Diffraction of cosine-Gaussian-correlated Schell-model beams

Liuzhan Pan; Chaoliang Ding; Haixia Wang

doi:10.1364/OE.22.011670

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

W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}) = \exp [- \frac{{x^{'}}_{1}^{2} + {x^{'}}_{2}^{2}}{w_{0}^{2}}] \cos [\frac{n \sqrt{2 π} ({x^{'}}_{2} - {x^{'}}_{1})}{σ}] \exp [- \frac{{({x^{'}}_{2} - {x^{'}}_{1})}^{2}}{2 σ_{}^{2}}],

where

({x^{'}}_{1}, {x^{'}}_{2})

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.

Fig. 1 Schematic illustration of CGSM beams diffracted by an aperture.

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Using the Huygens-Fresnel principle, we can obtain the spectral density of CGSM beams as follows.

\begin{array}{l} S (x, z) = W (x, x, z) = \frac{k}{2 π z} \int_{- a}^{a} \int_{- a}^{a} W^{(0)} ({x^{'}}_{1}, {x^{'}}_{2}, z = 0) \\ \times \exp {- \frac{i k}{2 z} [({x^{'}}_{1}^{2} - {x^{'}}_{2}^{2}) - 2 x ({x^{'}}_{1} - {x^{'}}_{2})]} d {x^{'}}_{1} d {x^{'}}_{2}_{2} . \end{array}

After the integration, the expression of the spectral density of CGSM beams can be derived.

\begin{array}{l} S (u, z) = \frac{i}{4} \sqrt{\frac{π}{Q_{2}}} \frac{z_{0}}{z} \times \int_{- δ}^{δ} \exp {- \frac{1}{4 {(σ / w_{0})}^{4} Q_{2}} [4 π \sqrt{2 π} n u \frac{z_{0}}{z} {(\frac{σ}{w_{0}})}^{3} - 4 π^{2} u^{2} {(\frac{z_{0}}{z})}^{2} {(\frac{σ}{w_{0}})}^{4} - 2 π n^{2} {(\frac{σ}{w_{0}})}^{2} \\ + 2 i (n \sqrt{2 π} \frac{σ}{w_{0}} + 2 π u \frac{z_{0}}{z} {(\frac{σ}{w_{0}})}^{2} + 4 π u \frac{z_{0}}{z} Q_{2} {(\frac{σ}{w_{0}})}^{4}) u^{'} + (1 - 4 Q_{1} Q_{2} {(\frac{σ}{w_{0}})}^{4}) {u^{'}}^{2}]} \\ \times {H_{1} \exp [\frac{i n \sqrt{2 π}}{{(σ / w_{0})}^{3} Q_{2}} u^{'}] + H_{2} \exp [\frac{2 \sqrt{2} n π^{3 / 2} (z_{0} / z)}{(σ / w_{0}) Q_{2}} u]} d u^{'}, \end{array}

where

\begin{array}{l} H_{1} = [\cos (\frac{u \sqrt{2 π} u^{'}}{σ / w_{0}}) + i \sin (\frac{u \sqrt{2 π} u^{'}}{σ / w_{0}})] \times {E r f [\frac{i u^{'} + \frac{σ}{w_{0}} (\sqrt{2 π} n - 2 π u \frac{z_{0}}{z} \frac{σ}{w_{0}}) - 2 i δ Q_{2} {(\frac{σ}{w_{0}})}^{2}}{2 {(σ / w_{0})}^{2} \sqrt{Q_{2}}}] \\ - E r f [\frac{i u^{'} + \frac{σ}{w_{0}} (\sqrt{2 π} n - 2 π u \frac{z_{0}}{z} \frac{σ}{w_{0}}) + 2 i δ Q_{2} {(\frac{σ}{w_{0}})}^{2}}{2 {(σ / w_{0})}^{2} \sqrt{Q_{2}}}]}, \end{array}

\begin{array}{l} H_{2} = [\cos (\frac{u \sqrt{2 π} u^{'}}{σ / w_{0}}) - i \sin (\frac{u \sqrt{2 π} u^{'}}{σ / w_{0}})] \times {E r f [\frac{- i u^{'} + \frac{σ}{w_{0}} (\sqrt{2 π} n + 2 π u \frac{z_{0}}{z} \frac{σ}{w_{0}}) - 2 i δ Q_{2} {(\frac{σ}{w_{0}})}^{2}}{2 {(σ / w_{0})}^{2} \sqrt{Q_{2}}}] \\ - E r f [\frac{- i u^{'} + \frac{σ}{w_{0}} (\sqrt{2 π} n + 2 π u \frac{z_{0}}{z} \frac{σ}{w_{0}}) + 2 i δ Q_{2} {(\frac{σ}{w_{0}})}^{2}}{2 {(σ / w_{0})}^{2} \sqrt{Q_{2}}}]}, \end{array}

δ = \frac{a}{w_{0}}, (t r u n c a t i o n p a r a m e t e r)

z_{0} = \frac{w_{0}^{2}}{λ},

u^{'} = \frac{x^{'}}{w_{0}}, (r e l a t i v e t r a n s v e r s a l c o o r d i n a t e a t z = 0 p l a n e)

u = \frac{x}{w_{0}}, (r e l a t i v e t r a n s v e r s a l c o o r d i n a t e a t z p l a n e)

Q_{1} = - 1 - \frac{1}{2 {(σ / w_{0})}^{2}} - i π \frac{z_{0}}{z},

Q_{2} = - 1 - \frac{1}{2 {(σ / w_{0})}^{2}} + i π \frac{z_{0}}{z},

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→∞.

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.

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. 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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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.

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.

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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.

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.

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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.

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→∞.

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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.

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→∞.

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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 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).

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Diffraction of cosine-Gaussian-correlated Schell-model beams

Abstract

1. Introduction

2. Theoretical model

3. Numerical results and analyses

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (11)

Optics Express