# Experimental demonstration of coupling of an electromagnetic Gaussian Schell-model beam into a single-mode optical fiber

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DOI: 10.1007/s00340-012-5176-5

- Cite this article as:
- Zhao, C., Dong, Y., Wu, G. et al. Appl. Phys. B (2012) 108: 891. doi:10.1007/s00340-012-5176-5

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## Abstract

We demonstrate experimentally the procedure of coupling of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam into a single-mode optical fiber. We find that the coupling efficiency depends closely on the coherence and polarization properties of the EGSM beam, which is consistent with theoretical prediction. Our results may find applications in connection with free-space optical communications and LIDARs, where coupling of a stochastic beam into an optical fiber is inevitable encountered.

## 1 Introduction

It has been recently asserted that the intimate relation between the states of coherence and polarization of random light [1] can be successfully exploited in optical systems, such as free-space communication systems, LIDARs, optical tweezers, and classic imaging systems [2–33]. A special yet broad class of the stochastic electromagnetic Gaussian Schell-model (EGSM) beams was introduced as a natural extension of the well-known stochastic scalar GSM beam and its generation, ghost imaging, detection, and interaction with various media are by now well understood [2–33]. It was revealed that EGSM beams have advantage over scalar GSM beams or coherent Gaussian beams in some applications, such as free-space optical communications, LIDARs, optical imaging, particle trapping, and remote sensing. Several theories have been proposed for describing the polarization and coherence properties of a 3D stochastic electromagnetic field, including the theory introduced by Wolf et al. [8, 28] and the theory introduced by Friberg et al. [29, 30], and two definitions of 3D degree of polarization were proposed, respectively. It was shown that a monotonic one-to-one correspondence exists between two definitions of 3D degree of polarization under certain condition [29]. More recently, Wang et al. [31] reported experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source. Zhang et al. [32] studied the effect of polarization on the degree of paraxiality of an EGSM beam. Wu and Cai [33] explored modulation of spectral intensity, polarization and coherence of an EGSM beam by a phase aperture.

In 1972, Cohen carried out an experimental study of the coupling of the GaAs injection laser beams into optical fibers [34]. Since then, the coupling of light into optical fibers has been investigated extensively due to its wide applications in optical communications, biomedical optics, LIDARs, stellar interferometry, and wavefront sensing. The majority of the previously published papers were devoted to coupling of coherent beams into optical fibers. In some practical applications, such as free-space optical communications and LIDARs, coupling of a stochastic beam into an optical fiber is inevitable encountered because the coherence of laser beam was degraded by the atmospheric turbulence during propagation [35, 36]. Thus, it is of great importance to study the coupling of a stochastic beam into an optical fiber. Recently, Salem et al. theoretically studied the effects of coherence and polarization on the coupling of an EGSM beam into optical fibers based on the theory of coherence and polarization introduced by Wolf et al. and found that the coupling efficiency is closely related to the states of coherence and polarization [37–39]. To our knowledge, no experimental results have been reported up until now on the coupling of an EGSM beams into optical fibers. In this article, we first offer the experimental results concerning the coupling of an EGSM beam into a single-mode fiber (SMF). We also use Wolf’s theory for describing the coherence and polarization properties of a stochastic electromagnetic beam to be consistent with the theoretical results of Refs. [37–39].

## 2 Theory

**r**

_{1}and

**r**

_{2}in the source plane (

*z*= 0), with elements [2–5]

*A*

_{α}is the square root of the spectral density of electric field component

*E*

_{α},

*B*

_{αβ}is the correlation coefficient between

*E*

_{α}and

*E*

_{β}with \( B_{xx} = B_{yy} = 1 \), σ

_{α}is the r.m.s. width of the intensity along α direction, δ

_{xx}, δ

_{yy}, and δ

_{xy}are the r.m.s. widths of the auto-correlation functions of the

*x*-component of the field, of the

*y*-component of the field, and of the mutual correlation function of

*x*and

*y*field components, respectively. The parameters

*A*

_{α}, σ

_{α},

*B*

_{αβ}, and δ

_{αβ}are assumed to be independent of position but may depend on oscillation frequency. The degree of polarization (DOP) of the EGSM beam at point

**r**is defined as [1–5]

*f*is the focal length of the coupling lens, \( W = {D \mathord{\left/ {\vphantom {D {2\sqrt 2 }}} \right. \kern-\nulldelimiterspace} {2\sqrt 2 }} \), with

*D*being the aperture diameter of the coupling lens;

*w*

_{α}is the width of the mode polarized along direction α in the optical fiber;

*C*

_{αα}is given by expression

**r**as

_{xx}, δ

_{yy}and

*P*

_{0}when the parameters of the fiber and the coupling lens are fixed. With the increase of both r.m.s. correlations δ

_{xx}and δ

_{yy}(i.e., with the increase of degree of coherence), η increases. For the case of \( A_{x} > A_{y} \), η decreases as

*P*

_{0}increases, and for the case of \( A_{x} < A_{y} \), η increases as

*P*

_{0}increases.

## 3 Experimental results

*y*(or

*x*) linearly polarized beam (λ = 532 nm) generated from LS

_{1}(or LS

_{2}) first passes through the NDF

_{1}(or NDF

_{2}) and the L

_{1}(or L

_{2}), then illuminates on the RGGD

_{1}(or RGGD

_{2}), producing

*y*(or

*x*) linearly polarized partially coherent beam. Then the polarization beam splitter combines two orthogonally polarized partially coherent beams, producing a stochastic electromagnetic beam. After passing through the L

_{3}and the GAF, the stochastic electromagnetic beam becomes an EGSM beam, whose intensity distribution and degree of coherence have Gaussian shapes. NDF

_{1}and NDF

_{2}are used for modulation of parameters

*A*

_{x}and

*A*

_{y}. The correlation coefficients δ

_{xx}and δ

_{yy}are determined by the roughness of the RGGD

_{1}, RGGD

_{2}and the focused beam spot sizes on the RGGD

_{1}and RGGD

_{2}. In our experiment, the roughness of the RGGD is fixed, while δ

_{xx}and δ

_{yy}are mainly modulated by the focused beam spot sizes which are controlled by the distances from L

_{1}to RGGD

_{1}and from L

_{2}to RGGD

_{2}. The L

_{3}is used to collimate the stochastic electromagnetic beam, and the GAF is used to transform its beam profile into Gaussian distribution.

_{1}(or LS

_{2}) and measuring its intensity distribution just behind GAF by use of a beam profile analyzer, we can obtain the values of

*A*

_{x}(or

*A*

_{y}) and σ

_{x}(or σ

_{y}). In our experiment, σ

_{x}and σ

_{y}are determined by the transmission function of the GAF, and both equal to 1.9 mm (see Fig. 2).

_{xx}, we first block the beam from LS

_{1}, in this case, only the element

*W*

_{xx}exists behind the GAF, then split the beam just behind the GAF into two beams by the BS

_{1}, the reflected beam is further split into two distinct imaging optical path by the BS

_{2}, the two beams from BS

_{2}pass to SPD

_{1}and SPD

_{2}(PMT120-OP), which scan the transverse planes

*u*and

*v*, respectively. Both the distances from the GAF to L

_{4}and from L

_{4}to SPD

_{1}and SPD

_{2}are 2

*f*(i.e., 2

*f*-imaging system). The output signals from SPD

_{1}and SPD

_{2}are sent to an electronic coincidence circuit (Flex02-01D [40]) to measure the fourth-order correlation function between two detectors (i.e., intensity correlation function). The fourth-order correlation function between SPD

_{1}and SPD

_{2}are expressed as

_{1}and SPD

_{2}, \( \tau \) denotes the delay time of the photon flux of two optical paths. Because the beam source in our experiment obeys the Gaussain statistics, with the help of Gausian moment theorem [41], \( g_{xx}^{(2)} (u_{1} - v_{1} ,\tau = 0) \) can be expressed as

_{2}at

*v*= 0, and SPD

_{1}scans along the plane

*u*. The coincidence circuit records the fourth-order correlation function between SPD

_{1}and SPD

_{2}. Then we can obtain the distribution of the normalized fourth-order correlation function \( g_{xx}^{(2)} (u_{1} ,\tau = 0) \). From the curve of the Gaussian fit for the experimental results, the value of δ

_{xx}can be obtained. If we block the beam from LS

_{2}, only the element

*W*

_{yy}exists behind the GAF. Then through a similar operation for obtaining δ

_{xx}, we can measure the value of δ

_{yy}. More detailed information about measuring the spatial coherence length of partially coherent beam can be found in [42].

After the beam parameters of the generated EGSM beam were measured, we can obtain the value of DOP of the generated EGSM beam and study the coupling of the beam into a SMF. In our experiment, the generated EGSM beam (i.e., the transmitted beam from BS_{1}) is coupled into the SMF with the objective lens. The numerical aperture (NA) of the objective lens equals to 0.1. The SMF with NA = 0.13 is made of fused silica (S460 HP produced by the THORLAB) and its operating wavelength ranges from 450 nm to 600 nm. The power meter is used to measure the power of the beam just before the objective lens and the power at the output of the SMF.

_{xx}and δ

_{yy}of the generated EGSM beam. Figure 4 shows our experimental results of the coupling efficiency versus the DOP of the generated EGSM beam. One finds from Figs. 3 and 4 that our experimental results agree reasonably well with the theoretical predictions. The coupling efficiency indeed increases with the increase of δ

_{xx}or δ

_{yy}(i.e., degree of coherence), decreases with the increase of DOP for the case \( A_{x} > A_{y} \), and increases with the increase of DOP for the case \( A_{x} < A_{y} \).

## 4 Conclusion

In conclusion, we have carried out experimental study of the coupling of an EGSM beam without anti-diagonal elements into a SMF. The dependency of the coupling efficiency on the correlation coefficients and the DOP of the generated EGSM beam were studied experimentally and were found to be consistent with the theoretical predictions. Our results are crucial for all applications relating to transmission of partially coherent beams through optical fibers.

## Acknowledgments

This study was supported by the National Natural Science Foundation of China under Grant Nos. 10904102 & 61008009 &11104195, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science Foundation of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009, the Key Project of Chinese Ministry of Education under Grant No. 210081, the Universities Natural Science Research Project of Jiangsu Province Grant Nos. 10KJB140011 & 11KJB140007, the Project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the National College Students Innovation Experiment Program under Grant No. 111028510. O. Korotkova’s research is funded by the US ONR (Grant N00189-12-P-0114).