In the free space optical communication system with circle polarization shift keying (CPolSK) modulation, the changes of polarization state of light beam have significant influence on the system performance. Keeping the state of polarization (SOP) unchanged on propagation can reduce the bit error rate. Based on the unified theory of coherence and polarization, we derive the sufficient condition for Gaussian Schell-model (GSM) beam to keep the SOP unchanged. We found that when the three spectral correlation widths (δxx, δyy and δxy) equal to each other and σx=σy, the GSM beam maintains the SOP on propagation. This conclusion can be helpful for the design of the transmitter in the CPolSK system.
©2009 Optical Society of America
Propagation of light beam in free space involves many research fields such as satellite optical communication, deep space optical relay and astronomical observation. It has received special attention and extensive investigation. We proposed a polarization modulation scheme called circle polarization shift keying (CPolSK) in recent work . For the free space optical communication system with CPolSK modulation, it is very important to study the changes of polarization properties on propagation. Traditionally, the polarization properties of the electromagnetic beams are considered to be unchanged as beams propagate in free space or in any linear medium. However, James demonstrated in 1994 that the degree of polarization (DOP) of a partially coherent light beam may experience changes on propagation, even in free space . Later, Wolf proposed the unified theory of coherence and polarization . Based on this theory, the generalized Stokes parameters were defined and used to study the propagation of light beam [4–6]. Meanwhile, the changes of the DOP of the partially polarized light on propagation in free space were studied . Sufficient conditions were obtained for the DOP of a light beam to keep invariance throughout the far zone and across the source plane [8–10].
The researches of the light propagation mainly concentrate on the DOP recently. Studies of the state of polarization (SOP) are restricted to the expression for the SOP of the partially polarized light and the propagations of some particular examples . According to the author’s knowledge, it has not been studied profoundly yet for the condition of polarization state invariance. Taking the Gaussian Schell-model (GSM), we calculate three normalized Stokes parameters based on the unified theory of coherence and polarization. Then the condition of polarization state invariance is obtained by theoretical analysis.
2. Theoretical derivation
Considering a stochastic, statistically stationary, electromagnetic beam generated by a source located in the plane z=0 (called the source plane), propagating close to the z direction and into the half-space z>0. The coherence and polarization properties of this beam can be characterized by the 2×2 cross-spectral density matrix (CSDM) :
where the asterisk denotes the complex conjugate and the angular brackets mean the ensemble average. x and y are two mutually orthogonal directions perpendicular to the beam axis. The elements of CSDM in the plane z=constant>0 can be obtained by the formula :
The Stokes parameters, which include the information of both the DOP and the SOP, can be expressed in terms of the elements of the CSDM by the formulas :
Then, the degree of polarization can be expressed by 
where Det and Tr denote the determinant and trace of the matrix, respectively. It is helpful to define the normalized Stokes parameters as:
Then the SOP can be expressed as a point s′1,s′2,s′3 in the Poincaré sphere . If the three normalized Stokes parameters don’t change, the SOP will keep invariance.
For the GSM beam, the CSDM in the source plane z=0 has elements :
The parameters Ai, Bij, σi, δij are independent of position and they satisfy the expressions :
Assuming σ x=σ y=σ, the normalized Stokes parameters and the SOP in the source plane can be expressed as
The elements of CSDM at a pair of points in any cross-sectional plane z=constant>0 are given by the expression 
In expressions above, k=2π/λ is the wave number of light beam. When the two points coincide, Eq. (12) simplifies to
Then the normalized Stokes parameters and the DOP in the plane z=constant>0 can be expressed as
If the normalized Stokes parameters satisfy
the SOP doesn’t change on propagation. Solving equations above, it is obtained
From the derivation above, we conclude that when the parameters of GSM beam satisfy
the normalized Stokes parameters will keep unchanged along the distance. So Eq. (19) is the sufficient condition for GSM beam to maintain the SOP on propagation in free space. Under this condition, there is no polarization state variation induced by the degree coherence and the spectral density. The use of light source under this condition is helpful for the simplification of polarization detection in CPolSK modulation system.
3. Analysis and discussion
In this section, propagations of GSM beams with different parameters are calculated. Changes of the SOP and the DOP along the distance z are obtained and the condition of polarization state invariance is exemplified.
For a GSM beam, the spectral correlation widths δij must correspond the realizable conditions as 
Considering there are three variables involved in Eq. (18), we select two situations with different beam parameters. For the first situation, we suppose δxx=δyy and change the value of cross spectral correlation width δxy. Variations of normalized Stokes parameters and the DOP along with propagation distance are calculated. In situation 2, δxx=δxy are assigned and δyy is changed. The results of two situations are shown in Fig. 1 and Fig. 2, respectively.
Figure 1 shows the behaviors of three normalized Stokes parameters and the DOP as functions of the distance z along the z-axis (ρ=0) in situation 1. When the parameters of source satisfy Eq. (19) such as beam (A), the normalized Stokes parameters and DOP keep unchanged during the propagation. For the other beams, s′2,s′3 and DOP increase with the distance z. The significant differences between δxy and δxx lead to quicker increments. It should be noted that curves of five beams’ s′1 are superposed and only one straight line can be seen in Fig. 1 (a). That is because when δxx=δyy the normalized Stokes parameter s′1 in Eq. (15) can be simplified as
Obviously, the result is independent of z and δxy.
The changes of both SOP and DOP of GSM beams along the z-axis in situation 2 are shown in Fig. 2. Satisfying the condition of polarization state invariance, both the SOP and the DOP of beam (A) don’t change on propagation. For the other beams, all the three normalized Stokes parameters and the DOP increase along the distance z.
The results show that the SOP and DOP keep unchanged for the beam satisfied Eq. (19). For the other beams, the normalized Stokes parameters and DOP increase along the distance z and they ultimately tend to certain values. These values are determined by parameters of GSM beams. Significant differences between the three spectral correlation widths induce noticeable differences between ultimate values and initial values. Because we have supposed two of the three spectral correlation widths are equal in calculations, results show monotone increasing trends of normalized Stokes parameters and DOP. If the three spectral correlation widths are mutually unequal to each other, downward trends can be obtained, such as some examples in .
In the polarization modulation schemes such as CPolSK, the changes of polarization state on propagation induce the increase of bit error rate. If we design the light source according with Eq. (19), there will be no polarization state change even through long distance propagation. Consequently, the reliability of the polarization modulation system can be greatly improved. The results are helpful for the wide extensive use of the free space optical communication system in telecommunication links.
Based on the unified theory of coherence and polarization, we derive the sufficient condition for GSM beam to maintain the SOP on propagation in free space. The condition states that the three spectral correlation widths equal to each other and σx=σy. For GSM beams satisfied this condition, both the SOP and the DOP keep unchanged on propagation. If the condition is not satisfied, normalized Stokes parameters and the DOP experience variations and ultimately tend to certain values. These results are useful for the design of light source in the free space optical communication system with polarization modulation.
This work is financially supported by Guangdong Natural Science Foundation (8151805707000004) and Development Program for Outstanding Young Teachers in Harbin Institute of Technology (HITQNJS.2008.60). The authors appreciate the help of Key Laboratory of Network Oriented Intelligent Computation, HIT.
References and links
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