Propagation of high-order Bessel–Gaussian beam through a misaligned first-order optical system

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Optics & Laser Technology 39 (2007) 1199–1203 www.elsevier.com/locate/optlastec

Propagation of high-order Bessel–Gaussian beam through a misaligned first-order optical system Chengliang Zhao, Ligang Wang, Xuanhui Lu, He Chen Institute of optics, Department of Physics, Zhejiang University, Hangzhou 310027, PR China Received 18 April 2006; received in revised form 10 August 2006; accepted 10 August 2006 Available online 16 October 2006

Abstract The generalized diffraction integralis used to derive a generalized formula for high-order Bessel–Gaussian beams (HBGBs) through a misaligned first-order ABCD optical system. It is found that, when a HBGB propagates through a misaligned optical system, the beam shape of the output beam is unchanged. However, the center of the output beam is deviated from the optical axis, forming a decentered HBGB. The position of the output beam may be controlled by adjusting the misaligned parameters. Based on the derived formula, the diffraction patterns of HBGBs propagating through a simple misaligned lens system have been calculated numerically. These results may be useful in the application of laser beams for trapping and manipulating a wide variety of particles. r 2006 Elsevier Ltd. All rights reserved. Keywords: Bessel–Gaussian beam; Misaligned optical system; Propagation

1. Introduction Optical beams with zero central intensity (also called dark hollow laser beams) have attracted much attention because of their increased application in guiding and manipulating in atoms [1–4]. Atoms can be trapped and guided in dark-hollow laser beams via the optical dipole force. Theoretical investigation of atomic guidance and trapping in dark-hollow beams have been reported [5–7]. Several models have been presented to describe darkhollow beams, the best-known example is a TEM01 beam (also known as a doughnut beam) [8]. Another alternative model with which to describe the dark-hollow beams is the high-order Bessel–Gaussian beams (HBGBs). Gori et al. [9] first introduced Bessel–Gaussian beams of zeroth order by solving the paraxial wave equation. Bagini et al. [10] and Palma et al. [11] presented generalized Bessel–Gaussian beams with higher orders. Various schemes have been proposed to produce Bessel–Gaussian modes [12–14]. Recently, HBGBs propagating through an idealized lens Corresponding author. Tel.: +86 057187953232; fax: +86 057187953231. E-mail addresses: [email protected] (C. Zhao), [email protected] (X. Lu).

0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.08.015

focusing system were analyzed in terms of Collins integral formula [15]. It was found that the HBGB is an ideal and convenient model to describe dark-hollow beams, and can be used for guiding, focusing and trapping ultracold atoms. Small perturbations of the optical system and the inevitable error of the adjustment of beam path etc, lead the misalignment of the optical system. Thus, it is of practical significance to investigate the propagation of beams through a misaligned optical system. Based on the generalized diffraction integral formula for a misaligned optical system [16], analyses of the propagation of various kinds of beams through misaligned optical systems have been presented [17–20]. In this paper, we consider the propagation of HBGBs through any misaligned ABCD system. An analytical propagation formula for an HBGB through any misaligned ABCD optical system is derived from the generalized diffraction integral. It is found that, when a HBGB propagates through a misaligned optical system, the beam shape of the output beam is unchanged. However, the center of the output beam is shifted from the optical axis, forming a decentered HBGB. Using this approach, an example of an HBGB propagating through the misaligned lens is presented.

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2. Propagation formula of HBGBs through a first-order misaligned optical system Consider an HBGB passing through a misaligned system as shown in Fig. 1, where PR1,2 are the alignment reference planes, PR1m, 2m are the misalignment reference planes, e the transverse offset and e0 the tilted angle. A, B, C and D are the transfer matrix elements of the aligned optical system from PR1 to PR2. In a cylindrical coordinate system, the generalized diffraction integral for any beam through paraxial misaligned optical systems can be expressed as [16] ZZ ik E ðr; y; zÞ ¼ expðikzÞ E 0 ðr1 ; y1 ; 0Þ 2pB  ik  2  exp  Ar1  2r1 r cosðy  y1 Þ 2B 2

þ Dr þ er1 cos y1 þ fr1 sin y1 þ gr cos y  þhr sin y r1 dr1 dy1 ,

ð1Þ

where E0 (r1, y1, 0) is the incident electric field at the input plane of PR1, and k ¼ 2p/l the wave number, l the wavelength. Other parameters e, f, g, and h are defined by   (2) e ¼ 2 aT  cos j þ bT arctanðcos j tan 0 Þ ,   f ¼ 2 aT  sin j þ bT arctanðsin j tan 0 Þ , 





g ¼ 2 BgT  DaT  cos j þ 2 BdT  DbT  arctanðcos j tan 0 Þ,

(3)  ð4Þ

    h ¼ 2 BgT  DaT  sin j þ 2 BdT  DbT  arctanðsin j tan 0 Þ.

ð5Þ

Here j is the misaligned azimuth angle of the optical elements in a cylindrical coordinate system (see the figure in Appendix), aT, bT, gT and dT represent the misaligned matrix elements are determined by aT ¼ 1  A; bT ¼ l  B; gT ¼ C; dT ¼ 1  D, PR1

PR2m

(6)

PR2

where l is the axial distance between the input plane and output plane. In Appendix, we give out how to derive e, f, g and h in cylindrical coordinate system. In order to describe the dark-hollow beams, we assume that the incident electric field has the form of an HBGB at the input plane of PR1 [9]:  2 r E 0 ðr1 ; y1 Þ ¼ E 0 J n ðar1 Þ exp  12 expðiny1 Þ w0 n ¼ 1; 2; 3       ,

ð7Þ

where Jn the Bessel function of the first kind of order n, and E0 is a constant, w0 is the waist of Gaussian beam, exp(iny1) is the phase term for the beam. By using the following integral formula [21]: h p i kr r 1 2 exp in  y2 J n 2 L Z n r r  o 1 2p 1 2 ¼ exp ik cosðy2  y1 Þ  iny1 dy1 , 2p 0 L

ð8Þ

substituted Eq. (7) into Eq. (1), we have ikE 0 E ðr; y; zÞ ¼ expðikzÞ 2pB   ik  2 Dr þ gr cos y þ hr sin y  exp  2B h p i Z 1  exp in  f J n ðar1 Þ 2 0   1 ikA 2  exp  2  r w0 2B 1 

1=2 ! kr1 ð2r cos y  eÞ2 þ ð2r sin y  f Þ2  Jn r1 dr1 , ð9Þ 2B where 2r cos y  e cos f ¼ 

1=2 , ð2r cos y  eÞ2 þ ð2r sin y  f Þ2 2r sin y  f sin f ¼ 

1=2 . ð2r cos y  eÞ2 þ ð2r sin y  f Þ2

PR1m A

B zm

C

D

The following integral formulas are used to resolve Eq. (9) [21]: Z 0

′

 z

1

 2 1 a þ b2 exp  x exp bx J m ðaxÞJ m ðgxÞ dx ¼ 2b 4b  ag  Im , ð10Þ 2b 

I m ðxÞ ¼ exp Fig. 1. Misaligned diagram for a two-dimensional forward-going system (online color).

2



  imp imp J m ðixÞ ¼ ð1Þm exp J m ðixÞ, 2 2 (11)

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where In(x) denotes the modified Bessel function of nth order. After trivial calculation, we finally obtain

E ðr; y; zÞ ¼

ikE 0

 expðikzÞ 2B w12 þ ikA 2B

0   ik  2 Dr þ gr cos y þ hr sin y  expðinfÞ  exp  2B 8 

9 < k2 ð2r cos y  eÞ2 þ ð2r sin y  f Þ2 = a2 þ

  exp 

: 4 1 þ ikA ; 16B2 w12 þ ikA 2B 2B w20 0 0 1 

2 2 1=2 Bika ð2r cos y  eÞ þ ð2rsin y  f Þ C ð12Þ  J n@ A. 1 ikA 4B w2 þ 2B



0

Eq. (12) provides the transformation rule for HBGBs propagating through any first-order misaligned ABCD optical system. This formula is a very powerful tool to deal with HBGBs passing through the first-order misaligned optical system. From Eq. (12), it is clear that an HBGB propagating through a misaligned system becomes a decentered beam, and the center of the output beam is shifted from the optical axis by an amount of 2e in xdirection and f2 in y-direction. The center of the beam can be adjusted by changing e, e0 and j.

Focus F

s

f′

z1

Fig. 2. Propagation of high-order Bessel–Gaussian beam passing through a lens (online colour).

1201

3. Application example Let us consider a HBGB propagating through a misaligned lens system as shown in Fig. 2. Based on the derived formula (12), the intensity distribution of HBGBs through a misaligned lens system can be obtained by I ¼ |E (r, y, z)|2. The matrix for the lens system is #    " z1 =f 0 z1 =f 0 s þ f 0 þ z1 A B ¼ (13) 1=f 0 1  s=f 0 C D where s is the axial distance between the beam waist and the thin lens, f0 is the focus length of the thin lens, and z1 is the distance from the output plane to the focus plane. Substituting the matrix in Eq. (13) into Eq. (12), the electric field distribution formula can be easily obtained. In the following numerical calculation, we choose the parameters:w0 ¼ 1 mm, n ¼ 10, a ¼ 1, l ¼ 1.06 mm, f0 ¼ 400 mm, s ¼ 300 mm. Fig. 3 shows the normalized intensity distributions of the HBGB with n ¼ 10 at the incident plane. At the input plane, the center of the beam is located on the optical axis. However, the situation changes as the HBGB propagates through the misaligned lens system. In Fig. 4 the normalized intensity distributions on the focusing plane under different misaligned parameters are plotted. It can be seen that, under the same transverse offset e and its misaligned azimuth angle j, the center of the output beam is shifted much more from the optical axis with increasing tilt angle e0 . Fig. 4 shows that, even for very small changes in the misaligned parameters, the position of the output beam will be greatly changed. Fig. 5 shows the normalized intensity distribution of the HBGB in the far field as it propagates through a misaligned lens system. Except for the position change of the beam center, the beam shape of the output intensity remains similar to that of the input even in the misaligned system. 4. Conclusion In conclusion, by using the generalized diffraction integral, a propagation formula of HBGBs propagating

Fig. 3. Normalized three-dimensional intensity distribution and its contour map of the input HBGBs (online colour).

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C. Zhao et al. / Optics & Laser Technology 39 (2007) 1199–1203

Fig. 4. Normalized three-dimensional intensity distributions and their corresponding contour maps of the output beam at the plane of z1 ¼ 0 mm with e ¼ 1 mm and j ¼ 301, (a) e0 ¼ 0 rad, and (b) e0 ¼ 8.7  103 rad (online colour).

Fig. 5. Normalized three-dimensional intensity distribution and its contour map of the output beam at z1 ¼ 5000 mm, with e0 ¼ 0 rad, e ¼ 1 mm and j ¼ 301 (online colour).

through a first-order misaligned optical system has been derived. It is found that the center of the HBGB is related to the misaligned parameters e, e0 and j, while its shape is unchanged. Therefore, the output beam of the HBGB through the misaligned optical system is a decentered HBGB. Based on the derived formula, an example for the HBGB through a misaligned lens system is numerically illustrated. Our formula is a very powerful tool to deal with the propagation of HBGBs through any first-order

misaligned optical system. In fact, the HBGB as one of dark-hollow beams has many potential advantages in the application of guiding and manipulating the cooling atoms, even Bose-Einstein condensates. Acknowledgements This work was supported by National Nature Science Foundation of China (No. 10334050 and 10547138).

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Appendix In this appendix, we present a few steps that the process of getting e, f, g and h in Eqs. (3–6). The e, f, g and h in Eqs. (3–6) in the cartesian coordinates system we can get in Ref. [16]:   e ¼ 2 aT x þ bT 0x , (A.1)

 f ¼ 2 aT y þ bT 0y ,

(A.2)

    g ¼ 2 BgT  DaT x þ 2 BdT  DbT 0x ,

(A.3)



   h ¼ 2 BgT  DaT y þ 2 BdT  DbT 0y .

(A.4)

Now we transform e, f, g and h from the cartesian coordinate system to the cylindrical coordinate system. From Fig. A1

we can get following equations:  tan 0 ¼ 0 , z

(A.5)

tan 0x ¼

x , z0

(A.6)

tan 0y ¼

y , z0

(A.7)

x ¼  cos j,

(A.8)

y ¼  sin j.

(A.9)

From Eqs. (A.5)–(A.9) we obtain 0x ¼ arctanðcos j tan 0 Þ,

(A.10)

0y ¼ arctanðsin j tan 0 Þ.

(A.11)

Substituting Eqs. (A.10), (A.11) into Eqs. (A.1–A.4) we obtain the Eqs. (3)–(6) as follows:   e ¼ 2 aT  cos j þ bT arctanðcos j tan 0 Þ , (A.12)   f ¼ 2 aT  sin j þ bT arctanðsin j tan 0 Þ , (A.13)

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    g ¼ 2 BgT  DaT  cos j þ 2 BdT  DbT  arctanðcos j tan 0 Þ,

ðA:14Þ

    h ¼ 2 BgT  DaT  sin j þ 2 BdT  DbT  arctanðsin j tan 0 Þ.

ðA:15Þ

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