Focal shift in Gaussian beams focused by a spherically aberrated bifocal lens

1 March 2001

Optics Communications 189 (2001) 47±53

www.elsevier.com/locate/optcom

Focal shift in Gaussian beams focused by a spherically aberrated bifocal lens Baida L u a,*, Xiaoling Ji a,b a

b

Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, People's Republic of China Department of Physics and Information Technology, Chongqing Normal University, Chongqing 400047, People's Republic of China Received 11 September 2000; accepted 12 December 2000

Abstract The focusing of Gaussian beams by a spherically aberrated bifocal lens and related focal shift are studied. It is shown that the axial irradiance distribution, maximum irradiance along the axis and its position of focused Gaussian beams depend upon the Fresnel number, spherical aberration and focal length of the bifocal lens in general. A detailed analysis of the e€ect of spherical aberration and bifocal length on focal shift is made. Ó 2001 Published by Elsevier Science B.V. Keywords: Focal shift; Spherical aberration; Bifocal lens; Gaussian beam

1. Introduction The laser beam focusing and the concept of the focal shift introduced by Wolf and Li [1±3] are subjects of current interest. Recently, a number of publications have been devoted to studying the focusing of Gaussian beams without focal shift and beyond conventional di€raction limit, where the e€ect of spherical aberration of a focusing lens has been considered [4±7]. Furthermore, the dependence of focal shift on Fresnel number and angular aperture has been reexamined [8], and the focal shift de®ned in terms of the second moment and its connection with the optical transfer function have been discussed [9]. The aim of the present paper is to study the focusing properties of Gaussian beams by a spherically aberrated bifocal lens and the focal * Corresponding author. Tel.: +86-28-541-2819; fax: +86-28540-3260. E-mail address: [email protected] (B. L u).

shift. Starting from the generalized Huygens± Fresnel di€raction integral [10], the irradiance distribution of Gaussian beams focused by an unapertured aberration-free bifocal lens and the third-order equation governing the position of maximum irradiance along the axis are derived. Numerical calculations are performed for both aberration-free and spherical aberration cases. It is found that for the aberration-free case the point of maximum irradiance zmax along the axis is located in the proximity of one waist position, but does not always coincide with it. The variation of the axial maximum irradiance Imax and its position zmax with spherical aberration and focal length is analyzed for the spherical aberration case, which shows that the focal shift can be away from the lens, and even exceed the larger geometrical focal point of the bifocal lens by a suitable choice of the positive spherical aberration coecient. The focusing beyond conventional di€raction limit can be achieved in use of a bifocal lens with negative aberration.

0030-4018/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 0 0 1 - X

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B. Lu, X. Ji / Optics Communications 189 (2001) 47±53

2. Theoretical model

Nwx ˆ

Assume that in the rectangular coordinate system a Gaussian beam with initial ®eld   x2 ‡ y 2 E0 …x; y; 0† ˆ exp …1† w20 is incident upon an unapertured bifocal lens with focal length fx and fy …fx > 0, fy > 0† at the z ˆ 0 plane, where w0 is the waist width of the beam, and the spherically aberrated bifocal lens is characterized by a transmission pro®le of the form [4,11], u…x; y† ˆ exp ‰ ikS1 …x4 ‡ y 4 †=w40 Š

…2†

where S1 denotes the spherical aberration coecient and k is the wave number related to the wavelength k by 2p kˆ k

…3†

For the sake of simplicity in Eq. (2) the case of two orthogonal but equal S1 cylindrical lenses is considered [12], however, the extension is straightforward. The substitution from Eqs. (1) and (2) into the generalized Huygens±Fresnel di€raction integral [10] leads to the ®eld distribution E…x; y; z† of the beam at the z-plane after passing through the bifocal lens   Z 1 i x20 E…x; y; z† ˆ exp kz 1 kfx Nwx ! 1 4  exp ikS1 2 x 2 0 k fx2 Nwx     ik z 2 1  exp 2x0 x dx0 x 2z fx 0   Z 1 y02  exp kfx Nwx 1 ! 1  exp ikS1 2 y4 2 0 k fx2 Nwx     ik z 2 1  exp 2y0 y dy0 y 2z fy 0 …4† where

w20 kfx

…5†

denotes the Fresnel number of the Gaussian beam in the x direction. The irradiance I…x; y; z† reads as I…x; y; z† ˆ j E…x; y; z†j

2

…6†

and the position of maximum irradiance of the focused Gaussian beam along the z-axis is determined by dI…0; 0; z† ˆ0 dz

…7†

The irradiance distribution for the aberrationfree case can be obtained from Eqs. (4) and (6) by letting S1 ˆ 0, and is given by I…x; y; z† ˆ 1  v" u  2  2 #" 2  2 # u z z z t 1 z ‡ ‡ 1 fx pfx Nwx fy pfx Nwx 2 3 6 6  exp 6 4

2

 kfx Nwx 1

2 6 6  exp 6 4

2

 kfx Nx 1

1 z fx

2 ‡

1 2

z fy

‡

2

kz p2 fx Nwx

2

kz p2 fx Nwx

7 7 x2 7 5 3

7 7 y27 5

…8†

It follows from Eq. (8) readily that the waist width w0j (j ˆ x, y unless otherwise stated), position of the waist width z0j and axial irradiance I…0; 0; z† at the z-plane are expressed as w0 fj …9† w0j ˆ q 2 2 fj ‡ p2 fx2 Nwx z0j ˆ fj

fj2

fj3 2 ‡ p2 fx2 Nwx

…10†

I…0; 0; z† 1  ˆv u" 2  2 #" 2  2 # u z z z t 1 z ‡ ‡ 1 fx pfx Nwx fy pfx Nwx

…11†

B. Lu, X. Ji / Optics Communications 189 (2001) 47±53

On substituting from Eq. (11) into Eq. (7), we obtain the following third-order equation governing the position of maximum irradiance zmax : " !#  1 1 1 1 2 ‡ ‡ z3max 2 2 fx2 p2 fx2 Nwx fy2 p2 fx2 Nwx    1 1 1 1 2 3 2 2 2 ‡ ‡ z p fx Nwx fx fy fx fy max " # 2 4 1 1 ‡ ‡ ‡ ‡ zmax 2 p2 fx2 Nwx fx fy fx2 fy2   1 1 ‡ ˆ0 …12† fx fy For the conventional case of a Gaussian beam focused by a circular lens (fx ˆ fy ˆ f , Nwx ˆ Nw ) Eq. (12) reduces to zmax p2 Nw2 ˆ f 1 ‡ p2 Nw2

…13†

The relative focal shift D ˆ …zmax f †=f is deduced from Eq. (13), and is given by

Dˆ

1 1 ‡ p2 Nw2

49

…14†

which is in agreement with Eq. (4.2) in Ref. [3].

3. Numerical results and analysis A large number of numerical calculations were performed by using above equations to illustrate the focusing properties and focal shift of Gaussian beams passing through a bifocal lens with or without spherical aberration, and in calculations k ˆ 1:06 lm, fx is ®xed and equal to 200 mm. Typical results for the aberration-free case are compiled in Figs. 1±3. Fig. 1 gives the aspect ratio w0x =w0y as a function of fy for Nwx ˆ 0:3, 0.5, 1.0, and 1.5, which indicates that w0x =w0y > 1, if fy < fx and vice versa, and w0x ˆ w0y for fy ˆ fx . The axial irradiance distribution I…0; 0; z† of a Gaussian beam focused by a bifocal lens is shown in Fig. 2. We see that I…0; 0; z† depends upon fy , if Nwx and fx

Fig. 1. Aspect ratio w0x =w0y of a Gaussian beam focused by a bifocal lens versus focal length fy .

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B. Lu, X. Ji / Optics Communications 189 (2001) 47±53

Fig. 2. Axial irradiance distribution I…0; 0; z† of a Gaussian beam focused by a bifocal lens, Nwx ˆ 1:5.

are ®xed. A comparison of the waist position z0y and point of maximum irradiance along the axis zmax is given in Fig. 3, from which it turns out that z0x < fx , z0y < fy , zmax < maxffx ; fy g, and zmax is located in the proximity of z0y , but does not always coincide with z0y , and slightly shifts towards z0x if fy < fx , whereas for the case of fy > fx zmax is positioned in the proximity of z0x and slightly shifts towards z0y . If fy ˆ fx , then zmax ˆ z0x ˆ z0y , which is easily seen from Eqs. (10) and (13). Numerical examples are z0y ˆ 127:6 mm; zmax z0y ˆ 3:7 mm for fy ˆ 130 mm; z0y ˆ 146:3 mm, zmax z0y ˆ 7:3 mm for fy ˆ 150 mm; z0x ˆ 191:4 mm, zmax z0x ˆ 11:4 mm for fy ˆ 250 mm, and z0x ˆ 191:4 mm, zmax z0x ˆ 9:9 mm for fy ˆ 300 mm; and if fy ˆ fx ˆ 200 mm, we have zmax ˆ z0x ˆ z0y ˆ 191:4 mm. Figs. 4±6 summarize numerical results for a Gaussian beam focused by a spherically aberrated bifocal lens. The axial irradiance distribution of a focused Gaussian beam is given in Fig. 4 for fy ˆ 180 mm, Nwx ˆ 1:5 and kS1 ˆ 0:1, 0, 0.1. As can be seen, the axial maximum irradiance for

kS1 ˆ 0:1 is larger than that for the aberrationfree case. Similar result for the unapertured circular lens …fy ˆ fx ˆ f † was reported in Ref. [5]. In addition, zmax ˆ 173, 181 and 187 mm, for kS1 ˆ 0:1, 0 and 0.1 respectively, i.e., zmax moves towards or away from the lens for kS1 ˆ 0:1 and 0.1 respectively, as compared with the aberrationfree case (kS1 ˆ 0). The results can be also clearly seen from Fig. 5(a) and (b), where the axial maximum irradiance Imax and its position zmax are plotted against kS1 for Nwx ˆ 1:0, 1.5, 2.0. In addition, Fig. 5(b) indicates that, if kS1 ˆ 0:01, Nwx ˆ 1:5, we have zmax ˆ fy ˆ 180 mm, and zmax ˆ fx ˆ 200 mm for kS1 ˆ 0:43, Nwx ˆ 1:5. It means that by a suitable choice of the spherical aberration, zmax can coincide with one of the geometrical focal point of the bifocal lens, thus the socalled focusing without focal shift [4] for the case of the bifocal lens is achieved. However, as showed by the numerical examples, the meaning is not the same. Moreover, if kS1 > 0:43, then zmax > fx , fy , i.e., zmax can even exceed the larger geometrical

B. Lu, X. Ji / Optics Communications 189 (2001) 47±53

51

Fig. 3. A comparison of the waist position z0y (- - -) and point of maximum irradiance along the axis zmax (±±), Nwx ˆ 1:5.

focal point. Therefore, the positive spherical aberration may result in a shift of zmax away from the lens, but in a decrement of Imax , whereas the negative spherical aberration may lead to a shift of zmax towards the lens, and to an increment of Imax . These e€ects can be also achieved by choosing the focal length fy , if fx and kS1 of the bifocal lens are ®xed. Fig. 6(a) and (b) give the corresponding results, where Imax and zmax is represented as a function of fy . From Fig. 6 we see that, for example, if Nwx ˆ 1:5, kS1 ˆ 0:4, fy ˆ 181 mm, we have zmax ˆ fx ˆ 200 mm and if 182:5 6 fy 6 217 mm, then zmax > fx , fy . 4. Conclusion The detailed study of focusing properties and focal shift of Gaussian beams passing through a spherically aberrated bifocal lens has been made. Generally, the axial irradiance distribution

I…0; 0; z†, maximum irradiance along the axis Imax and its position zmax are dependent upon the Fresnel number Nwx , spherical aberration kS1 , and focal length fy of the bifocal lens if fx is ®xed. As usual, by a suitable choice of kS1 , the positive spherical aberration results in a shift of zmax away from the lens, but in a decrement of Imax , whereas the negative spherical aberration leads to a shift of zmax towards the lens, and to an increment of Imax , as compared with the aberration-free case. In addition, the focal shift can be either o€set or away from the lens and even exceed the larger geometrical focal point, depending upon the choice of the spherical aberration, and/or focal length. For the case of focusing by a bifocal lens without aberration, zmax is located in the proximity of one waist position, but does not always coincide with it. zmax moves from one to another waist position by changing fy , however, zmax does not exceed the larger geometrical focal point of the bifocal lens. Finally, we would like to mention, that the

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B. Lu, X. Ji / Optics Communications 189 (2001) 47±53

Fig. 4. Axial irradiance distribution I…0; 0; z† of a Gaussian beam passing through a bifocal lens with and without spherical aberration, Nwx ˆ 1:5, fy ˆ 180 mm.

Fig. 5. (a) Axial maximum irradiance Imax of a Gaussian beam focused by a spherically aberrated bifocal lens as a function of spherical aberration kS1 . (b) Position of axial maximum irradiance zmax of a Gaussian beam focused by a spherically aberrated bifocal lens as a function of spherical aberration kS1 .

B. Lu, X. Ji / Optics Communications 189 (2001) 47±53

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Fig. 6. (a) Axial maximum irradiance Imax of a Gaussian beam focused by a spherically aberrated bifocal lens as a function of focal length fy . (b) Position of axial maximum irradiance zmax of a Gaussian beam focused by a spherically aberrated bifocal lens as a function of focal length fy .

extension of the method to studying astigmatic Gaussian beams focused by an aberrated bifocal lens is straightforward. The results obtained in this paper would be useful for the practical design and use of the bifocal lens in focusing and collimating Gaussian and astigmatic Gaussian beams.

Acknowledgements This work was supported by the National HiTech. Project of China and the Foundation of State Key Laboratory of Laser Technology.

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