May 1, 2014 / Vol. 39, No. 9 / OPTICS LETTERS
2549
Generation of a controllable optical cage by focusing a Laguerre–Gaussian correlated Schell-model beam Yahong Chen and Yangjian Cai* School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China *Corresponding author: [email protected] Received February 26, 2014; revised March 22, 2014; accepted March 23, 2014; posted March 24, 2014 (Doc. ID 207191); published April 17, 2014 We analyze the intensity of a Laguerre–Gaussian correlated Schell-model (LGCSM) beam focused by a thin lens near the focal region, and it is found that a controllable optical cage can be formed through varying the initial spatial coherence width. Furthermore, we carry out experimental measurement of the intensity of a focused LGCSM beam, and we observe that the optical cage is indeed formed in experiment. Our results will be useful for trapping particles or atoms. © 2014 Optical Society of America OCIS codes: (030.1640) Coherence; (030.1670) Coherent optical effects; (140.7010) Laser trapping. http://dx.doi.org/10.1364/OL.39.002549
In the past decades, partially coherent beams with conventional correlation functions (i.e., Gaussian correlated Schell-model functions) have been studied extensively [1,2] and have found wide applications, such as freespace optical communications [3–5], optical imaging [6], laser scanning [7], inertial confinement fusion [8], reduction of noise in photography [9], remote detection [10], second-harmonic generation [11], particle trapping [12,13], and optical scattering [14]. Recently, more and more attention has been paid to partially coherent beams with nonconventional correlation functions [15–26]. The sufficient condition for devising a genuine correlation function of partially coherent beams was established by Gori and collaborators [15,16]. Based on the pioneer work of Gori et al., a variety of partially coherent beams with special correlation functions, such as nonuniformly correlated Gaussian Schell-model beams [17–19], multiGaussian Schell-model beams [20–23], cosine-Gaussian Schell-model beams [24–26], special correlated partially coherent vector beams [27], and Laguerre–Gaussian correlated Schell-model (LGCSM) beams (also named Laguerre–Gaussian Schell-model beam) [28–31], were proposed. The LGCSM beam proposed in [28] was found to exhibit interesting propagation properties, such as far-field ring shaped profile formation. Propagation properties of a LGCSM beam in turbulent atmosphere were explored in [29] and [30], and it was found that a LGCSM beam has advantage over a GSM beam for reducing turbulence-induced degradation, which will be useful in free-space optical communications. In this Letter, we analyze the intensity of a LGCSM beam focused by a thin lens near the focal region, and we find that a controllable optical cage can be formed, which will be useful for trapping particles or atoms. Experimental observation of the optical cage is also carried out. In the space-time domain, the statistical properties of a scalar partially coherent beam are characterized by the mutual coherence function [1,2]. For a LGCSM beam, its mutual coherence function at z 0 is defined as [28]
r2 r2 r − r 2 r − r 2 Jr1 ; r2 exp − 1 2 2 − 1 2 2 L0n 1 2 2 ; (1) 4σ 0 2δ0 2δ0 0146-9592/14/092549-04$15.00/0
where ri ≡ xi ; yi is the position vector in the source plane, σ 0 and δ0 are the transverse beam width and the transverse coherence width of the LGCSM beam, respectively, L0n denotes the Laguerre polynomial of mode order n and 0. Within the validity of the paraxial approximation, the propagation of the mutual coherence function of a LGCSM beam through a stigmatic ABCD optical system can be studied with the help of the following extended Collins formula [32]: 1 ikD 2 2 ρ − ρ Jρ1 ; ρ2 exp − 2B 1 2 λB2 Z ∞Z ∞Z ∞Z ∞ ikA 2 2 r − r Jr1 ; r2 exp − × 2B 1 2 −∞ −∞ −∞ −∞ ik r1 · ρ1 − r2 · ρ2 d2 r1 d2 r2 ; (2) × exp B where ρi ≡ ρix ; ρiy is the position vector in the output plane, A, B, C, and D are the elements of a transfer matrix for the optical system, k 2π∕λ is the wavenumber with λ being the wavelength. Let us consider a LGCSM beam originating from the source plane (z 0) is focused by a thin lens with focal length f . The distance between the source plane and the thin lens is f and the distance between the thin lens and the output plane is z. Then the transfer matrix between the source plane and the output plane reads as A B 1 z 1 0 1 f C D 0 1 −1∕f 1 0 1 1 − z∕f f : (3) −1∕f 0 For the convenience of integration, we introduce the following “sum” and “difference” coordinates: rs
r1 r2 ; 2
Δr r1 − r2 :
(4)
Then applying Eqs. (1) and (4), Eq. (2) can be expressed in the following alternative form: © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014
Jρ1 ; ρ2
1 ikD 2 2 ρ exp − − ρ 2 2B 1 λB2 Z∞Z∞Z∞Z∞ Δr Δr P rs − gΔr P rs × 2 2 −∞ −∞ −∞ −∞ ik Δr Δr rs · ρ1 − rs − · ρ2 d2 Δrd2 rs ; × exp B 2 2 (5)
where Δr 1 ikA Δr 2 exp − 2 − rs ; (6) P rs 2 2 4σ 0 2B
Δr 1 ikA Δr 2 exp − r − − ; P rs − s 2 2 4σ 20 2B 2 Δr2 Δr gΔr exp − 2 L0n : 2δ0 2δ20
(7)
(8)
P rs Δr∕2 and Prs − Δr∕2 can be expressed in terms of their Fourier transforms P~ u1 ∕λB and ~ 2 ∕λB as follows: Pu P
Z∞Z∞ Δr 1 u rs P~ 1 λB 2 λB2 −∞ −∞ ik Δr r · u1 d2 u1 ; (9) × exp B s 2
Z∞Z∞ Δr 1 u e P 2 P rs − λB 2 λB2 −∞ −∞ ik Δr r − · u2 d2 u2 : × exp − B s 2
(10)
After tedious integration, we obtain the following expression for the mutual coherence function of the LGCSM in the output plane: Jρ1 ; ρ2 2
n−1
as Iρ Jρ; ρ. Note Eqs. (3) and (11) are applicable for any ideal thin lens. Applying Eqs. (3) and (11), we calculate in Fig. 1 the normalized intensity distribution of a LGCSM beam focused by a thin lens near the focal region in the ρx − z plane with ρy 0 for different values of the initial coherence width δ0 . The parameters used in the calculation are chosen as f 250 mm, λ 632.8 nm, n 1, and σ 0 1.0 mm. One finds that an optical cage is formed near the focal region when the initial coherence width is small [see Figs. 1(a) and 1(b)], and the size of the optical cage is larger for smaller value of δ0 . With the increase of δ0 , the optical cage disappears gradually, and intensity distribution of the focused LGCSM beam near the focal region looks like that of a focused GSM beam. The critical value of δ0 is about 2.0 mm when the optical cage disappears completely. One can explain this phenomenon by the fact that the Laguerre–Gaussian correlation function displays extraordinary properties and has strong influence on the focused intensity distribution when δ0 is small. When δ0 is large, the Laguerre–Gaussian correlation function does not display unique properties, and its influence on the focused intensity is almost the same with that of the Gaussian correlation function. The optical cage formed in our case will be useful for trapping particles whose refractive index is smaller than that of the ambient [33] or blue detuned atoms [34]. The principle for achieving the optical cage in this Letter is totally different from that reported in [33] and [34], where the optical cages are obtained through modulating the polarization or phase of the incident beam under the condition of tight focusing by high numerical aperture thin lens. The optical cage in this Letter is formed through modulating the correlation function of the incident beam, and is not related with the spiral phase because there is no spiral phase in Eq. (1). Our method can be realized in experiment easily. Now we carry out experimental verification of our theoretical prediction. We adopt the method proposed in [31] to generate a LGCSM beam with controllable coherence, where a LGCSM beam with order n can be formed from an incoherent dark hollow (DH) beam with intensity Iv v2 ∕ω20 n exp−2v2 ∕ω20 . Figure 2 shows
2 k σ 2 Bσ 2 Bδ02n2 B
× σ 2 B σ 2 B 2δ20 −n−1 ikD 2 ρ1 − ρ22 × exp − 2B 2 k 2 2 2 2 σ Bρ1 σ Bρ2 × exp − 2B 2 2 k σ Bρ1 σ 2 Bρ2 2 × exp 2B σ 2 B σ 2 B 2δ20 2 2 k σ Bρ1 σ 2 Bρ2 2 0 ; × Ln − 2B σ 2 B σ 2 B 2δ20
(11)
where σB 1∕4σ 20 − ikA∕2B−1∕2 . The average intensity of a LGCSM beam in the output plane is obtained
Fig. 1. Normalized intensity distribution of a LGCSM beam focused by a thin lens near the focal region in the ρx − z plane with ρy 0 for different values of the initial coherence width δ0 .
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Fig. 3. Experimental results of the square of the modulus of the generated LGCSM beam of n 1 with different values of δ0 just before the GAF. The solid curve is a result of the theoretical fit. Fig. 2. Experimental setup for generating a controllable optical cage by focusing a LGCSM beam and measuring its intensity. BE, beam expander; SLM, spatial light modulator; CA, circular aperture; L1 , L2 , L3 , thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; BPA, beam profile analyzer; PC1 , PC2 , personal computers.
our experimental setup for generating a controllable optical cage by focusing a LGCSM beam and measuring its intensity. After passing through a beam expander, the He–Ne laser beam (λ 632.8 nm) goes toward a spatial light modulator (SLM), which acts as a phase grating designed by the method of computer-generated holograms. The inset of Fig. 2 shows the distribution of the phase grating for generating a DH beam of n 1. The circular aperture is used to select out the first order of the beam from the SLM, and the transmitted beam is regarded as a DH beam of n 1. The generated DH beam first passes through a thin lens L1 , then it illuminates a rotating ground-glass disk (RGGD), producing an incoherent beam with DH beam profile. After passing through free space with length f 2 , a thin lens L2 with focal length f 2 and a Gaussian amplitude filter (GAF), the generated incoherent DH beam becomes a LGCSM beam of n 1 [31]. The generated LGCSM beam has a Gaussian beam spot with beam width σ 0 , which is determined by the transmission function of the GAF. In our experiment, σ 0 equals to 1 mm. The spatial coherence width δ0 of the generated LGCSM beam is approximated as λf ∕πω0 . In our experiment, the spatial coherence of the LGCSM beam is modulated by varying the size of the focused beam spot on the RGGD, which is controlled by the distance between the thin lens L1 and the RGGD. In order to obtain the value of the spatial coherence width δ0 of the generated LGCSM beam, we also carry out experimental measurement of the degree of coherence of the generated LGCSM beam by using the method developed in [35]. The generated LGCSM beam is split into two distinct imaging optical paths by a 50∶50 beam splitter, then the transmitted beam and the reflected beam go to two single-photon detectors, respectively. By measuring the fourth-order correlation function between two detectors, the distribution of the square of the degree of coherence of the generated LGCSM beam can be obtained with the help of the Gaussian moment theorem [1,31]. Through theoretical fit of the experimental data, we can obtain the value of δ0 . In our experiment, we generate three LGCSM beams with different values of δ0 . Figure 3 shows our experimental results of the square of the modulus of the generated LGCSM beam with different values of δ0 just before the GAF. Through theoretical fit of the
experimental data, we obtain δ0 0.2 mm, 0.5 mm, and 2 mm for Figs. 3(a), 3(b), and 3(c). The generated LGCSM beam from the GAF passes through the focusing system where the focal length is f 3 25 cm, then arrives at the beam profile analyzer, which is used to measure the intensity. Figures 4–6 show our experimental results of the intensity distribution of the generated LGCSM beam of n 1 near the focal region in the ρx − z plane and the transverse intensity distribution at several propagation distances for δ0 0.2 mm, 0.5 mm, and 2 mm, respectively. One finds from Figs. 4 and 5 that the optical cage is indeed formed in our experiment when δ0 is small as expected by Fig. 1, and the size of the optical cage is controlled by δ0 . For large δ0 , the optical cage disappears (see Fig. 6). In conclusion, we have analyzed the intensity of a focused LGCSM beam near the focal region, and we have found that an optical cage with controllable size can be formed by choosing suitable value of the spatial
Fig. 4. Experimental results of (a) the intensity distribution of the generated LGCSM beam of n 1 near the focal region in the ρx − z plane and (b) the transverse intensity distribution at several propagation distances with δ0 0.2 mm.
Fig. 5. Experimental results of (a) the intensity distribution of the generated LGCSM beam of n 1 near the focal region in the ρx − z plane and (b) the transverse intensity distribution at several propagation distances with δ0 0.5 mm.
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Fig. 6. Experimental results of (a) the intensity distribution of the generated LGCSM beam of n 1 near the focal region in the ρx − z plane and (b) the transverse intensity distribution at several propagation distances with δ0 2 mm.
coherence of the LGCSM beam. Furthermore, we have carried out experimental generation of an optical cage through focusing a LGCSM beam. Our results will be useful for trapping particles or atoms. Modulating the correlation function of a partially coherent beam provides a novel way for beam shaping. This work is supported by the National Natural Science Foundation of China under grant 11274005, the Huo Ying Dong Education Foundation of China under grant 121009, the Key Project of Chinese Ministry of Education under grant 210081, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References 1. L. Mandel and E. Wolf, eds., Optical Coherence and Quantum Optics (Cambridge, 1995). 2. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007). 3. J. C. Ricklin and F. M. Davidson, J. Opt. Soc. Am. A 19, 1794 (2002). 4. Y. Cai and S. He, Appl. Phys. Lett. 89, 041117 (2006). 5. X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, Opt. Lett. 38, 5323 (2013). 6. Y. Cai and S. Zhu, Phys. Rev. E 71, 056607 (2005).
7. D. Kermisch, J. Opt. Soc. Am. 65, 887 (1975). 8. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984). 9. A. Belendez, L. Carretero, and A. Fimia, Opt. Commun. 98, 236 (1993). 10. G. Wu and Y. Cai, Opt. Lett. 36, 1939 (2011). 11. Y. Cai and U. Peschel, Opt. Express 15, 15480 (2007). 12. C. Zhao and Y. Cai, Opt. Lett. 36, 2251 (2011). 13. Y. Dong, F. Wang, C. Zhao, and Y. Cai, Phys. Rev. A 86, 013840 (2012). 14. T. Van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 104, 173902 (2010). 15. F. Gori and M. Santarsiero, Opt. Lett. 32, 3531 (2007). 16. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009). 17. H. Lajunen and T. Saastamoinen, Opt. Lett. 36, 4104 (2011). 18. Z. Tong and O. Korotkova, Opt. Lett. 37, 3240 (2012). 19. Y. Gu and G. Gbur, Opt. Lett. 38, 1395 (2013). 20. S. Sahin and O. Korotkova, Opt. Lett. 37, 2970 (2012). 21. O. Korotkova, S. Sahin, and E. Shchepakina, J. Opt. Soc. Am. A 29, 2159 (2012). 22. S. Du, Y. Yuan, C. Liang, and Y. Cai, Opt. Laser Technol. 50, 14 (2013). 23. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, and H. T. Eyyuboğlu, Opt. Commun. 305, 57 (2013). 24. Z. Mei and O. Korotkova, Opt. Lett. 38, 2578 (2013). 25. Z. Mei, E. Schchepakina, and O. Korotkova, Opt. Express 21, 17512 (2013). 26. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, Opt. Lett. 39, 769 (2014). 27. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, Phys. Rev. A 89, 013801 (2014). 28. Z. Mei and O. Korotkova, Opt. Lett. 38, 91 (2013). 29. J. Cang, P. Xiu, and X. Liu, Opt. Laser Technol. 54, 35 (2013). 30. R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, Opt. Express 22, 1871 (2014). 31. F. Wang, X. Liu, Y. Yuan, and Y. Cai, Opt. Lett. 38, 1814 (2013). 32. Q. Lin and Y. Cai, Opt. Lett. 27, 216 (2002). 33. Y. Zhang, B. Ding, and T. Suyama, Phys. Rev. A 81, 023831 (2010). 34. P. Xu, X. He, J. Wang, and M. Zhan, Opt. Lett. 35, 2164 (2010). 35. F. Wang and Y. Cai, J. Opt. Soc. Am. A 24, 1937 (2007).
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Generation of a controllable optical cage by focusing a Laguerre–Gaussian correlated Schell-model beam Yahong Chen and Yangjian Cai* School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China *Corresponding author: [email protected] Received February 26, 2014; revised March 22, 2014; accepted March 23, 2014; posted March 24, 2014 (Doc. ID 207191); published April 17, 2014 We analyze the intensity of a Laguerre–Gaussian correlated Schell-model (LGCSM) beam focused by a thin lens near the focal region, and it is found that a controllable optical cage can be formed through varying the initial spatial coherence width. Furthermore, we carry out experimental measurement of the intensity of a focused LGCSM beam, and we observe that the optical cage is indeed formed in experiment. Our results will be useful for trapping particles or atoms. © 2014 Optical Society of America OCIS codes: (030.1640) Coherence; (030.1670) Coherent optical effects; (140.7010) Laser trapping. http://dx.doi.org/10.1364/OL.39.002549
In the past decades, partially coherent beams with conventional correlation functions (i.e., Gaussian correlated Schell-model functions) have been studied extensively [1,2] and have found wide applications, such as freespace optical communications [3–5], optical imaging [6], laser scanning [7], inertial confinement fusion [8], reduction of noise in photography [9], remote detection [10], second-harmonic generation [11], particle trapping [12,13], and optical scattering [14]. Recently, more and more attention has been paid to partially coherent beams with nonconventional correlation functions [15–26]. The sufficient condition for devising a genuine correlation function of partially coherent beams was established by Gori and collaborators [15,16]. Based on the pioneer work of Gori et al., a variety of partially coherent beams with special correlation functions, such as nonuniformly correlated Gaussian Schell-model beams [17–19], multiGaussian Schell-model beams [20–23], cosine-Gaussian Schell-model beams [24–26], special correlated partially coherent vector beams [27], and Laguerre–Gaussian correlated Schell-model (LGCSM) beams (also named Laguerre–Gaussian Schell-model beam) [28–31], were proposed. The LGCSM beam proposed in [28] was found to exhibit interesting propagation properties, such as far-field ring shaped profile formation. Propagation properties of a LGCSM beam in turbulent atmosphere were explored in [29] and [30], and it was found that a LGCSM beam has advantage over a GSM beam for reducing turbulence-induced degradation, which will be useful in free-space optical communications. In this Letter, we analyze the intensity of a LGCSM beam focused by a thin lens near the focal region, and we find that a controllable optical cage can be formed, which will be useful for trapping particles or atoms. Experimental observation of the optical cage is also carried out. In the space-time domain, the statistical properties of a scalar partially coherent beam are characterized by the mutual coherence function [1,2]. For a LGCSM beam, its mutual coherence function at z 0 is defined as [28]
r2 r2 r − r 2 r − r 2 Jr1 ; r2 exp − 1 2 2 − 1 2 2 L0n 1 2 2 ; (1) 4σ 0 2δ0 2δ0 0146-9592/14/092549-04$15.00/0
where ri ≡ xi ; yi is the position vector in the source plane, σ 0 and δ0 are the transverse beam width and the transverse coherence width of the LGCSM beam, respectively, L0n denotes the Laguerre polynomial of mode order n and 0. Within the validity of the paraxial approximation, the propagation of the mutual coherence function of a LGCSM beam through a stigmatic ABCD optical system can be studied with the help of the following extended Collins formula [32]: 1 ikD 2 2 ρ − ρ Jρ1 ; ρ2 exp − 2B 1 2 λB2 Z ∞Z ∞Z ∞Z ∞ ikA 2 2 r − r Jr1 ; r2 exp − × 2B 1 2 −∞ −∞ −∞ −∞ ik r1 · ρ1 − r2 · ρ2 d2 r1 d2 r2 ; (2) × exp B where ρi ≡ ρix ; ρiy is the position vector in the output plane, A, B, C, and D are the elements of a transfer matrix for the optical system, k 2π∕λ is the wavenumber with λ being the wavelength. Let us consider a LGCSM beam originating from the source plane (z 0) is focused by a thin lens with focal length f . The distance between the source plane and the thin lens is f and the distance between the thin lens and the output plane is z. Then the transfer matrix between the source plane and the output plane reads as A B 1 z 1 0 1 f C D 0 1 −1∕f 1 0 1 1 − z∕f f : (3) −1∕f 0 For the convenience of integration, we introduce the following “sum” and “difference” coordinates: rs
r1 r2 ; 2
Δr r1 − r2 :
(4)
Then applying Eqs. (1) and (4), Eq. (2) can be expressed in the following alternative form: © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014
Jρ1 ; ρ2
1 ikD 2 2 ρ exp − − ρ 2 2B 1 λB2 Z∞Z∞Z∞Z∞ Δr Δr P rs − gΔr P rs × 2 2 −∞ −∞ −∞ −∞ ik Δr Δr rs · ρ1 − rs − · ρ2 d2 Δrd2 rs ; × exp B 2 2 (5)
where Δr 1 ikA Δr 2 exp − 2 − rs ; (6) P rs 2 2 4σ 0 2B
Δr 1 ikA Δr 2 exp − r − − ; P rs − s 2 2 4σ 20 2B 2 Δr2 Δr gΔr exp − 2 L0n : 2δ0 2δ20
(7)
(8)
P rs Δr∕2 and Prs − Δr∕2 can be expressed in terms of their Fourier transforms P~ u1 ∕λB and ~ 2 ∕λB as follows: Pu P
Z∞Z∞ Δr 1 u rs P~ 1 λB 2 λB2 −∞ −∞ ik Δr r · u1 d2 u1 ; (9) × exp B s 2
Z∞Z∞ Δr 1 u e P 2 P rs − λB 2 λB2 −∞ −∞ ik Δr r − · u2 d2 u2 : × exp − B s 2
(10)
After tedious integration, we obtain the following expression for the mutual coherence function of the LGCSM in the output plane: Jρ1 ; ρ2 2
n−1
as Iρ Jρ; ρ. Note Eqs. (3) and (11) are applicable for any ideal thin lens. Applying Eqs. (3) and (11), we calculate in Fig. 1 the normalized intensity distribution of a LGCSM beam focused by a thin lens near the focal region in the ρx − z plane with ρy 0 for different values of the initial coherence width δ0 . The parameters used in the calculation are chosen as f 250 mm, λ 632.8 nm, n 1, and σ 0 1.0 mm. One finds that an optical cage is formed near the focal region when the initial coherence width is small [see Figs. 1(a) and 1(b)], and the size of the optical cage is larger for smaller value of δ0 . With the increase of δ0 , the optical cage disappears gradually, and intensity distribution of the focused LGCSM beam near the focal region looks like that of a focused GSM beam. The critical value of δ0 is about 2.0 mm when the optical cage disappears completely. One can explain this phenomenon by the fact that the Laguerre–Gaussian correlation function displays extraordinary properties and has strong influence on the focused intensity distribution when δ0 is small. When δ0 is large, the Laguerre–Gaussian correlation function does not display unique properties, and its influence on the focused intensity is almost the same with that of the Gaussian correlation function. The optical cage formed in our case will be useful for trapping particles whose refractive index is smaller than that of the ambient [33] or blue detuned atoms [34]. The principle for achieving the optical cage in this Letter is totally different from that reported in [33] and [34], where the optical cages are obtained through modulating the polarization or phase of the incident beam under the condition of tight focusing by high numerical aperture thin lens. The optical cage in this Letter is formed through modulating the correlation function of the incident beam, and is not related with the spiral phase because there is no spiral phase in Eq. (1). Our method can be realized in experiment easily. Now we carry out experimental verification of our theoretical prediction. We adopt the method proposed in [31] to generate a LGCSM beam with controllable coherence, where a LGCSM beam with order n can be formed from an incoherent dark hollow (DH) beam with intensity Iv v2 ∕ω20 n exp−2v2 ∕ω20 . Figure 2 shows
2 k σ 2 Bσ 2 Bδ02n2 B
× σ 2 B σ 2 B 2δ20 −n−1 ikD 2 ρ1 − ρ22 × exp − 2B 2 k 2 2 2 2 σ Bρ1 σ Bρ2 × exp − 2B 2 2 k σ Bρ1 σ 2 Bρ2 2 × exp 2B σ 2 B σ 2 B 2δ20 2 2 k σ Bρ1 σ 2 Bρ2 2 0 ; × Ln − 2B σ 2 B σ 2 B 2δ20
(11)
where σB 1∕4σ 20 − ikA∕2B−1∕2 . The average intensity of a LGCSM beam in the output plane is obtained
Fig. 1. Normalized intensity distribution of a LGCSM beam focused by a thin lens near the focal region in the ρx − z plane with ρy 0 for different values of the initial coherence width δ0 .
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Fig. 3. Experimental results of the square of the modulus of the generated LGCSM beam of n 1 with different values of δ0 just before the GAF. The solid curve is a result of the theoretical fit. Fig. 2. Experimental setup for generating a controllable optical cage by focusing a LGCSM beam and measuring its intensity. BE, beam expander; SLM, spatial light modulator; CA, circular aperture; L1 , L2 , L3 , thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; BPA, beam profile analyzer; PC1 , PC2 , personal computers.
our experimental setup for generating a controllable optical cage by focusing a LGCSM beam and measuring its intensity. After passing through a beam expander, the He–Ne laser beam (λ 632.8 nm) goes toward a spatial light modulator (SLM), which acts as a phase grating designed by the method of computer-generated holograms. The inset of Fig. 2 shows the distribution of the phase grating for generating a DH beam of n 1. The circular aperture is used to select out the first order of the beam from the SLM, and the transmitted beam is regarded as a DH beam of n 1. The generated DH beam first passes through a thin lens L1 , then it illuminates a rotating ground-glass disk (RGGD), producing an incoherent beam with DH beam profile. After passing through free space with length f 2 , a thin lens L2 with focal length f 2 and a Gaussian amplitude filter (GAF), the generated incoherent DH beam becomes a LGCSM beam of n 1 [31]. The generated LGCSM beam has a Gaussian beam spot with beam width σ 0 , which is determined by the transmission function of the GAF. In our experiment, σ 0 equals to 1 mm. The spatial coherence width δ0 of the generated LGCSM beam is approximated as λf ∕πω0 . In our experiment, the spatial coherence of the LGCSM beam is modulated by varying the size of the focused beam spot on the RGGD, which is controlled by the distance between the thin lens L1 and the RGGD. In order to obtain the value of the spatial coherence width δ0 of the generated LGCSM beam, we also carry out experimental measurement of the degree of coherence of the generated LGCSM beam by using the method developed in [35]. The generated LGCSM beam is split into two distinct imaging optical paths by a 50∶50 beam splitter, then the transmitted beam and the reflected beam go to two single-photon detectors, respectively. By measuring the fourth-order correlation function between two detectors, the distribution of the square of the degree of coherence of the generated LGCSM beam can be obtained with the help of the Gaussian moment theorem [1,31]. Through theoretical fit of the experimental data, we can obtain the value of δ0 . In our experiment, we generate three LGCSM beams with different values of δ0 . Figure 3 shows our experimental results of the square of the modulus of the generated LGCSM beam with different values of δ0 just before the GAF. Through theoretical fit of the
experimental data, we obtain δ0 0.2 mm, 0.5 mm, and 2 mm for Figs. 3(a), 3(b), and 3(c). The generated LGCSM beam from the GAF passes through the focusing system where the focal length is f 3 25 cm, then arrives at the beam profile analyzer, which is used to measure the intensity. Figures 4–6 show our experimental results of the intensity distribution of the generated LGCSM beam of n 1 near the focal region in the ρx − z plane and the transverse intensity distribution at several propagation distances for δ0 0.2 mm, 0.5 mm, and 2 mm, respectively. One finds from Figs. 4 and 5 that the optical cage is indeed formed in our experiment when δ0 is small as expected by Fig. 1, and the size of the optical cage is controlled by δ0 . For large δ0 , the optical cage disappears (see Fig. 6). In conclusion, we have analyzed the intensity of a focused LGCSM beam near the focal region, and we have found that an optical cage with controllable size can be formed by choosing suitable value of the spatial
Fig. 4. Experimental results of (a) the intensity distribution of the generated LGCSM beam of n 1 near the focal region in the ρx − z plane and (b) the transverse intensity distribution at several propagation distances with δ0 0.2 mm.
Fig. 5. Experimental results of (a) the intensity distribution of the generated LGCSM beam of n 1 near the focal region in the ρx − z plane and (b) the transverse intensity distribution at several propagation distances with δ0 0.5 mm.
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Fig. 6. Experimental results of (a) the intensity distribution of the generated LGCSM beam of n 1 near the focal region in the ρx − z plane and (b) the transverse intensity distribution at several propagation distances with δ0 2 mm.
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