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Analytical formula for a circular flattened Gaussian beam propagating through a misaligned paraxial ABCD optical system
Physics Letters A 360 (2006) 394–399 www.elsevier.com/locate/pla
Analytical formula for a circular flattened Gaussian beam propagating through a misaligned paraxial ABCD optical system Li Hu, Yangjian Cai ∗ Joint Research Center of Photonics of the Royal Institute of Technology and Zhejiang University, East Building No.5, Zijingang Campus, Zhejiang University, Hangzhou 310058, China Division of Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology, 10044 Stockholm, Sweden Received 17 April 2006; received in revised form 26 July 2006; accepted 9 August 2006 Available online 18 August 2006 Communicated by R. Wu
Abstract Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a misaligned paraxial ABCD optical system in the cylindrical coordinate system, analytical formula for a circular flattened Gaussian beam propagating through such optical system is derived. Furthermore, an approximate analytical formula is derived for a circular flattened Gaussian beam propagating through an apertured misaligned ABCD optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Numerical examples are given. © 2006 Elsevier B.V. All rights reserved. PACS: 42.25.Bs; 41.85.Ew Keywords: Flattened Gaussian beam; Misaligned ABCD optical system; Propagation
1. Introduction Laser beams with flat-topped spatial profiles are required in many applications, such as material thermal (uniform) processing, inertial confinement fusion, etc. Flattened Gaussian beam (FGB) proposed by Gori in 1994 is a typical and well-known model for describe a laser beam with flat-topped profile [1]. An FGB can be expressed as a finite sum of Laguerre–Gaussian modes or Hermite–Gaussian modes [1,2]. The propagation property of an FGB has been investigated widely. Amarande et al. studied the propagation factor and Kurtosis parameter of an FGB [3]. Santarsiero et al. compared an FBG with an SGB, and found that an FGB and a super Gaussian beam have nearly the same properties if they have the same propagation factor [4]. The focusing properties of an FGB such as the focal shift and integrated intensity were also studied [5,6]. We have studied the properties of an FGB in the fractional Fourier transform plane [7], and proposed the elliptical FGB in a ten* Corresponding author.
E-mail address: [email protected] (Y. Cai). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.029
sor form to describe a non-circular flat-topped beam [8]. Lü et al. have derived the approximate propagation equation for a circular FGB through an apertured aligned ABCD optical system [9]. Shen et al. have studied the propagation of a twodimensional FGB through a misaligned optical system with finite aperture [10]. Closed-form expressions for a two dimensional FGB propagating through an apertured ABCD optical system in the rectangular coordinate system were derived by Ibnchaikh et al. [11]. Zhou et al. have investigated the algorithms for two-dimensional FGBs passing through apertured and unapertured paraxial ABCD optical systems in the rectangular coordinate system [12]. Zheng have studied the propagation properties of an elliptical FGB in spatial-frequency domain [13]. More recently, Jiang et al. have studied the propagation characteristics of the rectangular flattened Gaussian beams through an apertured misaligned optical system [14]. In practice, most optical systems are slightly misaligned more or less. Thus, it is necessary to study the propagation of a laser beam through a slightly misaligned optical system [15,16]. In this Letter, we derive analytical formulas for a three-dimensional circular flattened Gaussian beam propa-
L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
gating through unapertured and apertured misaligned ABCD optical systems based on the generalized diffraction integral formulas for treating the propagation of a laser beam through such optical systems in the cylindrical coordinate system, and some numerical examples are given. 2. Propagation of a flattened Gaussian beam through a misaligned paraxial ABCD optical system Fig. 1 shows the misalignment diagram for a two-dimensional forward-going optical system [15]. In Fig. 1, RP1 and RP2 are alignment reference planes. RP1m and RP2m are misalignment reference planes, εx , εx , εy and εy denote the twodimensional slight misalignment parameters, εx and εy are the displacement element in x and y directions respectively, εx and εy are the tilted angles of the element in x and y directions respectively. a, b, c and d are the transfer matrix elements of an aligned optical system. z and zm are the aligned optical axis and misaligned optical axis, respectively. The generalized diffraction integral formula for treating the propagation of a laser beam through a misaligned paraxial ABCD optical system in the rectangular coordinate can be expressed as follows [15] E2 (x2 , y2 , z) ∞ ∞ ik E1 (x1 , y1 , 0) = 2πb −∞ −∞ ik 2 × exp − a x1 + y12 − 2(x1 x2 + y1 y2 ) + d x22 + y22 2b + ex1 + fy1 + gx2 + hy2 dx1 dy1 . (1) In Eq. (1), the phase factor exp(ikl0 ) along the axis between the two reference planes has been omitted. k = 2π/λ is the wave number, λ is the wavelength. 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: e = 2(αT εx + βT εx ),
f = 2(αT εy + βT εy ),
(2)
g = 2(bγT − dαT )εx + 2(bδT − dβT )εx , h = 2(bγT − dαT )εy + 2(bδT − dβT )εy ,
(3)
Fig. 1. Misalignment diagram for a two-dimensional forward-going system.
395
where αT , βT , γT and δT represent the misaligned matrix elements determined by αT = 1 − a,
βT = l − b,
γT = −c,
δT = 1 − d, (4) where l is the axial distance from the input plane and output plane. In the cylindrical coordinate system, we can express Eq. (1) as follows E2 (r2 , θ2 , z) ∞2π ik = E1 (r1 , θ1 , 0) 2πb 0 0 ik 2 × exp − ar − 2r1 r2 cos θ1 cos θ2 2b 1 − 2r1 r2 sin θ1 sin θ2 + dr22 + er1 cos θ1 + f r1 sin θ1 + gr2 cos θ2 + hr2 sin θ2 r1 dr1 dθ1 ,
(5)
where r1 , θ1 and r2 , θ2 are the radial and the azimuth angle coordinates in the input and output planes, respectively. Similar to the operation in Ref. [17], by introducing a new angle parameter φ, which satisfies following relations r2 cos θ2 − e/2 , cos φ = (r2 cos θ2 − e/2)2 + (r2 sin θ2 − f/2)2 r2 sin θ2 − f/2 sin φ = . (r2 cos θ2 − e/2)2 + (r2 sin θ2 − f/2)2
(6) (7)
Then Eq. (5) can also be expressed in following form E2 (r2 , θ2 , z) ∞2π ik E1 (r1 , θ1 , 0) = 2πb 0 0
ik 2 × exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b
ika 2 × exp − r 2b 1
ikr1 2 (e2 + f 2 ) × exp r2 − er2 cos θ2 − f r2 sin θ2 + b 4 × cos(θ1 − φ) r1 dr1 dθ1 . (8) In the cylindrical coordinate system, the electric field of a threedimensional circular flattened Gaussian beam (FGB) is expressed as follows [1,2] 2 n N
r12 r1 1 EN (r1 , θ1 , 0) = G0 (9) exp − , 2 2 n! w0N w0N n=0 √ where w0N = w0 / N + 1. N is the order of FGB, G0 is a constant and is set to unity in the following text. When N = 0, Eq. (9) reduces to a Gaussian beam with beam waist size w0 . Fig. 2 shows the three-dimensional (3D) normalized irradiance
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L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
Fig. 2. 3D normalized irradiance distribution of an FGB for two different values of beam order N . (a) N = 5, (b) N = 15.
distribution of a flattened Gaussian beam for two different values of beam order N with w0 = 1 mm. It is clear from Fig. 2 the beam profile becomes more flattened as N increases. Substituting Eq. (9) into Eq. (8), and applying following integral formulas [18] 2π
1 J0 (x) = 2π
(10)
exp(ix cos θ ) dθ, 0
∞
exp(−pt)t v/2+n Jv 2α 1/2 t 1/2 dt = n!α
p
In this section, we study the propagation of an FGB through a paraxial circularly apertured misaligned optical system. Applying Eq. (8), we obtain the following generalized diffraction integral formula for treating the propagation of a laser beam through a circularly apertured misaligned ABCD optical system E2 (r2 , θ2 , z)
0 v/2 −(n+v+1)
3. Propagation of a flattened Gaussian beam through a apertured misaligned paraxial ABCD optical system
exp(−α/p)Lvn (α/p).
(11)
ik = 2πb
ik 2 × exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b
ika 2 × exp − r 2b 1
ikr1 2 (e2 + f 2 ) × exp r2 − er2 cos θ2 − f r2 sin θ2 + b 4 × cos(θ1 − φ) r1 dr1 dθ1 , (13)
EN (r2 , θ2 , z)
× Ln
k2 4b2
w0N
r22 − er2 cos θ2 − f r2 sin θ2 + 1 ika 2 + 2b
e2 +f 2 4
.
(12)
w0N
Eq. (12) is an analytical formula for an FGB propagating through a misaligned paraxial ABCD optical system. When εx = εx = εy = εy = 0 (or e = f = g = h = 0), Eq. (12) reduces to the propagation formula for an FGB propagating through an aligned ABCD optical system. From Eq. (12), one sees that an FGB becomes decentered (or off-axis) after propagating through a misaligned optical system.
E1 (r1 , θ1 , 0) 0 0
After tedious but straightforward integration, we obtain
N ik 1 ika −n−1 1 = + 2 2b 2b w 2n w0N n=0 0N
ik 2 × exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b 2 2
k2 2 r − er2 cos θ2 − f r2 sin θ2 + e +f 4 4b2 2 × exp − 1 ika 2 + 2b
h1 2π
where h1 is the radius of the aperture. By introducing the following hard aperture function 1, |r1 | h1 , H (r1 ) = 0, |r1 | > h1 , Eq. (13) becomes E2 (r2 , θ2 , z) =
ik 2πb
∞2π E1 (r1 , θ1 , 0)H (r1 ) 0 0
ik 2 × exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b
(14)
L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
397
ika 2 × exp − r 2b 1
ikr1 2 (e2 + f 2 ) × exp r2 − er2 cos θ2 − f r2 sin θ2 + b 4 × cos(θ1 − φ) r1 dr1 dθ1 . (15) The hard aperture function can be expanded as the following sum (finite terms) of complex Gaussian functions [19–21] P
Bp 2 Ap exp − 2 r1 , H (r1 ) = (16) h1 p=1 where Ap and Bp are the expansion and Gaussian coefficients, which can be obtained by numerical optimization directly [19, 20], a table of Ap and Bp can be found in Refs. [19] and [20]. Numerical results have shown that the simulation accuracy improves as P increases. For a hard aperture, P = 10 assures a very good description of the diffracted beam (or a very good agreement with the straightforward diffraction integration) in the range from ∼0.12 times the Fresnel distance to the infinity, and discrepancies exist only in the extreme near field (<0.12 times the Fresnel distance) [19,20]. Substituting Eqs. (9) and (16) into Eq. (15), after tedious but straightforward integration, we obtain
× Ln
k2 4b2
w0N
r22
h1
− er2 cos θ2 − f r2 sin θ2 + 1 Bp ika 2 + 2b + 2 w0N
e2 +f 2 4
.
through a circularly apertured misaligned thin lens with a lateral displacement as shown in Fig. 3. The displacements and angle misalignment of the lens with respect to the optical axis of the system are εx , εy = εx = εy = 0. The thin lens is located at z = 0 and the output plane is located at z. The transfer matrix elements of the aligned optical system between the input plane and the output plane take the following form: a = 1 − z/f,
b = z,
c = −1/f,
d = 1,
(18)
where f is the focal length of the thin lens. The misalignment parameters αT , βT , γT and δT take the following form: αT =
EN (r2 , θ2 , z)
ik ik 2 = exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b 2b P N
Ap 1 ika Bp −n−1 × + + 2 2b h21 w 2n w0N n=0 p=1 0N 2 2
k2 2 r − er2 cos θ2 − f r2 sin θ2 + e +f 4 4b2 2 × exp − 1 Bp ika 2 + 2b + 2
Fig. 3. An optical system including a misaligned lens.
z , f
βT = 0,
γT =
1 , f
δT = 0
(19)
and the corresponding e, f , g and h are given by: e=2
(17)
h1
Eq. (17) is an approximate analytical propagation formula for a circular FGB passing through a circularly apertured misaligned ABCD optical system. When the radius of the aperture h1 → ∞, Eq. (17) reduces to the formula for an FGB passing through an unapertured misaligned ABCD optical system (Eq. (12) in Section 2). When εx = εx = εy = εy = 0 (or e = f = g = h = 0), Eq. (17) reduces to Eq. (6) of Ref. [9], which is the approximate propagation formula for an circular FGB propagating through an apertured aligned ABCD optical system. 4. A numerical example Eq. (12) and Eq. (17) are the main results of this Letter, they provide a convenient way for treating the propagation of a threedimensional circular FGB through an unapertured or apertured misaligned ABCD optical system. In this section, as a numerical example, we study the propagation properties of an FGB
zεx , f
f = 0,
g = 0,
h = 0.
(20)
Substituting Eqs. (18)–(20) into Eq. (17), we calculate in Fig. 4 the 3D normalized irradiance distribution of an FGB at z = 1.5f = 30 mm after passing through an apertured misaligned thin lens for different values of the aperture’s radius h1 with λ = 632.8 nm, N = 5, εx = 1 mm, f = 20 mm and w0 = 1 mm. In Fig. 5, we calculate the cross line (y = 0) of the normalized irradiance distribution of an FGB at several propagation distances after passing through an apertured misaligned thin lens with λ = 632.8 nm, h1 = 0.5 mm, N = 5, εx = 1 mm, f = 20 mm and w0 = 1 mm by using the analytical formula Eq. (17) (solid line). For comparison, the results calculated by integrating Eq. (13) numerically are also shown in Fig. 5 (dotted line), and the numerical calculations were carried out in the environment of MATLAB using the algorithm named successive adaptive Simpson rule where the truncation error is assumed to be 10−6 . One sees from Fig. 4 and Fig. 5 that the apertured misaligned thin lens has strong influences on the irradiance distribution of the focused FGB. The focused beam spot becomes off-axis, and the displacement of the beam’s center increases as the propagation distances increases. Under the influence of the aperture, the flattened beam profile disappears. When the radius of the aperture is enough small, diffraction pattern appears (see Fig. 4(a)). From Fig. 5, we also can find that the results calculated by using the analytical formula Eq. (17) are in a good agreement with that calculated by integrating Eq. (13) numerically. For convenience of quantitative comparison, similar to
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L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
(a)
(b)
(c)
(d)
Fig. 4. 3D normalized irradiance distribution of an FGB at z = 1.5f = 30 mm after passing through an apertured misaligned thin lens for different values of the aperture’s radius (a) h1 = 0.1 mm, (b) h1 = 0.5 mm, (c) h1 = 1.5 mm, (d) h1 = 10 mm.
(a)
(b)
Fig. 5. 3D normalized irradiance distribution of an FGB at several propagation distances after passing through an apertured misaligned thin lens (a) z = 1.5f = 30 mm, (b) z = 5f = 100 mm.
L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
that in Ref. [22], we introduce a new parameter called the average error percentage EA to denote the difference between the results calculated by the two methods, and EA is defined as follows S Is −Ias s=1 Is EA = (21) , S where S denotes the number of calculating points for plotting the lines in Fig. 5, Is and Ias denote the irradiances of an arbitrary point calculated by the direct numerical integration (Eq. (13)) and by approximate analytical formula (Eq. (17)), respectively. The average error percentage EA is about 1.87% for Fig. 5(a), and about 1.34% for Fig. 5(b). So our formulas provide a good description for the diffracted field. 5. Summary In conclusion, we have derived some analytical formula for an FGB propagating through a misaligned paraxial ABCD optical system in the cylindrical coordinate system based on the generalized diffraction integral formula. Furthermore, we have also derived an approximate analytical formula for an FGB propagating through an apertured misaligned ABCD optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. The results obtained by the approximate analytical formula are in a good agreement with those obtained by using the numerical integral calculation. Our formulas provide a convenient and effective way for treating the propagation of an FGB through an apertured or unapertured misaligned ABCD optical system.
399
Acknowledgements This work is partially supported by the National Basic Research Program (973) of China (2004CB719800). References [1] F. Gori, Opt. Commun. 107 (1994) 335. [2] V. Bagini, R. Borghi, F. Gori, A.M. Pacileo, M. Santarsiero, J. Opt. Soc. Am. A 13 (1996) 1385. [3] S.A. Amarande, Opt. Commun. 129 (1996) 311. [4] M. Santarsiero, R. Borghi, J. Opt. Soc. Am. A 16 (1999) 188. [5] R. Borghi, M. Santarsiero, S. Vicalvi, Opt. Commun. 154 (1998) 243. [6] D. Aiello, R. Borgh, M. Santarsiero, S. Vicalvi, Optik 109 (1998) 97. [7] Y. Cai, Q. Lin, J. Opt. A: Pure Appl. Opt. 5 (2003) 272. [8] Q. Lin, Y. Cai, Optik 113 (2002) 167. [9] B. Lü, S. Luo, J. Mod. Opt. 15 (2001) 2169. [10] M. Shen, S. Wang, D. Zhao, Optik 115 (2004) 193. [11] M. Ibnchaikh, A. Belafhal, Opt. Commun. 193 (2001) 73. [12] N. Zhou, G. Zeng, L. Hu, Opt. Commun. 240 (2004) 299. [13] C. Zheng, Optik 116 (2005) 563. [14] H. Jiang, D. Zhao, Z. Mei, Opt. Commun. 260 (2006) 1. [15] S. Wang, L. Ronchi, in: E. Wolf (Ed.), Progress in Optics, vol. 25, NorthHolland, Amsterdam, 1988. [16] Y. Cai, Q. Lin, J. Opt. Soc. Am. A 21 (2004) 1058. [17] G. Fan, D. Zhao, S. Wang, J. Opt. Mod. 52 (2005) 61. [18] A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms, McGraw–Hill, New York, 1954. [19] J.J. Wen, M.A. Breazeale, J. Acoust. Soc. Am. 83 (1988) 1752. [20] D. Ding, X. Liu, J. Opt. Soc. Am. A 16 (1999) 1286. [21] Y. Cai, L. Hu, Opt. Lett. 31 (2006) 685. [22] Z. Mei, D. Zhao, J. Opt. Soc. Am. A 21 (2004) 2375.
Analytical formula for a circular flattened Gaussian beam propagating through a misaligned paraxial ABCD optical system Li Hu, Yangjian Cai ∗ Joint Research Center of Photonics of the Royal Institute of Technology and Zhejiang University, East Building No.5, Zijingang Campus, Zhejiang University, Hangzhou 310058, China Division of Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology, 10044 Stockholm, Sweden Received 17 April 2006; received in revised form 26 July 2006; accepted 9 August 2006 Available online 18 August 2006 Communicated by R. Wu
Abstract Based on the generalized diffraction integral formula for treating the propagation of a laser beam through a misaligned paraxial ABCD optical system in the cylindrical coordinate system, analytical formula for a circular flattened Gaussian beam propagating through such optical system is derived. Furthermore, an approximate analytical formula is derived for a circular flattened Gaussian beam propagating through an apertured misaligned ABCD optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Numerical examples are given. © 2006 Elsevier B.V. All rights reserved. PACS: 42.25.Bs; 41.85.Ew Keywords: Flattened Gaussian beam; Misaligned ABCD optical system; Propagation
1. Introduction Laser beams with flat-topped spatial profiles are required in many applications, such as material thermal (uniform) processing, inertial confinement fusion, etc. Flattened Gaussian beam (FGB) proposed by Gori in 1994 is a typical and well-known model for describe a laser beam with flat-topped profile [1]. An FGB can be expressed as a finite sum of Laguerre–Gaussian modes or Hermite–Gaussian modes [1,2]. The propagation property of an FGB has been investigated widely. Amarande et al. studied the propagation factor and Kurtosis parameter of an FGB [3]. Santarsiero et al. compared an FBG with an SGB, and found that an FGB and a super Gaussian beam have nearly the same properties if they have the same propagation factor [4]. The focusing properties of an FGB such as the focal shift and integrated intensity were also studied [5,6]. We have studied the properties of an FGB in the fractional Fourier transform plane [7], and proposed the elliptical FGB in a ten* Corresponding author.
E-mail address: [email protected] (Y. Cai). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.029
sor form to describe a non-circular flat-topped beam [8]. Lü et al. have derived the approximate propagation equation for a circular FGB through an apertured aligned ABCD optical system [9]. Shen et al. have studied the propagation of a twodimensional FGB through a misaligned optical system with finite aperture [10]. Closed-form expressions for a two dimensional FGB propagating through an apertured ABCD optical system in the rectangular coordinate system were derived by Ibnchaikh et al. [11]. Zhou et al. have investigated the algorithms for two-dimensional FGBs passing through apertured and unapertured paraxial ABCD optical systems in the rectangular coordinate system [12]. Zheng have studied the propagation properties of an elliptical FGB in spatial-frequency domain [13]. More recently, Jiang et al. have studied the propagation characteristics of the rectangular flattened Gaussian beams through an apertured misaligned optical system [14]. In practice, most optical systems are slightly misaligned more or less. Thus, it is necessary to study the propagation of a laser beam through a slightly misaligned optical system [15,16]. In this Letter, we derive analytical formulas for a three-dimensional circular flattened Gaussian beam propa-
L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
gating through unapertured and apertured misaligned ABCD optical systems based on the generalized diffraction integral formulas for treating the propagation of a laser beam through such optical systems in the cylindrical coordinate system, and some numerical examples are given. 2. Propagation of a flattened Gaussian beam through a misaligned paraxial ABCD optical system Fig. 1 shows the misalignment diagram for a two-dimensional forward-going optical system [15]. In Fig. 1, RP1 and RP2 are alignment reference planes. RP1m and RP2m are misalignment reference planes, εx , εx , εy and εy denote the twodimensional slight misalignment parameters, εx and εy are the displacement element in x and y directions respectively, εx and εy are the tilted angles of the element in x and y directions respectively. a, b, c and d are the transfer matrix elements of an aligned optical system. z and zm are the aligned optical axis and misaligned optical axis, respectively. The generalized diffraction integral formula for treating the propagation of a laser beam through a misaligned paraxial ABCD optical system in the rectangular coordinate can be expressed as follows [15] E2 (x2 , y2 , z) ∞ ∞ ik E1 (x1 , y1 , 0) = 2πb −∞ −∞ ik 2 × exp − a x1 + y12 − 2(x1 x2 + y1 y2 ) + d x22 + y22 2b + ex1 + fy1 + gx2 + hy2 dx1 dy1 . (1) In Eq. (1), the phase factor exp(ikl0 ) along the axis between the two reference planes has been omitted. k = 2π/λ is the wave number, λ is the wavelength. 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: e = 2(αT εx + βT εx ),
f = 2(αT εy + βT εy ),
(2)
g = 2(bγT − dαT )εx + 2(bδT − dβT )εx , h = 2(bγT − dαT )εy + 2(bδT − dβT )εy ,
(3)
Fig. 1. Misalignment diagram for a two-dimensional forward-going system.
395
where αT , βT , γT and δT represent the misaligned matrix elements determined by αT = 1 − a,
βT = l − b,
γT = −c,
δT = 1 − d, (4) where l is the axial distance from the input plane and output plane. In the cylindrical coordinate system, we can express Eq. (1) as follows E2 (r2 , θ2 , z) ∞2π ik = E1 (r1 , θ1 , 0) 2πb 0 0 ik 2 × exp − ar − 2r1 r2 cos θ1 cos θ2 2b 1 − 2r1 r2 sin θ1 sin θ2 + dr22 + er1 cos θ1 + f r1 sin θ1 + gr2 cos θ2 + hr2 sin θ2 r1 dr1 dθ1 ,
(5)
where r1 , θ1 and r2 , θ2 are the radial and the azimuth angle coordinates in the input and output planes, respectively. Similar to the operation in Ref. [17], by introducing a new angle parameter φ, which satisfies following relations r2 cos θ2 − e/2 , cos φ = (r2 cos θ2 − e/2)2 + (r2 sin θ2 − f/2)2 r2 sin θ2 − f/2 sin φ = . (r2 cos θ2 − e/2)2 + (r2 sin θ2 − f/2)2
(6) (7)
Then Eq. (5) can also be expressed in following form E2 (r2 , θ2 , z) ∞2π ik E1 (r1 , θ1 , 0) = 2πb 0 0
ik 2 × exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b
ika 2 × exp − r 2b 1
ikr1 2 (e2 + f 2 ) × exp r2 − er2 cos θ2 − f r2 sin θ2 + b 4 × cos(θ1 − φ) r1 dr1 dθ1 . (8) In the cylindrical coordinate system, the electric field of a threedimensional circular flattened Gaussian beam (FGB) is expressed as follows [1,2] 2 n N
r12 r1 1 EN (r1 , θ1 , 0) = G0 (9) exp − , 2 2 n! w0N w0N n=0 √ where w0N = w0 / N + 1. N is the order of FGB, G0 is a constant and is set to unity in the following text. When N = 0, Eq. (9) reduces to a Gaussian beam with beam waist size w0 . Fig. 2 shows the three-dimensional (3D) normalized irradiance
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L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
Fig. 2. 3D normalized irradiance distribution of an FGB for two different values of beam order N . (a) N = 5, (b) N = 15.
distribution of a flattened Gaussian beam for two different values of beam order N with w0 = 1 mm. It is clear from Fig. 2 the beam profile becomes more flattened as N increases. Substituting Eq. (9) into Eq. (8), and applying following integral formulas [18] 2π
1 J0 (x) = 2π
(10)
exp(ix cos θ ) dθ, 0
∞
exp(−pt)t v/2+n Jv 2α 1/2 t 1/2 dt = n!α
p
In this section, we study the propagation of an FGB through a paraxial circularly apertured misaligned optical system. Applying Eq. (8), we obtain the following generalized diffraction integral formula for treating the propagation of a laser beam through a circularly apertured misaligned ABCD optical system E2 (r2 , θ2 , z)
0 v/2 −(n+v+1)
3. Propagation of a flattened Gaussian beam through a apertured misaligned paraxial ABCD optical system
exp(−α/p)Lvn (α/p).
(11)
ik = 2πb
ik 2 × exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b
ika 2 × exp − r 2b 1
ikr1 2 (e2 + f 2 ) × exp r2 − er2 cos θ2 − f r2 sin θ2 + b 4 × cos(θ1 − φ) r1 dr1 dθ1 , (13)
EN (r2 , θ2 , z)
× Ln
k2 4b2
w0N
r22 − er2 cos θ2 − f r2 sin θ2 + 1 ika 2 + 2b
e2 +f 2 4
.
(12)
w0N
Eq. (12) is an analytical formula for an FGB propagating through a misaligned paraxial ABCD optical system. When εx = εx = εy = εy = 0 (or e = f = g = h = 0), Eq. (12) reduces to the propagation formula for an FGB propagating through an aligned ABCD optical system. From Eq. (12), one sees that an FGB becomes decentered (or off-axis) after propagating through a misaligned optical system.
E1 (r1 , θ1 , 0) 0 0
After tedious but straightforward integration, we obtain
N ik 1 ika −n−1 1 = + 2 2b 2b w 2n w0N n=0 0N
ik 2 × exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b 2 2
k2 2 r − er2 cos θ2 − f r2 sin θ2 + e +f 4 4b2 2 × exp − 1 ika 2 + 2b
h1 2π
where h1 is the radius of the aperture. By introducing the following hard aperture function 1, |r1 | h1 , H (r1 ) = 0, |r1 | > h1 , Eq. (13) becomes E2 (r2 , θ2 , z) =
ik 2πb
∞2π E1 (r1 , θ1 , 0)H (r1 ) 0 0
ik 2 × exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b
(14)
L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
397
ika 2 × exp − r 2b 1
ikr1 2 (e2 + f 2 ) × exp r2 − er2 cos θ2 − f r2 sin θ2 + b 4 × cos(θ1 − φ) r1 dr1 dθ1 . (15) The hard aperture function can be expanded as the following sum (finite terms) of complex Gaussian functions [19–21] P
Bp 2 Ap exp − 2 r1 , H (r1 ) = (16) h1 p=1 where Ap and Bp are the expansion and Gaussian coefficients, which can be obtained by numerical optimization directly [19, 20], a table of Ap and Bp can be found in Refs. [19] and [20]. Numerical results have shown that the simulation accuracy improves as P increases. For a hard aperture, P = 10 assures a very good description of the diffracted beam (or a very good agreement with the straightforward diffraction integration) in the range from ∼0.12 times the Fresnel distance to the infinity, and discrepancies exist only in the extreme near field (<0.12 times the Fresnel distance) [19,20]. Substituting Eqs. (9) and (16) into Eq. (15), after tedious but straightforward integration, we obtain
× Ln
k2 4b2
w0N
r22
h1
− er2 cos θ2 − f r2 sin θ2 + 1 Bp ika 2 + 2b + 2 w0N
e2 +f 2 4
.
through a circularly apertured misaligned thin lens with a lateral displacement as shown in Fig. 3. The displacements and angle misalignment of the lens with respect to the optical axis of the system are εx , εy = εx = εy = 0. The thin lens is located at z = 0 and the output plane is located at z. The transfer matrix elements of the aligned optical system between the input plane and the output plane take the following form: a = 1 − z/f,
b = z,
c = −1/f,
d = 1,
(18)
where f is the focal length of the thin lens. The misalignment parameters αT , βT , γT and δT take the following form: αT =
EN (r2 , θ2 , z)
ik ik 2 = exp − dr2 + gr2 cos θ2 + hr2 sin θ2 2b 2b P N
Ap 1 ika Bp −n−1 × + + 2 2b h21 w 2n w0N n=0 p=1 0N 2 2
k2 2 r − er2 cos θ2 − f r2 sin θ2 + e +f 4 4b2 2 × exp − 1 Bp ika 2 + 2b + 2
Fig. 3. An optical system including a misaligned lens.
z , f
βT = 0,
γT =
1 , f
δT = 0
(19)
and the corresponding e, f , g and h are given by: e=2
(17)
h1
Eq. (17) is an approximate analytical propagation formula for a circular FGB passing through a circularly apertured misaligned ABCD optical system. When the radius of the aperture h1 → ∞, Eq. (17) reduces to the formula for an FGB passing through an unapertured misaligned ABCD optical system (Eq. (12) in Section 2). When εx = εx = εy = εy = 0 (or e = f = g = h = 0), Eq. (17) reduces to Eq. (6) of Ref. [9], which is the approximate propagation formula for an circular FGB propagating through an apertured aligned ABCD optical system. 4. A numerical example Eq. (12) and Eq. (17) are the main results of this Letter, they provide a convenient way for treating the propagation of a threedimensional circular FGB through an unapertured or apertured misaligned ABCD optical system. In this section, as a numerical example, we study the propagation properties of an FGB
zεx , f
f = 0,
g = 0,
h = 0.
(20)
Substituting Eqs. (18)–(20) into Eq. (17), we calculate in Fig. 4 the 3D normalized irradiance distribution of an FGB at z = 1.5f = 30 mm after passing through an apertured misaligned thin lens for different values of the aperture’s radius h1 with λ = 632.8 nm, N = 5, εx = 1 mm, f = 20 mm and w0 = 1 mm. In Fig. 5, we calculate the cross line (y = 0) of the normalized irradiance distribution of an FGB at several propagation distances after passing through an apertured misaligned thin lens with λ = 632.8 nm, h1 = 0.5 mm, N = 5, εx = 1 mm, f = 20 mm and w0 = 1 mm by using the analytical formula Eq. (17) (solid line). For comparison, the results calculated by integrating Eq. (13) numerically are also shown in Fig. 5 (dotted line), and the numerical calculations were carried out in the environment of MATLAB using the algorithm named successive adaptive Simpson rule where the truncation error is assumed to be 10−6 . One sees from Fig. 4 and Fig. 5 that the apertured misaligned thin lens has strong influences on the irradiance distribution of the focused FGB. The focused beam spot becomes off-axis, and the displacement of the beam’s center increases as the propagation distances increases. Under the influence of the aperture, the flattened beam profile disappears. When the radius of the aperture is enough small, diffraction pattern appears (see Fig. 4(a)). From Fig. 5, we also can find that the results calculated by using the analytical formula Eq. (17) are in a good agreement with that calculated by integrating Eq. (13) numerically. For convenience of quantitative comparison, similar to
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L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
(a)
(b)
(c)
(d)
Fig. 4. 3D normalized irradiance distribution of an FGB at z = 1.5f = 30 mm after passing through an apertured misaligned thin lens for different values of the aperture’s radius (a) h1 = 0.1 mm, (b) h1 = 0.5 mm, (c) h1 = 1.5 mm, (d) h1 = 10 mm.
(a)
(b)
Fig. 5. 3D normalized irradiance distribution of an FGB at several propagation distances after passing through an apertured misaligned thin lens (a) z = 1.5f = 30 mm, (b) z = 5f = 100 mm.
L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399
that in Ref. [22], we introduce a new parameter called the average error percentage EA to denote the difference between the results calculated by the two methods, and EA is defined as follows S Is −Ias s=1 Is EA = (21) , S where S denotes the number of calculating points for plotting the lines in Fig. 5, Is and Ias denote the irradiances of an arbitrary point calculated by the direct numerical integration (Eq. (13)) and by approximate analytical formula (Eq. (17)), respectively. The average error percentage EA is about 1.87% for Fig. 5(a), and about 1.34% for Fig. 5(b). So our formulas provide a good description for the diffracted field. 5. Summary In conclusion, we have derived some analytical formula for an FGB propagating through a misaligned paraxial ABCD optical system in the cylindrical coordinate system based on the generalized diffraction integral formula. Furthermore, we have also derived an approximate analytical formula for an FGB propagating through an apertured misaligned ABCD optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. The results obtained by the approximate analytical formula are in a good agreement with those obtained by using the numerical integral calculation. Our formulas provide a convenient and effective way for treating the propagation of an FGB through an apertured or unapertured misaligned ABCD optical system.
399
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