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

Muhammad Rafiq AbuturabDepartment of Physics, Maulana Azad College of Engineering and Technology, Patna 801113, India

a r t i c l e i n f o

Article history:Received 20 December 2013Received in revised form27 January 2014Accepted 14 February 2014Available online 6 March 2014

Keywords:LU decompositionAsymmetric color image cryptosystemGyrator transform

a b s t r a c t

A novel single-channel color information security system based on LU decomposition in gyrator transformdomains is proposed. The original color image to be encoded is separated into its red, green and bluechannels. They are modulated by corresponding random phase functions and then independently Fouriertransformed. The transformed images of red and green channels are multiplied and then inverse Fouriertransformed. The resulting image is phase- and amplitude truncated to obtain an encrypted image and anasymmetric decryption key, respectively. The encrypted image is multiplied by transformed image of bluechannel and then performed LU decomposition. Finally, L and U parts are individually gyrator transformedat different transformation angles, which can be assigned to two different authorized users. The proposedsingle-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 forthree channels and one asymmetric decryption key significantly improve the security and robustness ofthe proposed method. The encryption system can be realized digitally or optically. Numerical simulationsdemonstrate the feasibility and effectiveness of the suggested algorithms.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

With the rapid development of modern communication systems,information security has become a serious concern. Optical informa-tion techniques have been widely researched owing to their inherentadvantages of parallelism and high speed transmitting. The double-random phase-encoding (DRPE) is a well known technique which isbased on the 4-f optical correlator to encrypt a primary image intostationary white noise [1]. Since then various optical image encryptionsystems and algorithms have been investigated [2–8]. These encryp-tion techniques can be designed for compression operations simulta-neously in the spectral domain [9]. In all these encryption methods,as a monochromatic light is used to illuminate a real color image, colorinformation of a recovered image is lost. To include color information,a color image encryption method based on an indexed image anddouble phase randommasks has been introduced [10]. In this method,an RGB color image is converted into an indexed image format beforeencoding. During the decryption process, the color image is retrievedby converting the decrypted indexed image back into its RGB format.Many optical techniques and applications for the color image proces-sing have been further developed [11–20].

However, most of the existing methods belong to the sym-metric cryptosystem, in which the encryption and decryption keysare the same. From the cryptography point of view, a symmetric

cryptosystem would suffer from several problems in practicalapplication, particularly under the network environment. In orderto overcome these problems, an asymmetric cryptosystem basedon a phase-truncated Fourier transform (PTFT) approach has beenproposed [21]. In this method, with phase truncation in Fourierdomain, one can produce an asymmetric ciphertext as real-valuedand stationary white noise using two random phase keys as publickeys, while an authorized user can recover the plaintext employ-ing another two different private phase keys in the decryptionprocess. It has been demonstrated that an iterative amplitude-phase retrieval algorithm can decipher the PTFT-based asymmetriccryptosystems, due to the vulnerability of the decryption key pairs[22]. Based on asymmetric cryptosystem, a number of color imageencryption schemes have been proposed [23–25].

The aforementioned encryption techniques are based onmultiple-channel encryption systems. Compared with the multiple-channel encryption methods, single-channel encryption algorithmsare more simple and viable system. Recently, several single-channelimage encryption methods have been reported [26–29].

In this paper, for the first time to my knowledge, a novel single-channel color information security system by using LU decomposi-tion in gyrator transform domains is proposed. In the encryptionprocess, an original color image is segregated into its red, green andblue channels. They are then modulated by corresponding randomphase functions. The modulated channels are individually Fouriertransformed. The transformed red channel is multiplied with

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

Optics Communications

http://dx.doi.org/10.1016/j.optcom.2014.02.0610030-4018 & 2014 Elsevier B.V. All rights reserved.

E-mail address: rafiq.abuturab@gmail.com

Optics Communications 323 (2014) 100–109

transformed green channel. The product image is inverse Fouriertransformed. The Fourier spectrum is phase- and amplitude trun-cated to produce a ciphertext and a decryption key, respectively. Theasymmetric ciphertext is multiplied by transformed blue channel.The obtained image is separated by LU decomposition. Lastly, L and Uparts are independently gyrator transformed at different transforma-tion angles to get encrypted L and U parts. During the decryptionprocess, the encoded images are separately inverse gyrator trans-formed. They are then multiplied to each other and decryption key toreconstruct single-channel image, which is divided by transformedblue channel. The resulting image is Fourier transformed. The Fourierspectrum is multiplied by decryption keys for red and green channelsand the corresponding images are inverse Fourier transformed toretrieve red and green channels. To reconstruct blue channel, therecovered single-channel image is multiplied by decryption key forblue channel and obtained image is inverse Fourier transformed. Thesuggested encryption system can be performed digitally or optically.

The proposed single-channel encryption system is simpler thanthree-channel encryption systems and has three advantages over theearlier techniques [26–29]. First, the LU form is asymmetric. Second,two encoded L and U parts can be allocated to two different allowedusers for highly safe authentication. Third, the transformation angles ofGT are remarkably sensitive keys. Moreover, the ciphertexts are notcolor images but gray images which can bewilder the others.Numerical simulations exemplify the validity and effectiveness of theproposed technique.

2. Principle

2.1. LU decomposition

The LU (lower-and-upper) decomposition is a factorization tech-nique 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.

M.R. Abuturab / Optics Communications 323 (2014) 100–109 101

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

A¼ L� U )

a11 a12 … a1na21 a22 ⋯ a2n⋮ ⋮ ⋱ ⋮an1 an2 … ann

26664

37775¼

1 0 … 0l21 1 ⋯ 0⋮ ⋮ ⋱ ⋮ln1 ln2 … 1

26664

37775

u11 u12 … u1n

0 u22 ⋯ u2n

⋮ ⋮ ⋱ ⋮0 0 … unn

26664

37775

ð1Þwhere L is a row permutation of a lower triangular matrix havingidentity elements on the diagonal and the multipliers below thediagonal. U, on the other hand, is an upper triangular matrix havingsome coefficients on the diagonal and the multipliers above thediagonal such that L� U ¼ A [30]. The LU form is asymmetric in thiscase that L has always 1's on the diagonal, whereas U does not. Thisproperty can be investigated for image encryption.

2.2. Gyrator transform

The gyrator transform (GT) of a two-dimensional complex fieldfunction f iðxi; yiÞ with order α known as transformation angle, isdefined as [31]

f oðxo; yoÞ ¼ Gα½f iðxi; yiÞ�ðxo; yoÞ¼ 1j sin αj

Z Z þ1

�1f iðxi; yiÞ exp i2π

ðxoyoþxiyiÞ cos α�ðxiyoþxoyiÞsin α

� �dxi dyi

ð2Þ

where Gα½ � indicates GT operator. ðxi; yiÞ and ðxo; yoÞ are the inputand output coordinates, respectively. When αA ½0;2π�, the GT canbe implemented in an optical system composed of identical plano-convex cylindrical lenses. The transformation angle α is varied byproper rotation of these lenses [32]. Gα and G2π�α are reciprocaltransforms which can be exploited in optical information proces-sing. In this paper, the calculation of discrete gyrator transformalgorithm is obtained by using a convolution operation [33].

The optical image algorithms based on gyrator transform for colorimage [16–20,23–25] and gray image [34–38] have been reported.

3. Proposed technique

A single-channel color image security system by using LUdecomposition in GT domains is presented. The flow diagrams ofencryption and decryption algorithms have been displayed inFigs. 1(a) and 1(b), respectively.

The encryption process can be performed by the followingsteps.

1. A color image f ðxi; yiÞ to be encoded is separated into its red,green and blue channels f Rðxi; yiÞ, f Gðxi; yiÞ and f Bðxi; yiÞ,

CCD 1

LSLM1

f f

Computer System

SLM 2

L1 L2 L3

z z

CCD 2

SLM 3

L1 L2 L3

z z

CCD 3

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

M.R. Abuturab / Optics Communications 323 (2014) 100–109102

respectively, which is given as

f ðxi; yiÞ ¼ ½f Rðxi; yiÞ; f Gðxi; yiÞ; f Bðxi; yiÞ� ð3Þ

2. The red, green and blue channels are modulated by randomphase functions ϕRðxi; yiÞ, ϕGðxi; yiÞ and ϕBðxi; yiÞ, respectively

and then independently Fourier transformed.

FRðx; yÞ ¼ Fff Rðxi; yiÞ exp½iφRðxi; yiÞ�g ð4Þ

FGðx; yÞ ¼ Fff Gðxi; yiÞ exp½iφGðxi; yiÞ�g ð5Þ

FBðx; yÞ ¼ Fff Bfxi; yig exp½iφBðxi; yiÞ�g 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.

M.R. Abuturab / Optics Communications 323 (2014) 100–109 103

where FRðx; yÞ, FGðx; yÞ, and FBðx; yÞ denote Fourier transformedred, green and blue channels, respectively.

3. The transformed images for red and green channels are multi-plied and then inverse Fourier transformed. The obtainedimage is phase- and amplitude truncated to get first encryptedimage and an asymmetric decryption key, respectively.

E1ðxi; yiÞ ¼ PTfF �1½FRðx; yÞFGðx; yÞ�g ð7Þ

Pðxi; yiÞ ¼ ATfF �1½FRðx; yÞFGðx; yÞ�g ð8Þ

4. The encrypted image is multiplied by transformed image forblue channel and then performed LU decomposition.

E2ðxi; yiÞ ¼ E1ðxi; yiÞFBðx; yÞ ð9Þ

½Lðxi; yiÞ; Uðxi; yiÞ� ¼ lu ½E2ðxi; yiÞ� ð10Þwhere the function lu indicates LU decomposition. Lðxi; yiÞ andUðxi; yiÞ represent decomposed L and U parts.

5. Finally, encrypted L and U parts are individually gyratortransformed.

ELðx; yÞ ¼GαL ½Lðxi; yiÞ� ð11Þ

EUðx; yÞ ¼ GαU ½Uðxi; yiÞ� ð12Þwhere αL and αU are transformation angles of L and U parts,respectively.

The decryption process can be carried out by the following steps.

1. The encoded images are independently inverse gyrator trans-formed and then obtained images are multiplied to each otherand asymmetric decryption key to get single-channel image.

DLðxi; yiÞ ¼ G�αL ½ELðx; yÞ� ð13Þ

DU ðxi; yiÞ ¼ G�αU ½EUðx; yÞ� ð14Þ

Dðxi; yiÞ ¼DLðxi; yiÞDUðxi; yiÞPðxi; yiÞ ð15Þ

2. The reconstructed image Dðxi; yiÞ divided by FBðx; yÞ is Fouriertransformed.

DRGðx; yÞ ¼ FDðxi; yiÞFBðx; yÞ

� �ð16Þ

3. The resulting image DRGðx; yÞ is divided by decryption key forred channel 1=FGðx; yÞ and decryption key for green channel1=FRðx; yÞ. Then corresponding images are inverse Fouriertransformed to retrieve red and green channels.

f Rðx; yÞ ¼ F �1 DRGðx; yÞFGðx; yÞ

� �ð17Þ

f Gðx; yÞ ¼ F �1 DRGðx; yÞFRðx; 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) 100–109104

4. The obtained image Dðxi; yiÞ is multiplied by decryption key forblue channel 1=E1ðxi; yiÞ and then resulting image is inverseFourier transformed to recover blue channel.

FBðxi; yiÞ ¼Dðxi; yiÞE1ðxi; yiÞ

ð19Þ

f Bðxi; yiÞ ¼ F �1½FBðx; yÞ� ð20Þ

The red, green and blue channels are combined to decryptoriginal color image in the computer system.

Fourier transforming can be optically implemented by using asimple convergent lens L of focal length f [9]. The GT can be carriedout in the coherent optical system by using three generalized lenses(designated as L1, L2 and L3) with fixed equal distances z betweenthem. Each generalized lens corresponds to the assembled set oftwo identical plano-convex cylindrical lenses of the same power.The first and third identical lenses of focal length f 1 ¼ z are rotatedto vary the transformation angle α. The second lens L2 of a focallength f 2 ¼ z=2 is fixed [33].

The optoelectronic design for the optical encryption system isshown in Fig. 2. In the encryption process, Eqs. (3)–(6) are encodeddigitally by a computer system. FRðx; yÞ and FGðx; yÞ are multipliedand then displayed on spatial light modulator 1 (SLM 1). In-lineholography with CCD 1, the Fourier spectrum is recorded.The phase part of the spectrum is kept as a decryption key. Theamplitude part multiplied with FBðx; yÞ is decomposed by LU

decomposition digitally by the computer system. The encodedLðxi; yiÞ and Uðxi; yiÞ parts are, respectively, displayed on SLM 2 andSLM 3. They are separately illuminated by uniform plane wavesand then gyrator transformed at two different transformationangles. In-line holography with CCD 2 and CCD 3, the correspond-ing encrypted images are recorded. The SLMs and CCDs arecontrolled by the computer system.

4. Numerical simulation results

Numerical simulations have been performed on a Matlab 7.11.0(R2010b) platform to examine the viability of the proposedalgorithm. The color image comprising 512�512�3 pixels and8 bits, as shown in Fig. 3(a), is used as input image. The transfor-mation angles of the GT for αL and αU are, respectively, 0.101 and0.151. The decryption phase keys for red, green and blue channelsare demonstrated in Figs. 3(b), 3(c) and 3(d), respectively. Theasymmetric decryption key is shown in Fig. 3(e), The encrypted Land U parts, and decrypted image with all correct keys areillustrated in Figs. 3(f), 3(g), and 3(h), respectively.

The decrypted images obtained by using, only decryption keyfor red channel, only decryption key for green channel, onlydecryption key for blue channel, and no decryption keys aredepicted in Figs. 4(a), 4(b), 4(c) and 4(d), respectively. Theretrieved images by changing the transformation angles αL andαU 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 ischanged by 0.00011 while other keys are correct, (c) with reverse image multiplication order of LU, and (d) without asymmetric decryption key.

M.R. Abuturab / Optics Communications 323 (2014) 100–109 105

The reconstructed image with reverse image multiplication orderof LU is demonstrated in Fig. 5(c). The recovered image withoutasymmetric decryption key is illustrated in Fig. 5(d). It is clear thatthe proposed method is very sensitive to its parameters and

original image can only be obtained when all the parameters arecorrect.

A known-plaintext having a size of 512�512�3 pixels isshown Fig. 6(a). The fake decryption keys generated for red, green,

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 forblue 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.

M.R. Abuturab / Optics Communications 323 (2014) 100–109106

and blue channels are displayed in Figs. 6(b), 6(c) and 6(d),respectively. The recovered image by using fake L part but correctU part is illustrated in Fig. 6(e). The retrieved image by using fakeU part but correct L part is demonstrated in Fig. 6(f). Thereconstructed image by using fake U and L parts is depicted inFig. 6(g). The decrypted image by using fake decryption keys isshown in Fig. 6(h). Figs. 6(e)�6(h) are noise-like images, whichprove that the proposed system possesses the ability to resistknown-plaintext attack.

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

MSE¼ 1M � N

∑M

i ¼ 1∑N

j ¼ 1jIoðm;nÞ� Idðm;nÞj2 ð21Þ

where Ioðm;nÞ and Idðm;nÞ denote the original and decryptedimages at pixel position ðm;nÞ, respectively. M�N is the numberof image pixels.

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 with0.1 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 with0.1 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 noisewith 0.1 variance.

M.R. Abuturab / Optics Communications 323 (2014) 100–109 107

The MSE values of red, green and blue channels for encryptedimages of L and U parts with all accurate parameters are,respectively, ð2:0810� 104; 1:2867� 104; 1:1618� 104Þ andð6:0777� 1024; 6:0777� 1024; 6:0777� 1024Þ. These MSE valuesindicate that the primary image is fully veiled in the ciphertext asshown in Fig. 3(f) and (g). The MSE values of red, green and bluechannels for decrypted images with no decryption keys are,respectively, 7:0748� 1013; 7:0749� 1013, and 8:2303� 1017 asillustrated in Fig. 4(d). Thus decryption keys provide higher level ofsecurity.

In order to evaluate the sensitivity on transformation anglesαL and αU, their correct values are shifted by 0:0001 3 . Thecorresponding MSE values of red, green and blue channels areð1:7116� 106; 1:1648� 106; 9:5680� 104Þ, and ð2:3735�106; 9:5948� 105; 8:5832� 104Þ as displayed in Fig. 5(a) and(b). The MSE values of red, green and blue channels with reversedimage multiplication order of L and U parts, and without asym-metric decryption key are ð4:7919� 109; 2:6590� 109; 3:1989�108Þ and ð5:2448� 104; 3:0977� 104; 5:1024� 103Þ, respectivelyas demonstrated in Fig. 5(c) and (d). These values clearly indicatethat original image cannot be retrieved, with shifted transforma-tion angles by 0:0001 3 , with reversed image multiplication orderof L and U parts, or without asymmetric decryption key.

If an intruder generates fake keys by using, fake L part but correctU part, fake U part but correct L part, and fake decryption keys withcorrect parameters by employing known-plaintext and use toretrieve the original image. The corresponding MSE values of red,green and blue channels are ð3:5011� 109 2:2578� 109;

1:5243� 108Þ, ð3:1267� 1010; 1:4328� 1010; 1:5934� 109Þ and

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

0.599 0.5995 0.6 0.6005 0.601 0.60150

0.5

1

1.5

2

2.5 x 108

Transformation angle for L part of LU decomposition

MSE

RedGreenBlue

0.599 0.5995 0.6 0.6005 0.601 0.60150

0.5

1

1.5

2

2.5

3 x 108

Transformation angle for U part of LU decomposition

MSE

RedGreenBlue

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

0 100 200 300 400 500 6000

500

1000

1500

2000

2500

3000

3500

0 100 200 300 400 500 6000

2

4

6

8

10

12

14x 10

4

0 100 200 300 400 500 6000

2

4

6

8

10

12

14x 10

4

Fig. 9. Histogram analysis: (a) histogram of an original color image as shown inFig. 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).

M.R. Abuturab / Optics Communications 323 (2014) 100–109108

The noise interferences to the encrypted images by Gaussiannoise with 0.1 variance and speckle noise with 0.1 variance havealso been studied. The Gaussian noised images are shown in Fig. 7(a) and (b). The recovered image is depicted in Fig. 7(c). Thespeckle noised images are demonstrated in Fig. 7(d) and (e). Thereconstructed image is illustrated in Fig. 7(f).

As can be seen from Figs. 7(c) and (f), the decrypted images canbe recognized without difficulty.

The influence of transformation angles αL and αU has been studiedwith sampling interval and sampling length ð0:5990; 0:6010Þ and0:0001, respectively. The corresponding MSE values between originaland decrypted red, green and blue channels versus variation intransformation angle are shown in Fig. (8)(a) and (b). In both cases,MSE value becomes very high for a very small change in thetransformation angle. Therefore, the angle parameters αL and αU canbe used as strong keys in the proposed system.

The statistical analysis has been conducted on the proposedsecurity system. As can seen from Fig. (9)(b) and (c), the histo-grams of the encrypted L and U parts approximated by uniformdistribution are quite different from that of the plain image asshown in Fig.(9)(a). That means histograms of cipher imagescontain no statistical resemblance to that of the plain image. So,it is difficult to get any information about the original image withstatistical property.

5. Conclusion

A novel single-channel color information security system byusing LU decomposition in gyrator transform domains is pre-sented. The primary image is decomposed into its red, green andblue channels, modulated by corresponding random phase func-tions, and then separately Fourier transformed. The transformedred and green channels are multiplied and inverse Fourier trans-formed. The obtained image is phase- and amplitude truncated togenerate a ciphertext and an asymmetric key, respectively. Theciphertext is multiplied by transformed blue channel and sub-sequently performed LU decomposition. L and U parts are indepen-dently gyrator transformed at different transformation angles. Theencrypted L and U parts are assigned to two different authorizedusers as encryption keys. The two transformation angles of GT,three decryption keys for three channels and one asymmetric keyconsiderably enhance the security of the proposed system. The

encryption can be implemented digitally or optically. Numericalsimulations illustrate the viability and efficiency of the proposedalgorithms.

Acknowledgments

The author is indebted to Muhammad Waizul Haque andMehr-un-nisa for their inspiring supports.

References

[1] P. Refregier, B. Javidi, Opt. Lett. 20 (1995) 767.[2] G. Unnikrishnan, J. Joseph, K. Singh, Opt. Lett. 25 (2000) 887.[3] B. Hennelly, J.T. Sheridan, Opt. Lett. 28 (2003) 269.[4] G. Situ, J. Zhang, Opt. Lett. 29 (2004) 1584.[5] Z. Liu, S. Liu, Opt. Lett. 32 (2007) 2088.[6] H.-E. Hwang, H.T. Chang, W.-N. Lie, Opt. Express 17 (2009) 13700.[7] A. Alfalou, C. Brosseau, Opt. Lett. 35 (2010) 2185.[8] A. Alfalou, C. Brosseau, N. Abdallah, M. Jridi, Opt. Express 21 (2013) 8025.[9] A. Alfalou, C. Brosseau, Adv. Opt. Photon. 1 (2009) 589.[10] S.Q. Zhang, M.A. Karim, Microw. Opt. Technol. Lett. 21 (1999) 318.[11] L. Chen, D. Zhao, Opt. Express 14 (2006) 8552.[12] M. Joshi, C. Shakher, K. Singh, Opt. Commun. 279 (2007) 35.[13] Z. Liu, J. Dai, X. Sun, S. Liu, Opt. Laser Eng. 48 (2010) 800.[14] Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, S. Liu, Opt. Commun. 284 (2011) 123.[15] W. Chen, X. Chen, C.J.R. Sheppard, Opt. Express 20 (2012) 3853.[16] M.R. Abuturab, Appl. Opt. 51 (2012) 3006.[17] M.R. Abuturab, Opt. Lasers Eng. 50 (2012) 772.[18] M.R. Abuturab, Opt. Lasers Eng. 50 (2012) 1217.[19] M.R. Abuturab, Opt. Lasers Eng. 50 (2012) 1383.[20] M.R. Abuturab, Opt. Lasers Eng. 51 (2013) 317.[21] W. Qin, X. Peng, Opt. Lett. 35 (2010) 118.[22] X. Wang, D. Zhao, Opt. Commun. 285 (2012) 1078.[23] M.R. Abuturab, Appl. Opt. 52 (2013) 1555.[24] M.R. Abuturab, Appl. Opt 51 (2012) 7994.[25] M.R. Abuturab, Appl. Opt. 52 (2013) 5133.[26] N. Zhou, Y. Wang, L. Gong, H. He, J. Wu, Opt. Commun. 284 (2011) 2789.[27] X. Deng, D. Zhao, Opt. Laser Technol. 44 (2012) 136.[28] L. Sui, B. Gao, Opt. Laser Technol. 48 (2013) 117.[29] L. Sui, M. Xin, A. Tian, H. Jin, Opt. Lasers Eng. 51 (2013) 1297.[30] G. Strang, Linear Algebra and its Applications, 3rd ed., Thomson Learning, Inc.,

USA, 2006.[31] J.A. Rodrigo, T. Alieva, M.L. Calvo, Opt. Express 15 (2007) 2190.[32] J.A. Rodrigo, T. Alieva, M.L. Calvo, J. Opt. Soc. Am. A 24 (2007) 3135.[33] Z. Liu, D. Chen, J. Ma, S. Wei, Y. Zhang, J. Dai, S. Liu, Optik 122 (2011) 864.[34] N. Singh, A. Sinha, Opt. Laser Eng. 47 (2009) 539.[35] Z. Liu, L. Xu, C. Chin, S. Liu, Appl. Opt. 49 (2010) 5632.[36] Z. Liu, L. Xu, C. Chen, J. Dai, S. Liu, Opt. Lasers Eng. 49 (2011) 542.[37] Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, S. Liu, Opt. Commun. 285 (2012)

3921.[38] Z. Liu, S. Li, W. Liu, W. Liu, S. Liu, Opt. Laser Technol. 45 (2013) 198.

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