Approximate analytical representation of Wigner distribution function for Gaussian beams passing through ABCD optical systems with hard aperture

ARTICLE IN PRESS

Optik

Optics

Optik 116 (2005) 211–218 www.elsevier.de/ijleo

Approximate analytical representation of Wigner distribution function for Gaussian beams passing through ABCD optical systems with hard aperture Daomu Zhao, Haidan Mao, Dong Sun, Shaomin Wang Department of Physics, Zhejiang University, Hangzhou 310027, China Received 8 September 2004; accepted 7 January 2005

Abstract Based on the treatment that a rectangular function can be expanded as an approximate sum of complex Gaussian functions with finite numbers, the analytical expression of the Wigner distribution function for a Gaussian beam passing through a paraxial ABCD optical system with hard-edged aperture is obtained. By numerical simulation, it is shown that the effect of the aperture on the Wigner distribution function is prominent. By comparing the analytical results with the numerical integral results, it is shown that this method of expanding hard-edged aperture into Gaussian functions with finite numbers is proper and ascendant. This method could be extended to study the Wigner distribution functions of other light beams passing through a paraxial ABCD optical system with hard-edged aperture. r 2005 Elsevier GmbH. All rights reserved. Keywords: Wigner distribution function; Gaussian beams; Aperture; ABCD optical systems

1. Introduction The Wigner distribution function, originally discovered and used in the context of quantum mechanics [1], has become a powerful tool in the description of both coherent and partially coherent beams and their passage through first-order systems [2–8]. The Wigner function that corresponds to a Hermite–Gaussian mode has been known since the early days of quantum mechanics [9], and it takes a simple closed form in terms of Laguerre polynomials. Furthermore, the Wigner function that corresponds to a Laguerre–Gaussian mode is also given and takes a simple closed form in terms of Laguerre

Corresponding author. Tel.: +86 571 8886 3887.

E-mail address: [email protected] (D. Zhao). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.01.015

polynomials [10,11]. Recently, the Wigner distribution function of a circular or rectangular aperture was determined [12]. It is known that study on the propagation of Wigner distribution function for a light beam passing through an optical system with hard-edged aperture is a difficult and interesting problem, to the best of our knowledge, which has been paid little attention. In fact, hard-edged aperture always exists in practically optical systems and cannot be avoided, for instance, the finite size of a lens or an optical element. Therefore, study on the behavior of light beams through the apertured optical systems would be of practical interest. In this paper, taking the incidence of a Gaussian beam as an example, we will study the propagation of its Wigner distribution function through a paraxial ABCD optical system with hard-edged aperture.

ARTICLE IN PRESS 212

D. Zhao et al. / Optik 116 (2005) 211–218

This paper is organized as follows. The approximate analytical representations of Wigner distribution function for a Gaussian beam passage through an apertured optical system are derived in Section 2. Taking its propagation through an apertured free space and an apertured image forming system as examples, some numerical simulations are conducted in Section 3. Finally, a simple conclusion is outlined in Section 4.

  x0 y0 E x þ ;y þ 2 2 1 1   x0 y0 E x  ; y  2 2 Z1

 exp½2piðux0 þ vy0 Þ dx0 dy0 ,

Ap ðx; yÞ ¼

1;

jxjpax

and

jyjpay ;

0;

jxj4ax

or

jyjpay ;

! Bn 2 Bm 2 Ap ðx; yÞ ¼ An Am exp  2 x  2 y , ax ay n¼1 m¼1 N X N X

The Wigner distribution function that corresponds to a two-dimensional optical field amplitude Eðx; yÞ at a chosen transverse plane is defined by [1,2] W ðx; y; u; vÞ ¼

(

(4)

where ax and ay denote the half widths of the hardedged aperture in the x and y directions, respectively. Generally, the aperture function can be expanded as the sum of complex Gaussian functions with finite numbers and is given by

2. Approximate analytical representations of Wigner distribution function for a Gaussian beam passing through an apertured ABCD optical system

Z1

where ax ¼ 1=ð2w2x Þ; ay ¼ 1=ð2w2y Þ; bx ¼ ipu1 and by ¼ ipv1 : Introducing the rectangular hard-edged aperture function

ð1Þ

where x and y denote the position variables, u and v mean the spatial–frequency variables in phase space, respectively. It is well known that the fundamental mode Gaussian beam, which is a special solution of wave equation under slowly varying amplitude approximation, is a characteristic and interesting one in laser optics. Let us consider a two-dimensional Gaussian beam in Cartesian coordinate domains given by ! x21 y21 E 1 ðx1 ; y1 Þ ¼ exp  2  2 (2) wx wy is incident upon a rectangular hard-edged aperture followed by an ABCD optical system, where W x and W y are the beam waist sizes in the x and y directions, respectively. The corresponding Wigner distribution function of the input Gaussian beam is obtained as ! 2 b2x by 1=2 exp þ W 1 ðx1 ; y1 ; u1 ; v1 Þ ¼ pðax ay Þ ax ay ! 2x21 2y21  exp  2  2 wx wy ! 2x21 2y21 ¼ 2pwx wy exp  2  2 wx wy h i  exp 2p2 ðu21 w2x þ v21 w2y Þ , ð3Þ

(5)

where An;m and Bn;m are the expansion and Gaussian coefficients, respectively, which could be obtained by optimization–computation directly [13]. However, it should be noted that Eq. (5) is only an approximate expression for the hard-edged aperture function. It has been shown that the larger the N is, the higher is the simulation efficiency [14,15]. In the paraxial approximation, most of optical elements and optical systems can be characterized by ABCD transfer matrix and its determinant of the matrix is a constant for a linear optical system [16]. According to Collins’ diffraction integral formula in Fresnel approximation [17], the output field distribution for a Gaussian beam passing through an apertured ABCD optical system is obtained as follows: ik expðikL0 Þ 2pB Z1 Z1  E 1 ðx1 ; y1 ÞAp ðx1 ; y1 Þ

Eðx2 ; y2 Þ ¼ 

1 1



 exp

ik ½Aðx21 þ y21 Þ 2B

 2ðx1 x2 þ y1 y2 Þ þ Dðx22 þ y22 Þ dx1 dy1 , ð6Þ where L0 is a constant Eikonal function, k is the wavenumber. Substituting Eqs. (5) and (2) into (6), we obtain the output field distribution as Eðx2 ; y2 Þ ¼ 



N X N X ik ik An Am exp expðikL0 Þ Dðx22 þ y22 Þ 2B 2B n¼1 m¼1

ða0x a0y Þ1=2

! 02 2 b0 x b y exp 0 þ 0 , ax ay

ð7Þ

ARTICLE IN PRESS D. Zhao et al. / Optik 116 (2005) 211–218

where a0x ¼

1 Bn ikA ; þ  w2x a2x 2B

b0x ¼

ikx2 0 iky2 ;b ¼ . 2B y 2B

where a0y ¼

Bn þ B Z 1 þ ; ax ¼ 4a2x A2 2w2x A2

1 Bm ikA , þ  w2y a2y 2B

N X N X N X N k2 p X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W 2 ðx2 ; y2 ; u2 ; v2 Þ ¼  2 4B Z¼1 m¼1 n¼1 m¼1 a0 a0 a0 a0 x y x y ! 00 2 00 2   b x b y 00 2 00 2 00 00 1=2 exp 4ax x2  4ay y2 ða ay Þ exp þ 00 , ð9Þ a00x by

An Am A Z A m

where   k2 1 1 ¼ þ ; 16B2 a0x a0 x

a00y

! k2 1 1 . ¼ þ 16B2 a0y a0 y

  k2 x2 1 1 ikDx2 þ piu2 ,   2B 8B2 a0x a0 x ! k 2 x2 1 1 ikDy2 00 þ piv2 . by ¼   2B 8B2 a0y a0 y

(10)

b00x ¼

ð11Þ

For B ¼ 0 (corresponding to image forming systems), Eq. (6) seems divergent. Actually, this is not the case and the output field distribution becomes [16]

1 ikC 2 ðx þ y22 Þ Eðx2 ; y2 Þ ¼ expðikL0 Þ exp A 2A 2 n  x2 y o  Eðx1 ; y1 ÞAp ðx1 ; y1 Þd x1  ; y1  2 A " A  N X N X 1 1 Bn An Am exp  þ ¼ expðikL0 Þ A w2x a2x n¼1 m¼1 ! #  x22 1 Bm y22 ikC  2 2 x þ y2 ,  2 þ þ ð12Þ w2y a2y A2 2A 2 A where A means the transverse magnification factor, then the corresponding Wigner distribution function at the output plane becomes N X N X N X N 1 X An Am A Z A m A2 n¼1 m¼1 Z¼1 m¼1 ! 2 b2x by 1=2 2 2  expð4ax x2  4ay y2 Þpðax ay Þ exp þ , ð13Þ ax ay

W 2 ðx2 ; y2 ; u2 ; v2 Þ ¼



ay ¼

Bm þ Bm 1 þ , 4a2y A2 2w2y A2 (14)

ð8Þ

Substituting Eq. (7) into (1), after performing the tedious integration, we obtain the Wigner distribution function at the output plane as

a00x

213

bx ¼ by ¼

ðBn  B Z Þx2 2a2x A2 ðBm  B m Þy2 2a2y A2



ikCx2 þ piu2 , 2A



ikCy2 þ piv2 . 2A

ð15Þ

3. Numerical simulations Eqs. (9) and (13) are the main results of this paper, which are the approximate analytical propagation expressions of a two-dimensional Gaussian beam through an apertured paraxial ABCD optical system. By using Eq. (9), let us consider the case of a hard-edged aperture followed by a free space written in terms of the ray transfer matrix elements A ¼ D ¼ 1; B ¼ z; C ¼ 0; where z denotes the distance of the free space. In order to show its three-dimensional distribution graphically, we just choose the one-dimensional Gaussian beam given by expðx21 =w2x Þ as an input optical field. From Eq. (9), its Wigner distribution function at the output plane becomes N X N ik X An A Z ða0x a0 x Þ1=2 2z Z¼1 n¼1 ! 2 b0 x 00 2 00 1=2  expð4ax x2 Þ ðp=ax Þ exp 00 , ax

W 2 ðx2 ; u2 Þ ¼ 

ð16Þ

where   1 Bn ik k2 1 1 00 ; a þ  ¼ þ , x w2x a2x 2z 16z2 a0x a0 x   k2 x2 1 1 ikx2 00 þ piu2 .   bx ¼ 8z2 a0x a0 x 2z

a0x ¼

ð17Þ

Furthermore, an apertured image-forming system is also taken as an example. Where the image-forming system consists of a free space, a lens and a free space. The distances of the free space before and after the lens are both equal to 2f, where f is the focal length of the lens. An aperture is just located at the input plane and the lens is ideal. In the numerical calculation, f ¼ 200 mm, the wavelength l ¼ 1:06 mm; the waist size of Gaussian beam w0 ¼ 0.5 mm, the equivalent Fresnel number N f ¼ w20 =ðlzÞ: Fig. 1(a) shows the real and imaginary parts of the Gaussian expansion given by Eq. (5) for the aperture function and Fig. 1(b) shows the magnitude and phase

ARTICLE IN PRESS 214

D. Zhao et al. / Optik 116 (2005) 211–218

Fig. 1. (a) Real and imaginary parts of the Gaussian expansion for the aperture function, (b) magnitude and phase of the Gaussian expansion.

Fig. 2. The normalized Wigner distribution functions of a Gaussian beam at the input plane. (b) Projection of (a).

of the Gaussian expansion, valuated by the coefficients An and Bn with N ¼ 10 given by Wen and Breazeale [13]. One can see in the figure that the errors reach 5–6% and even 12% near the aperture edge. A hard-edged aperture can well be expanded into the sum of 10 or more complex Gaussian functions. Fig. 2 shows the normalized Wigner distribution functions of a Gaussian beam at the input plane. Figs. 3 and 4 show the normalized Wigner distribution functions at the output plane with Nf ¼ 1.0 of a Gaussian

beam passing through an apertured free space, calculated by the analytical expression and integral formula, respectively. Comparing these two figures, we find that the normalized Wigner distribution functions calculated by these two different methods keep well with each other in the centre, but some small deviations departing from the centre. In the case of a=w0 ¼ 3; their deviations are more significant than the case of a=w0 ¼ 1: Figs. 5 and 6 show the normalized Wigner distribution functions at the output plane with N f ¼ 0:1 of a Gaussian beam passing through an apertured free space, calculated by the analytical expression and integral formula, respectively. From Figs. 2–6, we find that the Fresnel diffraction corresponds to a shearing and these figures rotate clockwise with the decreasing of the equivalent Fresnel number. From the figures, we find that the effect of the hard-edged aperture on the Wigner distribution function is prominent and the diffraction effect of the hard-edge aperture can be neglected when a=w0 X3: From these two groups of figures, we also find that the results obtained by the approximate analytical expressions and by the direct numerical calculations are in accordance with each other very well. Moreover, its calculation efficiency of the former is far higher than that of the latter. Note that in the numerical calculation of Eq. (1) instead of Eq. (9), the truncation error is supposed to be 105. So this method of expanding hardedged aperture into Gaussian functions with finite numbers is proper and ascendant. Fig. 7 shows the normalized Wigner distribution functions at the output plane of a Gaussian beam passing through an apertured image-forming system, calculated by the approximate analytical expressions. Notice that the aperture is just located at the input plane and the lens is regarded as an ideal one. In the case of a=w0 ¼ 3; as shown by Fig. 7(c) and (d), the effect of aperture can be neglected and it can be regarded as the case of ideal image-forming system.

ARTICLE IN PRESS D. Zhao et al. / Optik 116 (2005) 211–218

Fig. 3. The normalized Wigner distribution functions at the output plane of a Gaussian beam passing through an apertured free space calculated by approximate analytical equation (9) where the equivalent Fresnel number N f ¼ 1: (a) a=W 0 ¼ 1; (c) a=W 0 ¼ 3; (b) and (d) are the projections of (a) and (c), respectively.

215

Fig. 4. The normalized Wigner distribution functions at the output plane of a Gaussian beam passing through an apertured free space calculated numerically where the equivalent Fresnel number N f ¼ 1: (a) a=W 0 ¼ 1; (c) a=W 0 ¼ 3; (b) and (d) are the projections of (a) and (c), respectively.

ARTICLE IN PRESS 216

D. Zhao et al. / Optik 116 (2005) 211–218

Fig. 5. The normalized Wigner distribution functions at the output plane of a Gaussian beam passing through an apertured free space calculated by approximate analytical equation (9) where the equivalent Fresnel number N f ¼ 0:1: (a) a=W 0 ¼ 1; (c) a=W 0 ¼ 3; (b) and (d) are the projections of (a) and (c), respectively.

Fig. 6. The normalized Wigner distribution functions at the output plane of a Gaussian beam passing through an apertured free space calculated numerically where the equivalent Fresnel number N f ¼ 0:1: (a) a=W 0 ¼ 1; (c) a=W 0 ¼ 3; (b) and (d) are the projections of (a) and (c), respectively.

ARTICLE IN PRESS D. Zhao et al. / Optik 116 (2005) 211–218

217

It should be pointed out that the Wigner distribution function of a Gaussian beam is positive and the that of other high-order Gaussian beam can have negative value [18].

4. Conclusions In conclusion, the approximate analytical representations of the Wigner distribution function for a Gaussian beam passing through an apertured ABCD optical system are obtained and some numerical calculations are also illustrated for its application. It provides a simple and powerful way to treat the propagation of the Wigner distribution function in apertured ABCD optical systems. It should be pointed out that only real-valued ABCD optical systems are treated in this paper; in fact, the results could be straightforwardly extended to complex ABCD optical systems. For example, Gaussian aperture could be regarded as a complex optical element, the presence of imaginary part in the matrix elements indicates that this optical element exhibits inhomogeneous loss or gain. This method of expanding hard-edged aperture into Gaussian beams with finite numbers used in this paper could also be applied to treating the Wigner distribution functions for other light beams, such as Hermite–Gaussian beams, Laguerre– Gaussian beams, Hermite-sinusoidal–Gaussian beams, passing through apertured ABCD optical systems.

Acknowledgements This work was supported by the joint foundation of National Natural Science Foundation of China and the Chinese Academy of Engineering Physics under Grant 10276034.

References

Fig. 7. The normalized Wigner distribution functions at the output plane of a Gaussian beam passing through an apertured image forming system calculate0d by approximate analytical eq. (13). (a) a=W 0 ¼ 1; (c) a=W 0 ¼ 3; (b) and (d) are the projections of (a) and (c), respectively.

[1] E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749–759. [2] M.J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Opt. Commun. 25 (1978) 26–30. [3] M.J. Bastiaans, Wigner distribution function and its application to first-order optics, J. Opt. Soc. Am. 69 (1979) 1710–1716. [4] M.J. Bastiaans, Application of the Wigner distribution function to partially coherent light, J. Opt. Soc. Am. A 3 (1986) 1227–1238. [5] R. Simon, E.C.G. Sudarshan, N. Mukunda, Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields, Phys. Rev. A 29 (1984) 3273–3279.

ARTICLE IN PRESS 218

D. Zhao et al. / Optik 116 (2005) 211–218

[6] R. Simon, E.C.G. Sudarshan, N. Mukunda, Anisotropic Schell-model beams: passage through optical systems and associated invariants, Phys. Rev. A 31 (1985) 2419–2434. [7] D. Dragoman, The Wigner distribution function in optics and optoelectronics, Prog. Opt. 37 (1997) 1–56. [8] D. Dragoman, Higher-order moments of the Wigner distribution function in first-order optical systems, J. Opt. Soc. Am. A 11 (1994) 2643–2646. [9] H.J. Groenewold, On the principles of elementary quantum mechanics, Physica 12 (1946) 405–460. [10] R. Gase, Representation of Laguerre–Gaussian modes by the Wigner distribution function, IEEE J. Quantum Electron. 31 (1995) 1811–1818. [11] R. Simon, G.S. Agarwal, Wigner representation of Laguerre–Gaussian beams, Opt. Lett. 25 (2000) 1313–1315. [12] M.J. Bastiaans, P.G.J. van de Mortel, Wigner distribution function of a circular aperture, J. Opt. Soc. Am. A 13 (1996) 1698–1703.

[13] J.J. Wen, M.A. Breazeale, A diffraction beam field expressed as the superposition of Gaussian beams, J. Acoust. Soc. Am. 83 (1988) 1752–1756. [14] D. Zhao, H. Mao, W. Zhang, S. Wang, Propagation of off-axial Hermite-cosine–Gaussian beams through an apertured and misaligned ABCD optical system, Opt. Commun. 224 (2003) 5–12. [15] D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, X. Wei, Propagation of Hermite–Gaussian beams in apertured fractional Fourier transforming systems, Optik 114 (2003) 504–508. [16] S. Wang, D. Zhao, Matrix Optics, CHEP-Springer, Beijing, 2000. [17] S.A. Collins, Lens-system diffraction integral written in terms of matrix optics, J. Opt. Soc. Am. 60 (1970) 1168–1177. [18] D. Dragoman, Phase-space interferences as the source of negative values of the Wigner distribution function, J. Opt. Soc. Am. A 17 (2000) 2481–2485.

BOOK REVIEW J. Eichler, H.J. Eichler, Laser — Bauforman, Strahlfu¨hrung, Anwednungen. 5., aktualisierte Auflage, Springer, Berlin, ISBN 3-540-00376-2, 2003 (XI/452 Seiten, 266 Abb., 57 Tab., gebunden. h 59,95).

Das vorliegende Buch ist aus Manuskripten von Vorlesungen entstanden und nun bereits in der 5. Auflage vorliegend. Das spricht fu¨r seinen Inhalt und seine Qualita¨t. Auch wenn der Stoff an vielen Stellen nicht immer sehr tief ist und nicht mit theoretischen Ableitungen untermauert wird, gibt der Text eine sehr gute U¨bersicht und eine exzellente Einfu¨hrung in das Gebiet. Ich kann das Buch nicht nur an Universita¨tsund Fachhochschulstudenten, sondern auch an Ingenieure und Lehrer empfehlen. Fu¨r diejenigen die noch ausfu¨hrlicher unterrichtet sein mo¨chten, dient eine sehr umfangreiche, wenn auch nicht vollsta¨ndige Liste von weiterfu¨hrenden Bu¨chern und Zeitschriften. Zuerst werden die Grundlagen der Laseroptik dargestellt, dann spezielle Lasertypen und –materialien beschrieben. Zuerst wird auf die Gaslaser eingegangen, bei denen das Licht von Atomen, Ionen und Moleku¨len emittiert wird. Gaslaser existieren in sehr verschiedenen Ausfu¨hrungsformen, und werden deshalb relativ ausfu¨hrlich in vier Kapiteln dargestellt. Weitere wichtige Lasertypen, die Festko¨rper-, Farbstoff-, Halbleiter-, Elektronenstrahl- und Ro¨ntgenlaser, werden ebenfalls detailliert beschrieben. Bei der Darstellung der verschie

denen Laser wird jeweils kurz auf typische Anwendungen eingegangen. Anschließend werden optische Laserkomponenten, wie Spiegel, Polarisatoren und Modulatoren beschrieben, mit denen Laser in verschiedenen Betriebsarten aufgebaut werden ko¨nnen. Von besonderem Interesse ist dabei der Pulsbetrieb, der Aufbau von frequenzstabilen, schmalbandigen Lasern sowie von abstimmbaren Lasern. In diesem Zusammenhang wird auch die externe Frequenzumsetzung durch nichtlineare optische Effekte kurz dargestellt. Außerdem werden Gera¨te zur Charakterisierung der Laserstrahlung, wie Photodetektoren, Energiemeßgera¨te, Spektralapparate und Anordnungen zur Koha¨renzmessung beschrieben. Die Ausbreitung von Laserlicht erfolgt in Form von Gaußstrahlen oder Gaußbu¨ndel. Diese werden aus Kugelwellen mit komplexem Quellpunkt entwickelt, wodurch die Benutzung von Differentialund Integralgleichungen in diesem Zusammenhang vermieden wird. Auch in den andern Teilen des Buches reicht gutes Abiturwissen in Mathematik zum Versta¨ndnis aus. Moderne Entwicklungen der Hochleistungsund Kurzpulslaser sowie ‘‘Blue’’ Laser und Quantenkaskadenlaser werden beru¨cksichtigt. Zahlreiche Abbildungen und Tabellen, sowie viele Aufgaben mit vollsta¨ndigen Lo¨sungswegen dienen der Festigung des Stoffes.

Gurtner Klaus