Single-channel color information security system using LU decomposition in gyrator transform domains

Optics Communications 323 (2014) 100–109

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Single-channel color information security system using LU decomposition in gyrator transform domains Muhammad Rafiq Abuturab Department of Physics, Maulana Azad College of Engineering and Technology, Patna 801113, India

ar t ic l e i nf o

a b s t r a c t

Article history: Received 20 December 2013 Received in revised form 27 January 2014 Accepted 14 February 2014 Available online 6 March 2014

A novel single-channel color information security system based on LU decomposition in gyrator transform domains is proposed. The original color image to be encoded is separated into its red, green and blue channels. They are modulated by corresponding random phase functions and then independently Fourier transformed. The transformed images of red and green channels are multiplied and then inverse Fourier transformed. The resulting image is phase- and amplitude truncated to obtain an encrypted image and an asymmetric decryption key, respectively. The encrypted image is multiplied by transformed image of blue channel and then performed LU decomposition. Finally, L and U parts are individually gyrator transformed at different transformation angles, which can be assigned to two different authorized users. The proposed single-channel encryption system is more compact than conventional three-channel encryption systems. Additionally, the ciphertexts are not color images but they are gray images which have obscure properties. The presented LU form is asymmetric. The two transformation angles of GT, three decryption keys for three channels and one asymmetric decryption key significantly improve the security and robustness of the proposed method. The encryption system can be realized digitally or optically. Numerical simulations demonstrate the feasibility and effectiveness of the suggested algorithms. & 2014 Elsevier B.V. All rights reserved.

Keywords: LU decomposition Asymmetric color image cryptosystem Gyrator transform

1. Introduction With the rapid development of modern communication systems, information security has become a serious concern. Optical information techniques have been widely researched owing to their inherent advantages of parallelism and high speed transmitting. The doublerandom phase-encoding (DRPE) is a well known technique which is based on the 4-f optical correlator to encrypt a primary image into stationary white noise [1]. Since then various optical image encryption systems and algorithms have been investigated [2–8]. These encryption techniques can be designed for compression operations simultaneously in the spectral domain [9]. In all these encryption methods, as a monochromatic light is used to illuminate a real color image, color information of a recovered image is lost. To include color information, a color image encryption method based on an indexed image and double phase random masks has been introduced [10]. In this method, an RGB color image is converted into an indexed image format before encoding. During the decryption process, the color image is retrieved by converting the decrypted indexed image back into its RGB format. Many optical techniques and applications for the color image processing have been further developed [11–20]. However, most of the existing methods belong to the symmetric cryptosystem, in which the encryption and decryption keys are the same. From the cryptography point of view, a symmetric E-mail address: rafi[email protected] http://dx.doi.org/10.1016/j.optcom.2014.02.061 0030-4018 & 2014 Elsevier B.V. All rights reserved.

cryptosystem would suffer from several problems in practical application, particularly under the network environment. In order to overcome these problems, an asymmetric cryptosystem based on a phase-truncated Fourier transform (PTFT) approach has been proposed [21]. In this method, with phase truncation in Fourier domain, one can produce an asymmetric ciphertext as real-valued and stationary white noise using two random phase keys as public keys, while an authorized user can recover the plaintext employing another two different private phase keys in the decryption process. It has been demonstrated that an iterative amplitudephase retrieval algorithm can decipher the PTFT-based asymmetric cryptosystems, due to the vulnerability of the decryption key pairs [22]. Based on asymmetric cryptosystem, a number of color image encryption schemes have been proposed [23–25]. The aforementioned encryption techniques are based on multiple-channel encryption systems. Compared with the multiplechannel encryption methods, single-channel encryption algorithms are more simple and viable system. Recently, several single-channel image encryption methods have been reported [26–29]. In this paper, for the first time to my knowledge, a novel singlechannel color information security system by using LU decomposition in gyrator transform domains is proposed. In the encryption process, an original color image is segregated into its red, green and blue channels. They are then modulated by corresponding random phase functions. The modulated channels are individually Fourier transformed. The transformed red channel is multiplied with

M.R. Abuturab / Optics Communications 323 (2014) 100–109

transformed green channel. The product image is inverse Fourier transformed. The Fourier spectrum is phase- and amplitude truncated to produce a ciphertext and a decryption key, respectively. The asymmetric ciphertext is multiplied by transformed blue channel. The obtained image is separated by LU decomposition. Lastly, L and U parts are independently gyrator transformed at different transformation angles to get encrypted L and U parts. During the decryption process, the encoded images are separately inverse gyrator transformed. They are then multiplied to each other and decryption key to reconstruct single-channel image, which is divided by transformed blue channel. The resulting image is Fourier transformed. The Fourier spectrum is multiplied by decryption keys for red and green channels and the corresponding images are inverse Fourier transformed to retrieve red and green channels. To reconstruct blue channel, the recovered single-channel image is multiplied by decryption key for blue channel and obtained image is inverse Fourier transformed. The suggested encryption system can be performed digitally or optically.

101

The proposed single-channel encryption system is simpler than three-channel encryption systems and has three advantages over the earlier techniques [26–29]. First, the LU form is asymmetric. Second, two encoded L and U parts can be allocated to two different allowed users for highly safe authentication. Third, the transformation angles of GT are remarkably sensitive keys. Moreover, the ciphertexts are not color images but gray images which can bewilder the others. Numerical simulations exemplify the validity and effectiveness of the proposed technique.

2. Principle 2.1. LU decomposition The LU (lower-and-upper) decomposition is a factorization technique for matrices in linear algebra. LU decomposes an n-by-n

Fig. 1. (a) Flow diagram of the proposed color image encryption process and (b) flow diagram of the proposed color image decryption process.

102

M.R. Abuturab / Optics Communications 323 (2014) 100–109

where Gα ½  indicates GT operator. ðxi ; yi Þ and ðxo ; yo Þ are the input and output coordinates, respectively. When α A ½0; 2π , the GT can be implemented in an optical system composed of identical planoconvex cylindrical lenses. The transformation angle α is varied by proper rotation of these lenses [32]. Gα and G2π  α are reciprocal transforms which can be exploited in optical information processing. In this paper, the calculation of discrete gyrator transform algorithm is obtained by using a convolution operation [33]. The optical image algorithms based on gyrator transform for color image [16–20,23–25] and gray image [34–38] have been reported.

non-singular square matrix A into a product of two matrices as 2

… ⋯

a11 6 a21 6 A¼LU ) 6 4 ⋮

a12 a22 ⋮

⋱

an1

an2

…

3

2

a1n 1 6 a2n 7 7 6 l21 7¼6 ⋮ 5 4 ⋮ ann ln1

0 1

… ⋯

⋮

⋱

ln2

…

32

0 u11 6 07 76 0 76 ⋮ 54 ⋮ 1 0

3

u12

…

u22 ⋮

⋯ ⋱

u1n u2n 7 7 7 ⋮ 5

0

…

unn

ð1Þ where L is a row permutation of a lower triangular matrix having identity elements on the diagonal and the multipliers below the diagonal. U, on the other hand, is an upper triangular matrix having some coefficients on the diagonal and the multipliers above the diagonal such that L  U ¼ A [30]. The LU form is asymmetric in this case that L has always 1's on the diagonal, whereas U does not. This property can be investigated for image encryption.

3. Proposed technique

2.2. Gyrator transform

A single-channel color image security system by using LU decomposition in GT domains is presented. The flow diagrams of encryption and decryption algorithms have been displayed in Figs. 1(a) and 1(b), respectively. The encryption process can be performed by the following steps.

The gyrator transform (GT) of a two-dimensional complex field function f i ðxi ; yi Þ with order α known as transformation angle, is defined as [31] f o ðxo ; yo Þ ¼ Gα ½f i ðxi ; yi Þðxo ; yo Þ ¼

1 j sin

Z Z

αj

þ1 1

  ðxo yo þ xi yi Þ cos α ðxi yo þ xo yi Þ dxi dyi f i ðxi ; yi Þ exp i2π sin α

1. A color image f ðxi ; yi Þ to be encoded is separated into its red, green and blue channels f R ðxi ; yi Þ, f G ðxi ; yi Þ and f B ðxi ; yi Þ,

ð2Þ

L

SLM1

CCD 1

f

f Computer System

L1

L2

L3

CCD 2

SLM 2

z

z L1

L2

L3

CCD 3

SLM 3 z

z

Fig. 2. Optoelectronic setup for the proposed color image encryption system.

M.R. Abuturab / Optics Communications 323 (2014) 100–109

and then independently Fourier transformed.

respectively, which is given as f ðxi ; yi Þ ¼ ½f R ðxi ; yi Þ; f G ðxi ; yi Þ; f B ðxi ; yi Þ

103

ð3Þ

2. The red, green and blue channels are modulated by random phase functions ϕR ðxi ; yi Þ, ϕG ðxi ; yi Þ and ϕB ðxi ; yi Þ, respectively

F R ðx; yÞ ¼ Fff R ðxi ; yi Þ exp½iφR ðxi ; yi Þg

ð4Þ

F G ðx; yÞ ¼ Fff G ðxi ; yi Þ exp½iφG ðxi ; yi Þg

ð5Þ

F B ðx; yÞ ¼ Fff B fxi ; yi g exp½iφB ðxi ; yi Þg

ð6Þ

Fig. 3. Simulation results of the proposal: (a) Original color image with 512  512 pixels, (b) decryption key for red channel, (c) decryption key for green channel, (d) decryption key for blue channel, (e) asymmetric decryption key, (f) encoded L part, (g) encoded U part, and (h) decrypted image with all correct keys.

104

M.R. Abuturab / Optics Communications 323 (2014) 100–109

where F R ðx; yÞ, F G ðx; yÞ, and F B ðx; yÞ denote Fourier transformed red, green and blue channels, respectively. 3. The transformed images for red and green channels are multiplied and then inverse Fourier transformed. The obtained image is phase- and amplitude truncated to get first encrypted image and an asymmetric decryption key, respectively. E1 ðxi ; yi Þ ¼ PTfF Pðxi ; yi Þ ¼ ATfF

1

1

½F R ðx; yÞF G ðx; yÞg

½F R ðx; yÞF G ðx; yÞg

ð7Þ ð8Þ

4. The encrypted image is multiplied by transformed image for blue channel and then performed LU decomposition. E2 ðxi ; yi Þ ¼ E1 ðxi ; yi ÞF B ðx; yÞ ½Lðxi ; yi Þ; Uðxi ; yi Þ ¼ lu ½E2 ðxi ; yi Þ

ð9Þ ð10Þ

where the function lu indicates LU decomposition. Lðxi ; yi Þ and Uðxi ; yi Þ represent decomposed L and U parts. 5. Finally, encrypted L and U parts are individually gyrator transformed. EL ðx; yÞ ¼ GαL ½Lðxi ; yi Þ

ð11Þ

EU ðx; yÞ ¼ GαU ½Uðxi ; yi Þ

ð12Þ

where αL and respectively.

αU are transformation angles of L and U parts,

The decryption process can be carried out by the following steps. 1. The encoded images are independently inverse gyrator transformed and then obtained images are multiplied to each other and asymmetric decryption key to get single-channel image. DL ðxi ; yi Þ ¼ G  αL ½EL ðx; yÞ

ð13Þ

DU ðxi ; yi Þ ¼ G  αU ½EU ðx; yÞ

ð14Þ

Dðxi ; yi Þ ¼ DL ðxi ; yi ÞDU ðxi ; yi ÞPðxi ; yi Þ

ð15Þ

2. The reconstructed image Dðxi ; yi Þ divided by F B ðx; yÞ is Fourier transformed.   Dðxi ; yi Þ ð16Þ DRG ðx; yÞ ¼ F F B ðx; yÞ 3. The resulting image DRG ðx; yÞ is divided by decryption key for red channel 1=F G ðx; yÞ and decryption key for green channel 1=F R ðx; yÞ. Then corresponding images are inverse Fourier transformed to retrieve red and green channels.   DRG ðx; yÞ ð17Þ f R ðx; yÞ ¼ F  1 F G ðx; yÞ f G ðx; yÞ ¼ F  1

  DRG ðx; yÞ F R ðx; yÞ

ð18Þ

Fig. 4. Decrypted results: (a) using only decryption key for red channel, (b) using only decryption key for green channel, (c) using only decryption key for blue channel, and (d) using no decryption keys.

M.R. Abuturab / Optics Communications 323 (2014) 100–109

4. The obtained image Dðxi ; yi Þ is multiplied by decryption key for blue channel 1=E1 ðxi ; yi Þ and then resulting image is inverse Fourier transformed to recover blue channel. F B ðxi ; yi Þ ¼

Dðxi ; yi Þ E1 ðxi ; yi Þ

f B ðxi ; yi Þ ¼ F  1 ½F B ðx; yÞ

ð19Þ

105

decomposition digitally by the computer system. The encoded Lðxi ; yi Þ and Uðxi ; yi Þ parts are, respectively, displayed on SLM 2 and SLM 3. They are separately illuminated by uniform plane waves and then gyrator transformed at two different transformation angles. In-line holography with CCD 2 and CCD 3, the corresponding encrypted images are recorded. The SLMs and CCDs are controlled by the computer system.

ð20Þ

The red, green and blue channels are combined to decrypt original color image in the computer system. Fourier transforming can be optically implemented by using a simple convergent lens L of focal length f [9]. The GT can be carried out in the coherent optical system by using three generalized lenses (designated as L1, L2 and L3) with fixed equal distances z between them. Each generalized lens corresponds to the assembled set of two identical plano-convex cylindrical lenses of the same power. The first and third identical lenses of focal length f 1 ¼ z are rotated to vary the transformation angle α. The second lens L2 of a focal length f 2 ¼ z=2 is fixed [33]. The optoelectronic design for the optical encryption system is shown in Fig. 2. In the encryption process, Eqs. (3)–(6) are encoded digitally by a computer system. F R ðx; yÞ and F G ðx; yÞ are multiplied and then displayed on spatial light modulator 1 (SLM 1). In-line holography with CCD 1, the Fourier spectrum is recorded. The phase part of the spectrum is kept as a decryption key. The amplitude part multiplied with F B ðx; yÞ is decomposed by LU

4. Numerical simulation results Numerical simulations have been performed on a Matlab 7.11.0 (R2010b) platform to examine the viability of the proposed algorithm. The color image comprising 512  512  3 pixels and 8 bits, as shown in Fig. 3(a), is used as input image. The transformation angles of the GT for αL and αU are, respectively, 0.101 and 0.151. The decryption phase keys for red, green and blue channels are demonstrated in Figs. 3(b), 3(c) and 3(d), respectively. The asymmetric decryption key is shown in Fig. 3(e), The encrypted L and U parts, and decrypted image with all correct keys are illustrated in Figs. 3(f), 3(g), and 3(h), respectively. The decrypted images obtained by using, only decryption key for red channel, only decryption key for green channel, only decryption key for blue channel, and no decryption keys are depicted in Figs. 4(a), 4(b), 4(c) and 4(d), respectively. The retrieved images by changing the transformation angles αL and αU through 0.00011 are shown in Figs. 5(a) and 5(b), respectively.

Fig. 5. Sensitivity results: (a) when the transformation angle αl of L part is changed by 0.00011 while other keys are correct, (b) when the transformation angle αu of U part is changed by 0.00011 while other keys are correct, (c) with reverse image multiplication order of LU, and (d) without asymmetric decryption key.

106

M.R. Abuturab / Optics Communications 323 (2014) 100–109

Fig. 6. Known-plaintext attack results: (a) known-plaintext, (b) fake decryption key for red channel, (c) fake decryption key for green channel, (d) fake decryption key for blue channel, (e) recovered image with fake L part but correct U part, (f) retrieved image with fake U part but correct L part, (g) reconstructed image with fake L and U parts, and (h) decrypted image with fake decryption keys.

The reconstructed image with reverse image multiplication order of LU is demonstrated in Fig. 5(c). The recovered image without asymmetric decryption key is illustrated in Fig. 5(d). It is clear that the proposed method is very sensitive to its parameters and

original image can only be obtained when all the parameters are correct. A known-plaintext having a size of 512  512  3 pixels is shown Fig. 6(a). The fake decryption keys generated for red, green,

M.R. Abuturab / Optics Communications 323 (2014) 100–109

107

Fig. 7. Results for noised encrypted data (a) encrypted L part contaminated by Gaussian noise with 0.1 variance, (b) encrypted U part contaminated by Gaussian noise with 0.1 variance, (c) recovered image with encrypted L and U parts contaminated by Gaussian noise with 0.1 variance, (d) encrypted L part contaminated by speckle noise with 0.1 variance, (e) encrypted U part contaminated by speckle noise with 0.1 variance, and (f) retrieved image with encrypted L and U parts contaminated by speckle noise with 0.1 variance.

and blue channels are displayed in Figs. 6(b), 6(c) and 6(d), respectively. The recovered image by using fake L part but correct U part is illustrated in Fig. 6(e). The retrieved image by using fake U part but correct L part is demonstrated in Fig. 6(f). The reconstructed image by using fake U and L parts is depicted in Fig. 6(g). The decrypted image by using fake decryption keys is shown in Fig. 6(h). Figs. 6(e)  6(h) are noise-like images, which prove that the proposed system possesses the ability to resist known-plaintext attack.

To determine the security measure of the proposed technique, the mean square error (MSE) is defined as MSE ¼

M N 1 ∑ ∑ jI o ðm; nÞ  I d ðm; nÞj2 M  Ni¼1j¼1

ð21Þ

where I o ðm; nÞ and I d ðm; nÞ denote the original and decrypted images at pixel position ðm; nÞ, respectively. M  N is the number of image pixels.

108

M.R. Abuturab / Optics Communications 323 (2014) 100–109

2.5

x 108

3500 Red Green Blue

2

3000 2500

MSE

1.5

2000 1

1500 0.5

1000

0 0.599

0.5995

0.6

0.6005

0.601

0.6015

500

Transformation angle for L part of LU decomposition 0 3

x 10

8

Red Green Blue

MSE

2.5

12

1.5

10

1

8

0.5

6

0.5995

0.6

0.6005

100

200

300

400

500

600

100

200

300

400

500

600

4

14

2

0 0.599

0

0.601

0.6015

x 10

4

Transformation angle for U part of LU decomposition Fig. 8. Transformation angle influence: (a) MSE versus variation in transformation angle of L part and (b) MSE versus variation in transformation angle of U part.

2 0

The MSE values of red, green and blue channels for encrypted images of L and U parts with all accurate parameters are, and respectively, ð2:0810  104 ; 1:2867  104 ; 1:1618  104 Þ ð6:0777  1024 ; 6:0777  1024 ; 6:0777  1024 Þ. These MSE values indicate that the primary image is fully veiled in the ciphertext as shown in Fig. 3(f) and (g). The MSE values of red, green and blue channels for decrypted images with no decryption keys are, respectively, 7:0748  1013 ; 7:0749  1013 , and 8:2303  1017 as illustrated in Fig. 4(d). Thus decryption keys provide higher level of security. In order to evaluate the sensitivity on transformation angles αL and αU, their correct values are shifted by 0:0001 3 . The corresponding MSE values of red, green and blue channels are and ð2:3735  ð1:7116  106 ; 1:1648  106 ; 9:5680  104 Þ, 106 ; 9:5948  105 ; 8:5832  104 Þ as displayed in Fig. 5(a) and (b). The MSE values of red, green and blue channels with reversed image multiplication order of L and U parts, and without asymmetric decryption key are ð4:7919  109 ; 2:6590  109 ; 3:1989  108 Þ and ð5:2448  104 ; 3:0977  104 ; 5:1024  103 Þ, respectively as demonstrated in Fig. 5(c) and (d). These values clearly indicate that original image cannot be retrieved, with shifted transformation angles by 0:0001 3 , with reversed image multiplication order of L and U parts, or without asymmetric decryption key. If an intruder generates fake keys by using, fake L part but correct U part, fake U part but correct L part, and fake decryption keys with correct parameters by employing known-plaintext and use to retrieve the original image. The corresponding MSE values of red, green and blue channels are ð3:5011  109 2:2578  109 ; 1:5243  108 Þ, ð3:1267  1010 ; 1:4328  1010 ; 1:5934  109 Þ and

0 4

14

x 10

12 10 8 6 4 2 0

0

100

200

300

400

500

600

Fig. 9. Histogram analysis: (a) histogram of an original color image as shown in Fig. 3(a), (b) histogram of encrypted L part as illustrated in Fig. 3(f), and (c) histogram of encrypted U part as depicted in Fig. 3(g).

ð1:3138  105 ; 9:7462  104 ; 2:9248  104 Þ as depicted in Figs. 6 (e), 6(f), and 6(h), respectively. It is obvious that no valuable information about the original image can be observed. Thus an intruder will fail to decrypt correct information.

M.R. Abuturab / Optics Communications 323 (2014) 100–109

The noise interferences to the encrypted images by Gaussian noise with 0.1 variance and speckle noise with 0.1 variance have also been studied. The Gaussian noised images are shown in Fig. 7 (a) and (b). The recovered image is depicted in Fig. 7(c). The speckle noised images are demonstrated in Fig. 7(d) and (e). The reconstructed image is illustrated in Fig. 7(f). As can be seen from Figs. 7(c) and (f), the decrypted images can be recognized without difficulty. The influence of transformation angles αL and αU has been studied with sampling interval and sampling length ð0:5990; 0:6010Þ and 0:0001, respectively. The corresponding MSE values between original and decrypted red, green and blue channels versus variation in transformation angle are shown in Fig. (8)(a) and (b). In both cases, MSE value becomes very high for a very small change in the transformation angle. Therefore, the angle parameters αL and αU can be used as strong keys in the proposed system. The statistical analysis has been conducted on the proposed security system. As can seen from Fig. (9)(b) and (c), the histograms of the encrypted L and U parts approximated by uniform distribution are quite different from that of the plain image as shown in Fig.(9)(a). That means histograms of cipher images contain no statistical resemblance to that of the plain image. So, it is difficult to get any information about the original image with statistical property. 5. Conclusion A novel single-channel color information security system by using LU decomposition in gyrator transform domains is presented. The primary image is decomposed into its red, green and blue channels, modulated by corresponding random phase functions, and then separately Fourier transformed. The transformed red and green channels are multiplied and inverse Fourier transformed. The obtained image is phase- and amplitude truncated to generate a ciphertext and an asymmetric key, respectively. The ciphertext is multiplied by transformed blue channel and subsequently performed LU decomposition. L and U parts are independently gyrator transformed at different transformation angles. The encrypted L and U parts are assigned to two different authorized users as encryption keys. The two transformation angles of GT, three decryption keys for three channels and one asymmetric key considerably enhance the security of the proposed system. The

109

encryption can be implemented digitally or optically. Numerical simulations illustrate the viability and efficiency of the proposed algorithms.

Acknowledgments The author is indebted to Muhammad Waizul Haque and Mehr-un-nisa for their inspiring supports. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

P. Refregier, B. Javidi, Opt. Lett. 20 (1995) 767. G. Unnikrishnan, J. Joseph, K. Singh, Opt. Lett. 25 (2000) 887. B. Hennelly, J.T. Sheridan, Opt. Lett. 28 (2003) 269. G. Situ, J. Zhang, Opt. Lett. 29 (2004) 1584. Z. Liu, S. Liu, Opt. Lett. 32 (2007) 2088. H.-E. Hwang, H.T. Chang, W.-N. Lie, Opt. Express 17 (2009) 13700. A. Alfalou, C. Brosseau, Opt. Lett. 35 (2010) 2185. A. Alfalou, C. Brosseau, N. Abdallah, M. Jridi, Opt. Express 21 (2013) 8025. A. Alfalou, C. Brosseau, Adv. Opt. Photon. 1 (2009) 589. S.Q. Zhang, M.A. Karim, Microw. Opt. Technol. Lett. 21 (1999) 318. L. Chen, D. Zhao, Opt. Express 14 (2006) 8552. M. Joshi, C. Shakher, K. Singh, Opt. Commun. 279 (2007) 35. Z. Liu, J. Dai, X. Sun, S. Liu, Opt. Laser Eng. 48 (2010) 800. Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, S. Liu, Opt. Commun. 284 (2011) 123. W. Chen, X. Chen, C.J.R. Sheppard, Opt. Express 20 (2012) 3853. M.R. Abuturab, Appl. Opt. 51 (2012) 3006. M.R. Abuturab, Opt. Lasers Eng. 50 (2012) 772. M.R. Abuturab, Opt. Lasers Eng. 50 (2012) 1217. M.R. Abuturab, Opt. Lasers Eng. 50 (2012) 1383. M.R. Abuturab, Opt. Lasers Eng. 51 (2013) 317. W. Qin, X. Peng, Opt. Lett. 35 (2010) 118. X. Wang, D. Zhao, Opt. Commun. 285 (2012) 1078. M.R. Abuturab, Appl. Opt. 52 (2013) 1555. M.R. Abuturab, Appl. Opt 51 (2012) 7994. M.R. Abuturab, Appl. Opt. 52 (2013) 5133. N. Zhou, Y. Wang, L. Gong, H. He, J. Wu, Opt. Commun. 284 (2011) 2789. X. Deng, D. Zhao, Opt. Laser Technol. 44 (2012) 136. L. Sui, B. Gao, Opt. Laser Technol. 48 (2013) 117. L. Sui, M. Xin, A. Tian, H. Jin, Opt. Lasers Eng. 51 (2013) 1297. G. Strang, Linear Algebra and its Applications, 3rd ed., Thomson Learning, Inc., USA, 2006. J.A. Rodrigo, T. Alieva, M.L. Calvo, Opt. Express 15 (2007) 2190. J.A. Rodrigo, T. Alieva, M.L. Calvo, J. Opt. Soc. Am. A 24 (2007) 3135. Z. Liu, D. Chen, J. Ma, S. Wei, Y. Zhang, J. Dai, S. Liu, Optik 122 (2011) 864. N. Singh, A. Sinha, Opt. Laser Eng. 47 (2009) 539. Z. Liu, L. Xu, C. Chin, S. Liu, Appl. Opt. 49 (2010) 5632. Z. Liu, L. Xu, C. Chen, J. Dai, S. Liu, Opt. Lasers Eng. 49 (2011) 542. Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, S. Liu, Opt. Commun. 285 (2012) 3921. Z. Liu, S. Li, W. Liu, W. Liu, S. Liu, Opt. Laser Technol. 45 (2013) 198.