Lensless theta decoder

Volume 60, number 4

OPTICS COMMUNICATIONS

15 November ! 986

LENSLESS T H E T A DECODER P. ANDRES ~, J. OJEDA-CASTAlqEDA and J. IBARRA. Instituto Nacional de Astrofisica, Optica y Electron6nica, Apdo. Postal No. 216, Puebla 72000 Mdxico.

Received 21 May 1986

The Lau effect at Fresnel distances is discussed in terms of the virtual Fourier transform, with the purpose of implementing a theta-modulation decoder, which does not require either any lens or any spatial filter, and works with spatially incoherent light. Experimental results are included.

1. Introduction The self-imaging phenomenon offers the possibility of implementing simple and yet versatile optical processors. For example, it can be applied for setting economical interferometers [ 1-5], or simple spectroscopes [ 6,17 ], and lensless spatial filters [ 8-10 ], or for performing optical correlations [ 11 ], and for implementing optical synthesizers [ 12,13 ]. Recently, the self-imaging phenomenon, in the form of the Lau effect [ 14 ], was employed to implement a theta-modulation decoder [ 15 ]. Our purpose here is to report a lensless version of the theta-modulation decoder. The present decoder works also under incoherent illumination (high signal to noise ratio). But in addition, it has the following advantages: (a) it does not require any lens, (b) it does not need any spatial filter, and (c) the set-up is quite compact. In section 2, we discuss, for the sake of completness, the basic theory. In section 3, we describe the fundamentals of the novel optical theta-decoder. In section 4, we show some experimental results.

domain. The mathematical treatment can be visualized, at finite distances, as the virtual Fourier transform. This conceptual tool offers the remarkable characteristics of relating, on the one hand, the Fraunhofer diffraction pattern of the object, and on the other hand, its Fresnel diffraction patterns. Thus, one is able to consider the self-imaging phenomenon, on free progation along the z-axis, while conveniently looking at the Fourier spectrum of the object at the plane of the source; see for example refs. [16-18]. In other words, let us consider an object that is periodic along the x-axis t(x)=

~

clexp(i2~zxl/d),

(1)

/=--zc

and that is illuminated by a spherical wavefront O(x) =exp[ - i ~ r ( x - s ) 2 / 2 Z o ] .

(2)

The complex amplitude just behind the object is u(x) =O(x)t(x).

(3)

At the plane of the source, the complex amplitude is

2. Basic theory The Fresnel diffraction patterns of periodical structures can be discussed easily in the Fourier Permanent address: Facult. de Ciencias Fisicas, Universidad de Valencia, Burjasot (Valencia), Spain. 206

v(~) = i u(x) - x .

×exp[ +irt(~--x) 2/2Zo]d x .

(4)

Of course, on free propagation the complex amplitude in eq. (4) will reconstruct the object at z = 0 , 0 030-401/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 60, number 4

OPTICS COMMUNICATIONS

Thus, it becomes clear from eqs. ( 5 ) and (6) that the Fresnel diffraction phenomenon can be described in terms of Fourier optics, by suitably considering the complex amplitude at the source plane. This formulation is employed next to design an optical decoder.

o)

\ u(x)

~(x')

3. Lau fringes at finite distances

OBJECT

F

-:o

15 November 1986

It is straightforward to show from eqs. (5), (6) and (7), that the Fresnel diffraction patterns of the object t(x) are

I

w(x') = i ~[(~-s)/2Zo] exp(i27t Ax'~/2Zo) - - zt3

X exp (ircB~2/);Zo) d~, u(x)

Mx')

w(x')= VIRTUAL FOURER TRANSFORM

OBJECT

(8a)

FRESNEL DIFFRACTION PATTERN

Fig. 1. Virtual Fourier transformation and Fresnel diffraction patterns.

j I=

c; exp[i2~(Ax'+Bs)l/d]

--oG

× exp (iztB2zo l 2/d2), where

A = ( l + z / z o ) -1, and the Fresnel diffraction patterns of the object at z # 0~ In other words, the complex amplitude 3f

w(x') = f v(~) ×exp[ - i n ( x ' - ~ ) 2/),(2+ go)] d ~

(5)

(8b)

B=(l+zo/z) -1

(9)

From eqs. (8) the four following interesting features become apparent: (i) There are planes at which the quadratic phase factors, in the integrand of eq. (8a), are irrelevant:

rcB~zo/d 2 =2zcm,

m = + 1,_+2,_+3 ....

(10a)

or equivalently becomes w ( x ' ) = u ( x ) at z = 0 and the Fresnel diffraction paterns o f u ( x ) at z # 0 . See fig. 1. To substitute the object, t(x), by v(~) in eq. (5) may appear irrelevant. There is, however, an important advantage. The complex amplitude in eq. (4) is essentially the Fourier transform of the object, namely

(ii) At these planes the object is self-imaged with magnification A - ' and a shift -Bs/A, i.e.

v(~) =exp[in(~ 2 -s2)/2Zo]7[(~-s)/2Zo] ,

ztBAzo/d 2 =2rrm,

(6)

where

~[(~-s)/2Zo] = i t(x) exp[-i2nx(~-s)/2Zo] d.x = ~ ct 6[(~-s)/2zo-l/d] . [=

--oO

(7)

( l/z+ 1/Zo) -J =2md2/2.

w(x') = t(Ax' + Bs).

(10b)

(11 )

If m=_+0.5,_+ 1.5,_+2.5 ....

the so-called negative self-images are obtained. As compared with the above - or positive - self-images a half-period shift is added. For m = _+0.5, _+1.5 ..... eq. (I 0b) also gives the negative self-image planes. (iii) Since the irradiance distribution at the selfimages, [w(x')[ 2, is periodic, the properly shifted 207

Volume 60, number 4

OPTICS C O M M U N I C A T I O N S

15 November 1986

Ar ~.lE(s )

self-images can be brought into consonance if

Jw(x' +nd/A) J 2 = Jw(x') J 2, n=_+l, _+2, _+3,....

(12)

(iv) An incoherent source that is composed of a number of line sources, which are mutually incoherent, N

E ( s ) = ~. E~c~(s-nd/B),

(13)

Fig. 2. Lensless theta decoder: sct-up.

n--N

is able to bring into consonance 2 N + 1 self-images, as in eq. (12). Note that the two conditions in eqs. (10b) and (13) can be joint in the following consonance condition

2zo/d=2md/b,

(14)

which can be interpreted as follows. The virtual Fourier spectrum of the object, eq. (7), must be in consonance with the positions of the line sources, eq. (13 ), to create Lau fringes at finite distances. These Lau fringes are incoherent superpositions of magnified self-images of the object t(x). If the codified line source is constructed so that N

E(s)=

~

E,O(s-nd,),

(15)

n= --N

where dl =2zo/d, all the positive and negative selfimage planes are Lau patterns since 2m in eq. (14) is in both cases an integer. In accordance with eq. (10b), if

Xo=2(mo+q)dZ/2

( 0 < q < 1)

the positions of both positive and negative self-image planes are z = zom/(mo + q - m ). Therefore, the number of real Lau fringe patterns at finite distances is equal to the integer part of the number 2(mo + q), and the general consonance - or spatial period of the onedimensional source-grating - is dt = 2 ( mo + q ) d. Of course, one can exploit this phenomenon to detect any lack of consonance, for instance, the mismatching between the virtual Fourier spectrum and the line sources that is created by misaligment [ 15 ]. Specifically, we expect that if t(x) is a theta-modulation object, then the section of t(x) that satisfies the condition in eq. (15 ) produces clear, Lau fringes. Otherwise, we will see uniform background, as is shown in fig. 2. The method does not require any lens or any spatial filter. Let us consider some experimental results. 208

Fig. 3. Gratings that were employed to create Lau fringes at finite distances: (a) source E(s), (b) object t(x). In our experiment q = 0, too= 2 and d = 0.2 m m . So, Zo= 25.3 cm and d t = 0.8 ram.

Volume 60, number 4

OPTICS C O M M U N I C A T I O N S

15 November 1986

4. Experimental verification We start by setting an experimental verification of the Lau fringes at finite distances. For that purpose, we employ as incoherent source the grating in fig. 3a, which was illuminated incoherently. The periodical object, t(x), was the grating in fig. 3b. Next we substitute the grating in fig. 3b by the object t(x) in fig. 2. Note that the different section~ of the theta-modulated object and the grating in fig. 3b have the same spatial period. As we rotate the source, we obtain the images that are shown in fig. 4. As can be appreciated the images are properly decoded and they are practically noise free.

5. Conclusions The incoherent superposition of self-images was analyzed in terms of the virtual Fourier transformation. This analyzis was employed to state a consonance condition between the Fourier spectrum of the object and the codified source. The failure, in the form of misaligment, to satisfy the consonance condition was exploited to implement a lensless theta-modulation decoder, which works under incoherent illumination and does not require either lenses or spatial filters.

Acknowledgement One of us (P.A.) gratefully acknowledges the financial support of the Conselleria de Cultura, Edicaci6n y Ciencia de la Generalidad Valenciana, Spain.

Fig.4. Lensless theta decoded images of the object in fig. 2. The photos were taken on the first negative self-image plane ( m = 0 . 5 ) . So, z = 8.4 cm.

209

Volume 60, number 4

OPTICS COMMUNICATIONS

References [1] A.W. Lohmann and D.A. Silvia, Optics Comm. 2 (1971) 417; 4(1972) 326. [2] S. Yokozeiki and T. Suzuki, Appl. Optics 10 (1971) 1575. [3] H.O. Bartelt and J. Jahns, Optics Comm. 30 (1979) 268. [4] H. Kaijun, J. Jahns and A.W. Lohmann, Optics Comm. 45 (1983) 295. [5] H.O. Bartelt and Y. Li, Optics Comm. 48 (1983) 1. [6] A.W. Lohmann, Proc. ICO: Conf. Opt. Instr., ed. K.J. Habell (Butterworth, London, 1961 ) p.58. [7] H. Klages, J. Phys. (Paris) 28 (1967) C2-40. [8] H. Dammann, G. Groh and M. Kock, Appl. Optics 10 (1971) 1454. [9] J. Ojeda-Castafieda and E.E. Sicre, Optics Comm. 47 (1983) 183.

210

15 November 1986

[ 10] A.W. Lohmann, J. Ojeda-Castafieda and E.E. Sicre, Optics Comm. 50 (1984) 388. [11] S. Jutamulia and T. Asakura, J. Optics (Paris) 16 (1985) 121. [12] Z. Jaroszewics and A. Kolodziejczyk, Optics Comm. 56 (1985) 73. [ 13 ] B. Packross, R. Eschabach and O. Bryngdahl, Optics Comm. 56 (1986) 394. [ 14] J. Jahns and A.W. Lohmann, Optics Comm. 28 (1979) 273. [15]J. Ojeda-Castafieda and E.E. Sicre, Optics Comm., submitted. [16] J. Jahns, A.W. Lohmann and J. Ojeda-Castafieda, Optica Acta 31 (1984) 313. [ 17 ] R. Jozwicki, Optica Acta 30 (1983 ) 73. [ 18] K. Patorski, Optica Acta 30 (1983) 745.