= 9 = 4 x 2 + 1, the predicated number of the different phase levels is 2 X 2 + 1 = 5, the practical number as given in Table 1 is 4, this is because of one unexpected redundancy in one of the phases. To consider in general this possible redundancy, corresponding to N= 4q, 4q+ 1,4q+ 2,4q+ 3, the numbers of the different multilevel phases are at most q+ 1,2q+ 1, q+ 1,2q+2, respectively. There are two distances 2,/N and ( 1 - 1/N)Z, at which a binary amplitude grating (with the opening ratio 2/N when N is even, 1IN when N is odd) will form a uniform irradiance distribution. It can be proved by using Liu’s equations [ 81 and Guigay’s equations [ 91, that the magnitudes of the phase distribution in one period at ( 1 - 1/N)Z, distance are equal to that at ( 1/N)Z, distance, but with a negative sign, for example, N= 5, the multilevel phase distribution in one period at ( l/5 ) Z,, as given in Table 1, is 0,4n/5,4rr/5,0,2n/5, and that at (1- l/N)Z, is calculated as 0, -47~15, -4~~15, 0, -2n/5, or equivalently into the (0, 2~) area as, 0, 6x15, 6x15, 0, 8x15, so it can be proved that if a coherent plane wave is retarded by a phase distribution given in Table 1 and Eqs. (14)-( 17), at (l/N)Z, distance, a binary amplitude grating will be generated; if a coherent plane wave is advanced by a phase distributiongiveninTablelandEqs.(14)-(17),at(l-l/ N)Z, distance, a binary amplitude grating will be generated. Arrizon et al. [ 41 have discussed in how many cases the binary phase grating can be used for the fractional Talbot array illumination; their conclusions are that there are only six cases for the binary phase gratings to realize the Talbot array illumination. In fact, these six cases are corresponding to N= 3,4,6 cases in Table 1, i.e., there are at (l/3)2,, (2/3)Z,, (l/4)2,, (3/4)Z,, ( l/6)2, and (5/6)Z, distances, respectively. From Table 1, we also know that there will be six cases for three phase gratings to realize the Talbot

array illumination,

43

i.e., N= 5,8, 10, corresponding

to

at (1/5)Z=, (4/5)Z,, (l/8)&, (7/8)Z,, (l/10)2,, (9/1O)Z, distances with the corresponding phase distributions. One more interesting characteristic of the multilevel phase distribution given in Table 1, which can be proved by the simple equation (14)-( 17), is t(m)=

f cos[@(k+m, k=l

=/3?

m=O,p,

10,

m=l,2,

.... p-1

N)-@(k,

.

N)]

(22)

where j?= N/2,

if N is even ,

=N,

if N is odd ,

(23)

and @(P+m, N) =@(m, N), m=O, 1, 2, .... j3. The above equation (22) has a clear physical meaning: only one unit in one period of the output discrete function is required to have light illumination, all the other units are expected to be zero in order to realize the 100% light efficiency for array illumination. To illustrate clearly what this phenomenon means, one example for N= 5 in Table 1 is given, @(k, N) in this case are 0, 47r/5, 4x15, 0, 2x15. t(O)=cos(0-0)+cos(47r/5-47r/5) +cos(47r/5-47l/5)+cos(o-o) +cos(2n/5-2x/5), =5, t( 1)=cos(47r/5-0)+cos(47r/5-47c/5) +cos(O-4rr/5)+cos(2n/5-0) +cos(O-2x/5), =o, itisalsoeasilyprovedthatt(2)=0,t(3)=0,t(4)=0, t ( 5 ) = t (0) = 5. It can also be proved that all the other cases given in Table 1 have the same characteristic. Using the simple equations given in Eqs. ( 14)- ( 17)) this characteristic can be analytically proved. For conciseness, the wordy proof is not given here. This characteristic has a clear physical meaning as we have noted, so it can be used to check whether the phase distribution obtained by any one of the above meth-

44

C. Zhou, L. Liu /Optics Communications 115 (1995) 40-44

ods is right or not (to check the calculation error), and what is of more importance is that it will help us to know more about the phase distribution for the fractional Talbot array illumination, thus may help us to study it further. In summary, simple equations to calculate the multilevel phase grating for fractional Talbot array illumination are given. Computer simulations show that these equations accord well with the previous equations given by Szwaykowski and Liu, but the computation complexity of the equations given in this paper is small, which may ease the design work for Talbot array illumination. The characteristics of the multilevel grating can also be easily analyzed with these simple equations, which may be helpful for design and further study of these gratings. With the designed multilevel phase grating, arbitrary-compression array illumination can be realized theoretically, although fabrication of such multilevel phase grating may be difftcult. The disadvantages of this Talbot il-

luminator are also obvious, which have been analyzed by many researchers [ 61, e.g., for the finite number of array illumination, light energy will be lost at the edge of array illumination plane, etc.. The authors acknowledge the support of Natural Science Foundation of China.

References [ 1] A.W. Lohmann, Optik 79 (1988) 41. [2] A.W. Lohmann and J.A. Thomas, Appl. Optics 29 ( 1990) 4337. [3] X.-Y. Da, Appl. Optics 31 (1992) 2983. [ 41 V. Arriz6n and J. Ojeda-Castaneda, Optics Lett. 18 ( 1993) [ 51 V. Arrizbn, Optics Lett. 18 ( 1993) 1205. [6] J.R. Leger and G.J. Swanson, Optics Lett. 15 (1990) 288. [ 71 P. Szwaykowski and V. Arrizbn, Appl. Optics 32 (1993) 1109. [8] L. Liu, Optics Lett. 14 (1989) 1312. [9] J.P. Guigay,OpticaActa 18 (1971) 677.