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Abstract
We present a class of broadband electromagnetic Gaussian Schell-model sources whose state of polarization is both uniform and identical for all frequencies, but whose far-zone polarization properties strongly depend on wavelength. Also, these sources can produce beams whose polarized portion is always linearly polarized but with a polarization angle that evolves on propagation. Our results offer new insights into the behavior of broadband partially coherent sources.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
One of the key insights of coherence theory is that field properties such as directionality, spectrum and degree of polarization are dependent on the statistical properties of the source [1]. In many researches Gaussian Schell-model (GSM) sources, either scalar or electromagnetic (EGSM), are used to illustrate these findings. These are planar, secondary sources whose spectral density and spectral degree of coherence both have a Gaussian form [2]. GSM sources can be divided into two categories. The first one consists of sources with an rms width of the spectral density (denoted $\sigma$) and an rms width of the spectral degree of coherence (denoted $\delta$) that are independent of the wavelength. Even in this restricted case a great variety of coherence-related phenomena like the Wolf shift [3] and a propagation-dependent state of polarization [4,5] can occur. The second category is formed by so-called spectral GSM sources for which $\sigma$ and $\delta$ explicitely depend on frequency [6–8]. (Non-Gaussian spectral sources were discussed in [9–12]). It is interesting to note that the studies dealing with spectral EGSM sources consider their behavior at only a single frequency. An exception is [13] in which the propagation of an EGSM beam through $2f$ and $4f$ systems is described.
In this study we present a new class of broadband spectral EGSM sources whose state of polarization is both uniform and the same at every frequency that is present in the source spectrum. However, as will be shown, their far-zone Stokes parameters strongly depend on wavelength. In particular, we investigate beam-generating sources and derive expressions for their on-axis state of polarization. It is found that these sources can produce beams whose polarized part on the axis is always linearly polarized, but with an angle that evolves differently for different frequencies.
In principle, all EGSM parameters can be taken to be frequency-dependent, with each one having a different functional dependence on the source frequency. In order to provide physical insight, we have chosen the simplest system that provides interesting polarization effects. In such a system only the coherence radius $\delta _{xy}$, defined below, needs to vary with wavelength. Furthermore, it will be seen that this parameter plays a central role in the so-called realizability conditions that the source must satisfy.
2. Electromagnetic Gausian Schell-model sources
A planar, secondary EGSM source (see Fig. 1) is described by a cross-spectral density matrix [2, Sec. 9.4.2]
Here ${\boldsymbol \rho }_1$ and ${\boldsymbol \rho }_2$ are two-dimensional position vectors in the source plane $z=0$, $S_i^{(0)}({\boldsymbol \rho };\omega )$ denotes the spectral density of the $i^\textrm {th}$ component of the electric field vector $\textbf {E}$ at frequency $\omega$, and $\mu ^{(0)}_{ij} ({\boldsymbol \rho }_2 - {\boldsymbol \rho }_1;\omega )$ represents the degree of correlation between $E_i$ at ${\boldsymbol \rho }_1$ and $E_j$ at ${\boldsymbol \rho }_2$. The superscript $(0)$ indicates quantities in the source plane. The spectral densities and the degrees of correlation are assumed to have a Gaussian form, i.e.,
As our notation indicates, the spectral amplitudes $A_i$ and the four coherence widths (or coherence radii) $\delta _{ij}$ are assumed to depend on the wavelength, but the rms widhts $\sigma _i$ and the correlation coefficient $B_{ij}$ do not. Furthermore, we take $\sigma _x= \sigma _y = \sigma$, and $A_x(\omega ) = A_y(\omega ) = A(\omega )$. The source parameters must satisfy certain constraints [2], namely
Additionally, the realizability conditions are [14],
We note that expression (5) is implied by (9). An EGSM source will produce a beam-like field if the inequality [15]
Fig. 1. A spectral electromagnetic, planar, secondary Gaussian Schell-model source occupies the plane $z=0$ and radiates a beam that propagates parallel to the $z$ axis. The vector ${\boldsymbol \rho }$ denotes a position in the $xy$ plane.
As the beam propagates to a transverse plane $z$, the cross-spectral density matrix at two coincident points on the axis $({\boldsymbol \rho }_1 = {\boldsymbol \rho }_2 = \textbf {0})$ evolves into [2, p. 184]
where and with the wavenumber $k=\omega /c$. As explained by Wolf [2], the matrix elements expand at different rates. This results in a change of the state of polarization as the beam propagates. Furthermore, this change can become quite complex when the source parameters are allowed to vary with frequency.The state of polarization (SOP) of an electromagnetic beam is characterized by the four spectral Stokes parameters, which can be expressed in terms of the cross-spectral density matrix as [2]
Their normalized versions are defined as $s_n({\boldsymbol \rho }; \omega ) = S_n({\boldsymbol \rho }; \omega )/S_0({\boldsymbol \rho }; \omega )$ for $n=1,2,3$. On making use of Eqs. (1)–(3), together with the previously mentioned assumptions $\sigma _x= \sigma _y = \sigma$, and $A_x(\omega ) = A_y(\omega ) = A(\omega )$, its readily found that in the source plane these parameters take on the form
We note that, since $B_{xy}$ is assumed to be constant, this implies that the SOP of the field in the plane $z=0$ is both uniform and identical at all frequencies at which the source radiates. The same holds true for the degree of polarization (DOP), which is defined as
In the far-zone, indicated by the superscript $(\infty )$, the first term on the right-hand side of Eq. (12) may be neglected, and the normalized Stokes parameters on the beam axis $({\boldsymbol \rho }= \textbf {0})$ are seen to be
Hence the far-zone, on-axis DOP equals
Let us first consider the case $\delta _{xx}(\omega )=\delta _{yy}(\omega )=\delta (\omega )$. It follows from the realizability condition (8) that then $\delta _{xy} (\omega )\ge \delta (\omega )$, and hence
On making use of (27) in Eqs. (23)–(26) we find that
In words, the SOP of the field has changed on propagation to the far-zone: for every frequency that is present in the source spectrum, the far-zone DOP is greater or equal than the DOP in the source plane.
An example is shown in Fig. 2 in which the parameters $s_2^{(\infty )}( \omega )$, $s_3^{(\infty )}( \omega )$ and the far zone DOP are plotted as a function of $\delta _{xy}$. It is seen that if the coherence radius $\delta _{xy}$ increases then so do the two normalized Stokes parameters, and hence the degree of polarization.
Fig. 2. The two non-zero normalized Stokes parameters in the far zone, $s_2^{(\infty )}$ (blue curve) and $s_3^{(\infty )}$ (green), together with the far-zone DOP $P^{(\infty )}$ (red) as a function of the coherence radius $\delta _{xy}$. When we set $\delta _{xx}=\delta _{yy}=2$ mm, $\sigma =1$ cm, and $B_{xy}=0.2+\textrm {i}0.25$, the parameter $\delta _{xy}$ is, according to Eq. (8), restricted to the interval $[2.0,3.5]$ mm.
When $\delta _{xx}(\omega ) \neq \delta _{yy}(\omega )$, the inequalities (29)–(31) are still valid. The proof of this is straightforward but somewhat tedious and is not presented here. It is readily seen from Eq. (23) that in this case $s_1^{(\infty )}( \omega ) \neq 0$. However, this quantity does not depend on the coherence radius $\delta _{xy}$. An example is presented in Fig. 3. It is again seen that the DOP increases with incresasing $\delta _{xy}$. This tendency can be understood by considering the quasi-homogeneous limit of $\sigma ^2 \gg \delta _{ij}^2$ for all $i,j=x,y$ [2]. In that case $\Omega ^2_{ij} \approx \delta _{ij}^2$, and both $s_2^{(\infty )}( \omega )$ and $s_3^{(\infty )}( \omega )$ are seen to increase with increasing $\delta _{xy}$. Furthermore, it then follows that if $\delta _{xy}$ nears its maximum value as given by Eq. (8), then $P^{(\infty )}(\omega )$ tends to $1$. Although the sources in Figs. 2 and 3 are stricly speaking not quasi-homogeneous, their far-zone DOPs do display this behavior. Note that at optical frequencies $(\sim 10^{15}\,\textrm {Hz})$ these sources satisfy the beam condition (10).
Fig. 3. The three normalized Stokes parameters in the far zone: $s_1^{(\infty )}$ (dashed black curve), $s_2^{(\infty )}$ (blue) and $s_3^{(\infty )}$ (green), together with the far-zone DOP $P^{(\infty )}$ (red) as a function of the coherence radius $\delta _{xy}$. For $\delta _{xx}=3.5$ mm, $\delta _{yy}=2$ mm, $\sigma =1$ cm, and $B_{xy}=0.2+\textrm {i}0.25$, the parameter $\delta _{xy}$ is, according to Eq. (8), restricted to the interval $[2.8,4.6]$ mm.
3. Spectral behavior of the degree of polarization
It follows from Eqs. (23)–(26) that in general the state of polarization of the far-zone field is wavelength dependent, even though this is not the case for the source field [as evidenced by Eqs. (18)–(22)]. We illustrate this spectral dependence of the SOP with several examples.
We begin with the case $\delta _{xx}=\delta _{yy}= C$, with $C$ a constant that does not depend on frequency. Furthermore, it is assumed that $\delta _{xy}$ has a frequency dependence given by
An example of the frequency dependence of the far-zone DOP is shown in Fig. 4. It is seen that, eventhough the degree of polarization of the source $P^{(0)}$ does not depend on frequency, $P^{(\infty )}$ strongly varies with wavelength. Near $\omega =\omega _0$ the beam is highly polarized, at other frequencies it is weakly polarized with $P^{(\infty )}< 0.4$. However, at all frequencies $P^{(\infty )} \ge P^{(0)}$.
Fig. 4. The on-axis, far-zone DOP $P^{(\infty )}(\omega )$ as a function of frequency. In this example $\delta _{xx}=\delta _{yy}=2$ mm, $\sigma =1$ cm, $B_{xy}=0.2+\textrm {i}0.25$. The frequency-dependent parameter $\delta _{xy}$ is given by Eq. (32), with $D= 1.5$ mm, $\omega _0=1.0 \times 10^{15} \textrm {s}^{-1}$, and $F = 0.05 \times 10^{15}\textrm {s}^{-1}$. The dashed line indicates $P^{(0)}$, the degree of polarization of the source.
We next consider the case that $\delta _{xx}$ and $\delta _{yy}$ are both constants, but not necessarily equal. Also, $\delta _{xy}$ is still given by (32), but with $C$ replaced by the left-most term of (8). An example is shown in Fig. 5. Although $s_1^{(\infty )}$ is independent of frequency, it is, unlike its counterpart in the source plane, non-zero. The two other Stokes parameters and the far-zone DOP reach their maximum value at $\omega =\omega _0$.
Fig. 5. The far-zone DOP $P^{(\infty )}(\omega )$ (blue curve), and the Stokes parameters $s_1^{(\infty )}(\omega )$ (orange), $s_2^{(\infty )}(\omega )$ (red) and $s_3^{(\infty )}(\omega )$ (green) as a function of frequency. In this example $\delta _{xx}=3$ mm, $\delta _{yy}=2$ mm, $\sigma =1$ cm, $B_{xy}=0.2+\textrm {i}0.25$. The parameter $\delta _{xy}$ is given by Eq. (32), with $D= 1.5$ mm, $\omega _0=1.0 \times 10^{15} \,\textrm {s}^{-1}$, and $F = 0.05 \times 10^{15}\,\textrm {s}^{-1}$.
An interesting polarization effect occurs when $\textrm {Im}\,[B_{xy}]=0$. In that case $s_3^{(0)}( \omega ) = 0$, i.e., the polarized portion of the field in the source plane is linearly polarized. Since $s_1^{(0)} = 0$, and $s_2^{(0)} = \textrm {Re} [B_{xy}]$ [see Eqs. (18) and (19)], this polarization is under an angle of $45^\circ$ with the $x$ axis. As the beam propagates $s_3(\omega )$ remains zero. However, this is not so for $s_1 (\omega )$ which will, in general, become non-zero. This means that, although the polarized part of the field remains linearly polarized, the angle of vibration of the electric field $\phi (\omega )$, defined by [2, p. 170],
rotates on propagation to the far zone, in a way that depends on frequency. Moreover, we find that this behavior is non-trivial. By using Eq. (11) without applying the far-zone approximation we can track the polarization angle $\phi (\omega )$ as the beam propagates. This is shown in Fig. 6. Although the polarized part of the beam remains lineraly polarized, the direction of vibration of the electric field is seen to rotate on propagation, with an angle that strongly depends on frequency. Furthermore, the angle of polarization does not always change monotonically, e.g., it can first decrease and then increase again (green curve). Also, $\phi (\omega )$ at infinity can either be larger or smaller than in the source plane. We note that $\phi (\omega )$ is also the orientation angle for which a linear polarizer inserted in the beam yields a maximum transmission. Finally, also the evolution of the degree of polarization strongly depends on wavelength. This is illustrated in Fig. 7. The red curve, corresponding to frequency $\omega _0$, is seen to rise considerably faster than the blue curve, which represents $0.8 \omega _0$.Fig. 6. Evolution of the angle of polarization $\phi (\omega )$ of the (linearly) polarized part of the field as the beam propagates. In this example $\sigma = 1$ cm, $B_{xy} = 0.1$. $\delta _{xx}=2.0$ mm, $\delta _{yy}=3.5$ mm, (red and blue curves); and $\delta _{xx}=3.5$ mm, $\delta _{yy}=2.0$ mm, (green and purple curves). The parameter $\delta _{xy}$ is given by Eq. (32), with $C=2.85$ mm, $D= 5.5$ mm, $\omega _0=1.0 \times 10^{15}\, \textrm {s}^{-1}$, and $F = 0.05 \times 10^{15}\,\textrm {s}^{-1}$. The coherence radius $\delta _{xy}$ is evaluated at $\omega _0$ (red and green curves), and at $0.8 \omega _0$ (blue and purple curves).
Fig. 7. Evolution of the degree of polarization on propagation for two different frequencies. The red curve is for $\omega _0$, the blue curve is for $0.8 \omega _0$. All other parameters are as in Fig. 6.
4. Conclusions
Spectral electromagnetic Gaussian Schell-model sources are characterized by parameters that are frequency dependent. Remarkably, to the best of our knowledge, the wavelength dependence of their state of polarization has up till now not been examined. We have presented a new class of broadband spectral EGSM sources whose state of polarization is the same at every source point, and the same at every wavelength. However, the Stokes parameters of the far-zone beams that they produce are shown to strongly vary with frequency. In particular the on-axis degree of polarization can be quite low $(P^{(\infty )} < 0.4)$ at certain frequencies and very high $(P^{(\infty )} > 0.9)$ at others. Examples were presented of beams whose polarized part is always linearly polarized, but with an angle of polarization that rotates non-trivially on propagation. Again, the precise nature of this behavior was seen to be frequency dependent. We have shown that the behavior of broadband partially coherent sources is more complicated than previously known. These results are relevant for the design of such sources with prescribed spectral polarization properties.
Funding
National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11904211, 11974218, 91750201, 11525418); Innovation Group of Jinan (2018GXRC010); .
Disclosures
The authors declare no conflicts of interest.
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