Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams

The unified theory of coherence and polarization proposed by Wolf is extended from stochastic stationary electromagnetic beams to stochastic spatially and spectrally partially coherent electromagnetic pulsed beams. Taking the stochastic electromagnetic Gaussian Schell-model pulsed (GSMP) beam as a typical example of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams, the expressions for the spectral density, spectral degree of polarization and spectral degree of coherence of stochastic electromagnetic GSMP beams propagating in free space are derived. Some special cases are analyzed. The illustrative examples are given and the results are interpreted physically.

where the coefficients A i , B i j and the variables σ i , δ i j are independent of position but may depend on the frequency. δ i j is related to the spatial correlation length, which represents the spatial correlation between the i and j components of the electric field vector in the source plane. T 0 is the pulse duration. T c describes the temporal coherence length of the pulse, which denotes the temporal correlation of the pulse. ω 0 is the carrier frequency of the pulse. By using the Fourier-transform we can derive the cross-spectral density matrix at the plane z = 0 where Equations (8) and (9) give the relation of the pulse duration T 0 , temporal coherence length T c , spectral width 0 and spectral coherence width c . The spectral coherence width c is a measure of the correlation between different frequency components of the pulse [13]. It is obvious from equations (7)-(9) that the spectrally fully coherent electromagnetic pulsed beams are obtained in the limit T c → ∞ ( c → ∞). If δ i j → ∞, we obtain spatially fully coherent electromagnetic pulsed beams. In the limit T 0 → ∞ ( c = 0, T c = √ 2/ 0 ) all frequency components become uncorrelated, thus we obtain stochastic stationary electromagnetic beams.
To simplify the analysis we take The elements of the cross-spectral density matrix of stochastic electromagnetic GSMP beams at the plane z = 0 are given by Therefore, the spectral density, the spectral degree of polarization and the spectral degree of coherence of stochastic electromagnetic GSMP beams at the plane z = 0 are expressed as [1] The cross-spectral density matrix of stochastic electromagnetic GSMP beams at the plane z > 0 in the free-space propagation is expressed as [15] W ↔ (r 1 , r 2 , z, ω 1 , ω 2 ) = ω 1 ω 2 5 The substitution of equations (12) and (13) into equation (17) yields where In accordance with the unified theory [1], the spectral density, the spectral degree of polarization and the spectral degree of coherence of stochastic electromagnetic GSMP beams at the plane z > 0 are given by where Note that the definition of the spectral degree of coherence is not unique in the literature. Another definition is found, for example, in [16].
Equations (14)- (16) and (22)-(24) are the main analytical results obtained in this paper, and describe the changes in the spectral density, the spectral degree of polarization and the spectral degree of coherence of stochastic electromagnetic GSMP beams from the z = 0 plane to the z-plane in free space, which depend on the pulse duration T 0 , temporal coherence length T c , spatial correlation length δ j j , coefficients A x , A y and propagation distance z.

Stochastic stationary electromagnetic beams
Letting the pulse duration T 0 → ∞, from equations (8) and (9) we obtain c = 0 and T c = √ 2/ 0 . As a result, the spectral components are completely uncorrelated and the mutual coherence matrix of stochastic stationary electromagnetic beams at the plane z = 0 reduces to On substituting equation (27) into equation (5), the cross-spectral density matrix of stochastic stationary electromagnetic beams is expressed as where with δ(·) being the Dirac delta function. Substituting equation (29) into equation (17) and making use of equation (11), the spectral density, the spectral degree of polarization and the spectral degree of coherence of stochastic stationary electromagnetic beams at the plane z > 0 are given by where 0 = √ 2/T c , equations (31)-(33) are consistent with equations (2)-(4) in [2].

Spectrally fully coherent electromagnetic pulsed beams
Letting the temporal coherence length T c → ∞ (i.e. c → ∞), the electric mutual coherence matrix of spectrally fully coherent electromagnetic GSMP beams at the plane z = 0 simplifies to From equations (5) and (34) the cross-spectral density matrix of spectrally fully coherent electromagnetic pulsed beams is given by where Substituting equation (36) into equation (17) and using equation (11), the spectral density, the spectral degree of polarization and the spectral degree of coherence of spectrally fully coherent electromagnetic pulsed beams at the plane z > 0 are expressed as where

Spatially fully coherent electromagnetic pulsed beams
Letting the spatial correlation length δ j j → ∞, the electric mutual coherence matrix of spatially fully coherent electromagnetic pulsed beams at the plane z = 0 becomes The cross-spectral density matrix of spatially fully coherent electromagnetic pulsed beams reads as where The spectral density, the spectral degree of polarization and the spectral degree of coherence of spatially fully coherent electromagnetic pulsed beams at the plane z > 0 are given by where = 1 + Equation (47) implies the invariance of polarization of spatially fully coherent electromagnetic pulses in the free-space propagation [17].
Equations (31) and (38) are formally the same as equation (22), but the fields are physically different because of the different spectral widths 0 = √ 2/T c , 0 = 1/T 0 and 0 = 1/T 2 0 + 2/T 2 c , respectively. Equations (32) and (39) are formally the same as equation (23), which implies that the temporal coherence length does not affect the spectral degree of polarization of stochastic electromagnetic GSMP beams provided that the temporal coherence length of the i component of the electric vector are the same as that of the j component of the electric vector, i.e. T ci = T c j = T c . However, it can be shown that for the case of T ci = T c j [18] the spectral degree of polarization depends on T ci (T c j ).

Illustrative examples
To illustrate the applications of the theory, numerical calculation results for stochastic spatially and spectrally partially coherent electromagnetic pulses propagating in free space are presented. Figure 1(a) shows the on-axis relative spectral shift δω/ω 0 of stochastic spatially and spectrally partially coherent electromagnetic pulses versus the propagation distance z for different values of the pulse duration T 0 = 3 fs, 5 fs and ∞. The calculation parameters are T c = 7 fs, σ = 1 mm, δ x x = 1 mm, δ yy = 2δ x x , A y = 1, P 0 = 0.2 and ω 0 = 2.36 rad fs −1 . The relative spectral shift δω/ω 0 is defined as δω/ω 0 = (ω max − ω 0 )/ω 0 , where ω max denotes the frequency at which the spectral density S(r, z, ω) takes the maximum value. It is shown that the spectrum is blue-shifted in free-space propagation. With increasing propagation distance z, the blue-shift increases and approaches an asymptotic value when z is large enough. For example, depending on the pulse duration, the asymptotic value is equal to 0.651, 0.407 and 0.234 for T 0 = 3 fs, 5 fs and ∞, respectively. The results can be interpreted as follows.
For large enough z, equations (25) and (26) can be approximately expressed as where Thus, the on-axis spectral density is written as By letting [∂S(0, z, ω)/∂ω] = 0, the on-axis relative spectral shift δω/ω 0 is given by On substituting T 0 = 3 fs, 5 fs and ∞ into equation (53) with T c = 7 fs and ω 0 = 2.36 rad fs −1 , we obtain δω/ω 0 = 0.651, 0.407 and 0.234, respectively, which are consistent with the above results in figure 1(a). Figure 1(b) represents the on-axis relative spectral shift δω/ω 0 of stochastic spatially and spectrally partially coherent electromagnetic pulses versus the propagation distance z for different values of the temporal coherence length T c = 5 fs, 7 fs, ∞ and T 0 = 5 fs. The other calculation parameters are the same as those in figure 1(a). As can be seen, with increasing z the relative spectral shift δω/ω 0 approaches asymptotic values 0.549, 0.407 and 0.230 for T c = 5 fs, 7 fs and ∞, respectively. The physical explanation of figure 1(b) is similar to figure 1(a), because the substitution from T c = 5 fs, 7 fs and ∞ into equation (53) with T 0 = 5 fs and ω 0 = 2.36 rad fs −1 yields δω/ω 0 = 0.549, 0.407 and 0.230, respectively. Figure 2(a) shows the changes of the on-axis spectral degree of polarization P of stochastic spatially and spectrally partially coherent electromagnetic pulses as a function of the propagation distance z and frequency ω/ω 0 , and the color-coded plot corresponding to figure 2(a) is shown in figure 2(b). The calculation parameters are δ x x = 1 mm, δ yy = 2δ x x , A y = 1, σ = 1 mm, P 0 = 0.2, T 0 = 5 fs and T c = 7 fs. It is seen that the frequency ω influences the distribution of on-axis spectral degree of polarization P of stochastic spatially and spectrally partially coherent electromagnetic pulses, and the position of the on-axis spectral degree of polarization P = const increases linearly with increasing ω, the physical reason is as follows.

Conclusion
In this paper, the unified theory of coherence and polarization proposed by Wolf has been extended from stochastic stationary electromagnetic beams to stochastic spatially and spectrally partially coherent electromagnetic pulsed beams. Taking the stochastic electromagnetic GSMP beam as a typical example of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams, the analytical expression for cross-spectral density matrix has been derived, and used to formulate the spectral density, spectral degree of polarization and spectral degree of coherence of stochastic electromagnetic GSMP beams propagating in free space. The stationary electromagnetic beams, spectrally fully coherent electromagnetic pulsed beams and spatially fully coherent electromagnetic pulsed beams can be regarded as special cases of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams by letting T 0 → ∞, T c → ∞ and δ j j → ∞, respectively. Numerical calculation examples have been presented to illustrate the applications of the theory. Finally, we would like to point out that, although the theoretical formulation has been made for stochastic electromagnetic GSMP beams to clarify the main physical aspects, the extension to other types of stochastic electromagnetic pulsed beams, for example, of stochastic electromagnetic cosh-Gaussian, Bessel-Gauss pulsed beams, etc are straightforward. Therefore, the results obtained in this paper would be useful for the study of more general types of stochastic electromagnetic pulsed beams in a unique way.