- Email: [email protected]
Beam propagation factor of hard-edge diffracted cosh-Gaussian beams
15 May 2000
Optics Communications 178 Ž2000. 275–281 www.elsevier.comrlocateroptcom
Beam propagation factor of hard-edge diffracted cosh-Gaussian beams Baida Lu¨ ) , Shirong Luo Department of Opto-Electronics, Science and Technology, Institute of Laser Physics and Chemistry, Sichuan UniÕersity, Chengdu 610064, China Received 21 December 1999; received in revised form 20 March 2000; accepted 29 March 2000
Abstract Based on a generalized truncated second-order irradiance moments definition, an analytical generalized beam propagation factor Ž MG2-factor. of hard-edge diffracted cosh-Gaussian beams is derived and illustrated numerically. It is shown that the advantage of the method is its consistency with the previous results. The relevant subjects of truncated beams are also discussed. q 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In recent years the laser beam characterization including the terminology of beam quality has been studied extensively. A general review is found in Ref. w1x. However, the theoretical formulation of the beam propagation factor, or called M 2-factor, and the propagation law of general laser beams hold true only for the non-truncated case in the strict sense w1,2x. In the ISO document the highly diffractive beams passing through a hard-edge aperture are excluded w3x, although in practical applications there are more or less aperture effects. As yet several groups have devoted themselves to studying the parametric characterization of hard-edge diffracted beams, using different definitions and methods, such as the generalized truncated second-order moments w4,5x, asymptotic approximation w6–8x, algebraic treatment w9x, and weighted moments based on the Wigner distribution function with a window w10x, to overcome or avoid the divergence problem due to the hard-edge aperture. For example, the basic idea of the asymptotic approach is that the truncated second-order moment defined in Refs. w6–8x approximately evolves according to a parabola, thus the beam propagation factor of truncated beams can be defined and calculated numerically by a least square parabolic fit. Experiments have shown w11,12x, that under certain conditions the M 2-factor remains nearly invariant, and the propagation law seems to be valid, whereas some questions still remain, in particular, the choice of the power fraction entering through the aperture w12x. The aim of the present paper is to study the M 2-factor of hard-edge diffracted cosh-Gaussian ŽChG.
)
Corresponding author. Tel.: q86-28-541-2819; fax: q86-28-541-0844; e-mail: [email protected]
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 6 2 - 3
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
276
beams, based on the generalized second-order irradiance moments definition proposed by Martinez-Herrero and co-workers w4,5x. The main advantage of this method is that a similar propagation formula for the truncated moments is obtained as for the non-truncated case. Also, a truncated-defined M 2-factor that is nearly invariant at beam propagation through a truncated ABCD optical system is obtained by using this definition. Furthermore, as far as the case of truncated ChG beams is concerned, it generalizes some previous results, a closed-form M 2-factor of truncated ChG beams is derived, which reduces to that for the non-truncated ChG beams, as the truncation parameter approaches infinite, and to that for truncated Gaussian beams, if the normalized modal parameter is zero. Numerical examples are given and illustrated.
2. M 2-factor of truncated ChG beams The field distribution EŽ x, z . of a one-dimensional ChG beam at the z s 0 plane reads w13,14x x2
ž /
E Ž x ,0 . s exp y
w 02
cosh Ž V 0 x .
Ž 1.
where w 0 denotes the waist width of the Gaussian amplitude distribution, and V 0 is the parameter associated with the cosh part. The amplitude factor in Eq. Ž1. is equal to unity for the sake of simplicity. Assume that a hard-edge aperture with full width 2 a is positioned at the z s 0 plane and the first-order moments are zero. In accordance with Refs. w4,5x, the generalized second-order irradiance moments at the z s 0 plane are defined as I
² x2:s ² u2 : s
² xu: s
a
Hya x
I0
1
2<
a
2
k I0
E < 2d x X 2
Hya< E <
1
a
2i kI0
Hya x
d xq
Ž 2. 4 2
< E Ž a . < 2 q < E Ž ya . < 2
Ž 3.
E Ž x . y xEX Ž x . E ) Ž x . 4 d x
Ž 4.
k I0 a
EX Ž x .
)
where ² x 2 :, ² u 2 : denote the irradiance second-order moments in the spatial domain and spatial-frequency domain, respectively, ² xu: is the cross second-order irradiance moment, the prime represents the derivative with respect to x, k is the wave number, U is the complex conjugate, I0 is the power entering through the aperture, expressed as a
I0 s
Hya< E Ž x ,0. <
2
dx
Ž 5.
which is related to the power fraction p by ps
I0 a
Hya
.
Ž 6.
< E Ž x ,0 . < 2 d x
Taking into account the definition of the error function w15x a
Hyaexp Ž yt
2
. d t s p 1r2 erf Ž b .
Ž 7.
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
277
the substitution from Eq. Ž1. into Eqs. Ž5. and Ž6. yields I0 s ps
w 0'2 p
s q 2erf Ž 2 1r2a .
8
Ž 8.
s q 2erf Ž 2 1r2a .
Ž 9.
b2
2 1 q exp
ž / 2
where w13,14x truncation parameter a s
a
Ž 10 .
w0
normalized modal parameter b s w 0 V 0
Ž 11 .
and
b2
s s exp
ž / 2
erf Ž 2 1r2a y 2y1r2b . q erf Ž 2 1r2a q 2y1r2b . .
Ž 12 .
On substituting from Eq. Ž8. into Eq. Ž1. and integrating by parts in the use of Eq. Ž7. we obtain ² x2:s
w 02 B
Ž 13 .
4 A
where A s s q 2erf Ž 2 1r2a . B s Ž 1 q b 2 . s q 2erf Ž 2 1r2a . y 8
Ž 14 . 2
1r2
ž / p
a ID y 2
2
ž / p
1r2
b sinh Ž 2 ba . exp Ž y2 a 2 .
Ž 15 .
and ID s exp Ž y2 a 2 . cosh2 Ž ba .
Ž 16 .
denotes the irradiance at the edge of aperture. Performing similar procedures as above, from Eqs. Ž2. and Ž3. we have ² u2 : s
1
C
w 02 k 2
A
Ž 17 .
² xu: s 0
Ž 18 .
where C s s q 2 Ž 1 y b 2 . erf Ž 2 1r2a . q 2
2
ž / p
1r2
b sinh Ž 2 ba . exp Ž y2 a 2 . q 8
2
1r2
ž / ž p
4
a
/
y a ID .
Ž 19 .
The physical meaning of Eq. Ž18. is evident, because w2,16x ² xu: s
² x2: R
Ž 20 .
with R being the effective radius of curvature of the phase front. The integration is performed at the waist plane, so that R approaches infinity.
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
278
The substitution from Eqs. Ž13., Ž17. and Ž18. into w2x ² x 2 :² u 2 : y ² xu: s
ž
1 2k
2
MG2
/
Ž 21 .
yields MG2 s
'BC
Ž 22 .
A
where A, B, C are given in Eqs. Ž14., Ž15. and Ž19., respectively, and MG2 is the generalized M 2-factor for the truncated case. It is noted that the advantage in the use of the generalized second-order irradiance moments given in Eqs. Ž2. – Ž4. is that by letting a ` Ž a `. the generalized M 2-factor Eq. Ž22. of truncated ChG beams reduces to that for the non-truncated case
™ ™
(
™`
MG2 s M2 s
b2
ž / Ž ž /
1 q b 2 q Ž 2 y b 4 . exp y
2
q 1 y b 2 . exp Ž yb 2 .
Ž 23 .
b2
a
1 q exp y
2
which is just the Eq. Ž8. in Ref. w17x, if d s b 2 is employed in place of b 2 . In addition, letting V 0 s 0 Ž b s 0. in Eq. Ž22., meaning the beam is a pure Gaussian one, leads to MG2 s
(
1q4
2
1r2
ž / ž p
2
a
y a erfy1 Ž 2 1r2a . exp Ž y2 a 2 . q
/
8 p
Ž a 2 y 4 . erfy2 Ž 2 1r2a . exp Ž y4a 2 . Ž 24 .
MG2 corresponds to the generalized M 2-factor of truncated Gaussian beams and essentially consistent with the result in Ref. w5x.
3. Numerical results and discussion Numerical calculations were performed to illustrate the above analytical results. Fig. 1 represents the power fraction p of ChG beams versus parameters a and b . It is well known that for Gaussian beams p depends only on the truncation parameter a . This case is readily obtained by letting b s 0 in Eq. Ž9. and gives p s erf Ž 2 1r2a .
Ž 25 .
p increases with increasing a . However, for ChG beams and other type of sinusoidal-Gaussian beams, p is not only a function of a , but also depends on b , as shown in Eq. Ž9., because, e.g., a ChG beam can be regarded as a superposition of two decentered Gaussian beams with the same waist width w 0 and in phase, whose irradiance distribution and also the beam width Ždefined based on the second-order moment. depend on both w 0 and V 0 w13,14x. By using Fig. 1 it is possible to choose values of p, say p G 85% w11,12x, to draw Fig. 2, where the MG2-factor of ChG beams is plotted against a with b being the parameter. From Fig. 2 it turns out that the MG2-factor decreases with increasing a under the condition that an appropriate enough large value of p is chosen, and then approaches the constant values corresponding to those for the non-truncated case ŽEq. Ž23.., namely M 2 s 1, 1.1, 1.5, 4.1 for b s 0, 1.5, 2, and 4, respectively. Thus, an interesting question arises: what
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
279
Fig. 1. The power fraction p of truncated ChG beams versus normalized modal parameter b for various truncation parameter a s 0.5, 0.8, 1, 1.5, 2, 2.5, `.
happens for truncated ChG beams with arbitrary value of p? The question was not explicitly answered in Refs. w4,5x; here, we would like only to take the numerical results given in Fig. 3 to illustrate it in part. As can be seen
Fig. 2. MG2-factor of truncated ChG beams as a function of truncation parameter a and Ža. b s 0, Žb. b s 1.5, Žc. b s 2, Žd. b s 4, where the power fraction p G 85%.
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
280
Fig. 3. MG2-factor of truncated ChG beams versus normalized modal parameter b , and a s 0.5, 0.8, 2, `.
from Fig. 3, the MG2-factor increases as b increases, but several cross points appear if small values of p are chosen. It means physically that for this case two ChG beams with different have a the same MG2-factor, e.g. at the cross point 1 Ž b s 1.7., two ChG beams with a s 0.5, 0.8 have the same MG2 s 2.5 and the corresponding power contents are p s 32%, 54%, respectively; at the cross point 2 Ž b s 3.2., MG2 s 4.1, a s 0.5, 2, p s 2%, 80%, and at the cross point 3 Ž b s 4.6., MG2 s 8.7, a s 0.8, 2, p s 0.15%, 29%.
4. Conclusion In this study based on the generalized second-order irradiance moments definition, the analytical MG2-factor expression for hard-edge diffracted ChG beams has been derived which is consistent with the previous results, and for the two limiting cases of a `, b s 0 reduces to those for non-truncated ChG beams and truncated Gaussian beams, respectively. Therefore, the formalism of the generalized second-order irradiance moments and MG2-factor is applicable to other type of beams apart from Gaussian beams. The extension to the two-dimensional ChG beams and other sinusoidal-Gaussian beams, such as cosine-Gaussian, sine-Gaussian beams, etc. are straightforward. In addition, the question of the power fraction has been discussed numerically. Certainly, it is difficult to handle, further study is necessary both theoretically and experimentally. However, our results have shown that for the strong truncation case Žlow power fraction. the MG2-factor is no more a useful parameter to characterize truncated ChG beams, whereas from a practical point of view the propagation of hard-diffracted beams remains nearly parabolic and the generalized MG2-factor formalism is valid, as long as p is chosen large enough, for example p G 85%. Finally, as mentioned in Section 1, there are different definitions and methods for truncated second-order moments and also beam propagation factor are used by different authors. For the same beam and truncation condition different results may be obtained for the truncated beam propagation parameter depending on its definition, an example can be found in Ref. w18x. Generally speaking, when the diffraction due to beam truncation comes into picture especially for the strong truncation case, unfortunately, as yet a general valid result has not been found.
™
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
281
Acknowledgements This work was supported by the Foundations of National Hi-Tech Laboratory and National Key Laboratory of Laser Technology under the contracts No. H98-01 and No. 9902. The authors are very grateful to the unknown referees for carefully reading the manuscript and insightful comments.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x
A.E. Siegman, OSA TOPS 17 Ž1998. 184. G. Nemes, J. Serna, OSA TOPS 17 Ž1998. 200. ISO Document, Terminology and test methods for lasers, ISOrTC 172rSC 9rWG 1 N80, 1995. R. Martinez-Herrero, P.M. Mejias, Opt. Lett. 18 Ž1993. 1669. R. Martinez-Herrero, P.M. Mejias, M. Arias, Opt. Lett. 20 Ž1995. 124. P.-A. Belanger, Y. Champagne, C. Par, Opt. Commun. 105 Ž1994. 233. C. Pare, P.-A. Belanger, Opt. Commun. 123 Ž1996. 679. C. Pare, P.-A. Belanger, Proc. SPIE 2870 Ž1996. 104. K. Witting, A. Giesen, H. Hugel, in: H. Weber, N. Reng, J. Ludtke, P.M. Mejias ŽEds.., Proc. 2nd Int. Workship on Laser Beam ¨ Characterization, 272, Festkorper-Laser-Institute, Berlin, 1994. ¨ M. Scholl, S. Mutter, O. Post, Proc. SPIE 2870 Ž1996. 112. ¨ M.A. Porras, Opt. Commun. 109 Ž1994. 5. R. Maestle, A. Giesen, Proc. SPIE 2870 Ž1996. 123. L.W. Casperson, D.G. Hall, A.A. Tovar, J. Opt. Soc. Am. A 14 Ž1997. 3341. B. Lu, ¨ H. Ma, B. Zhang, Opt. Commun. 164 Ž1999. 165. A. Erdelyi et al., Tables of Integral Transforms, vol. 2, McGraw-Hill, New York, 1954, p. 431. A.E. Siegman, IEEE J. Quantum Electron. 27 Ž1991. 1146. B. Lu, ¨ B. Zhang, H. Ma, Opt. Lett. 24 Ž1999. 640. B. Lu, ¨ S. Luo, Asymptotic approach to the truncated cosh-Gaussian beams, submitted for publication.
Optics Communications 178 Ž2000. 275–281 www.elsevier.comrlocateroptcom
Beam propagation factor of hard-edge diffracted cosh-Gaussian beams Baida Lu¨ ) , Shirong Luo Department of Opto-Electronics, Science and Technology, Institute of Laser Physics and Chemistry, Sichuan UniÕersity, Chengdu 610064, China Received 21 December 1999; received in revised form 20 March 2000; accepted 29 March 2000
Abstract Based on a generalized truncated second-order irradiance moments definition, an analytical generalized beam propagation factor Ž MG2-factor. of hard-edge diffracted cosh-Gaussian beams is derived and illustrated numerically. It is shown that the advantage of the method is its consistency with the previous results. The relevant subjects of truncated beams are also discussed. q 2000 Published by Elsevier Science B.V. All rights reserved.
1. Introduction In recent years the laser beam characterization including the terminology of beam quality has been studied extensively. A general review is found in Ref. w1x. However, the theoretical formulation of the beam propagation factor, or called M 2-factor, and the propagation law of general laser beams hold true only for the non-truncated case in the strict sense w1,2x. In the ISO document the highly diffractive beams passing through a hard-edge aperture are excluded w3x, although in practical applications there are more or less aperture effects. As yet several groups have devoted themselves to studying the parametric characterization of hard-edge diffracted beams, using different definitions and methods, such as the generalized truncated second-order moments w4,5x, asymptotic approximation w6–8x, algebraic treatment w9x, and weighted moments based on the Wigner distribution function with a window w10x, to overcome or avoid the divergence problem due to the hard-edge aperture. For example, the basic idea of the asymptotic approach is that the truncated second-order moment defined in Refs. w6–8x approximately evolves according to a parabola, thus the beam propagation factor of truncated beams can be defined and calculated numerically by a least square parabolic fit. Experiments have shown w11,12x, that under certain conditions the M 2-factor remains nearly invariant, and the propagation law seems to be valid, whereas some questions still remain, in particular, the choice of the power fraction entering through the aperture w12x. The aim of the present paper is to study the M 2-factor of hard-edge diffracted cosh-Gaussian ŽChG.
)
Corresponding author. Tel.: q86-28-541-2819; fax: q86-28-541-0844; e-mail: [email protected]
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 6 2 - 3
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
276
beams, based on the generalized second-order irradiance moments definition proposed by Martinez-Herrero and co-workers w4,5x. The main advantage of this method is that a similar propagation formula for the truncated moments is obtained as for the non-truncated case. Also, a truncated-defined M 2-factor that is nearly invariant at beam propagation through a truncated ABCD optical system is obtained by using this definition. Furthermore, as far as the case of truncated ChG beams is concerned, it generalizes some previous results, a closed-form M 2-factor of truncated ChG beams is derived, which reduces to that for the non-truncated ChG beams, as the truncation parameter approaches infinite, and to that for truncated Gaussian beams, if the normalized modal parameter is zero. Numerical examples are given and illustrated.
2. M 2-factor of truncated ChG beams The field distribution EŽ x, z . of a one-dimensional ChG beam at the z s 0 plane reads w13,14x x2
ž /
E Ž x ,0 . s exp y
w 02
cosh Ž V 0 x .
Ž 1.
where w 0 denotes the waist width of the Gaussian amplitude distribution, and V 0 is the parameter associated with the cosh part. The amplitude factor in Eq. Ž1. is equal to unity for the sake of simplicity. Assume that a hard-edge aperture with full width 2 a is positioned at the z s 0 plane and the first-order moments are zero. In accordance with Refs. w4,5x, the generalized second-order irradiance moments at the z s 0 plane are defined as I
² x2:s ² u2 : s
² xu: s
a
Hya x
I0
1
2<
a
2
k I0
E < 2d x X 2
Hya< E <
1
a
2i kI0
Hya x
d xq
Ž 2. 4 2
< E Ž a . < 2 q < E Ž ya . < 2
Ž 3.
E Ž x . y xEX Ž x . E ) Ž x . 4 d x
Ž 4.
k I0 a
EX Ž x .
)
where ² x 2 :, ² u 2 : denote the irradiance second-order moments in the spatial domain and spatial-frequency domain, respectively, ² xu: is the cross second-order irradiance moment, the prime represents the derivative with respect to x, k is the wave number, U is the complex conjugate, I0 is the power entering through the aperture, expressed as a
I0 s
Hya< E Ž x ,0. <
2
dx
Ž 5.
which is related to the power fraction p by ps
I0 a
Hya
.
Ž 6.
< E Ž x ,0 . < 2 d x
Taking into account the definition of the error function w15x a
Hyaexp Ž yt
2
. d t s p 1r2 erf Ž b .
Ž 7.
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
277
the substitution from Eq. Ž1. into Eqs. Ž5. and Ž6. yields I0 s ps
w 0'2 p
s q 2erf Ž 2 1r2a .
8
Ž 8.
s q 2erf Ž 2 1r2a .
Ž 9.
b2
2 1 q exp
ž / 2
where w13,14x truncation parameter a s
a
Ž 10 .
w0
normalized modal parameter b s w 0 V 0
Ž 11 .
and
b2
s s exp
ž / 2
erf Ž 2 1r2a y 2y1r2b . q erf Ž 2 1r2a q 2y1r2b . .
Ž 12 .
On substituting from Eq. Ž8. into Eq. Ž1. and integrating by parts in the use of Eq. Ž7. we obtain ² x2:s
w 02 B
Ž 13 .
4 A
where A s s q 2erf Ž 2 1r2a . B s Ž 1 q b 2 . s q 2erf Ž 2 1r2a . y 8
Ž 14 . 2
1r2
ž / p
a ID y 2
2
ž / p
1r2
b sinh Ž 2 ba . exp Ž y2 a 2 .
Ž 15 .
and ID s exp Ž y2 a 2 . cosh2 Ž ba .
Ž 16 .
denotes the irradiance at the edge of aperture. Performing similar procedures as above, from Eqs. Ž2. and Ž3. we have ² u2 : s
1
C
w 02 k 2
A
Ž 17 .
² xu: s 0
Ž 18 .
where C s s q 2 Ž 1 y b 2 . erf Ž 2 1r2a . q 2
2
ž / p
1r2
b sinh Ž 2 ba . exp Ž y2 a 2 . q 8
2
1r2
ž / ž p
4
a
/
y a ID .
Ž 19 .
The physical meaning of Eq. Ž18. is evident, because w2,16x ² xu: s
² x2: R
Ž 20 .
with R being the effective radius of curvature of the phase front. The integration is performed at the waist plane, so that R approaches infinity.
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
278
The substitution from Eqs. Ž13., Ž17. and Ž18. into w2x ² x 2 :² u 2 : y ² xu: s
ž
1 2k
2
MG2
/
Ž 21 .
yields MG2 s
'BC
Ž 22 .
A
where A, B, C are given in Eqs. Ž14., Ž15. and Ž19., respectively, and MG2 is the generalized M 2-factor for the truncated case. It is noted that the advantage in the use of the generalized second-order irradiance moments given in Eqs. Ž2. – Ž4. is that by letting a ` Ž a `. the generalized M 2-factor Eq. Ž22. of truncated ChG beams reduces to that for the non-truncated case
™ ™
(
™`
MG2 s M2 s
b2
ž / Ž ž /
1 q b 2 q Ž 2 y b 4 . exp y
2
q 1 y b 2 . exp Ž yb 2 .
Ž 23 .
b2
a
1 q exp y
2
which is just the Eq. Ž8. in Ref. w17x, if d s b 2 is employed in place of b 2 . In addition, letting V 0 s 0 Ž b s 0. in Eq. Ž22., meaning the beam is a pure Gaussian one, leads to MG2 s
(
1q4
2
1r2
ž / ž p
2
a
y a erfy1 Ž 2 1r2a . exp Ž y2 a 2 . q
/
8 p
Ž a 2 y 4 . erfy2 Ž 2 1r2a . exp Ž y4a 2 . Ž 24 .
MG2 corresponds to the generalized M 2-factor of truncated Gaussian beams and essentially consistent with the result in Ref. w5x.
3. Numerical results and discussion Numerical calculations were performed to illustrate the above analytical results. Fig. 1 represents the power fraction p of ChG beams versus parameters a and b . It is well known that for Gaussian beams p depends only on the truncation parameter a . This case is readily obtained by letting b s 0 in Eq. Ž9. and gives p s erf Ž 2 1r2a .
Ž 25 .
p increases with increasing a . However, for ChG beams and other type of sinusoidal-Gaussian beams, p is not only a function of a , but also depends on b , as shown in Eq. Ž9., because, e.g., a ChG beam can be regarded as a superposition of two decentered Gaussian beams with the same waist width w 0 and in phase, whose irradiance distribution and also the beam width Ždefined based on the second-order moment. depend on both w 0 and V 0 w13,14x. By using Fig. 1 it is possible to choose values of p, say p G 85% w11,12x, to draw Fig. 2, where the MG2-factor of ChG beams is plotted against a with b being the parameter. From Fig. 2 it turns out that the MG2-factor decreases with increasing a under the condition that an appropriate enough large value of p is chosen, and then approaches the constant values corresponding to those for the non-truncated case ŽEq. Ž23.., namely M 2 s 1, 1.1, 1.5, 4.1 for b s 0, 1.5, 2, and 4, respectively. Thus, an interesting question arises: what
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
279
Fig. 1. The power fraction p of truncated ChG beams versus normalized modal parameter b for various truncation parameter a s 0.5, 0.8, 1, 1.5, 2, 2.5, `.
happens for truncated ChG beams with arbitrary value of p? The question was not explicitly answered in Refs. w4,5x; here, we would like only to take the numerical results given in Fig. 3 to illustrate it in part. As can be seen
Fig. 2. MG2-factor of truncated ChG beams as a function of truncation parameter a and Ža. b s 0, Žb. b s 1.5, Žc. b s 2, Žd. b s 4, where the power fraction p G 85%.
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
280
Fig. 3. MG2-factor of truncated ChG beams versus normalized modal parameter b , and a s 0.5, 0.8, 2, `.
from Fig. 3, the MG2-factor increases as b increases, but several cross points appear if small values of p are chosen. It means physically that for this case two ChG beams with different have a the same MG2-factor, e.g. at the cross point 1 Ž b s 1.7., two ChG beams with a s 0.5, 0.8 have the same MG2 s 2.5 and the corresponding power contents are p s 32%, 54%, respectively; at the cross point 2 Ž b s 3.2., MG2 s 4.1, a s 0.5, 2, p s 2%, 80%, and at the cross point 3 Ž b s 4.6., MG2 s 8.7, a s 0.8, 2, p s 0.15%, 29%.
4. Conclusion In this study based on the generalized second-order irradiance moments definition, the analytical MG2-factor expression for hard-edge diffracted ChG beams has been derived which is consistent with the previous results, and for the two limiting cases of a `, b s 0 reduces to those for non-truncated ChG beams and truncated Gaussian beams, respectively. Therefore, the formalism of the generalized second-order irradiance moments and MG2-factor is applicable to other type of beams apart from Gaussian beams. The extension to the two-dimensional ChG beams and other sinusoidal-Gaussian beams, such as cosine-Gaussian, sine-Gaussian beams, etc. are straightforward. In addition, the question of the power fraction has been discussed numerically. Certainly, it is difficult to handle, further study is necessary both theoretically and experimentally. However, our results have shown that for the strong truncation case Žlow power fraction. the MG2-factor is no more a useful parameter to characterize truncated ChG beams, whereas from a practical point of view the propagation of hard-diffracted beams remains nearly parabolic and the generalized MG2-factor formalism is valid, as long as p is chosen large enough, for example p G 85%. Finally, as mentioned in Section 1, there are different definitions and methods for truncated second-order moments and also beam propagation factor are used by different authors. For the same beam and truncation condition different results may be obtained for the truncated beam propagation parameter depending on its definition, an example can be found in Ref. w18x. Generally speaking, when the diffraction due to beam truncation comes into picture especially for the strong truncation case, unfortunately, as yet a general valid result has not been found.
™
B. Lu, ¨ S. Luo r Optics Communications 178 (2000) 275–281
281
Acknowledgements This work was supported by the Foundations of National Hi-Tech Laboratory and National Key Laboratory of Laser Technology under the contracts No. H98-01 and No. 9902. The authors are very grateful to the unknown referees for carefully reading the manuscript and insightful comments.
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A.E. Siegman, OSA TOPS 17 Ž1998. 184. G. Nemes, J. Serna, OSA TOPS 17 Ž1998. 200. ISO Document, Terminology and test methods for lasers, ISOrTC 172rSC 9rWG 1 N80, 1995. R. Martinez-Herrero, P.M. Mejias, Opt. Lett. 18 Ž1993. 1669. R. Martinez-Herrero, P.M. Mejias, M. Arias, Opt. Lett. 20 Ž1995. 124. P.-A. Belanger, Y. Champagne, C. Par, Opt. Commun. 105 Ž1994. 233. C. Pare, P.-A. Belanger, Opt. Commun. 123 Ž1996. 679. C. Pare, P.-A. Belanger, Proc. SPIE 2870 Ž1996. 104. K. Witting, A. Giesen, H. Hugel, in: H. Weber, N. Reng, J. Ludtke, P.M. Mejias ŽEds.., Proc. 2nd Int. Workship on Laser Beam ¨ Characterization, 272, Festkorper-Laser-Institute, Berlin, 1994. ¨ M. Scholl, S. Mutter, O. Post, Proc. SPIE 2870 Ž1996. 112. ¨ M.A. Porras, Opt. Commun. 109 Ž1994. 5. R. Maestle, A. Giesen, Proc. SPIE 2870 Ž1996. 123. L.W. Casperson, D.G. Hall, A.A. Tovar, J. Opt. Soc. Am. A 14 Ž1997. 3341. B. Lu, ¨ H. Ma, B. Zhang, Opt. Commun. 164 Ž1999. 165. A. Erdelyi et al., Tables of Integral Transforms, vol. 2, McGraw-Hill, New York, 1954, p. 431. A.E. Siegman, IEEE J. Quantum Electron. 27 Ž1991. 1146. B. Lu, ¨ B. Zhang, H. Ma, Opt. Lett. 24 Ž1999. 640. B. Lu, ¨ S. Luo, Asymptotic approach to the truncated cosh-Gaussian beams, submitted for publication.