Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain

Optics & Laser Technology 34 (2002) 415 – 421

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Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain Yangjian Cai, Qiang Lin ∗ Institute of Optics & State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310028, China Received 30 January 2002; received in revised form 19 March 2002; accepted 10 April 2002

Abstract The propagation formulae of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain are investigated analytically by using tensor method. The derived formulae provide a powerful tool for treating the propagation and transformation of elliptical Gaussian beam through misaligned optical systems. As an application example, the propagation properties of elliptical Gaussian beam through a misaligned thin lens are illustrated. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Elliptical Gaussian beam; Misaligned optical systems; Propagation

1. Introduction It is well known that the propagation and transformation of a beam passing through aligned optical systems can be treated by the well-known di9raction integral formula derived by Collins [1]. This formula has been widely used in treating beam propagation and transformation [2,3]. Collins formula was extended to tensor form so as to make it applicable for general axially nonsymmetrical optical systems [4]. Recently, the Collins formula has been converted into spatial-frequency domain both for axially symmetrical and nonsymmetrical optical systems [5,6]. The Collins formula was also generalized to treat the propagation and transformation of partially coherent twisted Gaussian–Schell model beams [7]. As the misaligned optical systems always exist in practice, it is of practical signi?cance to study the propagation of beams through misaligned optical systems. The generalization of the di9raction integral formulae for misaligned optical systems in spatial domain has been done in [8], in spatial-frequency domain has been done in [9]. On the other hand, the extension of ?rst-order optics from the aligned to the misaligned case along the lines of canonical transform theory has been made [10]. The generalized beam matrices for beam propagation through misaligned complex optical systems have also been developed [11,12]. In all ∗

Corresponding author. Tel.=fax: +86-571-8827-3086. E-mail address: [email protected] (Q. Lin).

these works, however, only beams with circular transverse symmetry were treated. In this paper, the propagation and transformation of elliptical Gaussian beam through misaligned optical systems is investigated in terms of tensor method in both spatial and spatial-frequency domain. Based on the generalized integral formulae for misaligned optical systems, analytical propagation formulae for elliptical Gaussian beam through misaligned optical systems in spatial domain and spatial-frequency domain were derived. An application example was given. 2. Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain The complex electric ?eld of a generalized elliptical Gaussian beam can be expressed in tensor form as follows: 
ik T −1 E(r1 ; 0) = E0 exp − r1 Q1 r1 ; (1) 2 where E0 is a constant, which will be set to unit in the following. k =2= is the wave number,  is the wavelength, r1 is position vector in transverse plane given by r1T =(x1 ; y1 ). Q1−1 is the 2×2 complex curvature tensor for the generalized elliptical Gaussian beam given by [13,4,14]  −1 −1  qxy qxx −1 Q1 = : (2) −1 −1 qxy qyy

0030-3992/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 0 2 ) 0 0 0 4 0 - 3

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Y. Cai, Q. Lin / Optics & Laser Technology 34 (2002) 415–421

S1 a b  T
  c d  T
0 0 1  0 0 0 

Eq. (3) can be rearranged into a more compact form by using tensor method as following:

S2 T
'   T
'  0  1 

 ik E(r1 ) 2[det(B)]1=2 
ik T −1 −1 T −1 T ×exp − (r1 B Ar 1 − 2r1 B r2 + r2 DB r2 ) 2
 ik ×exp − (r1T B −1 ef + r2T B −1 gh ) dr1 ; (9) 2

E(r2 ) = −

Fig. 1. A misaligned optical system.

Let us consider a misaligned optical system with multi-element as shown in Fig. 1. The misaligned optical system can be characterized by the 4 × 4 augmented matrix, S1 and S2 represent the input and output reference plane, respectively. The generalized di9raction integral formula in spatial domain is given by [8]   ik ik E1 (x1 ; y1 ) exp − [a(x12 + y12 ) E2 (x2 ; y2 ) = − 2b 2b − 2(x1 x2 + y1 y2 ) + d(x22 + y22 )  + ex1 + fy1 + gx2 + hy2 ] d x1 dy1 :

where r1T = (x1 ; y1 ); r2T = (x2 ; y2 ); ef = (e; f); gh = (g; h); A; B; C and D take the following form:  A=  C=

a

0

0

a

c

0

0

c

(3)

e = 2(T x + T x );

(4)

f = 2(T y + T y );

(5)

g = 2(bT − dT )x + 2(bT − dT )x ;

(6)



 ;

D=

0

0

b

d

0

0

d

;  :

(10)

For forward-going optical elements, T is chosen for “+” sign; for backward-going ones, T is chosen for “−”sign.

(11)


  ik T −1 ik exp − r1 Q1 r1 E(r2 ) = − 2 2[det(B)]1=2 
ik ×exp − (r1T B −1 Ar 1 − 2r1T B −1 r2 + r2T DB −1 r2 ) 2
 ik ×exp − (r1T B −1 ef + r2T B −1 gh ) dr1 2

(7)

(8)

(DB −1 )T = DB −1 ;

Substituting Eq. (1) into Eq. (9), we obtain

T = l − b; T = ±1 − d:

B=



b

C − DB −1 A = −(B −1 )T :

where x ; x ; y and y denote the two-dimensional misalignment parameters, x and y are the displacement in x and y direction, respectively, x and y are the tilting angle of the element in x and y direction, respectively. T ; T ; T and T represent the misaligned matrix elements de?ned by [8]

T = −c;

;

(B −1 A)T = B −1 A;

=

T = 1 − a;



Because A; B; C and D are all diagonal matrices, they satisfy the following relations:

In Eq. (3), the phase factor exp(ikl0 ) along the axis between the two reference planes has been omitted. a; b; c and d are the transfer matrix elements of the aligned optical system. The parameters e; f; g and h take the following form:

h = 2(bT − dT )y + 2(bT − dT )y ;



1 1=2 [det(A + BQ −1 1 )]  ik ik ×exp − r2T Q2−1 r2 − r2T B −1 gh 2 2 ik −1 − r2T B −1T (A + BQ −1 ) e f 1 2
 ik T −1T −1 ×exp ef B (A + BQ −1 ) e f ; 1 8

(12)

where Q2−1 = DB −1 − B −1T (B −1 A + Q1−1 )−1 B −1 −1 −1 = (C + DQ −1 1 )(A + BQ 1 ) :

(13)

Y. Cai, Q. Lin / Optics & Laser Technology 34 (2002) 415–421

In of Eqs. (12) and (13), the integral formula

∞the derivation 2 exp(−ax ) d x = =a and Eq. (11) have been used. −∞ Eq. (12) is the general propagation formula of elliptical Gaussian beam passing through misaligned optical systems. Eq. (13) is the transformation law of complex curvature tensor through optical systems called tensor ABCD law, which is the same as aligned optical system. When the misalignment parameters take the values ef = (0; 0); gh = (0; 0), Eq. (12) is reduced to the general propagation formula of elliptical Gaussian beam passing through aligned optical system [4]  1 ik T −1 E(r2 ) = exp − r2 Q2 r2 : (14) 1=2 2 [det(A + BQ −1 1 )]

Taking the Fourier transformation of Eq. (1), we get A(C1 ) = −

− 2(u1 u2 + v1 v2 ) +

A(C2 ) =

where

  i c0 = exp − [d(e2 + f2 ) 4b2 c  +2(ge + fh) + a(g2 + h2 )] :

=−

1 T C BD−1 ef  1

 1 T −1 1 1 C1 B gh − CT2 B −1 ef − CT2 B −1 Agh ) dC1    ic0 exp[iCT2 Q2 C2 ] 1=2 [det(C + DQ −1 1 )]

×exp {iCT2 [C −1T (Q1 +C −1 D)−1 BD−1 −B −1 ]ef} ×exp{−iCT2 Q2 CB −1 gh } 
i T −1 T −1 −1 −1 ×exp − ef (BD ) (Q1 + C D) B gh 2
 i T −1 T −1 −1 −1 ×exp − ef (BD ) (Q1 + C D) BD ef 4
 i T −1T −1 −1 −1 ×exp − gh B (Q1 + C D) B gh ; (19) 4 where Q2 = AC −1 − C −1T (Q1 + C −1 D)−1 C −1 = (AQ 1 + B)(CQ 1 + D)−1 :

(16)

Eq. (15) also can be rearranged into a more compact form by using tensor method as follows:  ic0 A(C2 ) = A(C1 ) [det(C )]1=2
×exp i(CT1 C −1 DC1 − 2CT1 C −1 C2 1 1 T C1 BD−1 ef CT1 B −1 gh    1 T −1 1 T −1 − C2 B ef + C2 B Agh ) dC1 ;  

(i)2 c0 [det(C )] [det(Q1−1 )]1=2
 × exp i(CT1 Q1 v1 + CT1 C −1 Dv1

+

+

(15)

(18)

1=2

− 2CT1 C −1 C2 + CT2 AC −1 C2 +

1 (ed + g)u1 b

1 1 (fd + h)v1 − (ga + e)u2 b b  1 (ga + e)v2 du1 dv1 ; − b

i exp(iCT1 Q1 C1 ): [det(Q1−1 )]1=2

Substituting Eq. (18) into Eq. (17), we get

3. Propagation of elliptical Gaussian beam through misaligned optical systems in spatial-frequency domain The generalized di9raction integral formula for angular spectrum through the misaligned optical systems in spatial-frequency domain is given by [9]  i A2 (u2 ; v2 ) = c0 A1 (u1 ; v1 ) c  i ×exp [d(u12 + v12 ) + a(u22 + v22 ) c

417

+ CT2 AC −1 C2 +

(17)

where C1 = (u1 ; v1 ); C2 = (u2 ; v2 ) are the spatial-frequency vectors on the input and output plane, respectively.

(20)

In the derivation of Eqs. (18) and (19), the following relations has been used: (C −1 D)T = C −1 D; (AC −1 )T = AC −1 :

(−C −1 )T = (B − AC −1 D); (21)

Eq. (18) provides the transformation rule of the angular spectrum of general elliptical Gaussian beams through misaligned optical systems in spatial-frequency domain. Eq. (19) is the transformation law of complex curvature tensor through optical systems called tensor ABCD law in spatial-frequency domain. When the misalignment parameters take the values ef = (0 0); gh = (0 0), Eq. (19) is reduced to the general propagation formula of elliptical Gaussian beam passing through aligned optical system in

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4. An application example

fl


x z1

z-z1

Fig. 2. An optical system with a misaligned lens.

spatial-frequency domain [6] A(C2 ) = −

i exp[iCT2 Q2 C2 ]: 1=2 [det(C + DQ −1 )] 1

(22)

Let us consider a general elliptical Gaussian beam propagating through a misaligned optical system consisted of a transverse shifted lens as shown in Fig. 2. The displacements and angle misalignment of the lens with respect to the optical axis of the system are x ; y = x
= y
= 0. The thin lens is located at z = z1 and the exit plane is located at z. The ray transfer matrix of the optical system between the source plane and the exit plane take the following form:   1 − (z − z1 )=fl 0 A= ; 0 1 − (z − z1 )=fl   z − z1 (z − z1 )=fl 0 B= ; 0 z − z1 (z − z1 )=fl

Fig. 3. Contour graphs of 2-D intensity distribution of elliptical Gaussian beam before and after passing through a misaligned thin lens at several propagation distances in spatial domain: (a) z = 0; (b) z = 30 mm; (c) z = 60 mm; and (d) z = 150 mm.

Y. Cai, Q. Lin / Optics & Laser Technology 34 (2002) 415–421

 C=  D=

419



−1=fl

0

0

−1=fl

; 

1 − z1 =fl

0

0

1 − z1 =fl

;

(23)

where fl is the focal length of the thin lens. The misalignment parameters T ; T ; T and T take the following form: T =

z − z1 ; fl

T =

1 ; fl

T =

T =

z1 (z − z1 ) ; fl

z1 fl

(24)

and the corresponding parameters e; f; g and h are given by e=2

(z − z1 )x ; fl

f = 0;

g=

2z1 x ; fl

h = 0:

(25)

Substituting Eqs. (23) and (25) into Eqs. (12) and (13), we obtain the contour graphs of the 2-D normalized intensity distribution of the elliptical Gaussian beam at several propagation distances in spatial domain before and after passing through a misaligned thin lens as shown in Fig. 3. The parameters used in the calculation are  = 632:8 nm; z1 = 20 mm;

x = 0:5 mm; fl = 15 mm;   −0:20i −0:05i −1 Q1 = (m)−1 : −0:05i −0:08i

We should note that the intensity distribution of elliptical Gaussian beam when z ¡ z1 can be directly derived from Eq. (14) with A; B; C and D take the following form:     1 0 z 0 A= ; B= ; 0 1 0 z    0 0 10 C= ; D= : (26) 01 0 0 In order to compare the propagation properties of elliptical Gaussian beam through misaligned thin lens and through aligned thins lens, contour graphs of 2-D intensity distribution of elliptical Gaussian beam passing through an aligned thin lens at several propagation distances in spatial domain are also shown in Fig. 4. From Fig. 3, we can ?nd that the general elliptical Gaussian beam becomes a general decentered elliptical Gaussian beam after passing through the misaligned thin lens, and the peak position of the

Fig. 4. Contour graphs of 2-D intensity distribution of elliptical Gaussian beam passing through an aligned thin lens at several propagation distances in spatial domain: (a) z = 60 mm; and (b) z = 150 mm.

intensity distribution moves along with the propagation distance. From Fig. 4, we can ?nd that the peak position of the intensity distribution of elliptical Gaussian beam before and after passing through the aligned thin lens remains invariant. In the far ?eld, the intensity of elliptical Gaussian beam diverges along with propagation distance. Substituting Eqs. (23) and (25) into Eqs. (19) and (20), we obtain the contour graphs of 2-D intensity distribution of the elliptical Gaussian beam at several propagation distances in spatial-frequency domain before and after passing through a misaligned thin lens as shown in Fig. 5. Contour graphs of 2-D intensity distribution of elliptical Gaussian beam passing through an aligned thin lens at several

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Fig. 5. Contour graphs of 2-D intensity distribution of elliptical Gaussian beam before and after passing through a misaligned thin lens at several propagation distances in spatial-frequency domain: (a) z = 0; (b) z = 10 mm; (c) z = 30 mm; and (d) z = 60 mm.

propagation distances in spatial-frequency domain are also shown in Fig. 6. From Figs. 5 and 6, we can ?nd that the intensity distribution of general elliptical Gaussian beam remain invariant before passing through misaligned lens, after passing misaligned lens, the general elliptical Gaussian beam becomes a general decentered elliptical Gaussian beam, the peak position of the intensity distribution moves along with the propagation distance, and the beam spot rotates after passing through misaligned thin lens which are much di9erent from that in spatial domain as shown in Fig. 3, the peak position of the intensity distribution of elliptical Gaussian beam after passing through aligned thin lens remain invariant. We can also ?nd that the peak position of the intensity distribution deviates from the optical axis more seriously in spatial-frequency domain.

5. Conclusions Based on the generalized di9raction integral formulae for misaligned optical systems in spatial domain and spatial-frequency domain, the analytical propagation formulae for general elliptical Gaussian beams through misaligned optical systems in spatial domain and spatial-frequency domain are derived respectively by using vector integration. As an application example, we study the propagation properties of a general elliptical Gaussian beam passing through a misaligned thin lens. The results show that the general elliptical Gaussian beam becomes a decentered elliptical Gaussian beam after passing through a misaligned thin lens no matter in spatial domain and spatial-frequency domain. The peak position of the intensity distribution deviates from the optical axis more

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Acknowledgements This work is supported by National Natural Science Foundation of China (60078003) and Huo-Ying-Dong Education Foundation of China (71009).

References

Fig. 6. Contour graph of 2-D intensity distribution of elliptical Gaussian beam passing through an aligned thin lens at several propagation distances in spatial-frequency domain: (a) z = 30 mm; and (b) z = 60 mm.

seriously in spatial-frequency domain than in spatial domain.

[1] Collins SA. Lens-system di9raction integral written in terms of matrix optics. J Opt Soc Am 1970;60:1168–77. [2] Zhao D, Zhu J, Wang S. Azimuthally polarized Bessel–Gauss beam propagation through axisymmetric optical systems. J Opt 1997;28: 3–5. [3] Belafhal A, Dalil-Essakali L. Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system. Opt Commun 2001;177:181–8. [4] Lin Q, Wang S, Alda J, Bernabeu E. Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law. Optik 1990;85:67–72. [5] Liu Z, Wu X, Fan D. Collins formula in frequency-domain and fractional Fourier transforms. Opt Commun 1999;155:7–11. [6] Lin Q, Wang L. Collins formula and tensor ABCD law in spatial-frequency domain. Opt Commun 2000;185:263–9. [7] Lin Q, Cai Y. Tensor ABCD law for partially coherent twisted anisotropic Gaussian–Schell model beams. Opt Lett 2002;27: 216–8. [8] Wang S, Ronchi L. Principles and design of optical array. In: Wolf E, editor. Progress in optics, vol. 25, Amsterdam: Elsevier Science BV, 1988. p. 279. [9] Zhao D, Ge F, Wang S. Generalized di9raction integral formula for misaligned optical systems in spatial-frequency domain. Optik 2001;6:268–70. [10] Nazarathy M, Hardy A, Shamir J. Misaligned ?rst-order optics: canonical operator theory. J Opt Soc Am A 1986;3: 1360–9. [11] Tovar AA, Casperson LW. Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems. J Opt Soc Am A 1995;12:1522–33. [12] Tovar AA, Casperson LW. Generalized beam matrices. IV. optical system design. J Opt Soc Am A 1997;14:882–93. [13] Arnaud JA, Kogelink H. Gaussian light beams with general astigmatism. Appl Opt 1969;8:1687–93. [14] Alda J, Wang S, Bernabeu E. Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams. Opt Commun 1990;80:350–2.