Applied Physics B

, Volume 108, Issue 4, pp 891–895

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

Authors

  • Chengliang Zhao
    • School of Physical Science and TechnologySoochow University
  • Yuan Dong
    • School of Physical Science and TechnologySoochow University
  • Gaofeng Wu
    • School of Physical Science and TechnologySoochow University
  • Fei Wang
    • School of Physical Science and TechnologySoochow University
    • School of Physical Science and TechnologySoochow University
  • Olga Korotkova
    • Department of PhysicsUniversity of Miami Coral
Article

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

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 [233]. 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 [233]. 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 [3739]. 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. [3739].

2 Theory

The second-order statistical properties of an EGSM beam can be characterized by the 2 × 2 cross-spectral density (CSD) matrix specified at any two points with position vectors r1 and r2 in the source plane (z = 0), with elements [25]
$$ W_{\alpha \beta } ({\mathbf{r}}_{1} ,{\mathbf{r}}_{2} ) = A_{\alpha } A_{\beta } B_{\alpha \beta } \exp \left[ { - \frac{{{\mathbf{r}}_{1}^{2} }}{{4\sigma_{\alpha}^{2} }} - \frac{{{\mathbf{r}}_{2}^{2} }}{{4\sigma_{\beta }^{2} }} - \frac{{\left( {{\mathbf{r}}_{1} - {\mathbf{r}}_{2} } \right)^{2} }}{{2\delta_{\alpha \beta }^{2} }}} \right], $$
(1)
where \( \alpha ,\beta = x,y \), Aα is the square root of the spectral density of electric field component Eα, Bαβ is the correlation coefficient between Eα and Eβ with \( B_{xx} = B_{yy} = 1 \), σα is the r.m.s. width of the intensity along α direction, δxx, δyy, and δxy 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 Aα, σα, Bαβ, and δαβ 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 [15]
$$ P_{0} ({\mathbf{r}}) = \sqrt {1 - \frac{{4{\text{Det}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {W} ({\mathbf{r}},{\mathbf{r}})}}{{\left[ {{\text{Tr}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {W} ({\mathbf{r}},{\mathbf{r}})} \right]^{2} }}}. $$
(2)
According to [39], the coupling efficiency of an EGSM beam into a SMF may be approximately expressed as
$$ \eta = \left( {\frac{\pi }{\lambda fW}} \right)\frac{{\sum\nolimits_{\alpha = x,y} {A_{\alpha }^{2} w_{\alpha }^{2} \left[ {C_{\alpha \alpha }^{2} - (1/4\delta_{\alpha \alpha }^{4} )} \right]^{ - 1} } }}{{\sum\nolimits_{\alpha = x,y} {A_{\alpha }^{2} \sigma_{\alpha }^{2} \left( {W^{2} + 2\sigma_{\alpha }^{2} } \right)^{ - 1} } }}, $$
(3)
where 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; wα is the width of the mode polarized along direction α in the optical fiber; Cαα is given by expression
$$ C_{\alpha \alpha } = 1/4\sigma_{\alpha }^{2} + 1/2\delta_{\alpha \alpha } + \pi^{2} w_{\alpha }^{2} /\lambda^{2} f^{2} + 1/W^{2} . $$
(4)
From Eq. (3), one finds that the coupling efficiency is independent of the anti-diagonal elements of the CSD matrix. Thus in this letter, we will generate an EGSM beam without anti-diagonal elements and then couple it into a SMF. For an EGSM beam without anti-diagonal elements but with \( \sigma_{x} = \sigma_{y} \), applying Eqs. (1) and (2), we can express its DOP at point r as
$$ P_{0} \left( {\mathbf{r}} \right) = \frac{{\left| {1 - m} \right|}}{1 + m}, $$
(5)
with \( m = A_{y} /A_{x} \). It is important to note that the polarization properties of the considered EGSM beam are uniform across the source plane, and determined only by \( A_{y} /A_{x} \). Hence, the numerical results [3739] show that the coupling efficiency η is closely related with the parameters δxx, δyy and P0 when the parameters of the fiber and the coupling lens are fixed. With the increase of both r.m.s. correlations δxx and δyy (i.e., with the increase of degree of coherence), η increases. For the case of \( A_{x} > A_{y} \), η decreases as P0 increases, and for the case of \( A_{x} < A_{y} \), η increases as P0 increases.

3 Experimental results

To confirm experimentally the theoretical predictions, now we carry out experimental generation of an EGSM beam and then couple it into a SMF. Figure 1 shows our experimental setup for generating an EGSM beam without anti-diagonal elements, measuring the beam parameters and coupling of the EGSM beam into a SMF. The y (or x) linearly polarized beam (λ = 532 nm) generated from LS1 (or LS2) first passes through the NDF1 (or NDF2) and the L1 (or L2), then illuminates on the RGGD1 (or RGGD2), 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 L3 and the GAF, the stochastic electromagnetic beam becomes an EGSM beam, whose intensity distribution and degree of coherence have Gaussian shapes. NDF1 and NDF2 are used for modulation of parameters Ax and Ay. The correlation coefficients δxx and δyy are determined by the roughness of the RGGD1, RGGD2 and the focused beam spot sizes on the RGGD1 and RGGD2. In our experiment, the roughness of the RGGD is fixed, while δxx and δyy are mainly modulated by the focused beam spot sizes which are controlled by the distances from L1 to RGGD1 and from L2 to RGGD2. The L3 is used to collimate the stochastic electromagnetic beam, and the GAF is used to transform its beam profile into Gaussian distribution.
https://static-content.springer.com/image/art%3A10.1007%2Fs00340-012-5176-5/MediaObjects/340_2012_5176_Fig1_HTML.gif
Fig. 1

Experimental setup for generating an EGSM beam without anti-diagonal elements, measuring the beam parameters, and coupling the EGSM beam into a SMF. LS1, LS2 diode-pumped solid-state lasers, NDF1, NDF2 neutral density filters, M reflecting mirror, RGGD1, RGGD2 rotating ground-glass disks, PBS polarization beam splitter, GAF Gaussian amplitude filter, BS1, BS2 50:50 beam splitters; L1, L2, L3, L4 thin lenses, OL objective lens, SMF single-mode fiber, PM power meter, PC personal computer, SPD1, SPD2 single photon detectors, CC coincidence circuit

The parameters of the generated EGSM beam can be measured by the following procedure. Blocking the beam from LS1 (or LS2) and measuring its intensity distribution just behind GAF by use of a beam profile analyzer, we can obtain the values of Ax (or Ay) and σx (or σy). In our experiment, σx and σy are determined by the transmission function of the GAF, and both equal to 1.9 mm (see Fig. 2).
https://static-content.springer.com/image/art%3A10.1007%2Fs00340-012-5176-5/MediaObjects/340_2012_5176_Fig2_HTML.gif
Fig. 2

Experimental results of the intensity distribution (density plot) of the generated EGSM beam just behind the GAF and the corresponding cross line (dotted curve). The solid curve is a result of the Gaussian fit

In order to measure the value of δxx, we first block the beam from LS1, in this case, only the element Wxx exists behind the GAF, then split the beam just behind the GAF into two beams by the BS1, the reflected beam is further split into two distinct imaging optical path by the BS2, the two beams from BS2 pass to SPD1 and SPD2 (PMT120-OP), which scan the transverse planes u and v, respectively. Both the distances from the GAF to L4 and from L4 to SPD1 and SPD2 are 2f (i.e., 2f-imaging system). The output signals from SPD1 and SPD2 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 SPD1 and SPD2 are expressed as
$$ g_{xx}^{(2)} (u_{1} - v_{1} ,\tau ) = \frac{{\left\langle {I_{x} (u_{1} ,t)I_{x} (v_{1} ,t + \tau )} \right\rangle }}{{\left\langle {I_{x} (u_{1} ,t)} \right\rangle \left\langle {I_{x} (v_{1} ,t + \tau )} \right\rangle }}, $$
(6)
where \( I_{x} (u_{1} ,t) \) and \( I_{x} (v_{1} ,t + \tau ) \) are the instant intensities at SPD1 and SPD2, \( \tau \) 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
$$ g_{xx}^{(2)} (u_{1} - v_{1} ,\tau = 0) = 1 + \exp \left[ { - \frac{{(u_{1} - v_{1} )^{2} }}{{\delta_{xx}^{2} }}} \right]. $$
(7)
We fix SPD2 at v = 0, and SPD1 scans along the plane u. The coincidence circuit records the fourth-order correlation function between SPD1 and SPD2. 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 δxx can be obtained. If we block the beam from LS2, only the element Wyy exists behind the GAF. Then through a similar operation for obtaining δxx, we can measure the value of δyy. 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 BS1) 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.

Figure 3 shows our experimental results of the coupling efficiency versus the correlation coefficients δxx and δyy 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 δxx or δyy (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 \( A_{x} < A_{y} \).
https://static-content.springer.com/image/art%3A10.1007%2Fs00340-012-5176-5/MediaObjects/340_2012_5176_Fig3_HTML.gif
Fig. 3

Experimental results of the coupling efficiency versus the correlation coefficients δxx and δyy of the EGSM beam

https://static-content.springer.com/image/art%3A10.1007%2Fs00340-012-5176-5/MediaObjects/340_2012_5176_Fig4_HTML.gif
Fig. 4

Experimental result of the coupling efficiency versus the DOP of the EGSM beam

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).

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© Springer-Verlag 2012