Optical encryption by means of the Talbot array ...

Optical encryption by means of the Talbot array illuminator John Fredy Barrera a , Rodrigo Henao a , Zbigniew Jaroszewicz b, c , Andrzej Kolodziejczyk d a Instituto de Fisica, Universidad de Antioquia, calle 67 53-108, A.A. 1226 Medellin, Colombia b Institute of Applied Optics, Kamionkowska 18, 03-805 Warsaw, Poland c National Institute of Telecommunications, Szachowa 1, 04-894 Warsaw, Poland d Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland ABSTRACT The work presents the encryption process in the optical arrangement with the Talbot array illuminator (TAIL). The object to be encrypted is multiplicated, forming a periodical structure. Then the obtained periodical structure is placed in an optical set-up with a TAIL. The encryption is based on the Fresnel diffraction behind the TAIL. The encrypted image has a form resembling an elementary cell of the amplitude sampling filter. The decryption is based on the Talbot effect. Keywords: Optical encryption, Talbot array illuminator, Talbot effect, sampling filter

1. INTRODUCTION The paper presents an original application of the Talbot array illuminator (TAIL) for the optical encryption. TAILs are diffractive phase elements that transform a plane wave into an array of bright spots. These elements operate in the Fresnel diffraction regime. During the last years a substantial number of works has been devoted to TAILs’ design and their optical properties1-7. TAILs attract the attention of researchers because of potential applications for optical interconnections and optical computing. Lately was presented a new interpretation of the TAIL that underlines the imaging abilities of TAILs and makes it possible to characterize an imaging process in a way based on the theory of the thin lens8. According to this interpretation the TAIL is equivalent to a periodic combination of lenses modified by generalized pupil functions. The theoretical background of the paper is based on the approach given in Ref. 8. and the Talbot effect9. Our encryption method deals with so called conventional TAIL whose transmittance is defined by the complex amplitude of the diffractive field behind the amplitude grating3, 5. The optical encryption is an interesting example of the application of optical information processing for security purposes. Majority of optical encryption methods use random phase mask as key elements during encoding and decoding steps10-19. The main aim of the encryption is to recover hidden information by only authorized persons using proper keys. The important problem is a facility of recovering the information in the decryption process. The random mask have very complicated structure what is a serious disadvantage during a decoding step. In order to avoid the above disadvantage the implementation of structured phase mask for the optical encryption was lately proposed20. The TAIL belongs to the class of structured phase mask therefore its application for a real-time encryption in the optical set-up seems to be promising. The second advantage of the proposed encryption with the TAIL is its simplicity. The coding and encoding steps are realized by a propagation in a free space. The TAIL, coding and decoding keys are only used optical elements.

2. THEORETICAL BACKGROUND For the sake of simplicity, our analysis is limited to the conventional TAILs generating square arrays of square bright spots. The transmittance of the conventional TAIL is determined by a complex amplitude of a diffractive pattern behind the amplitude sampling filter (ASF). The ASF is a device consisting of pin-holes arranged in a square array of a period d in an absorbent black screen. The transmittance of the ASF can be expressed as follows:

Optical Security Systems, edited by Zbigniew Jaroszewicz, Sergei Y. Popov, Frank Wyrowski, Proc. of SPIE Vol. 5954, 59540I, (2005) · 0277-786X/05/$15 · doi: 10.1117/12.622095

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A( x, y ) =

1 d2

⎡ ⎤ ⎛ x y⎞ ⎢comb⎜ d , d ⎟ ⊗ g ( x, y )⎥ , ⎝ ⎠ ⎣ ⎦

(1)

where g(x,y) is a window function defined by a transmittance of the singular pin-hole of the ASF and the symbol ⊗ denotes the convolution operator. According to the theory of the Talbot effect9 some diffractive patterns behind the ASF illuminated by a plane wave, form self-images being exact images of the ASF. The self-images appear at the following distances behind the filter:

Zn =

d2

λ

n ,

(2)

where n is a natural number and λ is a wavelength of monochromatic illumination. Moreover, besides the selfimages there are formed behind the ASF the fractional Fresnel images being the superposition of mutually shifted self-images9. When the ASF consists of square pin-holes of the width b (g(x,y)=rect(x/b,y/b)) and the period d is a multiple value of the singular pin-hole width (d=Nb, N=2,3,4,...), then there exist planes behind the filter where the diffractive field has a uniform intensity. In this case, the phase of the function with a constant magnitude defines transmittance of the TAIL. The TAIL has the smallest feature size determined by the window function g(x,y). The planes with a uniform intensity distribution lie at distances behind the ASF, where the N-th fold compression of the singular window function occurs. The planes correspond to the following distances behind the ASF:

d 2 m1 . l= 2λ M 1

(3)

The numbers m1, M1 are connected with the number N as follows: M1 =N and m1 is even for N odd; M1 =N/2 and m1 is odd for N even8.

3. PRINCIPLE OF THE METHOD The object to be encrypted is an elementary cell of a two-dimensional periodical structure of the period d the same as the period of the used TAIL. The transmittance T(x,y) of the TAIL is defined by diffractive pattern of the ASF corresponding to the distance l described by Eq. (3). The encryption process is consisted of the following steps: 1) The superposition of the periodical object with the TAIL. It leads to the complex amplitude being the product U(x,y)T(x,y), where U(x,y) is a transmittance of the periodical object. 2) Propagation of the field defined by the complex amplitude U(x,y)T(x,y) along the distance Z1-l in a free space. 3) The superposition of the diffractive field with the phase mask defined by a transmittance F(x,y). After the above steps the object is encoded in diffractive element with the following transmittance:

⎧ ⎡ ik T1 ( x, y ) = ⎨[U ( x, y )T ( x, y )] ⊗ exp ⎢ x2 + y2 ⎣ 2(Z1 − l ) ⎩

(

where k=2π/λ.

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)⎤⎥ ⎫⎬F (x, y ) , ⎦⎭

(4)

The decryption process is consisted of the two following steps: 1) The superposition of the transmittance T1(x,y) with the complex conjugated phase mask described by the transmittance F*(x,y). 2) Propagation of the field defined by the product T1(x,y) F*(x,y) along a distance l in a free space. The decoding process can be realized in an optical set-up in a real-time. One part of the decoding key is a phase mask with the transmittance F*(x,y). It should be properly adjusted to the transmittance T1(x,y). Then the both, superposed elements should be illuminated by a plane wave with a proper wavelength λ. The encrypted image can be retrieved in an output plane, lying at the distance l behind the elements. The image of interest is an elementary cell of the reconstructed periodical structure. The knowledge of the propagation distance l is a second part of the decoding key. According to the theory of the TAIL, in a case of the uniform function U(x,y)=1 we obtain T1 ( x, y ) = A(x, y ) , where A(x,y) is a transmittance of the ASF8. It means, that the function T1(x,y) defines the element resembling the structure of the ASF with characteristic bright spots arranged in a square array. It can substantially utilize the superposition of the two elements in a decoding stage, especially when the transmittance F(x,y) describes the structured phase encoding mask.

4. NUMERICAL SIMULATIONS Usefulness of the TAIL for encryption purposes has been verified by numerical simulations. The simulations were conducted for the wavelength of He-Ne laser (λ=632.8 nm) using a diffractive modelling package working according to the modified convolution approach21 on a matrix 4096 x 4096 points with a sampling interval of 1 µm, covering an area of a square of the width about 4.1 mm. The transmittance of the TAIL

(

⎡ ik T ( x, y ) = A( x, y ) ⊗ exp ⎢ x 2 + y 2 ⎣ 2l

)⎤⎥ , ⎦

(4)

2 was defined by the following parameters: d=40 µm, N=10, l = d . As an encoding phase mask we have choosen 10λ also the same TAIL. Therefore T(x,y)=F(x,y) and the decoding key have belonged to the class of structured phase masks. The obtained results are presented in Fig. 1, 2.

5. CONCLUSIONS The work has presented application of TAILs for the optical encryption. The obtained numerical simulations are promising. The implementation of the structured phase mask lead to satisfactory results. The images retrieved without using the right key are completely spoiled and unrecognisable. The encryption method with the TAIL exhibit following advantages: - the mutual adjustment of the encrypted image and the encoding masks seems be easy - the decryption is based on the Talbot effect therefore a decoding key is the only optical element used in the decoding step Because of the above advantages the decryption process can be realized in real time in an optical arrangement.

ACKNOWLEDGEMENTS This work was supported by Warsaw University of Technology, the Network of Excellence on Micro-Optics (NEMO). Rodrigo Henao and John Fredy Barrera gratefully recognize the support of CODI-University of Antioquia and Colciencias.

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(a)

(b)

(b)

(d)

Fig. 1. Results of numerical simulations: (a) periodical structure with an elementary cell formed by the object to be encrypted, (b) encrypted image defined by the function T1(x,y), (c) retrieved image by the right decoding key, (d) image retrieved by only propagation without the use of the decoding key.

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(a)

(b)

(b)

(d)

Fig. 2. Results of numerical simulations: (a) periodical structure with an elementary cell formed by the object to be encrypted, (b) encrypted image defined by the function T1(x,y), (c) retrieved image by the right decoding key, (d) image retrieved by only propagation without the use of the decoding key.

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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

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