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The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams
Optik 113, No. 3 (2002) 145–148 ªXiaoling 2002 Urban & Fischer Ji, Baida Lu¨, Verlag The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams http://www.urbanfischer.de/journals/optik
145
International Journal for Light and Electron Optics
The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams Xiaoling Ji1, 2, Baida Lu¨2 1 2
College of Electronic Engineering, Sichuan Normal University, Chengdu 610066, China Institute of Laser Physics & Chemistry, Sichuan University, Chengdu 610064, China
Abstract: The propagation of the K parameter of Hermitecosh-Gaussian (HChG) beams passing through a spherically aberrated lens is studied by the straightforward integral of the Collins formula. Numerical calculations are made and the results show that for this case the K parameter changes upon propagation, and regardless of positive or negative aberration, spherical aberrations lead to an increment of the K parameter in comparison with the aberration-free case. The propagation properties of the K parameter of cosh-Gaussian (ChG) beams, and HermiteGaussian (H-G) beams are treated as special cases of HChG beams. Key words: Kurtosis parameter – Hermite-cosh-Gaussian (HChG) beam – spherical aberration
1. Introduction It is known that the second-order-moments-based beam propagation factor (M 2 factor) is a useful parameter characterizing different laser beams [1]. But it is insufficient, since no information concerning the shape features of a laser beam, such as the symmetry and degree of flatness, is inferred from the M 2 factor. The kurtosis parameter K related to the fourth- and second-order intensity moments is an important parameter, which describes the degree of sharpness of any beam [2–4]. Generally, the K parameter changes upon propagation through paraxial optical ABCD systems for all laser beams except for Gaussian and Hermit-Gaussian (H-G) beams, their shape remains unchanged upon propagation. In practical applications the propagation of the kurtosis parameter through ABCD optical systems is of interest [5], and the corresponding propagation formulae were given by Weber, Martinez-Herrero et al. [2, 3]. Piquero et al. studied sharpness changes in Gaussian beams induced by a spherically aberrated lens [6]. On the
Received 12 December 2001; accepted 3 March 2002. Correspondence to: B. Lu¨ Fax: ++86-28-540-3260 E-mail: [email protected]
other hand, the Hermite-sinusoidal-Gaussian beam introduced by Casperson et al. is a more general class of laser beams [7, 8]. Some familiar beams, such as the Gaussian, H-G and cosh-Gaussian (ChG) beams can be regarded as their special cases. The aim of the present paper is to deal with the prpagation of the kurtosis parameter of Hermite-cosh-Gaussian (HChG) beams. Starting from the Collins formula [9] instead of the propagation equation of the fourth-order moment in refs. [2] and [3], changes in the kurtosis parameter of HChG beams passing through the lens with or without spherical aberration are illustrated with detailed numerical examples. It is also shown that the results of ChG and H-G beams can be regarded as special cases of n ¼ 0 and W0 ¼ 0, respectively, treated in this paper.
2. Propagation of the kurtosis parameter of HChG beams passing through a spherically aberrated lens The field distribution of a one-dimensional HChG beam at the plane z ¼ 0 in the rectangular coordinate system is expressed as [7, 8] pffiffiffi x0 x20 cosh ðW0 x0 Þ exp 2 2 E0 ðx0 ; 0Þ ¼ Hn w0 w0 ð1Þ where w0 is the waist width of the Gaussian amplitude distribution, Hn is the nth Hermite polynomial, and W0 is the parameter associated with the cosh part. A spherically aberrated lens is characterized by a transmission profile of the form [9] Fðx0 Þ ¼ exp ðikC4 x40 Þ
ð2Þ
where C4 denotes the spherical aberration coefficient and k is the wave number related to the wavelength l by k¼
2p : l
ð3Þ 0030-4026/02/113/03-145 $ 15.00/0
146
Xiaoling Ji, Baida Lu¨, The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams
The propagation of HChG beams through a paraxial optical ABCD system obeys the Collins formula [10] rffiffiffiffiffiffi 1 ð i E0 ðx0 ; 0Þ Eðx; zÞ ¼ lB
þ Kmin ¼ Kmin ¼ 2:20 at the position of z=f ¼ 0:90. þ ¼ Kmax ¼ 2:30 for Another maximum Kmax C4 ¼ 3:0 104 mm3 is located at the focal plane 0 of z=f ¼ 1:0. The maxima Kmax ¼ 1:86,
1
ik ðAx20 2x0 x þ Dx2 Þ dx0 exp 2B
ð4Þ
where A; B; D are elements of the transfer matrix A B . Assume that the spherically aberrated lens B D of focal length f is located at the plane z ¼ 0, in consideration of eqs. (1) and (2), from eq. (4) we obtain rffiffiffiffiffi 1 ð pffiffiffi x0 i Hn Eðx; zÞ ¼ 2 cosh ðW0 x0 Þ w0 lz 1
x2 exp 02 exp ðikC4 x40 Þ w0 ik z 2 2 1 exp dx0 x0 2x0 x þ x 2z f ð5Þ where the propagation distance z is referred to the lens at the plane z ¼ 0. The kurtosis parameter K is defined as [4] K ¼ hx4 i=hx2 i2
ð6Þ
where 1 Ð
hxn i ¼
xn jEðx; zÞj2 dx
1 1 Ð
ðn ¼ 2; 4Þ :
ð7Þ
jEðx; zÞj2 dx
1
The substitution from eqs. (5) and (7) into eq. (6) delivers the propagation equation of the kurtosis parameter of the HChG beam through the spherically aberrated lens. Although it is difficult to derive a closedform propagation expression, numerical calculations are available. In addition, changes in the kurtosis parameter of ChG beams and H-G beams passing through the aberrated lens can be obtained readily by letting n ¼ 0 and W0 ¼ 0 in eqs. (5), (6) and (7), respectively. Typical numerical examples are compiled in figs. 1–3, where the calculation parameters are l ¼ 1:06 mm, w0 ¼ 0:5 mm and f ¼ 200 mm. Fig. 1 gives the propagation of the K parameter of a HChG beam with n ¼ 2, W0 ¼ 2 through a lens with or without spherical aberration. As can be seen, changes in the K parameter depend on the spherical aberration. For examþ ple, we have one maximum value of Kmax ¼ 2:38 located at the position of z=f ¼ 1:29, and one minimum þ ¼ 2:20 at the position of z=f ¼ 1:13 for Kmin C4 ¼ 3:0 104 mm3 . For C4 ¼ 3:0 104 mm3 , þ ¼ Kmax ¼ 2:38 at the there are one maximum Kmax position of z=f ¼ 0:82, and one minimum
Fig. 1. Propagation of the kurtosis parameter a HChG beam with n ¼ 2, W0 ¼ 2 passing through a lens with or without spherical aberration: a) C 4 ¼ 0, 3.0 104 mm3 , 5.0 104 mm3 ; b) C4 ¼ 0, 3.0 104 mm3 , 5.0 104 mm3 .
Xiaoling Ji, Baida Lu¨, The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams
Fig. 2. Propagation of the kurtosis parameter of a H-G beam with n ¼ 1 passing through a lens with or without spherical aberration. C 4 ¼ 0, 5.0 104 mm3 .
147
þ Kmax ¼ Kmax ¼ 3:90 are located at positions of z=f ¼ 1:0, 1.18 and 0.87 or cases of C4 ¼ 0; 5:0 104 mm3 , respectively. The effect of spherical aberrations on the K parameter of a H-G beam with n ¼ 1 is shown in fig. 2, which indicates that the K parameter is a constant of K 0 ¼ 1:67, while the H-G beam passes through an aberration-free lens ðC4 ¼ 0Þ: However, if the lens is spherically aberrated, the K parameter changes upon propagation. For C4 ¼ 5:0 104 mm3 , there are two maxima þ Kmax ¼ 2:25, 2.0 at positions of z=f ¼ 0:78, 1.34, and þ ¼ 1:8 at the position of z=f ¼ 1:08: one minimum Kmin For C4 ¼ 5:0 104 mm3 , there are two maxima þ ¼ Kmax ¼ 2:25, 2.0 at the positions of z=f ¼ 1:38, Kmax þ 0.81, and one minimum Kmin ¼ Kmin ¼ 1:80 at the position of z=f ¼ 0:93. Fig. 3 represents the variation of the K parameter of a ChG beam with W0 ¼ 4 passing through a lens with or without spherical aberration. 0 From fig. 3 it turns out that the maxima of Kmax ¼ 7:48, þ Kmax ¼ Kmax ¼ 12:89 appear at positions of z=f ¼ 1:0, 1.13 and 0.9 for C4 ¼ 0, 4:0 104 mm3 , respectively. In addition, from figs. 1–3 we see that, regardless of positive or negative aberration, spherical lead to an increment of the K parameter of HChG, ChG and HG beams, as compared with the aberration-free case.
3. Conclusion In this paper a detailed study of the effect of spherically aberrated lens on the K parameter of HChG, ChG and H-G beams has been performed by means of the straightforward integral of the Collins formula. It has been shown that positive and negative spherical aberrations affect the propagation of the K parameter of HChG, ChG and HG beams, and result in an increment of their K parameter in comparison with the aberration-free case. In particular, the K parameter of H-G beams changes after passing through the spherically aberrated lens, although it is a constant for the aberration-free case. This work was supported by the Foundations of National Hi-Tech Laboratory of Beam Control, and of State Key Laboratory of Laser Technology.
References
Fig. 3. Propagation of the kurtosis parameter of a ChG beam with W0 ¼ 4 passing through a lens with or without spherical aberration. C 4 ¼ 0, 4.0 104 mm3 .
[1] Siegman AE: New developments in laser resonators. SPIE 1224 (1990) 2–14 [2] Weber H: Propagation of hight-order intensity moments in quadratic-index media. Opt. Quantum Electron. 24 (1992) 1027–1048 [3] Martinez-Herrero R, Mejias PM, Sanchez M, Neira JLH: Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical. Opt. Quantum Electron. 24 (1992) 1021–1026 [4] Piquero G, Mejias PM, Martinez-Herrero R: On the propagation of the kurtosis parameter of general beams. Proceedings of the Workshop on Laser Beam Characterization, eds. Mejias PM, Weber H, Martinez-Herrero R, Gonzalez-Urena A (SEDO, Madrid, 1993) 141–148
148
Xiaoling Ji, Baida Lu¨, The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams
[5] Martinez-Herrero R, Piquero G, Mejias PM: On the propagation of the kurtosis parameter of general beams. Opt. Commun. 225 (1995) 225–232 [6] Piquero G, Mejias PM, Martinez-Herrero R: Sharpness changes of gaussian beams induced by spherically aberrated lenses. Opt Commun. 107 (1994) 179–183 [7] Casperson LW, Tovar AA: Hermite-sinusoidal-Gaussian beams in comlex optical systems. J. Opt. Soc. Am. A 15 (1998) 954–961
[8] Tovar AA, Casperson LW: Production and propagation of Hermite-sinusoidal-Gaussian laser beams. J. Opt. Soc. Am. A 15 (1998) 2425–2432 [9] Siegman AE: Analysis of laser beam quality degradation caused by quartic phase aberration. Appl. Opt. 32 (1993) 5893–5901 [10] Collins SA: Lens-system diffraction integral written terms of matrix optics. J. Opt. Soc. Am. 60 (1970) 1168–1177
145
International Journal for Light and Electron Optics
The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams Xiaoling Ji1, 2, Baida Lu¨2 1 2
College of Electronic Engineering, Sichuan Normal University, Chengdu 610066, China Institute of Laser Physics & Chemistry, Sichuan University, Chengdu 610064, China
Abstract: The propagation of the K parameter of Hermitecosh-Gaussian (HChG) beams passing through a spherically aberrated lens is studied by the straightforward integral of the Collins formula. Numerical calculations are made and the results show that for this case the K parameter changes upon propagation, and regardless of positive or negative aberration, spherical aberrations lead to an increment of the K parameter in comparison with the aberration-free case. The propagation properties of the K parameter of cosh-Gaussian (ChG) beams, and HermiteGaussian (H-G) beams are treated as special cases of HChG beams. Key words: Kurtosis parameter – Hermite-cosh-Gaussian (HChG) beam – spherical aberration
1. Introduction It is known that the second-order-moments-based beam propagation factor (M 2 factor) is a useful parameter characterizing different laser beams [1]. But it is insufficient, since no information concerning the shape features of a laser beam, such as the symmetry and degree of flatness, is inferred from the M 2 factor. The kurtosis parameter K related to the fourth- and second-order intensity moments is an important parameter, which describes the degree of sharpness of any beam [2–4]. Generally, the K parameter changes upon propagation through paraxial optical ABCD systems for all laser beams except for Gaussian and Hermit-Gaussian (H-G) beams, their shape remains unchanged upon propagation. In practical applications the propagation of the kurtosis parameter through ABCD optical systems is of interest [5], and the corresponding propagation formulae were given by Weber, Martinez-Herrero et al. [2, 3]. Piquero et al. studied sharpness changes in Gaussian beams induced by a spherically aberrated lens [6]. On the
Received 12 December 2001; accepted 3 March 2002. Correspondence to: B. Lu¨ Fax: ++86-28-540-3260 E-mail: [email protected]
other hand, the Hermite-sinusoidal-Gaussian beam introduced by Casperson et al. is a more general class of laser beams [7, 8]. Some familiar beams, such as the Gaussian, H-G and cosh-Gaussian (ChG) beams can be regarded as their special cases. The aim of the present paper is to deal with the prpagation of the kurtosis parameter of Hermite-cosh-Gaussian (HChG) beams. Starting from the Collins formula [9] instead of the propagation equation of the fourth-order moment in refs. [2] and [3], changes in the kurtosis parameter of HChG beams passing through the lens with or without spherical aberration are illustrated with detailed numerical examples. It is also shown that the results of ChG and H-G beams can be regarded as special cases of n ¼ 0 and W0 ¼ 0, respectively, treated in this paper.
2. Propagation of the kurtosis parameter of HChG beams passing through a spherically aberrated lens The field distribution of a one-dimensional HChG beam at the plane z ¼ 0 in the rectangular coordinate system is expressed as [7, 8] pffiffiffi x0 x20 cosh ðW0 x0 Þ exp 2 2 E0 ðx0 ; 0Þ ¼ Hn w0 w0 ð1Þ where w0 is the waist width of the Gaussian amplitude distribution, Hn is the nth Hermite polynomial, and W0 is the parameter associated with the cosh part. A spherically aberrated lens is characterized by a transmission profile of the form [9] Fðx0 Þ ¼ exp ðikC4 x40 Þ
ð2Þ
where C4 denotes the spherical aberration coefficient and k is the wave number related to the wavelength l by k¼
2p : l
ð3Þ 0030-4026/02/113/03-145 $ 15.00/0
146
Xiaoling Ji, Baida Lu¨, The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams
The propagation of HChG beams through a paraxial optical ABCD system obeys the Collins formula [10] rffiffiffiffiffiffi 1 ð i E0 ðx0 ; 0Þ Eðx; zÞ ¼ lB
þ Kmin ¼ Kmin ¼ 2:20 at the position of z=f ¼ 0:90. þ ¼ Kmax ¼ 2:30 for Another maximum Kmax C4 ¼ 3:0 104 mm3 is located at the focal plane 0 of z=f ¼ 1:0. The maxima Kmax ¼ 1:86,
1
ik ðAx20 2x0 x þ Dx2 Þ dx0 exp 2B
ð4Þ
where A; B; D are elements of the transfer matrix A B . Assume that the spherically aberrated lens B D of focal length f is located at the plane z ¼ 0, in consideration of eqs. (1) and (2), from eq. (4) we obtain rffiffiffiffiffi 1 ð pffiffiffi x0 i Hn Eðx; zÞ ¼ 2 cosh ðW0 x0 Þ w0 lz 1
x2 exp 02 exp ðikC4 x40 Þ w0 ik z 2 2 1 exp dx0 x0 2x0 x þ x 2z f ð5Þ where the propagation distance z is referred to the lens at the plane z ¼ 0. The kurtosis parameter K is defined as [4] K ¼ hx4 i=hx2 i2
ð6Þ
where 1 Ð
hxn i ¼
xn jEðx; zÞj2 dx
1 1 Ð
ðn ¼ 2; 4Þ :
ð7Þ
jEðx; zÞj2 dx
1
The substitution from eqs. (5) and (7) into eq. (6) delivers the propagation equation of the kurtosis parameter of the HChG beam through the spherically aberrated lens. Although it is difficult to derive a closedform propagation expression, numerical calculations are available. In addition, changes in the kurtosis parameter of ChG beams and H-G beams passing through the aberrated lens can be obtained readily by letting n ¼ 0 and W0 ¼ 0 in eqs. (5), (6) and (7), respectively. Typical numerical examples are compiled in figs. 1–3, where the calculation parameters are l ¼ 1:06 mm, w0 ¼ 0:5 mm and f ¼ 200 mm. Fig. 1 gives the propagation of the K parameter of a HChG beam with n ¼ 2, W0 ¼ 2 through a lens with or without spherical aberration. As can be seen, changes in the K parameter depend on the spherical aberration. For examþ ple, we have one maximum value of Kmax ¼ 2:38 located at the position of z=f ¼ 1:29, and one minimum þ ¼ 2:20 at the position of z=f ¼ 1:13 for Kmin C4 ¼ 3:0 104 mm3 . For C4 ¼ 3:0 104 mm3 , þ ¼ Kmax ¼ 2:38 at the there are one maximum Kmax position of z=f ¼ 0:82, and one minimum
Fig. 1. Propagation of the kurtosis parameter a HChG beam with n ¼ 2, W0 ¼ 2 passing through a lens with or without spherical aberration: a) C 4 ¼ 0, 3.0 104 mm3 , 5.0 104 mm3 ; b) C4 ¼ 0, 3.0 104 mm3 , 5.0 104 mm3 .
Xiaoling Ji, Baida Lu¨, The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams
Fig. 2. Propagation of the kurtosis parameter of a H-G beam with n ¼ 1 passing through a lens with or without spherical aberration. C 4 ¼ 0, 5.0 104 mm3 .
147
þ Kmax ¼ Kmax ¼ 3:90 are located at positions of z=f ¼ 1:0, 1.18 and 0.87 or cases of C4 ¼ 0; 5:0 104 mm3 , respectively. The effect of spherical aberrations on the K parameter of a H-G beam with n ¼ 1 is shown in fig. 2, which indicates that the K parameter is a constant of K 0 ¼ 1:67, while the H-G beam passes through an aberration-free lens ðC4 ¼ 0Þ: However, if the lens is spherically aberrated, the K parameter changes upon propagation. For C4 ¼ 5:0 104 mm3 , there are two maxima þ Kmax ¼ 2:25, 2.0 at positions of z=f ¼ 0:78, 1.34, and þ ¼ 1:8 at the position of z=f ¼ 1:08: one minimum Kmin For C4 ¼ 5:0 104 mm3 , there are two maxima þ ¼ Kmax ¼ 2:25, 2.0 at the positions of z=f ¼ 1:38, Kmax þ 0.81, and one minimum Kmin ¼ Kmin ¼ 1:80 at the position of z=f ¼ 0:93. Fig. 3 represents the variation of the K parameter of a ChG beam with W0 ¼ 4 passing through a lens with or without spherical aberration. 0 From fig. 3 it turns out that the maxima of Kmax ¼ 7:48, þ Kmax ¼ Kmax ¼ 12:89 appear at positions of z=f ¼ 1:0, 1.13 and 0.9 for C4 ¼ 0, 4:0 104 mm3 , respectively. In addition, from figs. 1–3 we see that, regardless of positive or negative aberration, spherical lead to an increment of the K parameter of HChG, ChG and HG beams, as compared with the aberration-free case.
3. Conclusion In this paper a detailed study of the effect of spherically aberrated lens on the K parameter of HChG, ChG and H-G beams has been performed by means of the straightforward integral of the Collins formula. It has been shown that positive and negative spherical aberrations affect the propagation of the K parameter of HChG, ChG and HG beams, and result in an increment of their K parameter in comparison with the aberration-free case. In particular, the K parameter of H-G beams changes after passing through the spherically aberrated lens, although it is a constant for the aberration-free case. This work was supported by the Foundations of National Hi-Tech Laboratory of Beam Control, and of State Key Laboratory of Laser Technology.
References
Fig. 3. Propagation of the kurtosis parameter of a ChG beam with W0 ¼ 4 passing through a lens with or without spherical aberration. C 4 ¼ 0, 4.0 104 mm3 .
[1] Siegman AE: New developments in laser resonators. SPIE 1224 (1990) 2–14 [2] Weber H: Propagation of hight-order intensity moments in quadratic-index media. Opt. Quantum Electron. 24 (1992) 1027–1048 [3] Martinez-Herrero R, Mejias PM, Sanchez M, Neira JLH: Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical. Opt. Quantum Electron. 24 (1992) 1021–1026 [4] Piquero G, Mejias PM, Martinez-Herrero R: On the propagation of the kurtosis parameter of general beams. Proceedings of the Workshop on Laser Beam Characterization, eds. Mejias PM, Weber H, Martinez-Herrero R, Gonzalez-Urena A (SEDO, Madrid, 1993) 141–148
148
Xiaoling Ji, Baida Lu¨, The effect of spherically aberrated lens on the kurtosis parameter of Hermite-cosh-Gaussian beams
[5] Martinez-Herrero R, Piquero G, Mejias PM: On the propagation of the kurtosis parameter of general beams. Opt. Commun. 225 (1995) 225–232 [6] Piquero G, Mejias PM, Martinez-Herrero R: Sharpness changes of gaussian beams induced by spherically aberrated lenses. Opt Commun. 107 (1994) 179–183 [7] Casperson LW, Tovar AA: Hermite-sinusoidal-Gaussian beams in comlex optical systems. J. Opt. Soc. Am. A 15 (1998) 954–961
[8] Tovar AA, Casperson LW: Production and propagation of Hermite-sinusoidal-Gaussian laser beams. J. Opt. Soc. Am. A 15 (1998) 2425–2432 [9] Siegman AE: Analysis of laser beam quality degradation caused by quartic phase aberration. Appl. Opt. 32 (1993) 5893–5901 [10] Collins SA: Lens-system diffraction integral written terms of matrix optics. J. Opt. Soc. Am. 60 (1970) 1168–1177