Approximate method for calculating the generalized beam propagation factor of truncated beams

Optics Communications 229 (2004) 1–10

www.elsevier.com/locate/optcom

Approximate method for calculating the generalizedbeam propagation factor of truncated beams

Bin Zhang *, Xiaoliang Chu, Qiao Wen, Qinggang Zeng

College of Electronics Information, Sichuan University, ChengDu 610064, China

Received 13 July 2003; received in revised form 18 September 2003; accepted 20 September 2003

Abstract

Based on the treatment that the rectangular function can be expanded into an approximate sum of complex

Gaussian functions with finite numbers, an approximate method for calculating the generalized beam propagation

factor (M2-factor) is proposed. A truncated flattened Gaussian beam (FGB) and a truncated cosh-Gaussian beam

(CGB) are taken as two examples of truncated laser beams. The analytical expressions for the generalized M2-factor are

derived. The typical numerical examples are given and compared to those obtained from numerical integration and

the known analytical method. A comparison of the results for the ideal beam models with the more realistic ones is

discussed.

� 2003 Elsevier B.V. All rights reserved.

PACS: 42.60.Jf; 42.25.Fx

Keywords: Generalized beam propagation factor; Truncated beams; Complex Gaussian functions; Flattened Gaussian beam; cosh-

Gaussian beam

1. Introduction

The beam-propagation factor (M2-factor) is a very useful beam parameter for characterizing laser beams

and can be regarded as a beam quality factor in many practical applications [1,2]. Because laser beams may

be more or less limited by the finite apertures of optical elements in many practical applications, study on

the truncated beams would be of practical interest. However, the M2-factor concept proposed by Siegmanet al. fails when the laser beams are truncated by the hard-edged optical elements, because the usual second-

order intensity moment in the space-frequency domain for a truncated beam becomes infinite. Martinez-

Herrero and Mejias [3,4] generalized the M2-factor concept to include the hard-edged diffraction case and

gave the integral formula with finite limit. In general, the analytical expressions for the generalized

* Corresponding author. Tel.: +86-028-85405363; fax: +86-028-85463871.

E-mail address: [email protected] (B. Zhang).

0030-4018/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2003.09.053

mail to: [email protected]

2 B. Zhang et al. / Optics Communications 229 (2004) 1–10

M2-factor of truncated beams are difficult to derive due to the finite integral limit, the numerical integral

calculation has to be performed and a large amount of computing time has to be taken. In this paper, based

on the treatment that the rectangular function can be expanded into an approximate sum of complex

Gaussian functions with finite numbers, an approximate calculation method for the generalized M2-factor

is proposed. A truncated flattened Gaussian beam (FGB) and a truncated cosh-Gaussian beam (CGB) are

taken as two examples of truncated laser beams. The analytical expressions for the generalized M2-factorare derived. The numerical calculation results are given and discussed.

2. Complex Gaussian function expansion of rectangular function

Wen and Breazeale [5] expanded the Circ function into a sum of complex Gaussian functions with finite

numbers in the cylindrical coordinate systems. In a similar way, the one-dimensional rectangular function

T ðxÞ ¼ 1 jxj < a;0 jxj > a

�ð1Þ

can also be expressed as a linear superposition of a set of complex Gaussian functions with finite numbers in

the rectangular coordinate systems, that is

T ðxÞ ¼XNj¼1

Aj exp

�� Bj

xa

� �2�; ð2Þ

where a is the half-width of the rectangular function, Aj and Bj are expansion and complex Gaussian

coefficients, respectively, and can be found by a computer optimization [5,6], N represents the number of

the expansion terms.

Eq. (2) implies that the rectangular function can be approximately expressed in terms of a superposition

of Gaussian function with different spot-size.

For the convenience of calculation, the rectangular function can also be written approximately as

T ðxÞ ¼ ReXNj¼1

Aj exp

�"� Bj

xa

� �2�#; ð3Þ

-1.5 -1.0 -0.5 0.0 0.5 1. .50.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

rect

real

T(x

)

Fig. 1. Real part of complex expansion for the rectangular function.

B. Zhang et al. / Optics Communications 229 (2004) 1–10 3

where Re represents the real part. Fig. 1 shows the real part of complex expansion for the rectangular

function evaluated by Eq. (3). The number of the complex Gaussian expansion terms is N ¼ 10, Aj and Bj

are shown in [6]. To examine the validity of this method, the curve of rectangular function was also given in

the figure.

3. Approximate method for calculating the generalized M2-factor of truncated beams

Let us assume that a laser beam is characterized by the field distribution whose domain is limited by a

hard-edged aperture with aperture width a. The second-order intensity moments in the spatial domain and

spatial frequency domain were given by Martinez-Herrero and Mejias [3,4].

hx2i ¼ 1

I0

Z a

�ax2jEðxÞj2 dx; ð4Þ

hu2i ¼ 1

k2I0

Z a

�ajE0ðxÞj2 dxþ 4ðjEðaÞj2 þ jEð�aÞj2Þ

k2I0a; ð5Þ

hxui ¼ 1

2ikI0

Z a

�ax½E0ðxÞ��EðxÞ�

� xE0ðxÞE�ðxÞdx; ð6Þ

where hx2i and hu2i are the second-order moments in the spatial and spatial frequency domains, respec-

tively, hxui denotes the second-order crossed moment, k is wave number, * denotes the complex conjugate,

the prime indicates derivation with respect to x and

I0 ¼Z a

�ajEðxÞj2 dx ð7Þ

is the total power entering through the aperture. It has been implicitly assumed that the first-order moments

are zero (this assumption can be simply realized by a shift in the coordinate system).

Recalling the definition of beam-quality parameter QG for the truncated beams introduced by Martinez-Herrero and Mejias [4]

QG ¼ hx2ihu2i � hxui2; ð8Þ
we can define the generalized M2-factor in the sense of generalized second-order moments from Eqs. (4)–(6)
in a way analogous to that with no hard-edged aperture

M2G ¼ 2k

ffiffiffiffiffiffiQG

p¼ 2k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihx2ihu2i � hxui2

q: ð9Þ

For the case of truncated beams, because of the finite integral limit in Eqs. (4)–(7), the analytical ex-

pressions for the generalized M2-factor of truncated beams are usually difficult to obtain by using the

conventional method. The numerical integral calculation has to be performed. However, the analytical

expressions for the generalized M2-factor of truncated beams can be obtained by using the complex

Gaussian function expansion of rectangular function and the integrability of the Gaussian function.

According to the mathematical method, Eqs. (4)–(7) can be written as the infinite integral of the productof the integrand in Eqs. (4)–(7) and the rectangular function T ðxÞ. On substituting from Eq. (3) into

Eqs. (4)–(7), we obtain

hx2i ¼ 1

I0Re

XNj¼1

Aj

Z 1

�1exp

�"� Bj

xa

� �2�x2jEðxÞj2 dx

#; ð10Þ


hu2i ¼ 1

k2I0Re

XNj¼1

Aj

Z 1

�1exp

�"� Bj

xa

� �2�jE0ðxÞj2 dx

#þ 4ðjEðaÞj2 þ jEð�aÞj2Þ

k2I0a; ð11Þ

hxui ¼ 1

2ikI0Re

XNj¼1

Aj

Z 1

�1exp

�"� Bj

xa

� �2�x E0ðxÞ� ��

EðxÞn

� xE0ðxÞE�ðxÞodx

#; ð12Þ

I0 ¼ ReXNj¼1

Aj

Z 1

�1exp

�"� Bj

xa

� �2�jEðxÞj2 dx

#: ð13Þ

It can be seen from Eqs. (10)–(13) that the second-order moments of truncated beams can be expressedas the infinite integral transform. By using the integrability of the Gaussian function, the analytical ex-

pressions of the generalized M2-factor of truncated laser beams can be derived.

4. Examples

4.1. Flattened Gaussian beams

The field distribution of a FGB in the plane of z ¼ 0 in the rectangular coordinate system is characterized

by [7] � �
EðxÞ ¼ exp
�� ðNFGB þ 1Þx2

w20

�XNFGB

k¼0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNFGB þ 1

pxw0

2k

k!; ð14Þ

where NFGB is the beam order and w0 is a positive beam parameter.Suppose that a hard-edged aperture with aperture width a is located at z ¼ 0. On substituting from

Eq. (14) into Eqs. (10)–(13), and using the integral formula [8]
Z þ1
0

tve�st dt ¼ v!s�v�1: ð15Þ

After some algebra calculation we get

I0 ¼w0

NFGB þ 1ð Þ1=2Re

XNj¼1

XNFGB

k¼0

XNFGB

l¼0

Aj

k!l!C k�"

þ lþ 1

2

�qðkþlþ1=2Þj

#; ð16Þ

hx2i ¼ w30

I0 NFGB þ 1ð Þ3=2Re

XNj¼1

XNFGB

k¼0

XNFGB

l¼0

Aj

k!l!C k�"

þ lþ 3

2

�qðkþlþ3=2Þj

#; ð17Þ

hu2i ¼ NFGB þ 1ð Þ1=2

k2I0w0

ReXNj¼1

XNFGB

k¼0

XNFGB

l¼0

Aj

k!l!4C k��"

þ lþ 3

2

�qðkþlþ3=2Þj � 8lC k

�þ lþ 1

2

�qðkþlþ1=2Þj

þ 4klC k�

þ l� 1

2

�qðkþl�1=2Þj

�þ 8

d NFGB þ 1ð Þ1=2exp

�� 2 NFGBð þ 1Þd2

�

�XNFGB

k¼0

XNFGB

l¼0

1

k!l!NFGBð þ 1Þkþld2ðkþlÞ

#; ð18Þ


hxui ¼ 0; ð19Þ

where C is the Gamma function,

qj ¼NFGB þ 1

Bj=d2 þ 2 NFGB þ 1ð Þ

ð20Þ

and

d ¼ a=w0 ð21Þ
is the beam truncation parameter.
On substituting from Eqs. (16)–(19) into Eq. (9), the generalized M2-factor of truncated FGBs passing

through the hard-edged apertures can be expressed as

M2G ¼ 2 Re

XNj¼1

XNFGB

k¼0

XNFGB

l¼0

Aj

k!l!C k�"(

þ lþ 1

2

�qðkþlþ1=2Þj

#)�1

� ReXNj¼1

XNFGB

k¼0

XNFGB

l¼0

Aj

k!l!C k�"(

þ lþ 3

2

�qðkþlþ3=2Þj

#)1=2

� ReXNj¼1

XNFGB

k¼0

XNFGB

l¼0

Aj

k!l!4C k��"(

þ lþ 3

2

�qðkþlþ3=2Þj � 8lC k

�þ lþ 1

2

�qðkþlþ1=2Þj

þ 4klC k�

þ l� 1

2

�qðkþl�1=2Þj

�þ 8

d NFGB þ 1ð Þ1=2exp

�� 2 NFGBð þ 1Þd2

�

�XNFGB

k¼0

XNFGB

l¼0

1

k!l!NFGBð þ 1Þkþld2ðkþlÞ

#)1=2

: ð22Þ

It can be shown from Eq. (22) that the generalized M2-factor of truncated FGBs not only depends on the

beam order NFGB, but also is related to the beam truncation parameter d.Two limiting case are considered:

(i) System with no aperture, i.e., d ! 1 Eq. (22) yields

qj ¼1

2; ð23Þ

M2G ¼ 2 Re

XNj¼1

XNFGB

k¼0

XNFGB

l¼0

Aj

k!l!2�ðkþlþ1=2ÞC k

�"(þ lþ 1

2

�#)�1

� ReXNj¼1

XNFGB

k¼0

XNFGB

l¼0

Aj

k!l!2�ðkþlþ3=2ÞC k

�"(þ lþ 3

2

�#)1=2

� ReXNj¼1

XNFGB

k¼0

XNFGB

l¼0

Aj

k!l!2�ðkþl�1=2ÞC k

��"(þ lþ 3

2

�

� 2�ðkþl�5=2ÞlC k�

þ lþ 1

2

�þ 2�ðkþl�5=2ÞklC k

�þ l� 1

2

��#)1=2

; ð24Þ

which corresponds to the generalized M2-factor of nontruncated FGBs.


It can be shown from Eq. (24) that the generalized M2-factor of unapertured FGBs only depends on the

beam order NFGB.

(ii) Truncated fundamental Gaussian beams, i.e. NFGB ¼ 0, Eq. (22) yields

M2G ¼ 2 Re

XNj¼1

Aj Bj=d2

"(24 þ 2��3=2

! XNj¼1

Aj Bj=d2

þ 2��3=2 þ 4 expð�2d2Þffiffiffi

pp

d

!#)1=235

ReXNj¼1

Aj Bj=d2

"",þ 2��1=2

##: ð25Þ

Eq. (25) indicates that the generalizedM2-factor of truncated fundamental Gaussian beams only depends

on the beam truncation parameter d.Furthermore, letting d ! þ1 in Eq. (25), we get

M2 ¼ 1; ð26Þ

which corresponds to the M2-factor of fundamental Gaussian beams in free space.

4.2. cosh-Gaussian beams

The field distribution of the two-dimensional CGBs at the plane of z ¼ 0 is characterized by [9]

EðxÞ ¼ exp

�� x2

w20

�coshðXxÞ; ð27Þ

where w0 is the waist width of the Gaussian amplitude distribution, X is the parameter associated with the

cosh part, and cosh denotes the hyperbolic cosine function.

Let us assume that a hard-edged aperture with aperture width a is placed in the plane of z ¼ 0. On

substituting from Eq. (27) into Eqs. (10)–(13), and using the definition of cosh function

coshðXxÞ ¼ eXx þ e�Xx

2ð28Þ

and integral formula (15), we obtain

I0 ¼ffiffiffip

pa

2Re

XNj¼1

Ajp1=2j

bdepjð

"þ 1Þ

#; ð29Þ

hx2i ¼ffiffiffip

pa3

4I0Re

XNj¼1

Ajp3=2j

b3d3epj 1 �"

þ 2pj�þ 1�#; ð30Þ

hu2i ¼ffiffiffip

pRe

XNj¼1

Ajp12j

bd2

epjð�""

� 1Þ þ dpjb3

epj 1 �

� 2b2 þ 2pj�þ 1��#

þ 8 cosh2 bdð Þ exp � 2d2

�#,k2I0a; ð31Þ

Fig. 2

order N


hxui ¼ 0; ð32Þ
where
pj ¼b2

Bj=d2 þ 2

; ð33Þ

b ¼ w0X ð34Þ
is the beam decentered parameter, and d is the beam truncation parameter that can be expressed as in
Eq. (21).

On substituting from Eqs. (29)–(32) into Eq. (9), the generalized M2-factor of truncated CGBs passing

through the hard-edged apertures can be expressed as

M2G ¼ 2 Re

XNj¼1

Ajp1=2j

bdepjð

"(þ 1Þ

#)�1

ReXNj¼1

Ajp3=2j

b3d3epj 1 �"(

þ 2pj�þ 1�#)1=2

� ReXNj¼1

Ajp1=2j

bd2

epjð�"(

� 1Þ þ dpjb3

epj 1 �

� 2b2 þ 2pj�þ 1��#

þ 8 cosh2 bdð Þ exp � 2d2

�)1=2

:

ð35Þ

It can be seen from Eq. (35) that the generalized M2-factor not only depends on the beam truncation

parameter d, but also is related to the beam decentered parameter b.For the case of b ¼ 0, the result for the truncated fundamental Gaussian beams can also be obtained

from Eq. (35), which is consistent with Eq. (25).

5. Numerical calculation results and analysis

Numerical calculations were performed by using Eqs. (22) and (35). Some illustrative examples are

compiled in Figs. 2–5. Fig. 2 gives the generalized M2-factor of truncated FGBs as a function of beam

0.0 0.5 1.0 1.5 2.00.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

NFGB= 20

NFGB= 5NFGB= 0MG

2

. Generalized M2-factor of truncated flattened Gaussian beams as a function of beam truncation parameter d for different beam

FGB.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

1

2

3

4

5

6

β=3.0

β=2.0

β=0

MG

2

Fig. 4. Generalized M2-factor of truncated cosh-Gaussian beams as a function of beam truncation parameter d for different beam

decentered parameter b.

0 5 10 15 20

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

δ=2.0

δ= 0.6

δ= 0.1

MG

2

Fig. 3. Generalized M2-factor of truncated flattened Gaussian beams as a function of beam order NFGB for different beam truncation

parameter d.


truncation parameter d for different beam order NFGB. Fig. 3 represents variation of the generalized M2-factor of truncated FGBs with beam order NFGB for different beam truncation parameter d. Fig. 4 denotes

the generalized M2-factor of truncated CGBs as a function of beam truncation parameter d for different

beam decentered parameter b. Fig. 5 gives variation of the generalized M2-factor of truncated CGBs with

beam decentered parameter b for various values of beam truncation parameter d. For the convenience of

comparison, the corresponding results from the numerical integral calculation method are given in figures,

which are represented by the dashed lines. Moreover, a comparison between the analytical method given in

[10] and our approximate method for the truncated CGBs is performed. The corresponding results from

the analytical method is also given in Figs. 4 and 5, which are plotted by the dotted lines. Because themodels for FGBs and CGBs used in this paper refer to ideal cases, to consider the more realistic cases,

noise is added to the intensity value of each sample as a random number with uniform distribution in the

0.0 0.5 1.0 1.5 2.0 2.5 3.0

1

2

3

4

5

MG

2

δ=0.1

δ=0.5

δ=2.0

Fig. 5. GeneralizedM2-factor of truncated cosh-Gaussian beams as a function of decentered parameter b for different beam truncation

parameter d.


interval (�eI0; eI0), where I0 is the maximum value of the intensity profile and e is a positive quantity [11].

Possible negative values of the corrupted intensity are set to zero. The corresponding results for the more

realistic cases (e ¼ 0:10) are also complied in Figs. 2–5, which are represented by the triangles. The number

of the complex Gaussian expansion terms using in our calculations is N ¼ 10, Aj and Bj are shown in [6].

It can be seen from Figs. 2–5 that the generalized M2-factors calculated by using Eqs. (22) and (35)are in keeping with the results from the numerical integral calculation method quite well when the

number of complex Gaussian terms N is chosen to be 10. For example, for the FGBs in Figs. 2 and 3,

the relative errors between the results calculated by our method and numerical integral calculation

method is less than 1.5%. For the CGBs in Figs. 4 and 5, the maximum relative errors between the

results calculated by our method and numerical integral calculation method is 0.8%. It can also be shown

from Figs. 4 and 5 that the results obtained by numerical integral method coincide with those obtained

by the known analytical method for the CGBs. It can also be seen from Figs. 2–5 that the difference

between the results for the ideal beam models and the more realistic ones is small. For example, therelative errors between the results for the ideal beam models and the more realistic ones is less than 2.0%

when e ¼ 0:10.Moreover, it can be shown from Figs. 2 and 3 that the generalized M2-factor of truncated FGBs in-

creases with the increasing of the beam order NFGB. For the small value of NFGB (For example, NFGB ¼ 5),

the generalized M2-factor of truncated FGBs increases with the decreasing of beam truncation parameter d.But for the larger NFGB (For example, NFGB ¼ 20), there is a minimum value of M2-factor when d increases,

i.e., the generalized M2-factor decreases with the increasing of d at first, then, it increases with d. Figs. 2 and

3 also show that for the case of d > 1:5, the diffraction effect due to the aperture can be ignored. Thecorresponding generalized M2-factor of truncated FGBs approaches to that of nontruncated FGBs. It can

be seen from Figs. 4 and 5 that the generalized M2-factor of truncated CGBs increases with the increasing

of beam decentered parameter b. In the case of small b, the generalized M2-factor of decreases with the

increasing of beams truncation parameter d. For the larger b, there is a maximum value of M2-factor as

beam truncation parameter d increases and the corresponding d increases with the beam decentered pa-

rameter b. Fig. 4 also indicates that the effect due to the aperture can be ignored when the beam truncation

parameter d is larger enough and the corresponding value of d increases with b. For example, d > 1:5 for

b ¼ 0, d > 2:5 for b ¼ 2:0, and d > 3:0 for b ¼ 3:0.


6. Concluding remarks

In this paper, based on the treatment that the rectangular function can be expanded into an approximate

sum of complex Gaussian functions with finite numbers, an approximate method for determining the

generalized beam propagation factor is proposed. A truncated flattened Gaussian beam and a truncatedcosh-Gaussian beam are taken as two examples of truncated laser beams. The analytical expressions for the

generalized M2-factor are derived. The typical numerical examples are given and compared to those ob-

tained from numerically integral calculation method and the known analytical method. A comparison of

the results for the ideal beam models with the more realistic ones is performed. It can be seen from our

study that the integral with finite limit often encountered in the use of conventional method proposed by

Martinez-Herrero et al. is simplified to the sum with finite numbers. The numerical calculation efficiency

can be improved significantly and a large amount of computing time can be saved. When the ten terms of

complex Gaussian functions are taken, the generalized M2-factors calculated by our approximate methodare in keeping with the results from the numerical integral calculation and the known analytical methods

quite well. It is worth pointing out that the treatment for the rectangular function in our method can also be

replaced by the model proposed by Li [12], in which the rectangular function was expressed approximately

in terms of the superposition of Gaussian function with different spot-size. However, further comparison

shows that our method is more efficient. Finally, we mention that the method in this paper can be extended

to study the generalized M2-factor of other types of truncated laser beams, such as higher-order Gaussian

beams, Hermite–Sinusiodal–Gaussian beams, Bessel beams Bessel–Gauss beams, Gaussian Shell-model

beams, and laser beams with amplitude modulations and phase fluctuations, etc.

Acknowledgements

This work was supported by the National Science Foundation of China (No. 60108004). Authors are

very thankful to the referees for the fruitful comments about our paper.

References

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Optical Society of America, Washington, DC, 1998, p. 184.

[3] R. Martinez-Herrero, P.M. Mejias, Opt. Lett. 18 (1993) 1669.

[4] R. Martinez-Herrero, P.M. Mejias, M. Arias, Opt. Lett. 20 (1995) 124.

[5] J.J. Wen, M.A. Breazeale, J. Acoust. Soc. Am. 83 (1988) 1752.

[6] D. Ding, X. Liu, J. Opt. Soc. Am. A 16 (1999) 1286.

[7] V. Bagini, R. Borghi, F. Gori, et al., J. Opt. Soc. Am. A 13 (1996) 1385.

[8] A. Erdelyi et al., Table of Integral Transforms, McGraw-Hill, New York, 1954.

[9] L.W. Casperson, D.G. Hall, J. Opt. Soc. Am. A 14 (1997) 3341.

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Approximate method for calculating the generalized beam propagation factor of truncated beams

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