Physics Letters A 360 (2006) 394–399

www.elsevier.com/locate/pla

Analytical formula for a circular flattened Gaussian beam propagatingthrough 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 2006Available 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 opticalsystem in the cylindrical coordinate system, analytical formula for a circular flattened Gaussian beam propagating through such optical systemis derived. Furthermore, an approximate analytical formula is derived for a circular flattened Gaussian beam propagating through an aperturedmisaligned ABCD optical system by expanding the hard aperture function as a finite sum of complex Gaussian functions. Numerical examplesare 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 inmany applications, such as material thermal (uniform) process-ing, inertial confinement fusion, etc. Flattened Gaussian beam(FGB) proposed by Gori in 1994 is a typical and well-knownmodel for describe a laser beam with flat-topped profile [1]. AnFGB can be expressed as a finite sum of Laguerre–Gaussianmodes or Hermite–Gaussian modes [1,2]. The propagationproperty of an FGB has been investigated widely. Amarandeet al. studied the propagation factor and Kurtosis parameterof an FGB [3]. Santarsiero et al. compared an FBG with anSGB, and found that an FGB and a super Gaussian beam havenearly the same properties if they have the same propagationfactor [4]. The focusing properties of an FGB such as the fo-cal shift and integrated intensity were also studied [5,6]. Wehave studied the properties of an FGB in the fractional Fouriertransform plane [7], and proposed the elliptical FGB in a ten-

* Corresponding author.E-mail address: yangjian_cai@yahoo.com.cn (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]. Lüet al. have derived the approximate propagation equation fora circular FGB through an apertured aligned ABCD opticalsystem [9]. Shen et al. have studied the propagation of a two-dimensional FGB through a misaligned optical system withfinite aperture [10]. Closed-form expressions for a two dimen-sional FGB propagating through an apertured ABCD opticalsystem in the rectangular coordinate system were derived by Ib-nchaikh et al. [11]. Zhou et al. have investigated the algorithmsfor two-dimensional FGBs passing through apertured and un-apertured paraxial ABCD optical systems in the rectangular co-ordinate system [12]. Zheng have studied the propagation prop-erties of an elliptical FGB in spatial-frequency domain [13].More recently, Jiang et al. have studied the propagation charac-teristics of the rectangular flattened Gaussian beams through anapertured misaligned optical system [14].

In practice, most optical systems are slightly misalignedmore or less. Thus, it is necessary to study the propagationof a laser beam through a slightly misaligned optical sys-tem [15,16]. In this Letter, we derive analytical formulas fora three-dimensional circular flattened Gaussian beam propa-

L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399 395

gating through unapertured and apertured misaligned ABCDoptical systems based on the generalized diffraction integralformulas for treating the propagation of a laser beam throughsuch optical systems in the cylindrical coordinate system, andsome numerical examples are given.

2. Propagation of a flattened Gaussian beam througha misaligned paraxial ABCD optical system

Fig. 1 shows the misalignment diagram for a two-dimen-sional forward-going optical system [15]. In Fig. 1, RP1 andRP2 are alignment reference planes. RP1m and RP2m are mis-alignment reference planes, εx , ε′

x , εy and ε′y denote the two-

dimensional slight misalignment parameters, εx and εy are thedisplacement element in x and y directions respectively, ε′

x andε′y are the tilted angles of the element in x and y directions re-

spectively. a, b, c and d are the transfer matrix elements of analigned optical system. z and zm are the aligned optical axisand misaligned optical axis, respectively. The generalized dif-fraction integral formula for treating the propagation of a laserbeam through a misaligned paraxial ABCD optical system inthe rectangular coordinate can be expressed as follows [15]

E2(x2, y2, z)

= ik

2πb

∞∫−∞

∞∫−∞

E1(x1, y1,0)

× exp

{− ik

2b

[a(x2

1 + y21

) − 2(x1x2 + y1y2) + d(x2

2 + y22

)(1)+ ex1 + fy1 + gx2 + hy2

]}dx1 dy1.

In Eq. (1), the phase factor exp(ikl0) along the axis between thetwo reference planes has been omitted. k = 2π/λ is the wavenumber, λ is the wavelength. a, b, c and d are the transfer ma-trix elements of the aligned optical system. The parameters e,f , g and h take the following form:

(2)e = 2(αT εx + βT ε′x), f = 2(αT εy + βT ε′

y),

g = 2(bγT − dαT )εx + 2(bδT − dβT )ε′x,

(3)h = 2(bγT − dαT )εy + 2(bδT − dβT )ε′y,

Fig. 1. Misalignment diagram for a two-dimensional forward-going system.

where αT , βT , γT and δT represent the misaligned matrix ele-ments determined by

(4)αT = 1 − a, βT = l − b, γT = −c, δT = 1 − d,

where l is the axial distance from the input plane and outputplane.

In the cylindrical coordinate system, we can express Eq. (1)as follows

E2(r2, θ2, z)

= ik

2πb

∞∫0

2π∫0

E1(r1, θ1,0)

× exp

{− ik

2b

[ar2

1 − 2r1r2 cos θ1 cos θ2

− 2r1r2 sin θ1 sin θ2 + dr22 + er1 cos θ1

(5)+ f r1 sin θ1 + gr2 cos θ2 + hr2 sin θ2]}

r1 dr1 dθ1,

where r1, θ1 and r2, θ2 are the radial and the azimuth angle co-ordinates in the input and output planes, respectively. Similar tothe operation in Ref. [17], by introducing a new angle parame-ter φ, which satisfies following relations

(6)cosφ = r2 cos θ2 − e/2√(r2 cos θ2 − e/2)2 + (r2 sin θ2 − f/2)2

,

(7)sinφ = r2 sin θ2 − f/2√(r2 cos θ2 − e/2)2 + (r2 sin θ2 − f/2)2

.

Then Eq. (5) can also be expressed in following form

E2(r2, θ2, z)

= ik

2πb

∞∫0

2π∫0

E1(r1, θ1,0)

× exp

[− ik

2b

(dr2

2 + gr2 cos θ2 + hr2 sin θ2)]

× exp

[− ika

2br2

1

]

× exp

[ikr1

b

√r2

2 − er2 cos θ2 − f r2 sin θ2 + (e2 + f 2)

4

(8)× cos(θ1 − φ)

]r1 dr1 dθ1.

In the cylindrical coordinate system, the electric field of a three-dimensional circular flattened Gaussian beam (FGB) is ex-pressed as follows [1,2]

(9)EN(r1, θ1,0) = G0

N∑n=0

1

n!(

r21

w20N

)n

exp

(− r2

1

w20N

),

where w0N = w0/√

N + 1. N is the order of FGB, G0 is a con-stant 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

396 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 val-ues of beam order N with w0 = 1 mm. It is clear from Fig. 2the beam profile becomes more flattened as N increases.

Substituting Eq. (9) into Eq. (8), and applying following in-tegral formulas [18]

(10)J0(x) = 1

2π

2π∫0

exp(ix cos θ) dθ,

∞∫0

exp(−pt)tv/2+nJv

(2α1/2t1/2)dt

(11)= n!αv/2p−(n+v+1) exp(−α/p)Lvn(α/p).

After tedious but straightforward integration, we obtain

EN(r2, θ2, z)

= ik

2b

N∑n=0

1

w2n0N

(1

w20N

+ ika

2b

)−n−1

× exp

[− ik

2b

(dr2

2 + gr2 cos θ2 + hr2 sin θ2)]

× exp

[−

k2

4b2

(r2

2 − er2 cos θ2 − f r2 sin θ2 + e2+f 2

4

)( 1

w20N

+ ika2b

) ]

(12)× Ln

[ k2

4b2

(r2

2 − er2 cos θ2 − f r2 sin θ2 + e2+f 2

4

)( 1

w20N

+ ika2b

) ].

Eq. (12) is an analytical formula for an FGB propagatingthrough a misaligned paraxial ABCD optical system. Whenεx = ε′

x = εy = ε′y = 0 (or e = f = g = h = 0), Eq. (12)

reduces to the propagation formula for an FGB propagatingthrough an aligned ABCD optical system. From Eq. (12), onesees that an FGB becomes decentered (or off-axis) after propa-gating through a misaligned optical system.

3. Propagation of a flattened Gaussian beam througha apertured misaligned paraxial ABCD optical system

In this section, we study the propagation of an FGB througha paraxial circularly apertured misaligned optical system. Ap-plying Eq. (8), we obtain the following generalized diffractionintegral formula for treating the propagation of a laser beamthrough a circularly apertured misaligned ABCD optical sys-tem

E2(r2, θ2, z)

= ik

2πb

h1∫0

2π∫0

E1(r1, θ1,0)

× exp

[− ik

2b

(dr2

2 + gr2 cos θ2 + hr2 sin θ2)]

× exp

[− ika

2br2

1

]

× exp

[ikr1

b

√r2

2 − er2 cos θ2 − f r2 sin θ2 + (e2 + f 2)

4

(13)× cos(θ1 − φ)

]r1 dr1 dθ1,

where h1 is the radius of the aperture.By introducing the following hard aperture function

(14)H(r1) ={

1, |r1| � h1,

0, |r1| > h1,

Eq. (13) becomes

E2(r2, θ2, z)

= ik

2πb

∞∫0

2π∫0

E1(r1, θ1,0)H(r1)

× exp

[− ik (

dr22 + gr2 cos θ2 + hr2 sin θ2

)]

2b

L. Hu, Y. Cai / Physics Letters A 360 (2006) 394–399 397

× exp

[− ika

2br2

1

]

× exp

[ikr1

b

√r2

2 − er2 cos θ2 − f r2 sin θ2 + (e2 + f 2)

4

(15)× cos(θ1 − φ)

]r1 dr1 dθ1.

The hard aperture function can be expanded as the followingsum (finite terms) of complex Gaussian functions [19–21]

(16)H(r1) =P∑

p=1

Ap exp

(−Bp

h21

r21

),

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 im-proves as P increases. For a hard aperture, P = 10 assuresa very good description of the diffracted beam (or a very goodagreement with the straightforward diffraction integration) inthe range from ∼0.12 times the Fresnel distance to the infinity,and discrepancies exist only in the extreme near field (<0.12times the Fresnel distance) [19,20].

Substituting Eqs. (9) and (16) into Eq. (15), after tedious butstraightforward integration, we obtain

EN(r2, θ2, z)

= ik

2bexp

[− ik

2b

(dr2

2 + gr2 cos θ2 + hr2 sin θ2)]

×N∑

n=0

P∑p=1

Ap

w2n0N

(1

w20N

+ ika

2b+ Bp

h21

)−n−1

× exp

[−

k2

4b2

(r2

2 − er2 cos θ2 − f r2 sin θ2 + e2+f 2

4

)( 1

w20N

+ ika2b

+ Bp

h21

)]

(17)× Ln

[ k2

4b2

(r2

2 − er2 cos θ2 − f r2 sin θ2 + e2+f 2

4

)( 1

w20N

+ ika2b

+ Bp

h21

)].

Eq. (17) is an approximate analytical propagation formula fora circular FGB passing through a circularly apertured mis-aligned ABCD optical system. When the radius of the aper-ture h1 → ∞, Eq. (17) reduces to the formula for an FGBpassing through an unapertured misaligned ABCD optical sys-tem (Eq. (12) in Section 2). When εx = ε′

x = εy = ε′y = 0 (or

e = f = g = h = 0), Eq. (17) reduces to Eq. (6) of Ref. [9],which is the approximate propagation formula for an circularFGB propagating through an apertured aligned ABCD opticalsystem.

4. A numerical example

Eq. (12) and Eq. (17) are the main results of this Letter, theyprovide a convenient way for treating the propagation of a three-dimensional circular FGB through an unapertured or aperturedmisaligned ABCD optical system. In this section, as a numer-ical example, we study the propagation properties of an FGB

Fig. 3. An optical system including a misaligned lens.

through a circularly apertured misaligned thin lens with a lat-eral displacement as shown in Fig. 3. The displacements andangle misalignment of the lens with respect to the optical axisof the system are εx , εy = ε′

x = ε′y = 0. The thin lens is located

at z = 0 and the output plane is located at z. The transfer matrixelements of the aligned optical system between the input planeand the output plane take the following form:

(18)a = 1 − z/f, b = z, c = −1/f, d = 1,

where f is the focal length of the thin lens. The misalignmentparameters αT , βT , γT and δT take the following form:

(19)αT = z

f, βT = 0, γT = 1

f, δT = 0

and the corresponding e, f , g and h are given by:

(20)e = 2zεx

f, f = 0, g = 0, h = 0.

Substituting Eqs. (18)–(20) into Eq. (17), we calculate in Fig. 4the 3D normalized irradiance distribution of an FGB at z =1.5f = 30 mm after passing through an apertured misalignedthin lens for different values of the aperture’s radius h1 withλ = 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 normal-ized irradiance distribution of an FGB at several propagationdistances after passing through an apertured misaligned thinlens 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 formulaEq. (17) (solid line). For comparison, the results calculated byintegrating Eq. (13) numerically are also shown in Fig. 5 (dottedline), and the numerical calculations were carried out in the en-vironment of MATLAB using the algorithm named successiveadaptive Simpson rule where the truncation error is assumedto be 10−6. One sees from Fig. 4 and Fig. 5 that the aperturedmisaligned thin lens has strong influences on the irradiance dis-tribution of the focused FGB. The focused beam spot becomesoff-axis, and the displacement of the beam’s center increases asthe propagation distances increases. Under the influence of theaperture, the flattened beam profile disappears. When the radiusof the aperture is enough small, diffraction pattern appears (seeFig. 4(a)). From Fig. 5, we also can find that the results cal-culated by using the analytical formula Eq. (17) are in a goodagreement with that calculated by integrating Eq. (13) numer-ically. For convenience of quantitative comparison, similar to

398 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 theaperture’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) 394–399 399

that in Ref. [22], we introduce a new parameter called the av-erage error percentage EA to denote the difference between theresults calculated by the two methods, and EA is defined as fol-lows

(21)EA =∑S

s=1

∣∣( Is−IasIs

)∣∣S

,

where S denotes the number of calculating points for plottingthe lines in Fig. 5, Is and Ias denote the irradiances of anarbitrary 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 formulasprovide a good description for the diffracted field.

5. Summary

In conclusion, we have derived some analytical formula foran FGB propagating through a misaligned paraxial ABCD op-tical system in the cylindrical coordinate system based on thegeneralized diffraction integral formula. Furthermore, we havealso derived an approximate analytical formula for an FGBpropagating through an apertured misaligned ABCD opticalsystem by expanding the hard aperture function as a finite sumof complex Gaussian functions. The results obtained by theapproximate analytical formula are in a good agreement withthose obtained by using the numerical integral calculation. Ourformulas provide a convenient and effective way for treatingthe propagation of an FGB through an apertured or unaperturedmisaligned ABCD optical system.

Acknowledgements

This work is partially supported by the National Basic Re-search Program (973) of China (2004CB719800).

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