1. Introduction

Gabor et al. [1] described optical image synthesis by complex amplitude addition and subtraction for the first time. With this technique, the composite image can be reconstructed from a recorded hologram that obtained by successive exposures of two or more complex diffraction patterns. Since then, optical information synthesis has become a popular topic in information optics and a few optical image synthesis techniques have also been proposed [2–6]. However, most of these proposed methods are costly and complicated and involve chemical or physical developing process after exposure.

With the rapid development of internet and computer technique, information security is becoming more and more important. How to protect the synthesized data from unauthorized use and counterfeiting also became an important subject for us. Since Refregier and Javidi proposed the double-random phase encoding technique [7], optical encryption techniques have been attracted increasing interests for their high processing speed and high security level, etc. A number of optical encryption methods have been proposed [8–18]. Because of physical limitations imposed by optical or electronic hardware, such as the complexity of their optical hardware, lack of flexibility, lack of low-cost optoelectronics devices, etc., most of the encryption methods proposed are limited to image encryption and difficult to be used in real world application. To overcome these problems, several encryption methods based on the concept of virtual optics have also been proposed [12, 13, 16].

In this paper we proposed a simple method to realize information synthesis and data encryption based on digital holography and virtual optics. No special devices such as grating, photosensitive crystal film and no need of filtering operation in the synthesis process. Digital recording of holograms enables us to store, transmit and reconstruct data digitally and has advantages over traditional methods in that no chemical or physical developing is needed after exposure [9, 10, 15]. To eliminate the inconvenience of calibrating phase shifter and increase the flexibility, we use a three-step phase-shifting interferometry [PSI] with arbitrary phase shifts [19] to record the synthesized digital hologram. For the purpose of information security, the digital data is to be encrypted by a virtual optical encryption system based on extended fractional Fourier transform [FRT], which was introduced by Hua et al. [20] with more parameters compared with the conventional FRT. As a result of the concept of “virtual optics”, there is no need to manufacture costly optical elements, no physical limitations and no difficulty in optical alignment. In this virtual system, we encrypt the input data by use of extended FRTs and computer generated random phase masks. Both the encryption and decryption processes are performed digitally.

2. Principle of information synthesis with digital holography technique

The proposed method for the synthesis of information by complex amplitude addition and subtraction is based on phase-shift digital holography technique. For clarity and simplicity sake, only two 2-D images are used to be synthesized in the following discuss. As shown in Fig. 1, A Fresnel diffraction of the first image can be recorded in CCD plane by use of an interference with an on-axis reference wave. The complex distribution of the object wave U ₁(x,y) in CCD plane after Fresnel diffraction can be written as

where x ₀, y ₀ and x, y are the coordinates of object plane and CCD plane, respectively. The function U ₀ (x ₀, y ₀) represents the original object complex wave in object plane. Let the on-axis reference wave in CCD plane at the jth (j = 1, 2, 3) exposure be expressed as

where δ_j is the phase shift introduced by phase retarders at the jth exposure, and A_r, φ_r are the constant amplitude and the phase of reference wave, respectively.

In this three-frame case we have

where we denote U ₁(x,y) = A ₀(x,y)exp[φ ₀(x,y)]. The digital hologram of the first image obtained from the three interference patterns can be expressed as [19]

where the constants A_r, φ_r are replaced with 1 and 0, respectively. In the same way, the corresponding digital hologram of the second image can be expressed as

where I′ ₁ , I′ ₂, I′ ₃ are the intensity distributions of the 1st, 2nd , 3rd interferograms respectively. Calculated from Eq. (6) and Eq. (7), a fused digital hologram can be written as

where the superscript f means the data is obtained during fusion. Similarly, a subtracted digital hologram can also be obtained from Eq. (6) and Eq. (7), which is given by

where the superscript S means the data is obtained during subtraction process. To obtain the synthesized image, a Fresnel inverse transform of the digital hologram has to be performed digitally. With digital holography technique, one can realize image synthesis by recording digital holograms U^f (x, y) and U^s (x, y) . When a half-wave plate and a quarter-wave plate are placed in the reference beam arm to generate phase shifts of 0 , $\frac{\pi }{2}$ ,π [9], the fused digital hologram obtained from Eq. (8) can be written as

And the subtracted digital hologram obtained from Eq. (9) can be written as

 

Fig. 1. Phase-shifting digital holography with three-step PSI.

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Though the real part and imaginary part of the fused or subtracted digital hologram can be stored and transmitted in communication channel, the fused digital hologram and subtracted digital hologram obtained from the different intensity patterns still need to be encoded for the security of the digital data [16]. In the following section, the recorded digital hologram will be divided into segments and each segment is to be encrypted independently.

3. Encryption of digital hologram based on the concept of “virtual optics”

Several encryption systems based on extended FRT have been proposed [8, 14, 18] to provide enhanced security by increasing the key size compared with those methods based on conventional FRT. In this paper, the encryption system is based on the concept of “virtual optics” and the key size will be enlarged further. As shown in Fig. 2(a), the recorded digital hologram is separated into independent n segments {S ₁,S ₂,S ₃,⋯,S_n } digitally and each segment S_k (k = 1,2,3,⋯n) is extended fractional Fourier transformed two times with a random phase mask placed at the output plane of the first extended FRT. The set of n encoded segments {$S_1^e$,$S_2^e$,$S_3^e$,⋯,$S_n^e$ } form the encrypted hologram. For clarity and convenience, the encryption process of the kth (k = 1,2,3 ⋯n) segment is used to illustrate this method and a one-dimensional representation is followed in this section. As shown as in Fig. 2(b), let x, x ₁ and x ₂ denote the coordinates of the input plane and the output plane of the first and second extended FRT, respectively. The complex distribution in output plane can be written as [8, 14, 18]

where the superscript e means the signal is obtained during encryption and a′_k, b′_k, φ′_k are the three parameters of the extended FRT that related to the physical parameters λ_k, d_k3, d_k4 and f_k2 . The function exp[iϕ_k (x ₁)] represents the random phase mask p_k , where ϕ_k (x ₁) is a random function distributed uniformly in the interval [0,2π] and g_k (x ₁) denotes the distribution in the output plane of the first extended FRT, which can be given by

where K is a complex constant and the function exp[iϕ′(x)] represents the random phase mask p′ placed in the input plane, where ϕ′ (x) is a random function distributed uniformly in the interval [0,2π] and statistically independent of ϕ_k (x ₁) . a_k, b_k, φ_k are the three parameters of the extended FRT that related to the virtual wavelength λ_k , the distances d _k1, d _k2, the focal length f _k1 and given by

One can calculate the values of the parameters a′_k, b′_k, φ′_k by substituting d _k3, d _k4, f _k2 for d _k1, d _k2 ,f _k1, respectively into Eqs. (14)–(16). It can be shown that the random phase mask p_k (x ₁) and the parameters a_k, b_k, φ_k, a′_k, b′_k, φ′_k related to λ_k , d _k1, d _k2, d _k3, d _k4 , f _k1 and f _k2 form the keys for decryption of the segment data S_k .

 

Fig. 2. (a) The schematic of the digital hologram encryption; (b) the virtual optical setup for encryption of the kth segment based on extended FRT.

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4. Decryption of digital hologram

The schematic of decryption system and the virtual optical setup for decryption are as shown in Fig. 3(a) and Fig. 3(b), respectively. The decryption is a reverse operation of the encryption process by taking the conjugate of the encrypted data. The resultant distribution function of decryption is given by

where the superscript d means the signal is obtained during decryption and $g_k^d$ (x′₁) denotes the distribution in the output plane of the first extended FRT during the decryption process that can be given by

where S′_k (x′₂) = [$S_k^e$ (x′₂)]^* . By substituting $S_k^e$ (x ₂) from Eq. (12) into Eq. (18) and $g_k^d$ (x′¹) from Eq. (18) into Eq. (17), one can show the decrypted function $h_k^d$ (x′) is equal to S_k (x). With all the correct decrypted digital segments, the fused or subtracted digital hologram can be retrieved.

 

Fig. 3. (a) The schematic of the digital hologram decryption; (b) the virtual optical setup for decryption of the kth segment.

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5. Numerical simulation results

Numerical simulations have been performed to verify the validity of the proposed idea in this paper. Two images with 256×256 pixels are chosen to be synthesized as shown in Fig. 4(a) and Fig. 4(b). The real part and imaginary part of the fused digital hologram are as shown in Fig. 5(a) and Fig. 5(b), respectively. In our numerical simulations, the digital hologram is split into four segments with equality in dimensions. Figure 5(c) and Fig. 5(d) show the real part and imaginary part of one of the four segments respectively. For the encryption of this segment, we set λ₁ =520nm, f ₁ =17cm, d ₁ = 16cm, f ₂ = 21cm, d ₂=22cm, d ₃=24cm, and d ₄ =29cm. The parameters of the extended FRT encryption system as calculated from Eqs. (14)–(16) are a ₁ =35.4111 + 35.4111i , b ₁ =15.8363-15.8363i , φ ₁=1.5708-0.1312i, a ₂ =39.2154 , b ₂ =24.0144 and φ ₂ =1.8063 . The real part and imaginary part of the encrypted results are as shown in Fig. 5(e) and Fig. 5(f), respectively. With all the correct keys, the real part and imaginary part of decrypted results of the digital hologram segment are shown in Fig. 5(g) and Fig. 5(h), respectively. Figure 5(i) and Fig. 5(j) show the real part and imaginary part of the decrypted results with correct random phase codes but incorrect parameters (a ₁ = 32.2990 + 32.2990i , b ₁=16.1495-16.1495i , φ ₁=1.5708-0.1174i , a ₂ =36.7115 , b ₂ = 27.3631 and φ ₂ =1.9316 corresponding to the case of λ ₁ =560nm, f ₁ = 17cm, d ₁ = 16cm, f ₂ = 19cm, d ₂ = 16cm, d ₃ =24cm, and d ₄ = 28cm). Figure 5(k) and Fig. 5(l) show the real part and imaginary part of the decrypted results with incorrect random phase codes but correct parameters. It can be shown that digital hologram segments cannot be retrieved without knowledge of the respective random phase codes and parameters.

 

Fig. 4. (a) and (b) The images to be synthesized and encrypted.

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Fig. 5. (a) Real part, (b) imaginary part of the recorded fused digital hologram; (c) real part, (d) imaginary part of one of digital hologram segments; (e) real part, (f) imaginary part of one of the encrypted digital hologram segments; decrypted results of one of the encrypted digital hologram segments with all correct keys (g) real part, (h) imaginary part; (i) real part, (j) imaginary part of decrypted results of the digital hologram segment with incorrect parameters but correct random phase codes; (k) real part, (l) imaginary part of decrypted results of the digital hologram segment with correct parameters but incorrect random phase codes.

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With all the segments are decrypted correctly, the real part and imaginary part of decrypted digital hologram can be shown in Fig. 6(a) and Fig. 6(b), respectively. Figure 6(c) shows the reconstructed results from the right decrypted data. The corresponding reconstructed image from the wrong decrypted data is as shown in Fig. 6(d).

 

Fig. 6. (a) Real part, (b) imaginary part of the decrypted digital hologram with correct codes; reconstructed results from (c) correct decrypted digital hologram, (d) incorrect decrypted digital hologram.

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The mean square error (MSE) between the decrypted image and the input image is defined as

where I ₁(i,j) and I ₂(i,j) denote the values of the original image and the decrypted image at the pixel (i, j) , respectively. The MSE between the real part of decrypted digital hologram as shown in Fig. 5(a) and the real part of input digital hologram as shown in Fig. 6(a) is about 3.54×10^-12. The corresponding MSE between the imaginary parts as shown in Fig. 5(b) and Fig. 6(b) is about 3.56×10^-12.

Figure 7(a) and Fig. 7(b) show the real part and imaginary part of the encrypted results of the subtracted digital hologram. From the right decrypted image of the digital hologram, the subtracted image can be reconstructed digitally as shown in Fig. 7(c).

 

Fig. 7. (a) Real part, (b) imaginary part of the encrypted results of the subtracted digital hologram; (c) reconstructed results from correct decrypted digital hologram.

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Three images, Figs. 8(a), 8(b) and 8(c), are chosen to be synthesis and encrypted. The real part and imaginary part of the encrypted fused digital hologram and the encrypted subtracted digital hologram are as shown in Figs. 8(d), 8(e), 8(f) and 8(g), respectively. In the same way, the reconstructed result from the correct decrypted fused digital hologram can be shown in Fig. 8(h). Figure 8(i) shows the corresponding synthesized image by subtracting Figs. 8(b) and 8(c) from Fig. 8(a).

 

Fig. 8. (a), (b) and (c) The images to be synthesized and encrypted; (d) real part, (e) imaginary part of encrypted fused digital hologram; (f) real part, (g) imaginary part of encrypted subtracted digital hologram; synthesized results reconstructed from (h) right decrypted fused digital hologram, (i) right decrypted subtracted digital hologram.

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6. Conclusions

In summary, we proposed a simple method to realize information synthesis without special and costly devices and chemical or physical developing process after exposure. For security, the synthesized digital hologram is encrypted into a white noise with the concept of “virtual optics”. The virtual wavelength λ_k , the random phase mask p_k , the parameters a_k, b_k, φ_k, a′_k, b′_k, φ′_k related to the focal length f _k1, f _k2 and the distances d _k1, d _k2 , d _k3, d _k4 (k = 1,2 ⋯n) form the keys for the decryption of the digital hologram. Only by use of the exact respective keys for each encrypted segments, the synthesized information can be retrieved digitally. Computer simulations have shown that 2-D images can be synthesized and encrypted successfully. With the proposed method, the synthesized information can be encrypted, stored and transmitted with high security. It can be believed that the key size still can be enlarged further by employing more lenses and random phase masks and the application of proposed method can be extended to the synthesis of 3-D information by recording a synthesized digital hologram of 3-D objects with digital holography technique.