Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams

Chaoliang Ding¹, Liuzhan Pan^2,3 and Baida Lü¹

¹ Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, People's Republic of China
² Department of Physics, Luoyang Normal College, Luoyang 471022, People's Republic of China

³ Author to whom any correspondence should be addressed.

E-mail: panliuzhan@263.net

Received 12 May 2009
Published 4 August 2009

Contents

• 1. Introduction
• 2. Theoretical formulation
• 3. Some special cases
  • 3.1. Stochastic stationary electromagnetic beams
  • 3.2. Spectrally fully coherent electromagnetic pulsed beams
  • 3.3. Spatially fully coherent electromagnetic pulsed beams
• 4. Illustrative examples
• 5. Conclusion
• Acknowledgments
• References

1. Introduction

The coherence and polarization of light beams are two aspects of statistical optics that can be treated in a unified manner. In 2003, Wolf proposed the unified theory of coherence and polarization of stochastic electromagnetic beams [1]–[3]. Since then, a lot of work has been done on the propagation of stochastic electromagnetic beams in free space [4], through turbulent atmosphere [5], through a gradient-index fiber [6] and through various complex optical systems [7]–[9]. The theoretical predictions were confirmed experimentally [10, 11]. However, all the above investigations have been restricted to stationary beams.

On the other hand, statistical optical pulses represent a wide class of partially coherent fields [12]. Recently, a scalar model of partially coherent pulses, in which the correlation between different frequency components is taken into consideration, was introduced by Pääkkönen et al  [13]. Lajunen et al  [14] described the coherent-mode representation for spatially and spectrally partially coherent Gaussian Schell-model scalar pulses. In this paper, taking the stochastic electromagnetic Gaussian Schell-model pulsed (GSMP) beam as a typical example of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams, we consider the characterization of stochastic electromagnetic GSMP beams. It is actually a generalization of Wolf's unified theory from stochastic stationary electromagnetic beams to stochastic spatially and spectrally partially coherent electromagnetic pulsed beams. In section 2, the expression for the cross-spectral density matrix of stochastic electromagnetic GSMP beams propagating in free space is derived, and used to formulate the spectral density (spectrum), the spectral degree of polarization and the spectral degree of coherence of stochastic electromagnetic GSMP beams. Some special cases are given in section 3. In section 4, numerical calculation examples are presented to illustrate the changes in the spectral density, the spectral degree of polarization and the spectral degree of coherence of stochastic electromagnetic GSMP beams propagating in free space. Section 5 concludes the main results obtained in this paper.

2. Theoretical formulation

In the space-time domain, the electric mutual coherence matrix of a stochastic spatially and temporally partially coherent electromagnetic pulse at the source plane z = 0 is given by

where E_i and E_j are the polarization components of the electric field at the plane z = 0, ρ_i(j) = (x_i(j), y_i(j)) is the position vector, t₁₍₂₎ is the time. The asterisk denotes the complex conjugate and the angular brackets denote the ensemble average. can be expressed as

where spectral densities and the degree of coherence between the i and j polarization components of the electric vector are given by [13]

where the coefficients A_i, B_ij and the variables σ_i, δ_ij are independent of position but may depend on the frequency. δ_ij 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. T₀ is the pulse duration. T_c describes the temporal coherence length of the pulse, which denotes the temporal correlation of the pulse. ω₀ 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 T₀, temporal coherence length T_c, spectral width Ω₀ and spectral coherence width Ω_c. The spectral coherence width Ω_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→∞(Ω_c→∞). If δ_ij→∞, we obtain spatially fully coherent electromagnetic pulsed beams. In the limit T₀→∞ (Ω_c = 0, ) all frequency components become uncorrelated, thus we obtain stochastic stationary electromagnetic beams.

To simplify the analysis we take [3]

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]

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 T₀, temporal coherence length T_c, spatial correlation length δ_jj, coefficients A_x, A_y and propagation distance z.

3. Some special cases

3.1. Stochastic stationary electromagnetic beams

Letting the pulse duration T₀→∞, from equations (8) and (9) we obtain Ω_c = 0 and . 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

where

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 , equations (31)–(33) are consistent with equations (2)–(4) in [2].

3.2. Spectrally fully coherent electromagnetic pulsed beams

Letting the temporal coherence length T_c→∞ (i.e. Ω_c→∞), the electric mutual coherence matrix of spectrally fully coherent electromagnetic GSMP beams at the plane z = 0 simplifies to

where

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

3.3. Spatially fully coherent electromagnetic pulsed beams

Letting the spatial correlation length δ_jj→∞, the electric mutual coherence matrix of spatially fully coherent electromagnetic pulsed beams at the plane z = 0 becomes

where

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

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 , Ω ₀ = 1/T₀ and , 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_cj = T_c. However, it can be shown that for the case of T_ci≠T_cj [18] the spectral degree of polarization depends on T_ci (T_cj).

4. 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/ω₀ of stochastic spatially and spectrally partially coherent electromagnetic pulses versus the propagation distance z for different values of the pulse duration T₀ = 3 fs, 5 fs and ∞. The calculation parameters are T_c = 7 fs, σ = 1 mm, δ_xx = 1 mm, δ_yy = 2δ_xx, A_y = 1, P⁰ = 0.2 and ω₀ = 2.36 rad fs^–1. The relative spectral shift δω/ω₀ is defined as δω/ω₀ = (ω_max–ω₀)/ω₀, 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 T₀ = 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/ω₀ is given by

On substituting T₀ = 3 fs, 5 fs and ∞ into equation (53) with T_c = 7 fs and ω₀ = 2.36 rad fs^–1, we obtain δω/ω₀ = 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/ω₀ 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 T₀ = 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/ω₀ 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 T₀ = 5 fs and ω₀ = 2.36 rad fs^-1 yields δω/ω₀ = 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/ω₀, and the color-coded plot corresponding to figure 2(a) is shown in figure 2(b). The calculation parameters are δ_xx = 1 mm, δ_yy = 2δ_xx, A_y = 1, σ = 1 mm, P⁰ = 0.2, T₀ = 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.

By letting on-axis spectral degree of polarization P = const, from equation (23) we obtain

If P = 0, we have

where

Equations (54) and (55) indicate that z is a linear function of ω, provided that σ and δ_jj are independent of the frequency, e.g. σ = 1 mm, δ_xx = 1 mm and δ_yy = 2 mm.

The variation of spectral degree of coherence |η(x,–x, z, ω₁, ω₂)| of stochastic spatially and spectrally partially coherent electromagnetic pulses versus the transverse coordinate x for different values of the spatial correlation length δ_xx = 1, 1.5 and 2 mm is plotted in figure 3(a). The calculation parameters are δ_yy = 2δ_xx, z = 1 m, y₁ = y₂ = 0, A_y = 1, P⁰ = 0.2, σ = 1 mm, T₀ = 5 fs, T_c = 7 fs, ω₁/ω₀ = 1.2 and ω₂/ω₀ = 0.8. It is seen that |η(x,–x, z, ω₁, ω₂)| decreases with increasing x, and δ_xx affects the distribution of |η(x,–x, z, ω₁, ω₂)| which has the same value for different δ_xx at x = 0 and for a fixed value x |η(x,–x, z, ω₁, ω₂)| increases with increasing spatial correlation length δ_xx. Therefore, the width at half maximum of |η(x,–x, z, ω₁, ω₂)| increases with an increase of δ_xx.

Figure 3(b) gives the spectral degree of coherence |η(x,–x, z, ω₁, ω₂)| of the stochastic spatially and spectrally partially coherent electromagnetic pulses as a function of the propagation distance z for different values of the temporal coherence length T_c = 3 fs, 7 fs, ∞ and x = 1 mm. The other calculation parameters are the same as those in figure 3(a). As can be seen, |η(x,–x, z, ω₁, ω₂)| increases with increasing z and approaches an asymptotic value, which increases with increasing T_c. The spectral degree of coherence |η(x,–x, z, ω₁, ω₂)| at a pair of points (x, z, ω₁) and (–x, z, ω₂) is expressed as

where

Equations (8), (9), (58)–(62) indicate that for fixed σ, δ_jj, z, ω₁, ω₂, A_x, A_y and T₀,|η(x,–x, z, ω₁, ω₂)| increases with an increase of T_c.

5. 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 T₀→∞, T_c→∞ and δ_jj→∞, 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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant no. 10874125 and the Program for New Century Excellent Talent in University of Henan Province No. 2006HANCET-09.

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