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Beam propagation factor of apertured super-Gaussian beams
Beam propagation factor of apertured super-Gaussian beams Baida Lu¨, Shirong Luo Institute of Laser Physics & Chemistry, Sichuan University, Chengdu, 610064, PR China
Abstract: Starting from the generalized truncated secondorder moments definition, an explicit expression for the beam propagation factor (M2-factor) of apertured superGaussian beams is derived and illustrated with numerical examples. Some special cases of our analytical result are discussed. Key words: Truncated second-order moment – beam propagation factor (M2-factor) – super-Gaussian beam
1. Introduction How to characterize laser beams including their beam quality is a widespread topic of current interest not only for laser scientists and engineers, but also for laser users and manufacturers. The beam propagation factor (M2-factor) has been proposed to characterize different laser beams [1]. However, the second-order-moments based M2-factor has a serious weakness, because the integral becomes divergent for the hard-edge apertured case, so that in the ISO document the highly diffracted beams, such as those produced by unstable resonators or passing through hard-edge apertures, are excluded [2]. As yet various methods [3]–[6] have been proposed to deal with hard-edge diffracted beams in order to limit the infinite integration interval, thus the calculation of the truncated second-order moments becomes possible. The purpose of the present paper is to study the M2-factor of apertured superGaussian beams. Based on the generalized truncated second-order moments definition [3], [4], a closed-form 2 expression for the M2-factor MG of apertured superGaussian beams is derived, and illustrated with numer2 describes the behaical examples. It is shown that MG vior of apertured super-Gaussian beams to a certain extent, and reduces to the well-known results for some special cases.
2. Analytical expression for the beam propagation factor of apertured super-Gaussian beams In the rectangular coordinate system the field distribution of super-Gaussian beams at the plane of z ¼ 0 is expressed as " # jxj N Eðx; 0Þ ¼ exp ; ð1Þ w0 where NðN 2Þ denotes the beam order and w0 is the waist width. Suppose that a hard-edge aperture of full width 2a is positioned at the plane of z ¼ 0, in accordance with the approach in refs. [3] and [4], the generalized second-order moments hx2 i, hu2 i in the spatial domain and spatial-frequency domain, and the cross second-order moment hxui are defined as hx2 i ¼
1 I0
hu2 i ¼
ða
x2 jEj2 dx ;
ð2Þ
a
1 k2 I0
ða
jE0 j2 dx þ
a
4 ½jEðaÞj2 þ jEðaÞj2 ; k2 I0 a ð3Þ
hxui ¼
1 2ikI0
ða
fx½E0 ðxÞ * EðxÞ xE0 ðxÞ E*ðxÞg dx ;
a
ð4Þ where all first-order moments vanish due to the symmetry of eq. (1), the prime represents the derivation with respect to x, * denotes the complex conjugate, k is the wave number, and I0 is the power entering through the aperture and is given by I0 ¼
Ða
jEðx; 0Þj2 dx
ð5Þ
a
relating to the power fraction p by Received 8 March 2001; accepted 17 June 2001. Correspondence to: B. Lu¨ Fax: ++86-28-540 3260 E-mail: [email protected]
Optik 112, No. 11 (2001) 503–506 ª 2001 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
p¼ 1 Ð
I0
:
ð6Þ
jEðx; 0Þj2 dx
1
0030-4026/01/112/11-503 $ 15.00/0
504
B. Lu¨, S. Luo, Beam propagation factor of apertured super-Gaussian beams
Eq. (16) gives an explicit expression for the M2-factor of apertured super-Gaussian beams in the generalized sense, which is determined by two parameters, namely, the beam order N and truncation parameter b, whereas the M2-factor of unapertured super-Gaussian beams depends only on the beam order N. Two special cases of eq. (16) are of interest: (1) For the unapertured case, i.e., by letting b ! 1ða ! 1Þ gða; xÞ reduces to the Gamma function GðaÞ, therefore eq. (16) simplifies to
Applying the definition of the incomplete Gamma function gða; xÞ [7] gða; xÞ ¼
Ðx
exp ðtÞ t a1 dt
ð7Þ
0
the substitution from eq. (1) into eqs. (5) and (6) delivers w0 11=N 1 N ; 2b 2 I0 ¼ g ; ð8Þ N N 1 1 ; 2bN G ; ð9Þ p¼g N N
2 MG
where G is the Gamma function and the truncation parameter b is defined as a : ð10Þ b¼ w0
w20 gð3=N; 2bN Þ ; 41=N gð1=N; 2bN Þ
hu2 i ¼
2 MG
g
1
2;
pffiffiffi x2 ¼ p erf ðxÞ
Eq. (13) is an expectant result, because hx2 i R
ð18Þ
ð19Þ
we obtain
ð13Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 2 2 8 b erf 1 ð21=2 bÞ exp ð2b2 Þ þ ðb2 4Þ erf 2 ð21=2 bÞ exp ð4b2 Þ : ¼ 1þ4 p b p
hxui ¼
ð17Þ
and the relation between the incomplete Gamma function and error function
241=N exp ð2bN Þ gð2 1=N; 2b Þ þ b N gð1=N; 2bN Þ (12)
hxui ¼ 0 :
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gð3=NÞ Gð2 1=NÞ ; Gð1=NÞ
gða þ 1; xÞ ¼ agða; xÞ xa exp ðxÞ
ð11Þ
N
N 2 41þ1=N k2 w20
¼M ¼
N
where M2 is the beam propagation factor of unapertured super-Gaussian beams, which is consistent with eq. (10) in ref. [8]. (2) For the apertured Gaussian beams (N ¼ 2), on substituting from N ¼ 2 into eq. (16) and recalling the iterative formula of the incomplete Gamma function
On substituting from eq. (1) into eqs. (2)–(4), after performing lengthy integral calculations with eq. (7) taken into account, the final results are expressed as hx2 i ¼
2
ð20Þ
Eq. (20) is essentially in agreement with the result in ref. [4]. ð14Þ
3. Numerical calculation examples and analysis
with R being the effective radius of curvature, and our integration is performed at the waist plane, so R approaches infinity. The substitution from eqs. (11), (12) and (13) into 2 =2kÞ2 hx2 ihu2 i hxui ¼ ðMG
ð15Þ
yields 2 ¼ MG vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #ffi u N 41=N u 2 exp ð2b Þ N tgð3=N; 2bN Þ gð2 1=N; 2bN Þþ b N
gð1=N; 2bN Þ
: (16)
Numerical calculations were performed to illustrate the above analytical results. Fig. 1 gives the variation of the power fraction p with truncation parameter b for different beam orders N ¼ 2; 4; 6 and 12. From fig. 1 it follows that for the smaller bðb 0:68Þ, p decreases with increasing N, whereas for the larger bðb 0:94Þ p increases with increasing N. This result can be explained using fig. 2, which shows that the irradiance distributions of super-Gaussian beams are determined by the beam order N, and the power fraction p depends on both beam order N and truncation parameter b. The irradiance distributions of super-Gaussian beams vary from a Gaussian profile to flattened profiles and the width D of the flat-top becomes larger with increasing N starting from N ¼ 2. For example,
B. Lu¨, S. Luo, Beam propagation factor of apertured super-Gaussian beams
505
Fig. 3. The beam propagation factor MG2 of apertured superGaussian beams for N = 2, 4, 6 and 12 versus truncation parameter b.
Fig. 1. The variation of the power fraction p with truncation parameter b for N = 2, 4, 6 and 12.
for N ¼ 4, D ¼ jxj=w0 0:20, for N ¼ 6 and N ¼ 12, D 0:34 and 0.58, respectively. It means that for the case of the smaller b the power concentricity and also p for the higher order N are less than that for the lower order N and vise versa. 2 The dependence of the MG -factor on b and N is shown in fig. 3, from which we see that: (1) For the larger b, i.e., b 1:34 for N ¼ 2, b 1:28 for N ¼ 4, b 1:20 for N ¼ 6, and b 1:12 for N ¼ 12, the M2-factor of apertured super-Gaussian beams is equal to that of unapertured super-Gaussian 2 ¼ 1, 1.17, 1.39 and 1.95, for beams, namely, MG N ¼ 2; 4; 6 and 12, respectively.
Fig. 2. Irradiance profiles I(x, 0) of super-Gaussian beams at the z = 0 plane for N = 2, 4, 6 and 12.
(2) For the smaller b, i.e. for the strong truncation case, e.g., b 0:04 for N ¼ 2, b 0:20 for N ¼ 4, b 0:34 for N ¼ 6, and b 0:58 for N ¼ 12 the M2factor of apertured super-Gaussian beams converges a value of 2.31. Following the above treatment and using the field distribution of apertured plane waves 1 xa Eðx; 0Þ ¼ : ð21Þ 0 x>a it is easily shown that the M2-factor of apertured plane waves is just equal to 2.31, independent of the aperture width a.
4. Concluding remarks In conclusion the closed-form expression for the beam 2 of apertured super-Gaussian propagation factor MG beams has been derived on the basis of the truncated second-order moments definition. It has been shown 2 that MG depends on the beam order N and truncation parameter b, and reduces to that of unapertured superGaussian beams and apertured Gaussian beams for the special cases of b ! 1 and N ¼ 2, respectively, and for 2 converges the value of the strong truncation case MG 2 -factor of apertured 2.31 corresponding to the MG plane waves. In comparison with refs. [3] and [4] our result has demonstrated that the method of the generalized truncated second-order moments is applicable not only for the simple apertured Gaussian beams, but also for the more complicated apertured beams, such as super-Gaussian beams, and the further extension is possible. However, the weakness of this approach, as pointed out in ref. [5], remains, so further study seems to be still necessary. This work was supported by the Foundation of Sichuan Province Commission of Science, and the Foundation of National High-Tech Key Laboratory of Beam Control.
506
B. Lu¨, S. Luo, Beam propagation factor of apertured super-Gaussian beams
References [1] Siegman AE: New developments in laser resonators. Proc. SPIE 1224 (1990) 2–14 [2] ISO Document, Terminology and test methods for lasers. ISO/TC 172/SC 9/WG 1 N80, 1995 [3] Martinez-Herrero R, Mejias PM: Second-order spatial characterization of hard-edge diffracted beams. Opt. Lett. 18 (1993) 1699–1671 [4] Martinez-Herrero R, Mejias PM, Arias M: Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures. Opt. Lett. 20 (1995) 124–126
[5] Pare C, Belanger P-A: Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam. Opt. Commun. 123 (1996) 679– 693 [6] Amarande S, Giesen A, Hu¨gel H: Propagation analysis of self-convergent beam width and characterization of hardedge diffracted beams. Appl. Opt. 39 (2000) 3914–3924 [7] Erdelyi A: Tables of Integral Transforms. Vol. 1. pp. 387. McGraw-Hill, New York 1954 [8] Luo S, Lu¨ B, Zhang B: A comparison study on the propagation characteristics of flattened Gaussian beams and super-Gaussian beams. Acta Phys. Sin. (China) 48 (1999) 1446–1451 (in Chinese)
Abstract: Starting from the generalized truncated secondorder moments definition, an explicit expression for the beam propagation factor (M2-factor) of apertured superGaussian beams is derived and illustrated with numerical examples. Some special cases of our analytical result are discussed. Key words: Truncated second-order moment – beam propagation factor (M2-factor) – super-Gaussian beam
1. Introduction How to characterize laser beams including their beam quality is a widespread topic of current interest not only for laser scientists and engineers, but also for laser users and manufacturers. The beam propagation factor (M2-factor) has been proposed to characterize different laser beams [1]. However, the second-order-moments based M2-factor has a serious weakness, because the integral becomes divergent for the hard-edge apertured case, so that in the ISO document the highly diffracted beams, such as those produced by unstable resonators or passing through hard-edge apertures, are excluded [2]. As yet various methods [3]–[6] have been proposed to deal with hard-edge diffracted beams in order to limit the infinite integration interval, thus the calculation of the truncated second-order moments becomes possible. The purpose of the present paper is to study the M2-factor of apertured superGaussian beams. Based on the generalized truncated second-order moments definition [3], [4], a closed-form 2 expression for the M2-factor MG of apertured superGaussian beams is derived, and illustrated with numer2 describes the behaical examples. It is shown that MG vior of apertured super-Gaussian beams to a certain extent, and reduces to the well-known results for some special cases.
2. Analytical expression for the beam propagation factor of apertured super-Gaussian beams In the rectangular coordinate system the field distribution of super-Gaussian beams at the plane of z ¼ 0 is expressed as " # jxj N Eðx; 0Þ ¼ exp ; ð1Þ w0 where NðN 2Þ denotes the beam order and w0 is the waist width. Suppose that a hard-edge aperture of full width 2a is positioned at the plane of z ¼ 0, in accordance with the approach in refs. [3] and [4], the generalized second-order moments hx2 i, hu2 i in the spatial domain and spatial-frequency domain, and the cross second-order moment hxui are defined as hx2 i ¼
1 I0
hu2 i ¼
ða
x2 jEj2 dx ;
ð2Þ
a
1 k2 I0
ða
jE0 j2 dx þ
a
4 ½jEðaÞj2 þ jEðaÞj2 ; k2 I0 a ð3Þ
hxui ¼
1 2ikI0
ða
fx½E0 ðxÞ * EðxÞ xE0 ðxÞ E*ðxÞg dx ;
a
ð4Þ where all first-order moments vanish due to the symmetry of eq. (1), the prime represents the derivation with respect to x, * denotes the complex conjugate, k is the wave number, and I0 is the power entering through the aperture and is given by I0 ¼
Ða
jEðx; 0Þj2 dx
ð5Þ
a
relating to the power fraction p by Received 8 March 2001; accepted 17 June 2001. Correspondence to: B. Lu¨ Fax: ++86-28-540 3260 E-mail: [email protected]
Optik 112, No. 11 (2001) 503–506 ª 2001 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
p¼ 1 Ð
I0
:
ð6Þ
jEðx; 0Þj2 dx
1
0030-4026/01/112/11-503 $ 15.00/0
504
B. Lu¨, S. Luo, Beam propagation factor of apertured super-Gaussian beams
Eq. (16) gives an explicit expression for the M2-factor of apertured super-Gaussian beams in the generalized sense, which is determined by two parameters, namely, the beam order N and truncation parameter b, whereas the M2-factor of unapertured super-Gaussian beams depends only on the beam order N. Two special cases of eq. (16) are of interest: (1) For the unapertured case, i.e., by letting b ! 1ða ! 1Þ gða; xÞ reduces to the Gamma function GðaÞ, therefore eq. (16) simplifies to
Applying the definition of the incomplete Gamma function gða; xÞ [7] gða; xÞ ¼
Ðx
exp ðtÞ t a1 dt
ð7Þ
0
the substitution from eq. (1) into eqs. (5) and (6) delivers w0 11=N 1 N ; 2b 2 I0 ¼ g ; ð8Þ N N 1 1 ; 2bN G ; ð9Þ p¼g N N
2 MG
where G is the Gamma function and the truncation parameter b is defined as a : ð10Þ b¼ w0
w20 gð3=N; 2bN Þ ; 41=N gð1=N; 2bN Þ
hu2 i ¼
2 MG
g
1
2;
pffiffiffi x2 ¼ p erf ðxÞ
Eq. (13) is an expectant result, because hx2 i R
ð18Þ
ð19Þ
we obtain
ð13Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 2 2 8 b erf 1 ð21=2 bÞ exp ð2b2 Þ þ ðb2 4Þ erf 2 ð21=2 bÞ exp ð4b2 Þ : ¼ 1þ4 p b p
hxui ¼
ð17Þ
and the relation between the incomplete Gamma function and error function
241=N exp ð2bN Þ gð2 1=N; 2b Þ þ b N gð1=N; 2bN Þ (12)
hxui ¼ 0 :
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gð3=NÞ Gð2 1=NÞ ; Gð1=NÞ
gða þ 1; xÞ ¼ agða; xÞ xa exp ðxÞ
ð11Þ
N
N 2 41þ1=N k2 w20
¼M ¼
N
where M2 is the beam propagation factor of unapertured super-Gaussian beams, which is consistent with eq. (10) in ref. [8]. (2) For the apertured Gaussian beams (N ¼ 2), on substituting from N ¼ 2 into eq. (16) and recalling the iterative formula of the incomplete Gamma function
On substituting from eq. (1) into eqs. (2)–(4), after performing lengthy integral calculations with eq. (7) taken into account, the final results are expressed as hx2 i ¼
2
ð20Þ
Eq. (20) is essentially in agreement with the result in ref. [4]. ð14Þ
3. Numerical calculation examples and analysis
with R being the effective radius of curvature, and our integration is performed at the waist plane, so R approaches infinity. The substitution from eqs. (11), (12) and (13) into 2 =2kÞ2 hx2 ihu2 i hxui ¼ ðMG
ð15Þ
yields 2 ¼ MG vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #ffi u N 41=N u 2 exp ð2b Þ N tgð3=N; 2bN Þ gð2 1=N; 2bN Þþ b N
gð1=N; 2bN Þ
: (16)
Numerical calculations were performed to illustrate the above analytical results. Fig. 1 gives the variation of the power fraction p with truncation parameter b for different beam orders N ¼ 2; 4; 6 and 12. From fig. 1 it follows that for the smaller bðb 0:68Þ, p decreases with increasing N, whereas for the larger bðb 0:94Þ p increases with increasing N. This result can be explained using fig. 2, which shows that the irradiance distributions of super-Gaussian beams are determined by the beam order N, and the power fraction p depends on both beam order N and truncation parameter b. The irradiance distributions of super-Gaussian beams vary from a Gaussian profile to flattened profiles and the width D of the flat-top becomes larger with increasing N starting from N ¼ 2. For example,
B. Lu¨, S. Luo, Beam propagation factor of apertured super-Gaussian beams
505
Fig. 3. The beam propagation factor MG2 of apertured superGaussian beams for N = 2, 4, 6 and 12 versus truncation parameter b.
Fig. 1. The variation of the power fraction p with truncation parameter b for N = 2, 4, 6 and 12.
for N ¼ 4, D ¼ jxj=w0 0:20, for N ¼ 6 and N ¼ 12, D 0:34 and 0.58, respectively. It means that for the case of the smaller b the power concentricity and also p for the higher order N are less than that for the lower order N and vise versa. 2 The dependence of the MG -factor on b and N is shown in fig. 3, from which we see that: (1) For the larger b, i.e., b 1:34 for N ¼ 2, b 1:28 for N ¼ 4, b 1:20 for N ¼ 6, and b 1:12 for N ¼ 12, the M2-factor of apertured super-Gaussian beams is equal to that of unapertured super-Gaussian 2 ¼ 1, 1.17, 1.39 and 1.95, for beams, namely, MG N ¼ 2; 4; 6 and 12, respectively.
Fig. 2. Irradiance profiles I(x, 0) of super-Gaussian beams at the z = 0 plane for N = 2, 4, 6 and 12.
(2) For the smaller b, i.e. for the strong truncation case, e.g., b 0:04 for N ¼ 2, b 0:20 for N ¼ 4, b 0:34 for N ¼ 6, and b 0:58 for N ¼ 12 the M2factor of apertured super-Gaussian beams converges a value of 2.31. Following the above treatment and using the field distribution of apertured plane waves 1 xa Eðx; 0Þ ¼ : ð21Þ 0 x>a it is easily shown that the M2-factor of apertured plane waves is just equal to 2.31, independent of the aperture width a.
4. Concluding remarks In conclusion the closed-form expression for the beam 2 of apertured super-Gaussian propagation factor MG beams has been derived on the basis of the truncated second-order moments definition. It has been shown 2 that MG depends on the beam order N and truncation parameter b, and reduces to that of unapertured superGaussian beams and apertured Gaussian beams for the special cases of b ! 1 and N ¼ 2, respectively, and for 2 converges the value of the strong truncation case MG 2 -factor of apertured 2.31 corresponding to the MG plane waves. In comparison with refs. [3] and [4] our result has demonstrated that the method of the generalized truncated second-order moments is applicable not only for the simple apertured Gaussian beams, but also for the more complicated apertured beams, such as super-Gaussian beams, and the further extension is possible. However, the weakness of this approach, as pointed out in ref. [5], remains, so further study seems to be still necessary. This work was supported by the Foundation of Sichuan Province Commission of Science, and the Foundation of National High-Tech Key Laboratory of Beam Control.
506
B. Lu¨, S. Luo, Beam propagation factor of apertured super-Gaussian beams
References [1] Siegman AE: New developments in laser resonators. Proc. SPIE 1224 (1990) 2–14 [2] ISO Document, Terminology and test methods for lasers. ISO/TC 172/SC 9/WG 1 N80, 1995 [3] Martinez-Herrero R, Mejias PM: Second-order spatial characterization of hard-edge diffracted beams. Opt. Lett. 18 (1993) 1699–1671 [4] Martinez-Herrero R, Mejias PM, Arias M: Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures. Opt. Lett. 20 (1995) 124–126
[5] Pare C, Belanger P-A: Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam. Opt. Commun. 123 (1996) 679– 693 [6] Amarande S, Giesen A, Hu¨gel H: Propagation analysis of self-convergent beam width and characterization of hardedge diffracted beams. Appl. Opt. 39 (2000) 3914–3924 [7] Erdelyi A: Tables of Integral Transforms. Vol. 1. pp. 387. McGraw-Hill, New York 1954 [8] Luo S, Lu¨ B, Zhang B: A comparison study on the propagation characteristics of flattened Gaussian beams and super-Gaussian beams. Acta Phys. Sin. (China) 48 (1999) 1446–1451 (in Chinese)