Diffraction of cosine-Gaussian-correlated Schell-model beams Liuzhan Pan,* Chaoliang Ding, and Haixia Wang Department of Physics, Luoyang Normal University, Luoyang 471022, China *[email protected]
Abstract: The expression of spectral density of cosine-Gaussian-correlated Schell-model (CGSM) beams diffracted by an aperture is derived, and used to study the changes in the spectral density distribution of CGSM beams upon propagation, where the effect of aperture diffraction is emphasized. It is shown that, comparing with that of GSM beams, the spectral density distribution of CGSM beams diffracted by an aperture has dip and shows dark hollow intensity distribution when the order-parameter n is big enough. The central intensity increases with increasing truncation parameter of aperture. The comparative study of spectral density distributions of CGSM beams with aperture and that of without aperture is performed. Furthermore, the effect of order-parameter n and spatial coherence of CGSM beams on the spectral density distribution is discussed in detail. The results obtained may be useful in optical particulate manipulation. © 2014 Optical Society of America OCIS codes: (030.1640) Coherence; (030.6600) Statistical optics; (260.0260) Physical optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999). E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, 1st ed. (Cambridge University, 2007). F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008). C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1–3), 113–121 (1996). H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). Z. S. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). Z. R. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). Z. R. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). Z. R. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21(22), 27246–27259 (2013). S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). Y. Y. Zhang and D. M. Zhao, “Scattering of multi-Gaussian Schell-model beams on a random medium,” Opt. Express 21(21), 24781–24792 (2013). Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013). Y. T. Zhang, L. Liu, C. L. Zhao, and Y. J. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014). Z. R. Mei, “Light sources generating self-focusing beams of variable focal length,” Opt. Lett. 39(2), 347–350 (2014). Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11670
18. S. W. Cui, Z. Y. Chen, L. Zhang, and J. X. Pu, “Experimental generation of nonuniformly correlated partially coherent light beams,” Opt. Lett. 38(22), 4821–4824 (2013). 19. C. H. Liang, F. Wang, X. L. Liu, Y. J. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). 20. Y. H. Chen, F. Wang, C. L. Zhao, and Y. J. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014). 21. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
1. Introduction During the past two decades, the propagation of partially coherent beam with classic Gaussian Schell-model correlations have been studied comprehensively due to its tractability and universality [1, 2]. Recently, the traditional family of Gaussian Schell-model sources has been augmented by other models, such as, the J0-correlated Schell-model sources [3, 4], the non-uniformly correlated sources [5, 6], the Bessel-Gaussian Schell-model sources, Laguerre-Gaussian Schell-model sources [7], the cosine-Gaussian Schell-model sources [8–10], the Multi-Gaussian Schell-model sources [11–15], the nonuniformly cosine-Gaussian sources [16], specially correlated radially polarized sources [17], where some sources have been generated experimentally [17–20]. Surprisingly, in propagation, many interesting and useful features have revealed by the beams generated by these sources. For instance, the intensity profile of beams originated by J0-correlated Schell-model sources have properties analogous to those of the Bessel-Gaussian beams but the degree of coherence does not preserve the J0(x) profile nor shift-invariance [4]; the beams generated by non-uniformly correlated light sources hold self-focusing and lateral shifts of the beam intensity maxima in free-space propagation [5]; the Bessel-Gaussian and Laguerre-Gaussian Schell-model sources are capable of producing far fields with ring-shaped intensities [7]; the far-field spectral density produced by the cosine-Gaussian Schell-model sources takes on the dark-hollow profile [8, 19]; both the Multi-Gaussian Schell-model (MGSM) beams in free-space propagation and MGSM beams scattered by random media can generate far fields with tunable flat profiles, whether circular [11, 14] or rectangular [21]; the nonuniformly cosine-Gaussian sources employs cosine function for modeling of the source degree of coherence, which can adjust the self-focusing focal length on propagation [16]; the modulation of the correlation functions of a specially correlated radially polarized beam in the source plane can lead to efficient control of its intensity distribution and its degree of polarization on propagation [17]. However, in the practical application of Laser, the various apertures are used to adjust and control light beams. In this paper, we consider the diffraction properties of cosine-Gaussian-correlated Schell-model (CGSM) beams incident an aperture. And the effect of aperture diffraction on the spectral density evolution of CGSM beams upon propagation is emphasized. 2. Theoretical model Consider CGSM beams generated by the cosine-Gaussian-correlated Schell-model source passing through an aperture shown in Fig. 1, which is located at the z = 0 plane. The cross-spectral density function of CGSM beams in front of the aperture is expressed as n 2π ( x2′ − x1′ ) x ′2 + x ′2 ( x2′ − x1′ ) 2 0 W ( ) ( x1′, x2′ ) = exp − 1 2 2 cos , exp − w0 σ 2σ 2
(1)
where ( x1′, x2′ ) are the transversal coordinates of two points at the z = 0 plane, w0 is the r.m.s. width, and σ is r.m.s. correlation width. We can see that the cross-spectral density function of CGSM beams reduce to that of the conventional Gaussian correlated Schell-model (GSM) beams for the order-parameter n = 0, while for n≠0 the function is modulated by the cosine function.
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11671
Fig. 1. Schematic illustration of CGSM beams diffracted by an aperture.
Using the Huygens-Fresnel principle, we can obtain the spectral density of CGSM beams as follows. S ( x , z ) = W ( x, x , z ) =
k
a
2π z
a
−a −a
W (0) ( x1′, x2′ , z = 0)
ik × exp − ( x1′2 − x2′ 2 ) − 2 x ( x1′ − x2′ ) dx1′dx2′ . 2z
(2)
After the integration, the expression of the spectral density of CGSM beams can be derived.
S ( u, z ) =
δ i π z0 1 × exp − 4 −δ 4 Q2 z 4( σ / w 0 ) Q2
3 4 2 2 σ z σ z σ 4π 2π nu 0 − 4π 2 u 2 0 − 2π n 2 z w0 z w0 w0
2 4 4 σ 2 z0 σ z0 σ σ ′ + 2i n 2π + 2π u + 4π u Q2 u + 1 − 4Q1Q2 u ′ w0 z w0 z w0 w0
2 2nπ 3/2 ( z0 / z ) in 2π ′ × H1 exp + u H exp u du ′ , 2 3 (σ / w0 ) Q2 (σ / w0 )Q2 (3)
where
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Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11672
u 2π u ′ u 2π u′ H1 = cos + i sin × Erf σ / w0 σ / w0
2 σ z σ σ − δ 2 i Q iu′ + 2π n − 2π u 0 2 w0 z w0 w0 2 2 (σ / w0 ) Q2
2 σ z σ σ + δ i Q 2 iu′ + 2π n − 2π u 0 2 w0 z w0 w0 − Erf 2 , 2 (σ / w0 ) Q2 (4)
u 2π u′ u 2π u ′ H 2 = cos − i sin × Erf σ / w0 σ / w0
2 σ z σ σ − δ 2 i Q −iu′ + 2π n + 2π u 0 2 w0 z w0 w0 2 2 (σ / w0 ) Q2
2 σ z0 σ σ −iu′ + 2π n + 2π u + 2iδ Q2 w0 z w0 w0 , − Erf 2 2 (σ / w0 ) Q2 (5)
δ=
a , ( truncation parameter ) w0
z0 = u′ =
w02
λ
,
(6)
(7)
x′ , ( relative transversal coordinate at z = 0 plane ) w0
(8)
x , ( relative transversal coordinate at z plane ) w0
(9)
u=
Q1 = −1 −
z 1 − iπ 0 , 2 z 2(σ / w0 )
(10)
Q2 = −1 −
z 1 + iπ 0 , 2 z 2(σ / w0 )
(11)
where λ is the central wavelength of CGSM beams, Erf is the error function. Equation (3) provides an expression for the spectral density of CGSM beams passing through an aperture. The far field results for the spectral density of CGSM beams can be obtained by letting z→∞.
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Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11673
3. Numerical results and analyses Numerical calculations were performed using Eq. (3) to illustrate the evolution of spectral density of CGSM beams passing through an aperture and to stress the influence of aperture diffraction on the changes of spectral density of CGSM beams upon propagation. In the following calculations we take λ = 632nm and w0 = 0.5mm.
Fig. 2. Normalized spectral density distribution S(u,z)/S(0,0) of CGSM beams as a function of propagation distance z and relative coordinate u for different values of order-parameter n (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 4 and (e), (f), (g), (h) the color-coded plot corresponding to (a), (b), (c), (d) respectively. The other parameters are δ = 0.4, σ/w0 = 0.5.
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Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11674
Figure 2 gives the normalized spectral density distribution S(u,z)/S(0,0) of CGSM beams as a function of propagation distance z and relative coordinate u for different values of order-parameter n (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 4. Figure 2(e), Fig. 2(f) and Fig. 2(g) and Fig. 2(h) are the color-coded plot corresponding to (a), (b), (c), (d) respectively. The other parameters are δ = 0.4, σ/w0 = 0.5. As shown in Fig. 2, for n = 0, i.e. the case of GSM beams, the spectral density distribution holds the Gaussian-like form upon propagation in the region 0.2≤z/z0≤1. For n = 1, i.e. the case of CGSM beams, the central intensity decreases and two peaks appear. With the increase of order-parameter n, the central intensity transforms into a dark hollow optical field distribution when n is big enough. And the dark area between two peaks broadens for the big n.
Fig. 3. Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of propagation distance (a) z/z0 = 0.2, (b) z/z0 = 0.3, (c) z/z0 = 0.4, (d) z/z0→∞. The other parameters are δ = 0.4, σ/w0 = 0.5, n = 2.
Figure 3 gives the normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of propagation distance (a) z/z0 = 0.2, (b) z/z0 = 0.3, (c) z/z0 = 0.4, (d) z/z0→∞. The other parameters are δ = 0.4, σ/w0 = 0.5, n = 2. As can be seen that, transverse spectral density distribution of CGSM beams has dip comparing with that of GSM beams. And the central intensity decreases with increasing propagation distance. Furthermore, the intensity sidelobe emerges because of the aperture diffraction. In the far field, the central intensity reach minimum.
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Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11675
Fig. 4. Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of truncation parameter. The other parameters are σ/w0 = 0.5, n = 2, (a) z/z0 = 0.3, (b) z/z0 = 0.4.
Figure 4 gives the normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of truncation parameter. The other parameters are σ/w0 = 0.5, n = 2, (a) z/z0 = 0.3, (b) z/z0 = 0.4. It is shown that in Fig. 4(a), for the free space case (δ = 20), transverse spectral density distribution has Gaussian-like form. With decresing truncation parameter, the central intensity decreases and the intensity dip appears. At the same time, two intensity peaks appears. For z/z0 = 0.4, there are always two intensity peaks in spite of free space case (δ = 20). And the central intensity decreases with decreasing truncation parameter. Thus the aperture diffraction plays an important role on the spectral density distribution of CGSM beams.
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Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11676
Fig. 5. Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of order-parameter n. The other parameters are σ/w0 = 0.5, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞.
Figure 5 gives the normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of order-parameter n. The other parameters are σ/w0 = 0.5, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞. It is clearly seen from Fig. 5(a) that, for the GSM beams case (n = 0), transverse spectral density distribution has Gaussian-like form. For n>0, i.e. the case of CGSM beams, the central intensity decreases and two peaks appear. And the area between the two peaks increases with increasing values of n. Furthermore, the value of n is only to change the area between the two peaks, without changing the width of the single peak. In the far field (see Fig. 5(b)), the similar properties can be seen and central intensity of CGSM beams becomes smaller for the same values of n.
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Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11677
Fig. 6. Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of coherence parameter σ/w0. The other parameters are n = 2, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞.
Figure 6 gives the normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of coherence parameter σ/w0. The other parameters are n = 2, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞. As can be seen from Fig. 6 (a) and Fig. 6 (b) that the coherence plays an important role on the spectral density evolution of CGSM beams. For the low coherence, there are two intensity peaks. With increasing coherence of CGSM, the area between the two peaks decreases. Finally, the spectral density distribution becomes Gaussian-like form. The results can be seen easily from Eq. (1) when σ is big enough. And, the value of σ/w0 not only changes the area between the two peaks, but also changes the width of the single peak. 4. Conclusion In this paper, we have studied the diffraction properties of CGSM beams incident an aperture. The expression of spectral density of CGSM beams diffracted by an aperture has been derived. The effect of truncation parameter of aperture, order-parameter and spatial coherence
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Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11678
of CGSM beams on the spectral density distribution is given. It is shown that the spectral density distribution of CGSM beams has dip and can transform into dark hollow intensity distribution comparing with that of GSM beams. The aperture diffraction plays an important role on the spectral density distribution of CGSM beams. The intensity sidelobe emerges because of the aperture diffraction. And the central intensity increases with increasing truncation parameter. When the truncation parameter is big enough, the intensity dip of spectral density distribution of CGSM beams can disappear under some conditon. In addition, the value of order-parameter n is only to change the area between two peaks of spectral density distribution. However, the value of spatial coherence parameter σ/w0 not only changes the area between two peaks, but also changes the width of the single peak. The results obtained may be useful in optical particulate manipulation. Acknowledgments This research is supported by the National Natural Science Foundation of China under Grant Nos. 61275150, 61078077 and 61108090, the Education Department of Henan Province Project 13A140797, the Program for Science & Technology Innovation Talents in Universities of Henan Province (13HASTIT048) and the Program for Innovative Research Team (in Science and Technology) in University of Henna Province (Grant No. 13IRTSTHN020).
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11679
Abstract: The expression of spectral density of cosine-Gaussian-correlated Schell-model (CGSM) beams diffracted by an aperture is derived, and used to study the changes in the spectral density distribution of CGSM beams upon propagation, where the effect of aperture diffraction is emphasized. It is shown that, comparing with that of GSM beams, the spectral density distribution of CGSM beams diffracted by an aperture has dip and shows dark hollow intensity distribution when the order-parameter n is big enough. The central intensity increases with increasing truncation parameter of aperture. The comparative study of spectral density distributions of CGSM beams with aperture and that of without aperture is performed. Furthermore, the effect of order-parameter n and spatial coherence of CGSM beams on the spectral density distribution is discussed in detail. The results obtained may be useful in optical particulate manipulation. © 2014 Optical Society of America OCIS codes: (030.1640) Coherence; (030.6600) Statistical optics; (260.0260) Physical optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999). E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, 1st ed. (Cambridge University, 2007). F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008). C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1–3), 113–121 (1996). H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). Z. S. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). Z. R. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). Z. R. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). Z. R. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21(22), 27246–27259 (2013). S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). Y. Y. Zhang and D. M. Zhao, “Scattering of multi-Gaussian Schell-model beams on a random medium,” Opt. Express 21(21), 24781–24792 (2013). Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, and H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013). Y. T. Zhang, L. Liu, C. L. Zhao, and Y. J. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014). Z. R. Mei, “Light sources generating self-focusing beams of variable focal length,” Opt. Lett. 39(2), 347–350 (2014). Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11670
18. S. W. Cui, Z. Y. Chen, L. Zhang, and J. X. Pu, “Experimental generation of nonuniformly correlated partially coherent light beams,” Opt. Lett. 38(22), 4821–4824 (2013). 19. C. H. Liang, F. Wang, X. L. Liu, Y. J. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). 20. Y. H. Chen, F. Wang, C. L. Zhao, and Y. J. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014). 21. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
1. Introduction During the past two decades, the propagation of partially coherent beam with classic Gaussian Schell-model correlations have been studied comprehensively due to its tractability and universality [1, 2]. Recently, the traditional family of Gaussian Schell-model sources has been augmented by other models, such as, the J0-correlated Schell-model sources [3, 4], the non-uniformly correlated sources [5, 6], the Bessel-Gaussian Schell-model sources, Laguerre-Gaussian Schell-model sources [7], the cosine-Gaussian Schell-model sources [8–10], the Multi-Gaussian Schell-model sources [11–15], the nonuniformly cosine-Gaussian sources [16], specially correlated radially polarized sources [17], where some sources have been generated experimentally [17–20]. Surprisingly, in propagation, many interesting and useful features have revealed by the beams generated by these sources. For instance, the intensity profile of beams originated by J0-correlated Schell-model sources have properties analogous to those of the Bessel-Gaussian beams but the degree of coherence does not preserve the J0(x) profile nor shift-invariance [4]; the beams generated by non-uniformly correlated light sources hold self-focusing and lateral shifts of the beam intensity maxima in free-space propagation [5]; the Bessel-Gaussian and Laguerre-Gaussian Schell-model sources are capable of producing far fields with ring-shaped intensities [7]; the far-field spectral density produced by the cosine-Gaussian Schell-model sources takes on the dark-hollow profile [8, 19]; both the Multi-Gaussian Schell-model (MGSM) beams in free-space propagation and MGSM beams scattered by random media can generate far fields with tunable flat profiles, whether circular [11, 14] or rectangular [21]; the nonuniformly cosine-Gaussian sources employs cosine function for modeling of the source degree of coherence, which can adjust the self-focusing focal length on propagation [16]; the modulation of the correlation functions of a specially correlated radially polarized beam in the source plane can lead to efficient control of its intensity distribution and its degree of polarization on propagation [17]. However, in the practical application of Laser, the various apertures are used to adjust and control light beams. In this paper, we consider the diffraction properties of cosine-Gaussian-correlated Schell-model (CGSM) beams incident an aperture. And the effect of aperture diffraction on the spectral density evolution of CGSM beams upon propagation is emphasized. 2. Theoretical model Consider CGSM beams generated by the cosine-Gaussian-correlated Schell-model source passing through an aperture shown in Fig. 1, which is located at the z = 0 plane. The cross-spectral density function of CGSM beams in front of the aperture is expressed as n 2π ( x2′ − x1′ ) x ′2 + x ′2 ( x2′ − x1′ ) 2 0 W ( ) ( x1′, x2′ ) = exp − 1 2 2 cos , exp − w0 σ 2σ 2
(1)
where ( x1′, x2′ ) are the transversal coordinates of two points at the z = 0 plane, w0 is the r.m.s. width, and σ is r.m.s. correlation width. We can see that the cross-spectral density function of CGSM beams reduce to that of the conventional Gaussian correlated Schell-model (GSM) beams for the order-parameter n = 0, while for n≠0 the function is modulated by the cosine function.
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11671
Fig. 1. Schematic illustration of CGSM beams diffracted by an aperture.
Using the Huygens-Fresnel principle, we can obtain the spectral density of CGSM beams as follows. S ( x , z ) = W ( x, x , z ) =
k
a
2π z
a
−a −a
W (0) ( x1′, x2′ , z = 0)
ik × exp − ( x1′2 − x2′ 2 ) − 2 x ( x1′ − x2′ ) dx1′dx2′ . 2z
(2)
After the integration, the expression of the spectral density of CGSM beams can be derived.
S ( u, z ) =
δ i π z0 1 × exp − 4 −δ 4 Q2 z 4( σ / w 0 ) Q2
3 4 2 2 σ z σ z σ 4π 2π nu 0 − 4π 2 u 2 0 − 2π n 2 z w0 z w0 w0
2 4 4 σ 2 z0 σ z0 σ σ ′ + 2i n 2π + 2π u + 4π u Q2 u + 1 − 4Q1Q2 u ′ w0 z w0 z w0 w0
2 2nπ 3/2 ( z0 / z ) in 2π ′ × H1 exp + u H exp u du ′ , 2 3 (σ / w0 ) Q2 (σ / w0 )Q2 (3)
where
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Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11672
u 2π u ′ u 2π u′ H1 = cos + i sin × Erf σ / w0 σ / w0
2 σ z σ σ − δ 2 i Q iu′ + 2π n − 2π u 0 2 w0 z w0 w0 2 2 (σ / w0 ) Q2
2 σ z σ σ + δ i Q 2 iu′ + 2π n − 2π u 0 2 w0 z w0 w0 − Erf 2 , 2 (σ / w0 ) Q2 (4)
u 2π u′ u 2π u ′ H 2 = cos − i sin × Erf σ / w0 σ / w0
2 σ z σ σ − δ 2 i Q −iu′ + 2π n + 2π u 0 2 w0 z w0 w0 2 2 (σ / w0 ) Q2
2 σ z0 σ σ −iu′ + 2π n + 2π u + 2iδ Q2 w0 z w0 w0 , − Erf 2 2 (σ / w0 ) Q2 (5)
δ=
a , ( truncation parameter ) w0
z0 = u′ =
w02
λ
,
(6)
(7)
x′ , ( relative transversal coordinate at z = 0 plane ) w0
(8)
x , ( relative transversal coordinate at z plane ) w0
(9)
u=
Q1 = −1 −
z 1 − iπ 0 , 2 z 2(σ / w0 )
(10)
Q2 = −1 −
z 1 + iπ 0 , 2 z 2(σ / w0 )
(11)
where λ is the central wavelength of CGSM beams, Erf is the error function. Equation (3) provides an expression for the spectral density of CGSM beams passing through an aperture. The far field results for the spectral density of CGSM beams can be obtained by letting z→∞.
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11673
3. Numerical results and analyses Numerical calculations were performed using Eq. (3) to illustrate the evolution of spectral density of CGSM beams passing through an aperture and to stress the influence of aperture diffraction on the changes of spectral density of CGSM beams upon propagation. In the following calculations we take λ = 632nm and w0 = 0.5mm.
Fig. 2. Normalized spectral density distribution S(u,z)/S(0,0) of CGSM beams as a function of propagation distance z and relative coordinate u for different values of order-parameter n (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 4 and (e), (f), (g), (h) the color-coded plot corresponding to (a), (b), (c), (d) respectively. The other parameters are δ = 0.4, σ/w0 = 0.5.
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11674
Figure 2 gives the normalized spectral density distribution S(u,z)/S(0,0) of CGSM beams as a function of propagation distance z and relative coordinate u for different values of order-parameter n (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 4. Figure 2(e), Fig. 2(f) and Fig. 2(g) and Fig. 2(h) are the color-coded plot corresponding to (a), (b), (c), (d) respectively. The other parameters are δ = 0.4, σ/w0 = 0.5. As shown in Fig. 2, for n = 0, i.e. the case of GSM beams, the spectral density distribution holds the Gaussian-like form upon propagation in the region 0.2≤z/z0≤1. For n = 1, i.e. the case of CGSM beams, the central intensity decreases and two peaks appear. With the increase of order-parameter n, the central intensity transforms into a dark hollow optical field distribution when n is big enough. And the dark area between two peaks broadens for the big n.
Fig. 3. Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of propagation distance (a) z/z0 = 0.2, (b) z/z0 = 0.3, (c) z/z0 = 0.4, (d) z/z0→∞. The other parameters are δ = 0.4, σ/w0 = 0.5, n = 2.
Figure 3 gives the normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of propagation distance (a) z/z0 = 0.2, (b) z/z0 = 0.3, (c) z/z0 = 0.4, (d) z/z0→∞. The other parameters are δ = 0.4, σ/w0 = 0.5, n = 2. As can be seen that, transverse spectral density distribution of CGSM beams has dip comparing with that of GSM beams. And the central intensity decreases with increasing propagation distance. Furthermore, the intensity sidelobe emerges because of the aperture diffraction. In the far field, the central intensity reach minimum.
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11675
Fig. 4. Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of truncation parameter. The other parameters are σ/w0 = 0.5, n = 2, (a) z/z0 = 0.3, (b) z/z0 = 0.4.
Figure 4 gives the normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of truncation parameter. The other parameters are σ/w0 = 0.5, n = 2, (a) z/z0 = 0.3, (b) z/z0 = 0.4. It is shown that in Fig. 4(a), for the free space case (δ = 20), transverse spectral density distribution has Gaussian-like form. With decresing truncation parameter, the central intensity decreases and the intensity dip appears. At the same time, two intensity peaks appears. For z/z0 = 0.4, there are always two intensity peaks in spite of free space case (δ = 20). And the central intensity decreases with decreasing truncation parameter. Thus the aperture diffraction plays an important role on the spectral density distribution of CGSM beams.
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11676
Fig. 5. Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of order-parameter n. The other parameters are σ/w0 = 0.5, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞.
Figure 5 gives the normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of order-parameter n. The other parameters are σ/w0 = 0.5, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞. It is clearly seen from Fig. 5(a) that, for the GSM beams case (n = 0), transverse spectral density distribution has Gaussian-like form. For n>0, i.e. the case of CGSM beams, the central intensity decreases and two peaks appear. And the area between the two peaks increases with increasing values of n. Furthermore, the value of n is only to change the area between the two peaks, without changing the width of the single peak. In the far field (see Fig. 5(b)), the similar properties can be seen and central intensity of CGSM beams becomes smaller for the same values of n.
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11677
Fig. 6. Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of coherence parameter σ/w0. The other parameters are n = 2, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞.
Figure 6 gives the normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of coherence parameter σ/w0. The other parameters are n = 2, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞. As can be seen from Fig. 6 (a) and Fig. 6 (b) that the coherence plays an important role on the spectral density evolution of CGSM beams. For the low coherence, there are two intensity peaks. With increasing coherence of CGSM, the area between the two peaks decreases. Finally, the spectral density distribution becomes Gaussian-like form. The results can be seen easily from Eq. (1) when σ is big enough. And, the value of σ/w0 not only changes the area between the two peaks, but also changes the width of the single peak. 4. Conclusion In this paper, we have studied the diffraction properties of CGSM beams incident an aperture. The expression of spectral density of CGSM beams diffracted by an aperture has been derived. The effect of truncation parameter of aperture, order-parameter and spatial coherence
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11678
of CGSM beams on the spectral density distribution is given. It is shown that the spectral density distribution of CGSM beams has dip and can transform into dark hollow intensity distribution comparing with that of GSM beams. The aperture diffraction plays an important role on the spectral density distribution of CGSM beams. The intensity sidelobe emerges because of the aperture diffraction. And the central intensity increases with increasing truncation parameter. When the truncation parameter is big enough, the intensity dip of spectral density distribution of CGSM beams can disappear under some conditon. In addition, the value of order-parameter n is only to change the area between two peaks of spectral density distribution. However, the value of spatial coherence parameter σ/w0 not only changes the area between two peaks, but also changes the width of the single peak. The results obtained may be useful in optical particulate manipulation. Acknowledgments This research is supported by the National Natural Science Foundation of China under Grant Nos. 61275150, 61078077 and 61108090, the Education Department of Henan Province Project 13A140797, the Program for Science & Technology Innovation Talents in Universities of Henan Province (13HASTIT048) and the Program for Innovative Research Team (in Science and Technology) in University of Henna Province (Grant No. 13IRTSTHN020).
#208915 - $15.00 USD (C) 2014 OSA
Received 25 Mar 2014; accepted 29 Apr 2014; published 6 May 2014 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011670 | OPTICS EXPRESS 11679
