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Multiple interaction bandstop filters based on the Talbot effect
Volume 49, number 6
OPTICS COMMUNICATIONS
15 April 1984
MULTIPLE INTERACTION BANDSTOP FILTERS BASED ON THE TALBOT EFFECT A.W. LOHMANN, J. OJEDA-CASTANEDA * and E.E. SICRE ** Physikalisches Institut, 8520 Erlangen, Fed. Rep. Germany
Received 30 January 1984
We report on two tunable, bandstop spatial filters, which are based on the Talbot effect. The light interacts with more than two gratings, arranged in series. The techniques may be considered as the analogues of the Fabry-Perot interferometer and the Lyot-Ohman filter, respectively. Some experimental results are included.
1. Introduction Spatial fikering can be performed at several locations: with a spatial filter in the Fraunhofer plane, with a thick hologram immediately behind the object, or with a mask at a Fresnel distance behind the object. This latter approach to spatial filtering seems not to be so general as the Fraunhofer approach. However, if it is applicable, the technique can be quite simple. For example, based on the Talbot effect one can implement Fourier spectrometers [ 1,2], and interferometers [ 3 - 5 ] , and the Bragg e f f e c t can be applied to implement filtering and derivative operations [6,7]. Recently, a bandstop filtering technique was reported [8], which exploits the same basic result as used to implement Fourier spectrometers [1,2]. A nice feature o f the method in ref. [8] is that, for certain kind of objects [9], a variable bandstop operation is implemented by only moving a mirror. The rejection bandwidth of this method is, however, wide. Here, our aim is to show that if the Talbot effect is exploited in a multiple-interaction fashion, as suggested previously for Fourier spectrometry [1 ], then the selectivity can be improved. Specifically, we consider two line transmission analogues: one related to the Fabry-Perot interferometer, and the other to the Lyot-Ohman filter [ 1 0 - 1 2 ] . * Permanent address: Instituto Nacional de Astrofisica, Optica y Electr6nica, Puebla, M6xico. ** Permanent address: Centro de Investigaciones Opticas, La Plata, Argentina. 388
In section 2, we discuss the basic theory, and in section 3, we show some experimental results.
2. Basic theory Any tuning device (based on the Talbot effect) for selecting temporal frequencies can also be employed for selecting spatial frequencies. In fact, from the Talbot-formulae Z n = 2np2/X,
n = 1, 2, 3 ....
(1)
it is clear that either the grating period p, or the wavelength X may act equally well as independent variable. Furthermore, the sensitivity of the resonance distance Z n is higher for the spatial period dZ n = 2 Z n dp/p,
(2)
than for the wavelength d Z n = - Z n dX/X.
(3)
Hence, we expect not only that the set-ups suggested before, for Fourier spectrometry [ 1 ], will work also for spatial filtering, but in addition, they should have better performance, as indicated by the factor 2 in eq. (2). Encouraged by this argument on sensitivity we now study the theory of multi-Talbot interactions as spatial filtering devices. At the plane z = 0, we assume that the object transmittance, u(x), can be expressed as the superposition of binary members Un(X), i.e.
Volmne 49, number 6
OPTICS COMMUNICATIONS
N
15 April 1984
Un(X,z=(2rn+ 1)ZT) = [1
u(x) = ~
anU,,(x),
n=l
(4a)
where each member is the binary version of the nth order Rademacher function [9]"
Un(X) = [1 + Rn(x)] /2,
n> 1
-Rn(x)l/2.
(6d)
Third, it makes evident that one and only one, Radereacher function satisfies eq. (6b). In fact, for r 4: n
Ur(X, z = (2m + I)ZT) = exp [-i(2m + 1)7r4 n - r ] Rr(x ).
(7a)
Thus, only member Un(X) is self-imaged with reversedcontrast. The other members ur(x ) are self-imaged with positive contrast, only if r < n:
Rn(X ) = sign [sin(2 n - 1rrx/p)] e~
C2s+l exp(i2rr[2n-l(2s + 1)lx/p}.
= ~
(4b)
Ur(X, z = (2m + 1)ZT) = [ 1 + Rr(x)]/2
= Ur(X),
r < n. This representation has the following advantages. First, the object is expressed in terms of the members of an orthogonal basis in the interval (--p, p), the Rademacher functions: N
(7b)
For r 2> n, the Rademacher functions are self-imaged with a phase-difference exp(-iTr/4). That is, under self-image inversion the members in eq. (4) are independent. This justifies our choice Of Un, as in eq. (4), for the present discussion.
u(x) = ~3 an[1 +Rn(X)]/2 n=l
2. 1. Fabry Perot analogue
N =a + ~ n=l
b,,n,,(x).
(5)
Second, it shows clearly that on free-propagation, under the paraxial approximation:
Rn(X, z)
Bandstop filtering operations employing a FabryPerot interferometer had been proposed previously by Indebetouw [13]. Here we propose a line-transmission analogue based in the Talbot-effect. It is shown, in fig. 1, a longitudinally periodic distribution of gratings, each having transmittance t(x) Un(X), Thus, if in this system a plane wave illuminates the initial screen at z = 0, then the complex amplitude arriving to the second screen at z = Z T is, according to eq. (6), =
oo
= ~
C2s+lexp{i2n[2n-l(2s + 1)Ix/p}
X exp (-irrz [ 2 n - l ( 2 s + 1)] 2X/p2},
(6a)
the Rademacher functions are self-imaged with negative sign
Rn(X, z) = exp [-i(2m + l)Trl Rn(x ) = -Rn(X ),
(6b)
at the "negative Talbot planes": z = ( 2 m + l)[21-n/(2s + 1)]2p2/X = (2m + 1)Z T.
(6c)
Therefore, some parts of the object are contrast-reversed. (An overall factor exp(ikz) has been suppressed in eqs. (6)):
ufi(x, ZT) = [1 -- Rn(x)]/2,
(8a)
I I 1 4 I I I z:O
BZ
z=z r
Z=2Z T
z=mz T
z:(m*1)z T
Fig. 1. Schematic representation of the in-line multiple interaction method, which is analog to the Fabry-Peiot interferometer.
389
Volume 49, number 6
OPTICS COMMUNICATIONS
and just behind the second screen it becomes zero:
z T) = t(x)u;(x, z T) = [1 - R 2 ( x ) l [4 = O.
(8b)
This consideration indicates how simple it is to filterout one element Un(X). Of course, if the initial wavefield, or the initial screen, at z = 0 if not an exact element Un(X) but it has a spatial frequency error:
On(X)=[l +Qn(x)]/2=2 (l + ~
2.2. Lyot-Ohman analog In fig. 2 we show an analog of the polarization filters designed by Lyot [10] and Ohman [1 1]. We assume again that the initial wavefield, or the initial screen, at z = 0 is the function On(X) in eq. (9). At z = Z T the transmittance of the screen is t(x) = On(X), then the complex amplitude just behind the screen is V+n(X,ZT) as in eq. (10). Now, if until z = 3Z y
on(x,
oo
15 April 1984
3ZT) = [i exp(--i0/2)sin 0/2]
= [i exp(--i0/2)sin 0/2] [1 + X exp[i2rt(2s +
1×2n-lip
+6a)x])
(9a)
then the filter out operation in eq. (8) is not exact. In fact, the spatial frequency error, 6o, produces a phase variation, 0 = 27r(21-np6a) "~ 27r,
= [1 --
O+(X,ZT)
=
exp(-iO)Qn(x)l/2,
On(X, 2Z T) =
[i
2ZT)= [i
3ZT) : 21[1 --
Qn(x)] On(X, 3ZT)
× [i exp(--i0)sin 0] [1 --
(10)
(13)
7ZT) = [i exp(-i0/2)sin 0/2]
exp(-i40)Qn(x)] ]2,
and
v~(x,
7ZT) : ½ [1 +
Qn(X)] On(X, 7ZT)
= [i exp(--i0/2)sin 0/2] [i exp(--i0)sin 0]
exp(-iO /2 )sin( O[2) ] On(X, ZT ), exp(--iO/2)sin(O[2)] 2On(X),
Qn(x)]/2.
In a successive fashion at z = 7Z T we place a positive version of On(X) . Then
× [i exp(-i0)sin 0] [1 --
Vn(X).
exp(-i20)an(x)]/2,
= [i exp(--i0/2)sin 0/2]
on(x,
Consequently, at the next screen z = 2Z T, eq. (10) becomes
oS(x,
oS(x,
On(X)Pn(x , Z T)
= [i exp(-i0/2)sin(0/2)]
2ZT)
we place the reverse-contrasted version (or negative) o f e q . (9), On(X) = [1 -- Qn(X)]/2, then
(9b)
at each Fourier component. Thus, from eqs. (6) and (9), it is easy to show that on free propagation the analog to eq. (8) is
On(X, ZT)
on(x,
C2s+l
× [i exp(--iZ0)sin 20] [1
(1 1)
+Qn(X)]/2.
(14)
Therefore, after m-interactions the mathematical forand of course at z = mZ T we have that
v+(x, mZT ) =
[i exp(--i0/2)sin(0/2)]
m On(X),
I I I
In(X, mZ T) = hu+(x, mZT)L 2 = [sin(0/2)]
2ml)n(X ).
(12)
Hence, it is evident that as the number of interaction increases, the selectivity of the method increases as [sin0r 21-np6o) ] 2m. This property is not a unique property of the set-up shown in fig. 1. Let us consider another possibility. 390
r~ o
2
+
I I I z=O ~---z T
Z:Z T
-!_--2z~-
z:Tz T
Zz~Z T
.],
--~zT
Fig. 2. Analog to the Lyot-Ohman filter.
.I
Volume 49, number 6
OPTICS COMMUNICATIONS
15 April 1984
mulae for the complex amplitude and the irradiance are
i~i~~~
• . . . . . . iiliI ii~i!
![l[.,i i~i¸ :~: i ~!~ !~ii~!f'~l !:il ~!!i~~:i~ ~ ~i~!! i~:i~i i
o+(x, (1 - 2 m +I)ZT/(1 -- 2)) = [i exp(--i0/2)sin 0/2]
i ............ ~~:,;i,l~i,i~ii,~}~i,i,li
× [i exp(--i0)sin 0] ... [i exp(--i2 m - 1 0 )
ii !iii!i!!iii i
× sin(2m-10)] [1 _+ Qn(x)]/2,
ii
(15a)
ii!i! i
'
and ~ ~i'~i ~i~ii!iii'~i' !i
/n(X, (1 - 2m+l)ZT/(1 -- 2)) =
sin2S-10
[1 ±Qn(x)]/2,
(15b)
where the symbol I1 denotes the product of m terms, and the +- signs before Qn indicate that for m = even we have a positive version o f vn, while for m = odd we have the negative version of un. The two above methods have increasing selectivity as the number o f interactions increases, but while the former shows a clear rejection [sin(Tr21-npso)] the latter has a better resolution [sin(0/2) ... sin(2m-10)] 2, if the same number o f gratings are used. F r o m the viewpoint of experimental implementation, the former method is indeed simpler than the latter one.
2m,
3. Experimental
ill ,i'qi
f![
,
i
implementation
The set-up employed to implement one o f the above proposals, the analog to the Fabry-Perot interferometer, is similar to the one employed in ref. [8]. An increasing frequency grating G and a duplicate of it were placed as shown in fig. 3. The distance be-
G
//7
..
j "';
G
set-up
employed
2 ,
i!
M Fig. 4. Bandstop experiments performed o n a g r a t i n g w i t h increasing spatial frequency. (a) I n i t i a l w a v e f i e l d . ( b ) W a v e f i e l d after one interaction. (c) After three interactions.
:
I
rl; ~ F i g . 3. E x p e r i m e n t a l
;?
for the method
in fig. 1,
tween the two gratings was set to Z T = 16 mm, and behind the second grating a mirror M was located at z = ZT/2 to produce a triple interaction. The experimental results are shown in fig. 4. It can be seen that from one interaction to three there is a clear increasement in the selectivity. However, we would like to 391
Volume 49, number 6
OPTICS COMMUNICATIONS
indicate that there was also an increasement of noise, which was caused by dust or imperfections on the surfaces of G, the mirror M, and the beam splitter. Thus, one may expect that as the number of surfaces increases, the selectivity will increase, but also the amount of artifact noise.
4. Conclusions In this study we have shown our "multi-Talbot stucture" to act as a spatial frequency filter or as a Fourier filter. We could have presented our study also in another way: instead of acting upon Fourier frequencies (cosine pattern), our Talbot structure acts more directly upon "Rademacher frequencies" (+1 square wave pattern). More specifically, a pair of Ronchi gratings with periods d = separated by a distance z like in eq. (6c), will suppress the Rademacher frequency pattern [1 + Based on this effect we have discussed two arrangements: one related to the in-line analogue of the Fabry-Perot interferometer, the other to the Lyot-Ohman spectral filter.
22-np
Rn(X)]/2.
Acknowledgment Two of the authors, JOC and EES, gratefully ac-
392
15 April 1984
knowledge the financial support of the Alexander yon Humboldt-Foundation.
References [1] A.W. Lohmann, Proc. ICO: Conf. Opt. Instr., ed. K.J. Habell (Butterworth, London, 1961) p. 58. [2] H. Klages, J. de Physique, colloque 2 supplement au no. 3-4, 28 (1967) C2-40. [31 A.W. Lohmann and D.E. Silva, Optics Comm. 2 (1971) 413;4 (1972) 326. [41 S. Yokozeki and T. Suzuki, Appl. Optics 10 (1971) 1575, 1690. [5] C.S. Lira and V. Srinivasan, Optics Comm. 44 (1983) 219. [61 D. Peri and A.A. Friesem, Optics Lett. 3 (1978) 124. [71 S.K. Case, Optics Lett. 4 (1979) 286. [81 J. Ojeda-Castafiedaand E.E. Sicre, Optics Comm. 47 (1983) 183. {91 K.G. Beauchamp, Walsh functions and their applications (Academic Press, New York, 1975). [101 B. Lyot, Ann. Astrophys. 7 (1944) 31. [111 Y. Ohman, Nature 141 (1938) 157. 1121 R.W. Ditchburn, Light (Blackie and Son Limited, London, 1963) p. 500. [131 G. Indebetouw, Appl. Optics 19 (1980) 761; SPIE 232 (1980) International Computing Conference, p. 224.
OPTICS COMMUNICATIONS
15 April 1984
MULTIPLE INTERACTION BANDSTOP FILTERS BASED ON THE TALBOT EFFECT A.W. LOHMANN, J. OJEDA-CASTANEDA * and E.E. SICRE ** Physikalisches Institut, 8520 Erlangen, Fed. Rep. Germany
Received 30 January 1984
We report on two tunable, bandstop spatial filters, which are based on the Talbot effect. The light interacts with more than two gratings, arranged in series. The techniques may be considered as the analogues of the Fabry-Perot interferometer and the Lyot-Ohman filter, respectively. Some experimental results are included.
1. Introduction Spatial fikering can be performed at several locations: with a spatial filter in the Fraunhofer plane, with a thick hologram immediately behind the object, or with a mask at a Fresnel distance behind the object. This latter approach to spatial filtering seems not to be so general as the Fraunhofer approach. However, if it is applicable, the technique can be quite simple. For example, based on the Talbot effect one can implement Fourier spectrometers [ 1,2], and interferometers [ 3 - 5 ] , and the Bragg e f f e c t can be applied to implement filtering and derivative operations [6,7]. Recently, a bandstop filtering technique was reported [8], which exploits the same basic result as used to implement Fourier spectrometers [1,2]. A nice feature o f the method in ref. [8] is that, for certain kind of objects [9], a variable bandstop operation is implemented by only moving a mirror. The rejection bandwidth of this method is, however, wide. Here, our aim is to show that if the Talbot effect is exploited in a multiple-interaction fashion, as suggested previously for Fourier spectrometry [1 ], then the selectivity can be improved. Specifically, we consider two line transmission analogues: one related to the Fabry-Perot interferometer, and the other to the Lyot-Ohman filter [ 1 0 - 1 2 ] . * Permanent address: Instituto Nacional de Astrofisica, Optica y Electr6nica, Puebla, M6xico. ** Permanent address: Centro de Investigaciones Opticas, La Plata, Argentina. 388
In section 2, we discuss the basic theory, and in section 3, we show some experimental results.
2. Basic theory Any tuning device (based on the Talbot effect) for selecting temporal frequencies can also be employed for selecting spatial frequencies. In fact, from the Talbot-formulae Z n = 2np2/X,
n = 1, 2, 3 ....
(1)
it is clear that either the grating period p, or the wavelength X may act equally well as independent variable. Furthermore, the sensitivity of the resonance distance Z n is higher for the spatial period dZ n = 2 Z n dp/p,
(2)
than for the wavelength d Z n = - Z n dX/X.
(3)
Hence, we expect not only that the set-ups suggested before, for Fourier spectrometry [ 1 ], will work also for spatial filtering, but in addition, they should have better performance, as indicated by the factor 2 in eq. (2). Encouraged by this argument on sensitivity we now study the theory of multi-Talbot interactions as spatial filtering devices. At the plane z = 0, we assume that the object transmittance, u(x), can be expressed as the superposition of binary members Un(X), i.e.
Volmne 49, number 6
OPTICS COMMUNICATIONS
N
15 April 1984
Un(X,z=(2rn+ 1)ZT) = [1
u(x) = ~
anU,,(x),
n=l
(4a)
where each member is the binary version of the nth order Rademacher function [9]"
Un(X) = [1 + Rn(x)] /2,
n> 1
-Rn(x)l/2.
(6d)
Third, it makes evident that one and only one, Radereacher function satisfies eq. (6b). In fact, for r 4: n
Ur(X, z = (2m + I)ZT) = exp [-i(2m + 1)7r4 n - r ] Rr(x ).
(7a)
Thus, only member Un(X) is self-imaged with reversedcontrast. The other members ur(x ) are self-imaged with positive contrast, only if r < n:
Rn(X ) = sign [sin(2 n - 1rrx/p)] e~
C2s+l exp(i2rr[2n-l(2s + 1)lx/p}.
= ~
(4b)
Ur(X, z = (2m + 1)ZT) = [ 1 + Rr(x)]/2
= Ur(X),
r < n. This representation has the following advantages. First, the object is expressed in terms of the members of an orthogonal basis in the interval (--p, p), the Rademacher functions: N
(7b)
For r 2> n, the Rademacher functions are self-imaged with a phase-difference exp(-iTr/4). That is, under self-image inversion the members in eq. (4) are independent. This justifies our choice Of Un, as in eq. (4), for the present discussion.
u(x) = ~3 an[1 +Rn(X)]/2 n=l
2. 1. Fabry Perot analogue
N =a + ~ n=l
b,,n,,(x).
(5)
Second, it shows clearly that on free-propagation, under the paraxial approximation:
Rn(X, z)
Bandstop filtering operations employing a FabryPerot interferometer had been proposed previously by Indebetouw [13]. Here we propose a line-transmission analogue based in the Talbot-effect. It is shown, in fig. 1, a longitudinally periodic distribution of gratings, each having transmittance t(x) Un(X), Thus, if in this system a plane wave illuminates the initial screen at z = 0, then the complex amplitude arriving to the second screen at z = Z T is, according to eq. (6), =
oo
= ~
C2s+lexp{i2n[2n-l(2s + 1)Ix/p}
X exp (-irrz [ 2 n - l ( 2 s + 1)] 2X/p2},
(6a)
the Rademacher functions are self-imaged with negative sign
Rn(X, z) = exp [-i(2m + l)Trl Rn(x ) = -Rn(X ),
(6b)
at the "negative Talbot planes": z = ( 2 m + l)[21-n/(2s + 1)]2p2/X = (2m + 1)Z T.
(6c)
Therefore, some parts of the object are contrast-reversed. (An overall factor exp(ikz) has been suppressed in eqs. (6)):
ufi(x, ZT) = [1 -- Rn(x)]/2,
(8a)
I I 1 4 I I I z:O
BZ
z=z r
Z=2Z T
z=mz T
z:(m*1)z T
Fig. 1. Schematic representation of the in-line multiple interaction method, which is analog to the Fabry-Peiot interferometer.
389
Volume 49, number 6
OPTICS COMMUNICATIONS
and just behind the second screen it becomes zero:
z T) = t(x)u;(x, z T) = [1 - R 2 ( x ) l [4 = O.
(8b)
This consideration indicates how simple it is to filterout one element Un(X). Of course, if the initial wavefield, or the initial screen, at z = 0 if not an exact element Un(X) but it has a spatial frequency error:
On(X)=[l +Qn(x)]/2=2 (l + ~
2.2. Lyot-Ohman analog In fig. 2 we show an analog of the polarization filters designed by Lyot [10] and Ohman [1 1]. We assume again that the initial wavefield, or the initial screen, at z = 0 is the function On(X) in eq. (9). At z = Z T the transmittance of the screen is t(x) = On(X), then the complex amplitude just behind the screen is V+n(X,ZT) as in eq. (10). Now, if until z = 3Z y
on(x,
oo
15 April 1984
3ZT) = [i exp(--i0/2)sin 0/2]
= [i exp(--i0/2)sin 0/2] [1 + X exp[i2rt(2s +
1×2n-lip
+6a)x])
(9a)
then the filter out operation in eq. (8) is not exact. In fact, the spatial frequency error, 6o, produces a phase variation, 0 = 27r(21-np6a) "~ 27r,
= [1 --
O+(X,ZT)
=
exp(-iO)Qn(x)l/2,
On(X, 2Z T) =
[i
2ZT)= [i
3ZT) : 21[1 --
Qn(x)] On(X, 3ZT)
× [i exp(--i0)sin 0] [1 --
(10)
(13)
7ZT) = [i exp(-i0/2)sin 0/2]
exp(-i40)Qn(x)] ]2,
and
v~(x,
7ZT) : ½ [1 +
Qn(X)] On(X, 7ZT)
= [i exp(--i0/2)sin 0/2] [i exp(--i0)sin 0]
exp(-iO /2 )sin( O[2) ] On(X, ZT ), exp(--iO/2)sin(O[2)] 2On(X),
Qn(x)]/2.
In a successive fashion at z = 7Z T we place a positive version of On(X) . Then
× [i exp(-i0)sin 0] [1 --
Vn(X).
exp(-i20)an(x)]/2,
= [i exp(--i0/2)sin 0/2]
on(x,
Consequently, at the next screen z = 2Z T, eq. (10) becomes
oS(x,
oS(x,
On(X)Pn(x , Z T)
= [i exp(-i0/2)sin(0/2)]
2ZT)
we place the reverse-contrasted version (or negative) o f e q . (9), On(X) = [1 -- Qn(X)]/2, then
(9b)
at each Fourier component. Thus, from eqs. (6) and (9), it is easy to show that on free propagation the analog to eq. (8) is
On(X, ZT)
on(x,
C2s+l
× [i exp(--iZ0)sin 20] [1
(1 1)
+Qn(X)]/2.
(14)
Therefore, after m-interactions the mathematical forand of course at z = mZ T we have that
v+(x, mZT ) =
[i exp(--i0/2)sin(0/2)]
m On(X),
I I I
In(X, mZ T) = hu+(x, mZT)L 2 = [sin(0/2)]
2ml)n(X ).
(12)
Hence, it is evident that as the number of interaction increases, the selectivity of the method increases as [sin0r 21-np6o) ] 2m. This property is not a unique property of the set-up shown in fig. 1. Let us consider another possibility. 390
r~ o
2
+
I I I z=O ~---z T
Z:Z T
-!_--2z~-
z:Tz T
Zz~Z T
.],
--~zT
Fig. 2. Analog to the Lyot-Ohman filter.
.I
Volume 49, number 6
OPTICS COMMUNICATIONS
15 April 1984
mulae for the complex amplitude and the irradiance are
i~i~~~
• . . . . . . iiliI ii~i!
![l[.,i i~i¸ :~: i ~!~ !~ii~!f'~l !:il ~!!i~~:i~ ~ ~i~!! i~:i~i i
o+(x, (1 - 2 m +I)ZT/(1 -- 2)) = [i exp(--i0/2)sin 0/2]
i ............ ~~:,;i,l~i,i~ii,~}~i,i,li
× [i exp(--i0)sin 0] ... [i exp(--i2 m - 1 0 )
ii !iii!i!!iii i
× sin(2m-10)] [1 _+ Qn(x)]/2,
ii
(15a)
ii!i! i
'
and ~ ~i'~i ~i~ii!iii'~i' !i
/n(X, (1 - 2m+l)ZT/(1 -- 2)) =
sin2S-10
[1 ±Qn(x)]/2,
(15b)
where the symbol I1 denotes the product of m terms, and the +- signs before Qn indicate that for m = even we have a positive version o f vn, while for m = odd we have the negative version of un. The two above methods have increasing selectivity as the number o f interactions increases, but while the former shows a clear rejection [sin(Tr21-npso)] the latter has a better resolution [sin(0/2) ... sin(2m-10)] 2, if the same number o f gratings are used. F r o m the viewpoint of experimental implementation, the former method is indeed simpler than the latter one.
2m,
3. Experimental
ill ,i'qi
f![
,
i
implementation
The set-up employed to implement one o f the above proposals, the analog to the Fabry-Perot interferometer, is similar to the one employed in ref. [8]. An increasing frequency grating G and a duplicate of it were placed as shown in fig. 3. The distance be-
G
//7
..
j "';
G
set-up
employed
2 ,
i!
M Fig. 4. Bandstop experiments performed o n a g r a t i n g w i t h increasing spatial frequency. (a) I n i t i a l w a v e f i e l d . ( b ) W a v e f i e l d after one interaction. (c) After three interactions.
:
I
rl; ~ F i g . 3. E x p e r i m e n t a l
;?
for the method
in fig. 1,
tween the two gratings was set to Z T = 16 mm, and behind the second grating a mirror M was located at z = ZT/2 to produce a triple interaction. The experimental results are shown in fig. 4. It can be seen that from one interaction to three there is a clear increasement in the selectivity. However, we would like to 391
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OPTICS COMMUNICATIONS
indicate that there was also an increasement of noise, which was caused by dust or imperfections on the surfaces of G, the mirror M, and the beam splitter. Thus, one may expect that as the number of surfaces increases, the selectivity will increase, but also the amount of artifact noise.
4. Conclusions In this study we have shown our "multi-Talbot stucture" to act as a spatial frequency filter or as a Fourier filter. We could have presented our study also in another way: instead of acting upon Fourier frequencies (cosine pattern), our Talbot structure acts more directly upon "Rademacher frequencies" (+1 square wave pattern). More specifically, a pair of Ronchi gratings with periods d = separated by a distance z like in eq. (6c), will suppress the Rademacher frequency pattern [1 + Based on this effect we have discussed two arrangements: one related to the in-line analogue of the Fabry-Perot interferometer, the other to the Lyot-Ohman spectral filter.
22-np
Rn(X)]/2.
Acknowledgment Two of the authors, JOC and EES, gratefully ac-
392
15 April 1984
knowledge the financial support of the Alexander yon Humboldt-Foundation.
References [1] A.W. Lohmann, Proc. ICO: Conf. Opt. Instr., ed. K.J. Habell (Butterworth, London, 1961) p. 58. [2] H. Klages, J. de Physique, colloque 2 supplement au no. 3-4, 28 (1967) C2-40. [31 A.W. Lohmann and D.E. Silva, Optics Comm. 2 (1971) 413;4 (1972) 326. [41 S. Yokozeki and T. Suzuki, Appl. Optics 10 (1971) 1575, 1690. [5] C.S. Lira and V. Srinivasan, Optics Comm. 44 (1983) 219. [61 D. Peri and A.A. Friesem, Optics Lett. 3 (1978) 124. [71 S.K. Case, Optics Lett. 4 (1979) 286. [81 J. Ojeda-Castafiedaand E.E. Sicre, Optics Comm. 47 (1983) 183. {91 K.G. Beauchamp, Walsh functions and their applications (Academic Press, New York, 1975). [101 B. Lyot, Ann. Astrophys. 7 (1944) 31. [111 Y. Ohman, Nature 141 (1938) 157. 1121 R.W. Ditchburn, Light (Blackie and Son Limited, London, 1963) p. 500. [131 G. Indebetouw, Appl. Optics 19 (1980) 761; SPIE 232 (1980) International Computing Conference, p. 224.