- Email: [email protected]
Focal switch of spherically aberrated Laguerre–Gaussian beams
ARTICLE IN PRESS
Optics & Laser Technology 39 (2007) 430–434 www.elsevier.com/locate/optlastec
Focal switch of spherically aberrated Laguerre–Gaussian beams Wang Chun-Fang, Tao Xiang-Yang, Nie Yi-You, Zeng zhao-Yang Institute of Physics & Communication and Electronics, Jiangxi Normal University, Nanchang 330027, China Received 28 September 2004; received in revised form 31 March 2005; accepted 11 June 2005 Available online 1 August 2005
Abstract Based on the Collins formula, we study the focal switch of spherically aberrated Laguerre–Gaussian (L–G) beams passing through an optical system with the aperture and lens separated. Our results demonstrate that the behavior of the focal switch of spherically aberrated L–G beams is generally dependent on the spherical aberration coefficient, truncation parameter, Fresnel number and mode indices. The spherical aberration affects the relative focal shift, critical position of the focal switch and the relative transition. We also illustrate that there exists a minimum in addition to a maximum in the truncation parameter. Only inside the region between the minimum and the maximum the focal switch effect can be expected. The region depends on the spherical aberration coefficient as well as the Fresnel number. r 2005 Elsevier Ltd. All rights reserved. Keywords: Focal shift and focal switch; Spherical aberration; Laguerre–Gaussian beam
1. Introduction It is well known that the point of the maximum irradiance of a particular beam along the axis is not exactly located at the geometrical focus when the beam is focused by an aperture lens, but rather somewhat closer to the aperture lens. This phenomenon is called focal shift [1] and has been widely studied in the literature [2–4] since 1981. It is shown that the spherical aberration affects the qualities [5] of beams as well as the focal shift, even the focal shift will vanish [6] when using a specific spherical aberration lens. However, in 1996, Martinez and Climent found that, when a converging spherical wave illuminates an axially superresolving diffraction screen, simultaneously with the focal shift, an effective permutation [7] of the focal point would appear. This kind of novel effect is referred to as focal switch. Furthermore, it was found that the phenomenon could also appear when a beam illuminates a Fresnel zone plate [8]. The above optical systems and beams used to study the focal switch are free of spherical Corresponding author.
E-mail address: [email protected] (W. Chun-Fang). 0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.06.030
aberration. However, due to the fabrication imperfections, there exist more or less phase aberrations. The purpose of the present work is to investigate the focal switch effect of the spherically aberrated L–G beams. It is shown that the focal switch effect appears when spherically aberrated L–G beams pass through an optical system with the aperture and lens separated. The influence of spherical aberration on focal switch is studied in detail.
2. Background knowledge Fig.1 shows the optical system under consideration, comprising a square aperture with full width 2a and a lens with focal length f. The separation between the aperture and lens is s. The incident L–G beam with initial field in the cylindrical coordinate system may take the form [9] pffiffiffi !l 2 2 2r0 r 2r0 Eðr0 ; j0 ; z0 ¼ 0Þ ¼ exp 02 Llp w0 w0 w20 expðilj0 Þ,
ð1Þ
ARTICLE IN PRESS W. Chun-Fang et al. / Optics & Laser Technology 39 (2007) 430–434
where w0 is the waist width corresponding to the fundamental Gaussian beam, r0 represents the radial distance in incidence plane, LlP denotes Laguerre polynomials with mode indices p and l, j0 is the initial phase and l is the wavelength. According to the Collins formula, the field Eðr; j; zÞ of the spherically aberrated L–G beam passing through the system can be expressed as [10] Z 2p Z a i Eðr; j; zÞ ¼ Eðr0 ; j0 ; z0 Þ exp½ikC 4 r40 lB 0 0 ip ½Ar20 2r0 r cosðj0 jÞ þ Dr2 exp lB r0 dr0 dj0 ,
3. Numerical results and analysis A great deal of numerical calculations were performed by using the above equations to illustrate the focusing properties and focal switch effect of spherically aberrated L–G beams passing through the optical system shown in Fig. 1. In the following calculations only the case l ¼ 0 is taken into consideration, which gives rise to the non-zero axial intensity and is of interest for the present work. In addition, we take l ¼ 1:06 103 mm and w0 ¼ 1 mm. Fig. 2 represents the relative focal shift
ð2Þ
where k is the wave number related to the wavelength l by k ¼ 2p=l, C 4 is the spherical aberration coefficient [5], and the axial distance z is referred to the back geometrical focal plane F of the lens. The parameters A; B and D are determined by ! 1 0 1 s A B 1 f þz . (3) ¼ 1=f 1 0 1 C D 0 1
0
z0
f
(4)
0.8 C4=10-3mm-3
lf
.
(5)
TEM10
0.4 ∆ zf
Nw ¼
z
Fig. 1. An optical system with the aperture and lens separated.
and the Fresnel number associated with the corresponding spherically aberrated L–G beam by w20
C4=5×10-4mm-3
0
Substituting Eqs. (1) and (3) into Eq. (2), the axial irradiance is expressed as Ið0; 0; zÞ ¼ jEð0; 0; zÞj2 ,
jf þ zmax j jRj , jRj
0
1.5
2
2.5
s/f 0.8 C4=5×10-4mm-3
TEM20
(7) 0 C4=0
(8) C4=10-3mm-3
(9)
C4=10-4mm-3
-0.8 0 (b)
s2 þ p2 f 2 N 2w . s ðs2 =f þ p2 fN 2w Þ
1
0.5
(a)
where R is the position of the geometrical focal point of un-truncated or weakly truncated L–G beams focused by lens, given by R¼
C4=0
-0.4
∆zf
dIð0; 0; zÞ ¼ 0. dz The relative focal shift is defined as [11]
C4=10-4mm-3
(6)
where we have assumed that r ¼ 0, j ¼ 0 and l ¼ 0. The position of axial maximum intensity zmax can be determined by
Dzf ¼
P
F
2a
s
For convenience, we define the truncation parameter by a2 d¼ 2 w0
431
0.5
1
1.5
2
s/f
Fig. 2. The relative focal shift Dzf versus relative separations s=f for different values of C 4 : (a) TEM10 -mode, (b) TEM20 -mode.
ARTICLE IN PRESS W. Chun-Fang et al. / Optics & Laser Technology 39 (2007) 430–434
0.6 TEM10 0.3
∆ zf
Dzf versus the relative separation s=f for different values of the spherical aberration coefficient C 4 , where d ¼ 1, N w ¼ 4. As can be seen, the relative focal shift demonstrates a transition from a negative value to a positive value as the relative separation s=f passes through a turning position sc =f , an apparent focal switch effect. However, for the TEM10 -mode L–G beam, C 4 ¼ 103 mm3 , in Fig. 2(a), the sign of Dzf uniformly turns to be positive around the turning position sc =f . Compared with the aberration-free case, where the turning position is always located at sc =f ¼ 1 [12], our results for spherical aberration reveal an important fact that the critical position sc =f is not definitely 1. The critical position sc =f has a strong dependence on the mode indices and the spherical aberration coefficient C 4 . For the TEM10 -mode L–G beam in Fig. 2(a), the critical position sc =f increases with increasing C 4 , whereas for the TEM20 -mode L–G beam in Fig. 2(b), sc =f decreases as C 4 increases. Besides the turning position, the relative transition height Dzsw defined as Dzsw ¼ Dzmax =jRj, with Dzmax being the transition height at the turning point (absolute value), is also dependent on C 4 . The relative focal shift Dzf of TEM10 - and TEM20 mode L–G beams versus s=f for different truncation parameters d is plotted in Fig. 3 for C 4 ¼ 104 mm3 and N w ¼ 4. One can also find an apparent focal switch effect; moreover, the critical point and the relative transition height now depend strongly on the truncation parameter d. In Fig. 4 we depict the relative transition height Dzsw of TEM10 - and TEM20 -mode L–G beams as a function of the truncation parameter d for different values of N w and different C 4 . One sees that the relative transition height Dzsw depends on the aberration coefficient C 4 , truncation parameter d, Fresnel number N w and mode indices in general. We find an interesting phenomenon in Fig. 4; there appears a critical minimum dc;min in addition to a critical maximum dc;max of the truncation parameter. Only within the region dc;min ododc;max , the focal switch effect can be seen. The results also show that even for the aberration-free case, a critical minimum value for the truncation parameter still exists. Nevertheless, the region Dd ¼ dc;max dc;min shrinks in the presence of spherical aberration. Considering first the TEM10 -mode with N w ¼ 4, one observes that 0:64odo4:7 in the aberration-free case C 4 ¼ 0, whereas 0:69odo2:2 when C 4 ¼ 104 mm3 . Dd decreases from 4.06 to 1.51. For the TEM20 -mode with N w ¼ 4, one sees that 0:34odo7:7 in the case C 4 ¼ 0 and 0:4odo3:05 when C 4 ¼ 104 mm3 . Dd is also reduced from 7.36 to 2.65. In Fig. 4(a) and (b), one sees approximately the same 0:64odo4:7 for the TEM10 -mode and 0:34odo7:7 for the TEM20 -mode for different Fresnel numbers N w . The observation supports the fact that the value Dd has a trivial dependence on the Fresnel number N w for some
0
δ=2
-0.3
δ=1.5 δ =1
-0.6 0
0.5
(a)
1
1.5
2
s/f 0.6 TEM20 0.3 δ=1.5
∆ zf
432
0 δ=2 -0.3 δ =1 -0.6
0
(b)
0.5
1
1.5
2
s/f
Fig. 3. The relative focal shift Dzf versus relative separation s=f for different values of d: (a) TEM10 -mode, (b) TEM20 -mode.
specific beams in the aberration-free case. However, it cannot be expected in the aberration case. From Fig. 4 (c) and (d) we find a different situation: the value Dd decreases with increase in the Fresnel number N w . We plotted in Fig. 5 the relation of turning position sc =f with the truncation parameter d for different Fresnel numbers N w . The spherical aberration coefficient C 4 is taken as 104 mm3 . One sees that the turning position sc =f varies with the truncation parameter d. It is no longer always located at sc =f ¼ 1 [12], as in the aberration-free case. For the TEM10 -mode in Fig. 5 (a), sc =f decreases with increase in the truncation parameter d, whereas for the TEM20 -mode L–G beam in Fig. 5 (b), the dependence of sc =f on d is not monotonous, which is relevant to different intensity distributions of TEM10 - and TEM20 -mode L–G beams. Moreover, sc =f decreases as the Fresnel number N w increases.
4. Conclusion In this paper the focal switch of the spherical aberration L–G beams diffracted in an aperture and
ARTICLE IN PRESS W. Chun-Fang et al. / Optics & Laser Technology 39 (2007) 430–434
433
1.2
1.2
TEM10
TEM10 Nw=3
Nw=2
0.8
0.8 Nw=3 sc /f
∆zsw
Nw=4
0.4
0
0.4
0
1
3
2
(a)
0 0.5
5
4
Nw=4
1
1.5
(a)
2
2.5
3
3.5
1.2
TEM20
TEM20
1.6
Nw=2
Nw=3 ∆ zsw
0.8
Nw=3
sc /f
Nw=4 0.8
Nw=4
0.4
0
0
4
2
(b)
8
6
0
10
0
1
(b)
2
3
4
1.8
TEM10
Fig. 5. The turning position sc =f of spherical aberration L–G beams versus d: (a) TEM10 -mode, (b) TEM20 -mode.
Nw=2 1.2
∆ zsw
Nw=3 Nw=4 0.6
0 0
1
0.5
1.5
2.5
2
3
(c) 3
lens separated system has been studied in detail. The main results are as follows: (1) the critical position sc =f is not definitely 1, depending strongly on the mode indices, the spherical aberration coefficient and the truncation parameter: (2) apart from a critical maximum truncation parameter, there also exists a critical minimum truncation parameter, and only inside the region dc;min ododc;max , the focal switch can be expected.
TEM20 Nw=2
Acknowledgements
2
∆zsw
Nw=3
Nw=4
This work was supported by the Nature Science Foundation of Jiangxi Province of China.
1
References 0 0 (d)
1
2
3
4
Fig. 4. The relative transition height Dzsw versus relative separation d for different values of d: (a) TEM10 -mode, C 4 ¼ 0; (b) TEM20 -mode, C 4 ¼ 0; (c) TEM10 -mode, C 4 ¼ 104 mm3 ; (d) TEM20 -mode, C 4 ¼ 104 mm3 .
[1] Li Y, Wolf E. Focal shifts in diffracted converging spherical waves. Opt Commun 1981;39:211–5. [2] Li Y, Wolf E. Focal shift in focused truncated Gaussian beams. Opt Commun 1982;42:151–6. [3] Poon T-C. Focal shift in focused annular beams. Opt Commun 1988;65:401–6. [4] Kathuria YP. Focal shift in converging annular beam: a corollary. J Opt (Paris) 1989;20:141–4.
ARTICLE IN PRESS 434
W. Chun-Fang et al. / Optics & Laser Technology 39 (2007) 430–434
[5] Siegman AE. The beam quality of annular lasers and related problems. Appl Opt 1993;32:5893–901. [6] Yoshida A, Asakura T. Focusing a Gaussian laser beams without focal shift. Opt Commun 1994;109:368–74. [7] Martinez-Corral M, Clinment V. Focal switch: a new effect in low-Fresnel-number systems. Appl Opt 1996;35: 24–7. [8] Li Y. Focal shift and focal switch in dual-focus systems. J Opt Soc Am A 1997;149:1297–304.
[9] Saghafi S, Sheppard CJR. Near and far field of elegant Hermiteand Laguerre-Gaussian modes. J Mod Opt 1998;45(10): 1999–2009. [10] Collins SA. Lens-system diffraction integral written in terms of matrix optics. J Opt Soc Am 1970;60;1168–70. [11] Li Y. Propagation of focal shift through axisymmetrical optical system. Opt Commun 1993;95:13–7. [12] Bai-Da Lu¨, Run-Wu Peng. Focal shift and focal switch in systems with the aperture and lens separated. Appl Opt 2002;4:601–5.
Optics & Laser Technology 39 (2007) 430–434 www.elsevier.com/locate/optlastec
Focal switch of spherically aberrated Laguerre–Gaussian beams Wang Chun-Fang, Tao Xiang-Yang, Nie Yi-You, Zeng zhao-Yang Institute of Physics & Communication and Electronics, Jiangxi Normal University, Nanchang 330027, China Received 28 September 2004; received in revised form 31 March 2005; accepted 11 June 2005 Available online 1 August 2005
Abstract Based on the Collins formula, we study the focal switch of spherically aberrated Laguerre–Gaussian (L–G) beams passing through an optical system with the aperture and lens separated. Our results demonstrate that the behavior of the focal switch of spherically aberrated L–G beams is generally dependent on the spherical aberration coefficient, truncation parameter, Fresnel number and mode indices. The spherical aberration affects the relative focal shift, critical position of the focal switch and the relative transition. We also illustrate that there exists a minimum in addition to a maximum in the truncation parameter. Only inside the region between the minimum and the maximum the focal switch effect can be expected. The region depends on the spherical aberration coefficient as well as the Fresnel number. r 2005 Elsevier Ltd. All rights reserved. Keywords: Focal shift and focal switch; Spherical aberration; Laguerre–Gaussian beam
1. Introduction It is well known that the point of the maximum irradiance of a particular beam along the axis is not exactly located at the geometrical focus when the beam is focused by an aperture lens, but rather somewhat closer to the aperture lens. This phenomenon is called focal shift [1] and has been widely studied in the literature [2–4] since 1981. It is shown that the spherical aberration affects the qualities [5] of beams as well as the focal shift, even the focal shift will vanish [6] when using a specific spherical aberration lens. However, in 1996, Martinez and Climent found that, when a converging spherical wave illuminates an axially superresolving diffraction screen, simultaneously with the focal shift, an effective permutation [7] of the focal point would appear. This kind of novel effect is referred to as focal switch. Furthermore, it was found that the phenomenon could also appear when a beam illuminates a Fresnel zone plate [8]. The above optical systems and beams used to study the focal switch are free of spherical Corresponding author.
E-mail address: [email protected] (W. Chun-Fang). 0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.06.030
aberration. However, due to the fabrication imperfections, there exist more or less phase aberrations. The purpose of the present work is to investigate the focal switch effect of the spherically aberrated L–G beams. It is shown that the focal switch effect appears when spherically aberrated L–G beams pass through an optical system with the aperture and lens separated. The influence of spherical aberration on focal switch is studied in detail.
2. Background knowledge Fig.1 shows the optical system under consideration, comprising a square aperture with full width 2a and a lens with focal length f. The separation between the aperture and lens is s. The incident L–G beam with initial field in the cylindrical coordinate system may take the form [9] pffiffiffi !l 2 2 2r0 r 2r0 Eðr0 ; j0 ; z0 ¼ 0Þ ¼ exp 02 Llp w0 w0 w20 expðilj0 Þ,
ð1Þ
ARTICLE IN PRESS W. Chun-Fang et al. / Optics & Laser Technology 39 (2007) 430–434
where w0 is the waist width corresponding to the fundamental Gaussian beam, r0 represents the radial distance in incidence plane, LlP denotes Laguerre polynomials with mode indices p and l, j0 is the initial phase and l is the wavelength. According to the Collins formula, the field Eðr; j; zÞ of the spherically aberrated L–G beam passing through the system can be expressed as [10] Z 2p Z a i Eðr; j; zÞ ¼ Eðr0 ; j0 ; z0 Þ exp½ikC 4 r40 lB 0 0 ip ½Ar20 2r0 r cosðj0 jÞ þ Dr2 exp lB r0 dr0 dj0 ,
3. Numerical results and analysis A great deal of numerical calculations were performed by using the above equations to illustrate the focusing properties and focal switch effect of spherically aberrated L–G beams passing through the optical system shown in Fig. 1. In the following calculations only the case l ¼ 0 is taken into consideration, which gives rise to the non-zero axial intensity and is of interest for the present work. In addition, we take l ¼ 1:06 103 mm and w0 ¼ 1 mm. Fig. 2 represents the relative focal shift
ð2Þ
where k is the wave number related to the wavelength l by k ¼ 2p=l, C 4 is the spherical aberration coefficient [5], and the axial distance z is referred to the back geometrical focal plane F of the lens. The parameters A; B and D are determined by ! 1 0 1 s A B 1 f þz . (3) ¼ 1=f 1 0 1 C D 0 1
0
z0
f
(4)
0.8 C4=10-3mm-3
lf
.
(5)
TEM10
0.4 ∆ zf
Nw ¼
z
Fig. 1. An optical system with the aperture and lens separated.
and the Fresnel number associated with the corresponding spherically aberrated L–G beam by w20
C4=5×10-4mm-3
0
Substituting Eqs. (1) and (3) into Eq. (2), the axial irradiance is expressed as Ið0; 0; zÞ ¼ jEð0; 0; zÞj2 ,
jf þ zmax j jRj , jRj
0
1.5
2
2.5
s/f 0.8 C4=5×10-4mm-3
TEM20
(7) 0 C4=0
(8) C4=10-3mm-3
(9)
C4=10-4mm-3
-0.8 0 (b)
s2 þ p2 f 2 N 2w . s ðs2 =f þ p2 fN 2w Þ
1
0.5
(a)
where R is the position of the geometrical focal point of un-truncated or weakly truncated L–G beams focused by lens, given by R¼
C4=0
-0.4
∆zf
dIð0; 0; zÞ ¼ 0. dz The relative focal shift is defined as [11]
C4=10-4mm-3
(6)
where we have assumed that r ¼ 0, j ¼ 0 and l ¼ 0. The position of axial maximum intensity zmax can be determined by
Dzf ¼
P
F
2a
s
For convenience, we define the truncation parameter by a2 d¼ 2 w0
431
0.5
1
1.5
2
s/f
Fig. 2. The relative focal shift Dzf versus relative separations s=f for different values of C 4 : (a) TEM10 -mode, (b) TEM20 -mode.
ARTICLE IN PRESS W. Chun-Fang et al. / Optics & Laser Technology 39 (2007) 430–434
0.6 TEM10 0.3
∆ zf
Dzf versus the relative separation s=f for different values of the spherical aberration coefficient C 4 , where d ¼ 1, N w ¼ 4. As can be seen, the relative focal shift demonstrates a transition from a negative value to a positive value as the relative separation s=f passes through a turning position sc =f , an apparent focal switch effect. However, for the TEM10 -mode L–G beam, C 4 ¼ 103 mm3 , in Fig. 2(a), the sign of Dzf uniformly turns to be positive around the turning position sc =f . Compared with the aberration-free case, where the turning position is always located at sc =f ¼ 1 [12], our results for spherical aberration reveal an important fact that the critical position sc =f is not definitely 1. The critical position sc =f has a strong dependence on the mode indices and the spherical aberration coefficient C 4 . For the TEM10 -mode L–G beam in Fig. 2(a), the critical position sc =f increases with increasing C 4 , whereas for the TEM20 -mode L–G beam in Fig. 2(b), sc =f decreases as C 4 increases. Besides the turning position, the relative transition height Dzsw defined as Dzsw ¼ Dzmax =jRj, with Dzmax being the transition height at the turning point (absolute value), is also dependent on C 4 . The relative focal shift Dzf of TEM10 - and TEM20 mode L–G beams versus s=f for different truncation parameters d is plotted in Fig. 3 for C 4 ¼ 104 mm3 and N w ¼ 4. One can also find an apparent focal switch effect; moreover, the critical point and the relative transition height now depend strongly on the truncation parameter d. In Fig. 4 we depict the relative transition height Dzsw of TEM10 - and TEM20 -mode L–G beams as a function of the truncation parameter d for different values of N w and different C 4 . One sees that the relative transition height Dzsw depends on the aberration coefficient C 4 , truncation parameter d, Fresnel number N w and mode indices in general. We find an interesting phenomenon in Fig. 4; there appears a critical minimum dc;min in addition to a critical maximum dc;max of the truncation parameter. Only within the region dc;min ododc;max , the focal switch effect can be seen. The results also show that even for the aberration-free case, a critical minimum value for the truncation parameter still exists. Nevertheless, the region Dd ¼ dc;max dc;min shrinks in the presence of spherical aberration. Considering first the TEM10 -mode with N w ¼ 4, one observes that 0:64odo4:7 in the aberration-free case C 4 ¼ 0, whereas 0:69odo2:2 when C 4 ¼ 104 mm3 . Dd decreases from 4.06 to 1.51. For the TEM20 -mode with N w ¼ 4, one sees that 0:34odo7:7 in the case C 4 ¼ 0 and 0:4odo3:05 when C 4 ¼ 104 mm3 . Dd is also reduced from 7.36 to 2.65. In Fig. 4(a) and (b), one sees approximately the same 0:64odo4:7 for the TEM10 -mode and 0:34odo7:7 for the TEM20 -mode for different Fresnel numbers N w . The observation supports the fact that the value Dd has a trivial dependence on the Fresnel number N w for some
0
δ=2
-0.3
δ=1.5 δ =1
-0.6 0
0.5
(a)
1
1.5
2
s/f 0.6 TEM20 0.3 δ=1.5
∆ zf
432
0 δ=2 -0.3 δ =1 -0.6
0
(b)
0.5
1
1.5
2
s/f
Fig. 3. The relative focal shift Dzf versus relative separation s=f for different values of d: (a) TEM10 -mode, (b) TEM20 -mode.
specific beams in the aberration-free case. However, it cannot be expected in the aberration case. From Fig. 4 (c) and (d) we find a different situation: the value Dd decreases with increase in the Fresnel number N w . We plotted in Fig. 5 the relation of turning position sc =f with the truncation parameter d for different Fresnel numbers N w . The spherical aberration coefficient C 4 is taken as 104 mm3 . One sees that the turning position sc =f varies with the truncation parameter d. It is no longer always located at sc =f ¼ 1 [12], as in the aberration-free case. For the TEM10 -mode in Fig. 5 (a), sc =f decreases with increase in the truncation parameter d, whereas for the TEM20 -mode L–G beam in Fig. 5 (b), the dependence of sc =f on d is not monotonous, which is relevant to different intensity distributions of TEM10 - and TEM20 -mode L–G beams. Moreover, sc =f decreases as the Fresnel number N w increases.
4. Conclusion In this paper the focal switch of the spherical aberration L–G beams diffracted in an aperture and
ARTICLE IN PRESS W. Chun-Fang et al. / Optics & Laser Technology 39 (2007) 430–434
433
1.2
1.2
TEM10
TEM10 Nw=3
Nw=2
0.8
0.8 Nw=3 sc /f
∆zsw
Nw=4
0.4
0
0.4
0
1
3
2
(a)
0 0.5
5
4
Nw=4
1
1.5
(a)
2
2.5
3
3.5
1.2
TEM20
TEM20
1.6
Nw=2
Nw=3 ∆ zsw
0.8
Nw=3
sc /f
Nw=4 0.8
Nw=4
0.4
0
0
4
2
(b)
8
6
0
10
0
1
(b)
2
3
4
1.8
TEM10
Fig. 5. The turning position sc =f of spherical aberration L–G beams versus d: (a) TEM10 -mode, (b) TEM20 -mode.
Nw=2 1.2
∆ zsw
Nw=3 Nw=4 0.6
0 0
1
0.5
1.5
2.5
2
3
(c) 3
lens separated system has been studied in detail. The main results are as follows: (1) the critical position sc =f is not definitely 1, depending strongly on the mode indices, the spherical aberration coefficient and the truncation parameter: (2) apart from a critical maximum truncation parameter, there also exists a critical minimum truncation parameter, and only inside the region dc;min ododc;max , the focal switch can be expected.
TEM20 Nw=2
Acknowledgements
2
∆zsw
Nw=3
Nw=4
This work was supported by the Nature Science Foundation of Jiangxi Province of China.
1
References 0 0 (d)
1
2
3
4
Fig. 4. The relative transition height Dzsw versus relative separation d for different values of d: (a) TEM10 -mode, C 4 ¼ 0; (b) TEM20 -mode, C 4 ¼ 0; (c) TEM10 -mode, C 4 ¼ 104 mm3 ; (d) TEM20 -mode, C 4 ¼ 104 mm3 .
[1] Li Y, Wolf E. Focal shifts in diffracted converging spherical waves. Opt Commun 1981;39:211–5. [2] Li Y, Wolf E. Focal shift in focused truncated Gaussian beams. Opt Commun 1982;42:151–6. [3] Poon T-C. Focal shift in focused annular beams. Opt Commun 1988;65:401–6. [4] Kathuria YP. Focal shift in converging annular beam: a corollary. J Opt (Paris) 1989;20:141–4.
ARTICLE IN PRESS 434
W. Chun-Fang et al. / Optics & Laser Technology 39 (2007) 430–434
[5] Siegman AE. The beam quality of annular lasers and related problems. Appl Opt 1993;32:5893–901. [6] Yoshida A, Asakura T. Focusing a Gaussian laser beams without focal shift. Opt Commun 1994;109:368–74. [7] Martinez-Corral M, Clinment V. Focal switch: a new effect in low-Fresnel-number systems. Appl Opt 1996;35: 24–7. [8] Li Y. Focal shift and focal switch in dual-focus systems. J Opt Soc Am A 1997;149:1297–304.
[9] Saghafi S, Sheppard CJR. Near and far field of elegant Hermiteand Laguerre-Gaussian modes. J Mod Opt 1998;45(10): 1999–2009. [10] Collins SA. Lens-system diffraction integral written in terms of matrix optics. J Opt Soc Am 1970;60;1168–70. [11] Li Y. Propagation of focal shift through axisymmetrical optical system. Opt Commun 1993;95:13–7. [12] Bai-Da Lu¨, Run-Wu Peng. Focal shift and focal switch in systems with the aperture and lens separated. Appl Opt 2002;4:601–5.