- Email: [email protected]
The fractional Fourier transform of Airy beams using Lohmann and quadratic optical systems
Optics & Laser Technology 44 (2012) 1463–1467
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
The fractional Fourier transform of Airy beams using Lohmann and quadratic optical systems Dahai Han n, Chuntang Liu, Xinyuan Lai State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, P.O. Box 128, #10 Xitucheng Road, Haidian District, Beijing 100876, China
a r t i c l e i n f o
abstract
Article history: Received 18 November 2011 Received in revised form 8 December 2011 Accepted 8 December 2011 Available online 30 December 2011
Based on the generalized Huygens–Fresnel integral, the analytical expressions for Airy beams through three types of fractional Fourier transform (FRFT) optical systems (Lohmann I, Lohmann II and quadratic graded-index systems) are derived. Using the two analytical expressions, the propagation properties of Airy beams for the three types of FRFT optical systems are discussed in detail with numerical examples. Results indicate that the intensity of Airy beams periodically changes and remains form-invariant with the changing of fractional order p for the three types of FRFT optical systems. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Airy beams Fractional Fourier transform
1. Introduction Diffraction-free beams have generated considerable interests for their unique propagation properties. The most well known propagation-invariant beam, bessel beam, was first predicted theoretically [1] and observed experimentally [2] by Durnin et al. More recently, another kind of diffraction-free beams, whose lateral profile is governed by Airy function, has been observed within the framework of optics [3]. Diffraction-free Airy beams exhibit another unusual feature, which they tend to freely accelerate during propagation even in the absence of any external force [3]. Then, the acceleration dynamics of finite energy Airy beams was investigated in detail [4,5], which indicated the local acceleration dynamics of Airy beam relating to its nonlinear lateral shift [5]. The self-healing properties of Airy beams were investigated by Broky et al. [6]. Brandres and Gutie´rrez-Vega studied the propagation of generalized Airy-Gauss beams through ABCD system [7]. The propagation properties of spatially partial coherent Airy beam were first investigated by Morris et al. [8]. Using the vectorial Rayleigh–Sommerfeld diffraction integral, the nonparaxial propagation properties of Airy and SAiry beams were analyzed [9]. Deng and Guo presented a novel family of paraxial laser beams, i.e., Airy complex variable function Gausian beams [10]. The unique propagation properties of Airy generated considerable interest in further exploring the peculiarities of Airy beams in other media [11,12].
n
Corresponding author. E-mail address: [email protected] (D. Han).
0030-3992/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2011.12.017
On the other hand, the fractional Fourier transform (FRFT), as the generalization of a conventional Fourier transform, has been widely studied and advances into application areas, such as signal processing, beam shaping and optical security [13] since it was introduced into optics [14–16]. Since then, the fractional Fourier transform properties of all kinds of laser beams have been investigated with considerable interest [17–23]. The FRFT of partially coherent beams has been studied based on the mutual intensity function [24], the Wigner distribution function [25] and tensor method [26]. Cai et al. investigated the FRFT of partially coherent and partially polarized Gaussian Schell model beams using the tensor method [27]. The FRFT of pulses was investigated by Dragoman et al. [28]. The experimental results indicated that the source coherence of the partially coherent beam has influence on the intensity of Gaussian Schell model beams in the FRF plane [29,30]. However, the FRFT of Airy beams has not been investigated, which is the aim of this paper. Based on the generalized Huygens–Fresnel integral, an analytical expression for FRFT of Airy beams is obtained, and its properties are illustrated by numerical examples.
2. Theoretical model The three kinds of optical system for performing the FRFT are shown in Fig. 1. Fig. 1(a)–(c) denotes the Lohmann I system, Lohmann II system and GRIN system, respectively. f is the standard focal length, z is the axial distance along the optic axis, f ¼ pp=2 and p is the FRFT order. As known, the transfer matrices of the Lohmann I optical system and Lohmann II optical system are equivalent and can be described by the following transfer
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D. Han et al. / Optics & Laser Technology 44 (2012) 1463–1467
Fig. 1. Three kinds of optical system for performing the FRFT (a) Lohmann I system, (b) Lohmann II system and (c) GRIN system.
the Airy beam. Fig. 2 shows the intensity distribution of Airy beams as a function of x for different x0. After some calculations, one obtains an analytical expression for an Airy beams at the output plane [8]: ! ! 2 1 x B2 a0 B kCx EðxÞ ¼ pffiffiffi Ai 2 2 þi exp i 2 Ax0 4A k x4 2A A Akx0 0 ! 2 3 2 a0 B a0 x a0 B B Bx 2 2 i þ i þ i exp : ð5Þ 3 2 3 Ax0 2A k x4 2A2 kx 12A3 k x6 2Akx
0.6 a0=0.1x 0=0.01 a0=0.1x 0=0.015 a0=0.1x 0=0.02
Average Intensity
0.4
0.2
0
0
-0.2
-0.4 -0.2
0
0.2
x (m) Fig. 2. The intensity distribution of Airy beams as a function of x.
matrix [21]: 0 cos f A B B ¼@ 1 sin f C D f
f sin f cos f
1 C A:
ð1Þ
The transfer matrix for the GRIN system, with quadratic index variation nðrÞ ¼ n0 ð1r 2 =2a2 Þ, can be written as [23,32] 0 1 cosðz=aÞ a sinðz=aÞ A B A: ð2Þ ¼@ 1 sinðz=aÞ cosðz=aÞ C D a The term a denotes the radius of the GRIN medium. The propagation of laser beams passing through a paraxial ABCD system obeys the generalized Huygens–Fresnel integral of the form: rffiffiffiffiffiffiffiffiffi Z ik ik 0 Eðx1 Þ ¼ ðAx0 22x1 x0 þDx21 Þ Eðx0 Þ dx , exp ð3Þ 2pB 2B where Eðx0 Þ and Eðx1 Þ are the fields in the source and final transverse planes, respectively. Let us consider an Airy beams represented by [11] 0 x a0 x0 Eðx0 Þ ¼ Ai exp , ð4Þ x0 x0 where Ai denotes the Airy function. For real values of x, the Airy R1 function is defined by the integral: AiðxÞ ¼ ð1=pÞ 0 cos 13 t 3 þ xt dt [31]. The terms x0 and a0 are the characteristic length and aperture coefficient, respectively. And a0 ¼ w20 =4x20 with w0 being the waist of the Gaussian laser beam and x0 being the length of
0
0
0
For Lohmann I and Lohmann II systems, the output field is given by " # 1 x ðf sin fÞ2 a0 f sin f EðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi Ai þi 2 x0 cos f 4 cos2 fk2 x4 cos f cos fkx0 0 " # " 2 kx sin f a0 x a0 ðf sin fÞ2 exp i exp x0 cos f 2 cos f2 k2 x4 2f cos f 0 # a20 f sin f ðf sin fÞ3 fx sin f þ i þ i i : ð6Þ 3 2 3 12 cos3 fk x60 2 cos fkx0 2 cos2 fkx0 For GRIN system, the output field is given by ( ) 1 x ½a sinðz=aÞ2 a0 a sinðz=aÞ þi EðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ai 2 cosðz=aÞx0 4 cosðz=aÞ2 k2 x4 cosðz=aÞ cosðz=aÞkx0 0 ( k sinðz=aÞx2 a0 x a0 ½a sinðz=aÞ2 exp exp i cosðz=aÞx0 2 cosðz=aÞ2 k2 x4 2a cosðz=aÞ 0 ) a20 a sinðz=aÞ ½a sinðz=aÞ3 a sinðz=aÞx þ i þ i i : 3 2 3 2 cosðz=aÞ2 kx0 12 cosðz=aÞ3 k x60 2 cosðz=aÞkx0 ð7Þ
3. Numerical calculations and analysis In the following, the properties of Airy beams in the FRFT plane are investigated using the formula derived in Section 2. Fig. 3 plots the intensity distribution of Airy beams in the FRFT planes with different fractional orders for Lohmann I and Lohmann II optical systems. The calculation parameters are f ¼ 1 m, l ¼ 1064 nm, a0 ¼ 0:1, x0 ¼ 0:012 m. As indicated by Fig. 3, the intensity of Airy beams in the FRFT plane varies as the fractional order p varies. However, the profile of the intensity of Airy beams in FRFT plane does not change with the changing of fractional order p which can be seen from Eq. (5). Comparing Eq. (5) with Eq. (4), the Airy beams remain Airy function form when propagating through ABCD optical system. Therefore, Airy beams remain form-invariant under paraxial transformations [7], which are not valid for nonparaxial cases [9]. As can be seen from Fig. 3(a) and (d), the intensity distributions of Airy beams in FRFT plane for p ¼ 0:4 and p ¼ 1:6 are symmetric with x ¼ 0. Also, the intensity distributions of Airy beams in FRFT plane for p ¼ 0:8 and p ¼ 1:2 are symmetric with x ¼ 0. As shown by Fig. 3(d) and (e),
D. Han et al. / Optics & Laser Technology 44 (2012) 1463–1467
0.3
1465
0.8 p=0.4
Intensity
Intensity
P=0.8
0
0 -0.2
0
0.2
-0.2
x(m)
0
0.2
x(m)
0.8
0.3 p=1.2
Intensity
Intensity
p=1.6
0 -0.2
0 x(m)
0 -0.2
0.2
0.3
0 x(m)
0.2
0.8 p=2.8
Intensity
Intensity
p=2.4
0
0 -0.2
0 x(m)
0.2
-0.2
0.8
0 x(m)
0.2
0.3 p=3.2
Intensity
Intensity
p=3.6
0
0 -0.2
0 x(m)
0.2
-0.2
0 x(m)
0.2
Fig. 3. The intensity distribution of Airy beams in the FRFT planes with different fractional orders for Lohmann I and Lohmann II optical systems.
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D. Han et al. / Optics & Laser Technology 44 (2012) 1463–1467
the intensity distributions of Airy beams in FRFT plane for p ¼ 1:6 are the same as that of p ¼ 2:4. As a whole, the intensity distribution of Airy beams in FRFT plane periodically varies when 0.3
Intensity
a0=0.1 a0=0.2 a0=0.3
the fractional order p varies with form-invariant. The intensity distributions of Airy beams in the FRFT planes for different a0 with p ¼ 2:4 (Lohmann I and Lohmann II) are depicted in Fig. 4. The other calculation parameters are the same as those of Fig. 3. As shown by Fig. 4, the intensity distributions of Airy beams in FRFT plane still remain Airy-form and weaken with the uniform extend as the parameter a0 increases. In the following, we discuss the propagation properties of Airy beams in the GRIN medium. Fig. 5 plots the intensity distribution of Airy beams in the FRFT planes with different fractional order in the GRIN medium. The calculation parameters are a ¼ 1, l ¼ 1064 nm, a0 ¼ 0:1 and x0 ¼ 0:02 m. The intensity distribution of Airy beams periodically changes and remains form-invariant with the propagation distance z increasing. This can be explained as follows: Eqs. (6) and (7) are completely equivalent if setting a ¼ f and f ¼ z=a. Therefore, the fractional order p in GRIN medium is equal to 2z=ðpaÞ [23]. The changing of propagation distance z is equivalent to the changing of fractional order p.
4. Conclusions 0 -0.2
0
0.2
x (m) Fig. 4. The intensity distribution of Airy beams as a function of x for different a0 with p ¼ 2:4 (Lohmann I and Lohmann II).
In conclusion, the propagation properties of Airy beams through three types of FRFT optical systems have been investigated. The effect of a0 and fractional order p on the propagation characteristics of Airy beams through FRFT optical systems is discussed in detail. Results show that the intensity of Airy beams periodically changes and remains form-invariant with the
0.4
0.4
z=2.198
Intensity
Intensity
z=0.942
0
0 -0.2
0
0.2
-0.2
x(m)
0
0.4
0.4 z=4.082
Intensity
z=5.338
Intensity 0 -0.2
0.2
x(m)
0 x(m)
0.2
0.2
0 -0.2
0 x(m)
Fig. 5. The intensity distribution of Airy beams in the FRFT planes with different fractional orders in the GRIN medium.
0.2
D. Han et al. / Optics & Laser Technology 44 (2012) 1463–1467
increasing of fractional order p through three types of FRFT optical systems. The intensity of Airy beams is weakened uniform with the increasing of x0. Acknowledgements This work has been supported in part by 973 Program (2010CB328204), NSFC project (61101110 and 61008049), RFDP project (20100005120015) of China, and Youth Research and Innovation Program of BUPT (no. 2011RC0308). References [1] Durnin J. Exact solutions for nondiffracting beams. I. The scalar theory. Journal of the Optical Society of America A 1987;4:651. [2] Durnin J, Miceli Jr JJ, Eberly JH. Diffraction-free beams. Physical Review Letters 1987;58:1499. [3] Siviloglu GA, Broky J, Dogariu A, Christodoulides DN. Observation of accelerating Airy beams. Physical Review Letters 2007;99:213901. [4] Siviloglou GA, Christodoulides DN. Accelerating finite energy Airy beams. Optics Letters 2007;32:979. [5] Besieris IM, Shaarawi AM. A note on an accelerating finite energy Airy beam. Optics Letters 2007;32:2477. [6] Broky J, Siviloglou GA, Dogariu A, Christodoulides DN. Self-healing properties of optical Airy beams. Optics Express 2008;16:12880. [7] Bandres MA, Gutie´rrez-Vega JC. Airy-Gauss beams and their transformation by paraxial optical systems. Optics Express 2007;15:16719. ˘ iz˘ma´r T, Dholakia K. Propagation [8] Morris JE, Mazilu M, Baumgartl J, C characteristics of Airy beams: dependence upon spatial coherence and wavelength. Optics Express 2009;17:13236. [9] Carretero L, Acebal P, Blaya S, Garcia C, Fimia A, Madrigal R, et al. Nonparaxial diffraction analysis of Airy and SAiry beams. Optics Express 2009;19:22432. [10] Deng D, Guo Q. Airy complex variable function Gaussian beams. New Journal of Physics 2009;11:103029. [11] Gu Y, Gbur G. Scintillation of Airy beam arrays in atmospheric turbulence. Optics Letters 2010;35:3456. [12] Polynkin P, Kolesik M, Moloney J. Filamentation of femtosecond laser Airy beams in water. Physical Review Letters 2009;103:123902. [13] Torre A. The fractional Fourier transform and some of its applications to optics. Progress in Optics 2002;43:531.
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[14] Mendlovic D, Ozaktas HM. Fractional Fourier transformations and their optical implementation. I. Journal of the Optical Society of America A 1993;10:1875. [15] Ozaktas HM, Mendlovic D. Fractional Fourier transforms and their optical implementation. II. Journal of the Optical Society of America A 1993;10:2522. [16] Lohmann AW. Image rotation, Wigner rotation, and the fractional Fourier transform. Journal of the Optical Society of America A 1993;10:2181. [17] Cai Y, Lin Q. Properties of flattened Gaussian beam in the fractional Fourier transform plane. Journal of Optics A: Pure and Applied Optics 2003;5:272. [18] Zhao D, Mao H, Liu H, Wang S, Jing F, Wei X. Propagation of Hermite-coshGaussian beams in apertured fractional Fourier transforming systems. Optics Communications 2004;236:225. [19] Du X, Zhao D. Fractional Fourier transform of truncated elliptical Gaussian beams. Applied Optics 2006;45:9049. [20] Zheng C. Fractional Fourier transform for a hollow Gaussian beam. Physics Letters A 2006;355:156. [21] Zhou G. Fractional Fourier transform of Lorentz–Gauss beams. Journal of the Optical Society of America A 2009;26:350. [22] Zhou G. Fractional Fourier transform of a higher-order cosh-Gaussian beam. Journal of Modern Optics 2009;56:886. [23] Gao Y, Zhu B, Liu D, Lin Z. Fractional Fourier transform of flat-topped multiGaussian beams. Journal of the Optical Society of America A 2010;27:358. [24] Erden MF, Ozaktas HM, Mendlovic D. Synthesis of mutual intensity distribution using the fractional Fourier transform. Optics Communications 1996;125:288. [25] Yoshimura H, Iwai T. Properties of the Gaussian Schell-model source field in a fractional Fourier plane. Journal of the Optical Society of America A 1997;14:3388. [26] Lin Q, Cai Y. Fractional Fourier transform for partially coherent Gaussian Schell-model beams. Optics Letters 2002;27:1672. [27] Cai Y, Ge D, Lin Q. Fractional Fourier transform for partially coherent and partially polarized Gaussian-Schell model beams. Journal of Optics A 2003;5:453. [28] Dragoman D, Dragoman M, Brenner KH. Variant fractional Fourier transformer for optical pulses. Optics Letters 1999;24:933. [29] Wang F, Cai Y. Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics. Journal of the Optical Society of America A 2007;24:1937. [30] Wang F, Cai Y, Lin Q. Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam. Journal of the Optical Society of America A 2008;25:2001. [31] Valle´e O, Soares M. Airy functions and applications to physics. London: Imperial College Press; 2004. [32] McMullin JN. The ABCD matrix in arbitrarily tapered quadratic-index waveguides. Applied Optics 1986;25:2184.
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
The fractional Fourier transform of Airy beams using Lohmann and quadratic optical systems Dahai Han n, Chuntang Liu, Xinyuan Lai State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, P.O. Box 128, #10 Xitucheng Road, Haidian District, Beijing 100876, China
a r t i c l e i n f o
abstract
Article history: Received 18 November 2011 Received in revised form 8 December 2011 Accepted 8 December 2011 Available online 30 December 2011
Based on the generalized Huygens–Fresnel integral, the analytical expressions for Airy beams through three types of fractional Fourier transform (FRFT) optical systems (Lohmann I, Lohmann II and quadratic graded-index systems) are derived. Using the two analytical expressions, the propagation properties of Airy beams for the three types of FRFT optical systems are discussed in detail with numerical examples. Results indicate that the intensity of Airy beams periodically changes and remains form-invariant with the changing of fractional order p for the three types of FRFT optical systems. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Airy beams Fractional Fourier transform
1. Introduction Diffraction-free beams have generated considerable interests for their unique propagation properties. The most well known propagation-invariant beam, bessel beam, was first predicted theoretically [1] and observed experimentally [2] by Durnin et al. More recently, another kind of diffraction-free beams, whose lateral profile is governed by Airy function, has been observed within the framework of optics [3]. Diffraction-free Airy beams exhibit another unusual feature, which they tend to freely accelerate during propagation even in the absence of any external force [3]. Then, the acceleration dynamics of finite energy Airy beams was investigated in detail [4,5], which indicated the local acceleration dynamics of Airy beam relating to its nonlinear lateral shift [5]. The self-healing properties of Airy beams were investigated by Broky et al. [6]. Brandres and Gutie´rrez-Vega studied the propagation of generalized Airy-Gauss beams through ABCD system [7]. The propagation properties of spatially partial coherent Airy beam were first investigated by Morris et al. [8]. Using the vectorial Rayleigh–Sommerfeld diffraction integral, the nonparaxial propagation properties of Airy and SAiry beams were analyzed [9]. Deng and Guo presented a novel family of paraxial laser beams, i.e., Airy complex variable function Gausian beams [10]. The unique propagation properties of Airy generated considerable interest in further exploring the peculiarities of Airy beams in other media [11,12].
n
Corresponding author. E-mail address: [email protected] (D. Han).
0030-3992/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2011.12.017
On the other hand, the fractional Fourier transform (FRFT), as the generalization of a conventional Fourier transform, has been widely studied and advances into application areas, such as signal processing, beam shaping and optical security [13] since it was introduced into optics [14–16]. Since then, the fractional Fourier transform properties of all kinds of laser beams have been investigated with considerable interest [17–23]. The FRFT of partially coherent beams has been studied based on the mutual intensity function [24], the Wigner distribution function [25] and tensor method [26]. Cai et al. investigated the FRFT of partially coherent and partially polarized Gaussian Schell model beams using the tensor method [27]. The FRFT of pulses was investigated by Dragoman et al. [28]. The experimental results indicated that the source coherence of the partially coherent beam has influence on the intensity of Gaussian Schell model beams in the FRF plane [29,30]. However, the FRFT of Airy beams has not been investigated, which is the aim of this paper. Based on the generalized Huygens–Fresnel integral, an analytical expression for FRFT of Airy beams is obtained, and its properties are illustrated by numerical examples.
2. Theoretical model The three kinds of optical system for performing the FRFT are shown in Fig. 1. Fig. 1(a)–(c) denotes the Lohmann I system, Lohmann II system and GRIN system, respectively. f is the standard focal length, z is the axial distance along the optic axis, f ¼ pp=2 and p is the FRFT order. As known, the transfer matrices of the Lohmann I optical system and Lohmann II optical system are equivalent and can be described by the following transfer
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D. Han et al. / Optics & Laser Technology 44 (2012) 1463–1467
Fig. 1. Three kinds of optical system for performing the FRFT (a) Lohmann I system, (b) Lohmann II system and (c) GRIN system.
the Airy beam. Fig. 2 shows the intensity distribution of Airy beams as a function of x for different x0. After some calculations, one obtains an analytical expression for an Airy beams at the output plane [8]: ! ! 2 1 x B2 a0 B kCx EðxÞ ¼ pffiffiffi Ai 2 2 þi exp i 2 Ax0 4A k x4 2A A Akx0 0 ! 2 3 2 a0 B a0 x a0 B B Bx 2 2 i þ i þ i exp : ð5Þ 3 2 3 Ax0 2A k x4 2A2 kx 12A3 k x6 2Akx
0.6 a0=0.1x 0=0.01 a0=0.1x 0=0.015 a0=0.1x 0=0.02
Average Intensity
0.4
0.2
0
0
-0.2
-0.4 -0.2
0
0.2
x (m) Fig. 2. The intensity distribution of Airy beams as a function of x.
matrix [21]: 0 cos f A B B ¼@ 1 sin f C D f
f sin f cos f
1 C A:
ð1Þ
The transfer matrix for the GRIN system, with quadratic index variation nðrÞ ¼ n0 ð1r 2 =2a2 Þ, can be written as [23,32] 0 1 cosðz=aÞ a sinðz=aÞ A B A: ð2Þ ¼@ 1 sinðz=aÞ cosðz=aÞ C D a The term a denotes the radius of the GRIN medium. The propagation of laser beams passing through a paraxial ABCD system obeys the generalized Huygens–Fresnel integral of the form: rffiffiffiffiffiffiffiffiffi Z ik ik 0 Eðx1 Þ ¼ ðAx0 22x1 x0 þDx21 Þ Eðx0 Þ dx , exp ð3Þ 2pB 2B where Eðx0 Þ and Eðx1 Þ are the fields in the source and final transverse planes, respectively. Let us consider an Airy beams represented by [11] 0 x a0 x0 Eðx0 Þ ¼ Ai exp , ð4Þ x0 x0 where Ai denotes the Airy function. For real values of x, the Airy R1 function is defined by the integral: AiðxÞ ¼ ð1=pÞ 0 cos 13 t 3 þ xt dt [31]. The terms x0 and a0 are the characteristic length and aperture coefficient, respectively. And a0 ¼ w20 =4x20 with w0 being the waist of the Gaussian laser beam and x0 being the length of
0
0
0
For Lohmann I and Lohmann II systems, the output field is given by " # 1 x ðf sin fÞ2 a0 f sin f EðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi Ai þi 2 x0 cos f 4 cos2 fk2 x4 cos f cos fkx0 0 " # " 2 kx sin f a0 x a0 ðf sin fÞ2 exp i exp x0 cos f 2 cos f2 k2 x4 2f cos f 0 # a20 f sin f ðf sin fÞ3 fx sin f þ i þ i i : ð6Þ 3 2 3 12 cos3 fk x60 2 cos fkx0 2 cos2 fkx0 For GRIN system, the output field is given by ( ) 1 x ½a sinðz=aÞ2 a0 a sinðz=aÞ þi EðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ai 2 cosðz=aÞx0 4 cosðz=aÞ2 k2 x4 cosðz=aÞ cosðz=aÞkx0 0 ( k sinðz=aÞx2 a0 x a0 ½a sinðz=aÞ2 exp exp i cosðz=aÞx0 2 cosðz=aÞ2 k2 x4 2a cosðz=aÞ 0 ) a20 a sinðz=aÞ ½a sinðz=aÞ3 a sinðz=aÞx þ i þ i i : 3 2 3 2 cosðz=aÞ2 kx0 12 cosðz=aÞ3 k x60 2 cosðz=aÞkx0 ð7Þ
3. Numerical calculations and analysis In the following, the properties of Airy beams in the FRFT plane are investigated using the formula derived in Section 2. Fig. 3 plots the intensity distribution of Airy beams in the FRFT planes with different fractional orders for Lohmann I and Lohmann II optical systems. The calculation parameters are f ¼ 1 m, l ¼ 1064 nm, a0 ¼ 0:1, x0 ¼ 0:012 m. As indicated by Fig. 3, the intensity of Airy beams in the FRFT plane varies as the fractional order p varies. However, the profile of the intensity of Airy beams in FRFT plane does not change with the changing of fractional order p which can be seen from Eq. (5). Comparing Eq. (5) with Eq. (4), the Airy beams remain Airy function form when propagating through ABCD optical system. Therefore, Airy beams remain form-invariant under paraxial transformations [7], which are not valid for nonparaxial cases [9]. As can be seen from Fig. 3(a) and (d), the intensity distributions of Airy beams in FRFT plane for p ¼ 0:4 and p ¼ 1:6 are symmetric with x ¼ 0. Also, the intensity distributions of Airy beams in FRFT plane for p ¼ 0:8 and p ¼ 1:2 are symmetric with x ¼ 0. As shown by Fig. 3(d) and (e),
D. Han et al. / Optics & Laser Technology 44 (2012) 1463–1467
0.3
1465
0.8 p=0.4
Intensity
Intensity
P=0.8
0
0 -0.2
0
0.2
-0.2
x(m)
0
0.2
x(m)
0.8
0.3 p=1.2
Intensity
Intensity
p=1.6
0 -0.2
0 x(m)
0 -0.2
0.2
0.3
0 x(m)
0.2
0.8 p=2.8
Intensity
Intensity
p=2.4
0
0 -0.2
0 x(m)
0.2
-0.2
0.8
0 x(m)
0.2
0.3 p=3.2
Intensity
Intensity
p=3.6
0
0 -0.2
0 x(m)
0.2
-0.2
0 x(m)
0.2
Fig. 3. The intensity distribution of Airy beams in the FRFT planes with different fractional orders for Lohmann I and Lohmann II optical systems.
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the intensity distributions of Airy beams in FRFT plane for p ¼ 1:6 are the same as that of p ¼ 2:4. As a whole, the intensity distribution of Airy beams in FRFT plane periodically varies when 0.3
Intensity
a0=0.1 a0=0.2 a0=0.3
the fractional order p varies with form-invariant. The intensity distributions of Airy beams in the FRFT planes for different a0 with p ¼ 2:4 (Lohmann I and Lohmann II) are depicted in Fig. 4. The other calculation parameters are the same as those of Fig. 3. As shown by Fig. 4, the intensity distributions of Airy beams in FRFT plane still remain Airy-form and weaken with the uniform extend as the parameter a0 increases. In the following, we discuss the propagation properties of Airy beams in the GRIN medium. Fig. 5 plots the intensity distribution of Airy beams in the FRFT planes with different fractional order in the GRIN medium. The calculation parameters are a ¼ 1, l ¼ 1064 nm, a0 ¼ 0:1 and x0 ¼ 0:02 m. The intensity distribution of Airy beams periodically changes and remains form-invariant with the propagation distance z increasing. This can be explained as follows: Eqs. (6) and (7) are completely equivalent if setting a ¼ f and f ¼ z=a. Therefore, the fractional order p in GRIN medium is equal to 2z=ðpaÞ [23]. The changing of propagation distance z is equivalent to the changing of fractional order p.
4. Conclusions 0 -0.2
0
0.2
x (m) Fig. 4. The intensity distribution of Airy beams as a function of x for different a0 with p ¼ 2:4 (Lohmann I and Lohmann II).
In conclusion, the propagation properties of Airy beams through three types of FRFT optical systems have been investigated. The effect of a0 and fractional order p on the propagation characteristics of Airy beams through FRFT optical systems is discussed in detail. Results show that the intensity of Airy beams periodically changes and remains form-invariant with the
0.4
0.4
z=2.198
Intensity
Intensity
z=0.942
0
0 -0.2
0
0.2
-0.2
x(m)
0
0.4
0.4 z=4.082
Intensity
z=5.338
Intensity 0 -0.2
0.2
x(m)
0 x(m)
0.2
0.2
0 -0.2
0 x(m)
Fig. 5. The intensity distribution of Airy beams in the FRFT planes with different fractional orders in the GRIN medium.
0.2
D. Han et al. / Optics & Laser Technology 44 (2012) 1463–1467
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