Experimental generating the partially coherent and partially polarized electromagnetic source

Andrey S. Ostrovsky; Gustavo Rodríguez-Zurita; Cruz Meneses-Fabián; Miguel Á. Olvera-Santamaría; Carolina Rickenstorff-Parrao

doi:10.1364/OE.18.012864

1. Introduction

During the last decade the substantial efforts were made in the developing the vector coherence theory of electromagnetic fields (for a brief analysis of the state of art in this area see, e.g., Ref. 1). To realize any experimental investigation on this subject, one needs a genuine partially coherent and partially polarized electromagnetic source with known statistical properties. Nevertheless, the source commonly used in optical practice, is the gas laser, whose radiation is almost completely coherent and completely polarized. Thus, the problem of the controlled altering the coherence and polarization properties of the laser radiation arises. In Ref. 2 it was shown that such an alteration can be done by means of the computer controlled liquid crystal (LC) spatial light modulator (SLM). Somewhat later this technique has been improved with using two LC SLMs placed at the opposite arms of the Mach-Zehnder interferometer [3]. Recently we have proposed an alternative technique employing two crossed parallel aligned LC SLMs [4]. Unfortunately, the mentioned techniques have not been yet realized in practice because of the lack of commercial LC SLMs with the desired characteristics.

In the present paper we propose a very simple technique for modulating the coherence and polarization of laser radiation which does not need any LC SLM. This technique bears some resemblance with the one reported in Ref. 5, but differs from it as by another action principle so by the simplicity. Due to the last and the wide-scale availability of the required optical components we have succeeded in physical realization of the proposed technique and have demonstrated its efficiency in physical experiment.

Once the desired secondary partially coherent and partially polarized source has been created, it must be subject to experimental characterization. The idea of such a determination is well known [6–8], but its physical realization has not been yet reported. Here we propose also a rather simple technique for characterizing the coherence of electromagnetic source, which was used in our experiments.

2. Background

As well known, the second-order statistical properties of a random planar (primary or secondary) electromagnetic source can be completely described by the cross-spectral density matrix (for brevity we omit the explicit dependence of the considered quantities on frequency $ν$ ) given by the formula [9–11]

W (x_{1}, x_{2}) = [\begin{matrix} 〈 E_{x}^{*} (x_{1}) E_{x} (x_{2}) 〉 & 〈 E_{x}^{*} (x_{1}) E_{y} (x_{2}) 〉 \\ 〈 E_{y}^{*} (x_{1}) E_{x} (x_{2}) 〉 & 〈 E_{y}^{*} (x_{1}) E_{y} (x_{2}) 〉 \end{matrix}],

where

E_{x} (x)

and

E_{y} (x)

are the orthogonal components of the electric field vector

E (x),

asterisk denotes the complex conjugate, and the angle brackets denote the average over the statistical ensemble. Using this matrix, Wolf defines the following three fundamental statistical characteristics of the source:the power spectrum

S (x) = Tr W (x, x),

the spectral degree of coherence

μ (x_{1}, x_{2}) = \frac{Tr W (x_{1}, x_{2})}{{[Tr W (x_{1}, x_{1}) Tr W (x_{2}, x_{2})]}^{1 / 2}},

and the spectral degree of polarization

P (x) = {[1 - \frac{4 Det W (x, x)}{[Tr W (x, x)]^{2}}]}^{1 / 2} .

In Eqs. (2)–(4) Tr stands for the trace and Det denotes the determinant.

It is not out of place to mention here that, equally with Wolf’s definition of the degree of coherence, there are known the other definitions [12–14]. But, for our following analysis, the definition given by Eq. (3) proves to be quite sufficient.

3. Technique for generation

As a primary source (PS) we consider the single-mode gas laser whose radiation is linearly polarized and in some plane normal to the direction of propagation, under certain conditions, can be characterized by the cross-spectral density matrix

W_{PS} (x_{1}, x_{2}) = S_{0} \exp (- \frac{x_{1}^{2} + x_{2}^{2}}{4 ε^{2}}) [\begin{matrix} \cos^{2} θ & \sin θ \cos θ \\ \sin θ \cos θ & \sin^{2} θ \end{matrix}],

where

S_{0}

is the value of power spectrum at the origin of source plane, ε is the effective (rms) size of the source, and θ is the angle that the direction of the linear polarized electric field makes with the x axis. On substituting from Eq. (5) into Eqs. (3) and (4), it may be readily verified that in this case

| μ_{PS} (x_{1}, x_{2}) | = 1

and

P_{PS} (x) = 1,

i.e. that such a source is completely coherent and completely polarized. Let us assume that the radiation from this source passes through a Mach-Zehnder interferometer sketched schematically in Fig. 1 . The polarizing beam splitter PBS separates the orthogonal field components

E_{x} (x)

and

E_{y} (x)

so that each of them can be independently changed by means of two rotating ground glass plates GGP₁ and GGP₂ placed at the opposite arms of the interferometer.

Fig. 1 Schematic illustration of the technique for generating the partially coherent and partially polarized source: BS, beam splitter; PBS, polarizing beam splitter; M, mirror; GGP₁, GGP₂, rotating ground glass plates.

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Disregarding the negligible changes of the coherence and polarization properties of the electrical field induced by a free space propagation within the interferometer, one can represent the considering system as a thin polarization dependent phase screen with an amplitude transmittance given by the Jones matrix

T (x) = [\begin{matrix} \exp [i φ_{1} (x)] & 0 \\ 0 & \exp [i φ_{2} (x)] \end{matrix}],

where

φ_{1} (x)

and

φ_{2} (x)

are the position-dependent random variables associated with the roughness of the ground glass plates. Following Ref. 2, we assume that the random variables

φ_{1} (x)

and

φ_{2} (x)

possess the same Gaussian probability density

p [φ_{1 (2)} (x)] = \frac{1}{\sqrt{2 π} σ} exp [- \frac{φ_{1(2)}^{2} (x)}{{2σ}^{2}}]

and are Gauss-correlated for two different points

x_{1}

and

x_{2}

separated by a distance ξ, i.e.

〈 φ_{1 (2)} (x_{1}) φ_{1 (2)} (x_{2}) 〉 = σ^{2} exp (- \frac{ξ^{2}}{{2γ}^{2}}) .

Moreover, taking into account that both variables $φ_{1} (x)$ and $φ_{2} (x)$ are originated by two different ground glass plates, we can assume that they are statistically independent, i.e.

p [φ_{1} (x), φ_{2} (x)] = p [φ_{1} (x)] p [φ_{2} (x)] .

The cross-spectral density matrix of the secondary source (SS) created just behind the interferometer can be calculated as follows:

W_{SS} (x_{1}, x_{2}) = 〈 T^{†} (x_{1}) W_{PS} (x_{1}, x_{2}) T (x_{2}) 〉,

where the dagger denotes the Hermitian conjugation. On substituting from Eqs. (5) and (6) into Eq. (10) and making use of Eqs. (A8) and (A9), we find

W_{SS} (x_{1}, x_{2}) = S_{0} e x p (- \frac{x_{1}^{2} + x_{2}^{2}}{4 ε^{2}})

\times [\begin{matrix} exp {- σ^{2} [1 - exp (- \frac{ξ^{2}}{{2γ}^{2}})]} \cos^{2} θ & exp (- σ^{2}) \sin θ \cos θ \\ exp (- σ^{2}) \sin θ \cos θ & exp {- σ^{2} [1 - exp (- \frac{ξ^{2}}{{2γ}^{2}})]} \sin^{2} θ \end{matrix}] .

We note here that for a ground glass plate the root-mean-square phase retardation σ is much greater than $2 π$ rad. Hence, the following approximations can be accepted (see, e.g., Ref. 2):

exp (- σ^{2}) \approx 0

and

exp {- σ^{2} [1 - exp (- \frac{ξ^{2}}{{2γ}^{2}})]} \approx \exp (- \frac{ξ^{2}}{2 η^{2}}),

where

η = γ / σ

. These approximations reduce Eq. (11) to

W_{SS} (x_{1}, x_{2}) = S_{0} \exp (- \frac{x_{1}^{2} + x_{2}^{2}}{4 ε^{2}}) \exp (- \frac{ξ^{2}}{2 η^{2}}) [\begin{matrix} \cos^{2} θ & 0 \\ 0 & \sin^{2} θ \end{matrix}] .

Then substituting this result into definitions given by Eqs. (3) and (4), we obtain respectively,

μ_{SS} (x_{1}, x_{2}) = \exp (- \frac{ξ^{2}}{2 η^{2}}),

P_{SS} (x) = | 1 - 2 c o s^{2} θ | .

Equations (15) and (16) show that the generated source is partially coherent and partially polarized. The transverse coherence length $η$ of this source depends on the diffusion properties of the used ground glass plates characterized by parameter γ, and its degree of polarization can change in the range from 1 to 0 with a proper choice of polarization angle θ.

4. Technique for characterization

Once the desired secondary partially coherent and partially polarized source has been created, it must be subject to experimental characterization, i.e. the elements $W_{i j}^{SS}$ of the matrix $W_{SS}$ have to be experimentally determined. Taking into account the symmetry of the cross-spectral density given by Eq. (11), for this purpose the technique sketched schematically in Fig. 2 can be used.

Fig. 2 Schematic illustration of the technique for characterizing the generated secondary source: BS, beam splitter; M, mirror; TP, translating pinhole; P₁, P₂, polarizers; R₁, R₂; polarization rotators.

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This technique represents a modified version of well known two-pinhole Young’s experiment. The Mach-Zehnder interferometer is employed here to extend the effective distance between the pinholes so that the radiation emerged from each pinhole can be processed independently by finite size optical components. The polarizers P₁ and P₂ serve to cut off only one of the orthogonal field components. The removable rotators R₁ and R₂ serve to produce the rotation of one of the transmitted field component through 90°. For such a purpose a suitably oriented half-wave birefringent plate can be used. The operation description of the technique is given bellow.

The determination of the elements $W_{i j}^{SS}$ of the matrix $W_{SS}$ can be realized by means of the following four experiments. In the first experiment the polarizers P₁ and P₂ are aligned to transmit only x component of the incident field without any subsequent rotation of the plane of polarization. In the second experiment P₁ and P₂ are aligned to transmit only y components of the incident field again without any subsequent rotation of the plane of polarization. In the third and the fourth experiments the polarizers P₁ and P₂ cut off the different orthogonal components of the incident field and the corresponding polarization rotator R₁ or R₂ serves to allow the interference of these components.

The power spectrum of the field observed at the output of the interference system in each experiment can be described by the spectral interference law [8]

S_{i j} (x^{'}) = S_{i}^{SS} (\frac{ξ}{2}) + S_{j}^{SS} (\frac{ξ}{2}) + 2 | W_{i j}^{SS} (\frac{ξ}{2}, - \frac{ξ}{2}) | \cos [\frac{k ξ}{z_{0}} x^{'} + α_{i j} (\frac{ξ}{2}, - \frac{ξ}{2})] (i, j = x, y),

where

S_{i}^{SS}

and

S_{j}^{SS}

are the power spectra of the field components in the pinhole position

ξ / 2

, k is the wave number,

z_{0}

is the geometrical path between the pinhole plane and the observation plane, and

α_{i j} = \arg W_{i j}^{SS} .

As well known, the measure of the contrast of the interference fringes is the so-called visibility coefficient defined as

V_{i j}^{} (ξ) = \frac{S_{i j}^{max} (x^{'}) - S_{i j}^{min} (x^{'})}{S_{i j}^{max} (x^{'}) + S_{i j}^{min} (x^{'})} .

On substituting from Eq. (17) with $\cos (.) = \pm 1$ into Eq. (18), we find that

| W_{i j}^{SS} (ξ) | = \frac{1}{2} [S_{i}^{SS} (\frac{ξ}{2}) + S_{j}^{SS} (\frac{ξ}{2})] V_{i j}^{} (ξ)
.

The spectra $S_{i}^{SS}$ and $S_{j}^{SS}$ can be easily measured when one of the pinholes is covered by an opaque screen. The phase $α_{i j}$ can be measured by determining the location of maxima in the interference pattern. Hence, measuring in each experiment the visibility $V_{i j}^{},$ power spectra $S_{i (j)}^{SS},$ and phase $α_{i j}$ , one can determine all the elements $W_{i j}^{SS}$ of the matrix $W_{SS}$ . The degree of coherence and the degree of polarization of the generated source can be then calculated using definitions given by Eqs. (3) and (4).

5. Experiments and results

To verify the efficiency of the proposed technique in practice, we realized a physical experiment sketched in Fig. 3 . The principal part of the experimental setup was composed of two interferometers shown in Figs. 1 and 2. Besides, to reduce the power loss, we used the output beam splitter of the first interferometer as the input beam splitter of the second interferometer. As the primary source we used the expanded well collimated linearly polarized beam generated by the He-Ne laser. To generate the secondary source with two different values of the transverse coherence length, we used two pairs of ground glass plates with the diffusion angles of 10° and 30°. The interference pattern at the first output of the second interferometer was registered by the CCD camera connected to the computer and the corresponding power spectra were measured by the photodiode with optical power meter located at the second output.

Fig. 3 Experimental setup: L, laser; BE, beam expander; ZL, zoom-lens; PD, photodiode; the other abbreviations are just the same as in Figs. 1 and 2.

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The results of the experiments for different pairs of ground glass plates and polarization angle $θ = 45^{\circ}$ are plotted in Fig. 4 together with their theoretical interpolations in accordance with Eq. (15). The calculated spectral degree of polarization changed in the full range from 1 to 0 when varying the polarization angle θ from 0° to $45^{\circ}$ in both experiments. As can be seen, the obtained experimental results are in a good correspondence with the theoretical predictions.

Fig. 4 Results for the experimental characterization of generated sources for two different pairs of ground glass plates with the diffusion angles of 10° and 30°, and θ = 45°. Theoretical results are plotted by solid curves.

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6. Conclusions

We have proposed a rather simple technique for generating the partially coherent and partially polarized electromagnetic source using two rotating ground glass plates placed at the opposite arms of a Mach-Zehnder interferometer. We would like to stress particularly that such a source represents a special case of the well known Gaussian Schell-model uniformly polarized electromagnetic source [10], which appears here not as some handy mathematical construction but as a true product of physical experiment. Besides, the diagonal form of the cross-spectral density matrix given by Eq. (14), permits to find in a closed form the coherent-mode structure [15], a fact that results in considerable simplification when analyzing optical systems with such an illumination [16].

The efficiency of the proposed technique has been confirmed with the physical experiment. We consider that the proposed technique for generating the partially coherent and partially polarized electromagnetic source, as well as the employed technique of its characterization, can be easily reproduced in any advanced optics laboratory and, hence, will serve for subsequent developing the experimental researches on coherence and polarization of electromagnetic fields.

Appendix: To derivation of Eq. (11)

Taking into account Eq. (7), one can state the relation

〈 \exp [\pm iφ (x)] 〉 = \frac{1}{\sqrt{2 π} σ} \int_{- \infty}^{\infty} \exp [\pm iφ (x)] exp [- \frac{φ^{2} (x)}{{2σ}^{2}}] dφ .

Then, making use of well known Fourier-transform relation

\int_{- \infty}^{\infty} \exp (- π a^{2} φ^{2}) \exp (\pm i 2 πφ u) dφ = \frac{1}{a} \exp (- π \frac{u^{2}}{a^{2}}),

we find

〈 \exp [\pm iφ (x)] 〉 = exp (- \frac{σ^{2}}{2}) .

Now we will introduce a random variable

ψ(x_{1}, x_{2}) = φ(x_{2}) - φ(x_{1}) .

Employing Eq. (7), one can write the probability distribution of this variable as

p [ψ(x_{1}, x_{2})] = \frac{1}{\sqrt{2 π} β} exp [- \frac{ψ_{}^{2} (x_{1}, x_{2})}{2 β^{2}}],

where

β^{2} = 〈 ψ^{2} (x_{1}, x_{2}) 〉 = 〈 {[φ(x_{2}) - φ(x_{1})]}^{2} 〉 .

Calculating the square in Eq. (A6) and applying Eq. (8), we find

β^{2} = 2 σ^{2} [1 - \exp (- \frac{ξ^{2}}{2 γ^{2}})] .

Then, by analogy with derivation of Eq. (A3), but this time for argument ψ, we find

〈 \exp {i[φ(x_{2}) - φ(x_{1})]} 〉 = exp {- σ^{2} [1 - exp (- \frac{ξ^{2}}{{2γ}^{2}})]} .

Finally, on making use of Eq. (A3) and the assumption given by Eq. (9) in the main text, we find

〈 \exp {{i[φ}_{1 (2)} (x_{2}) - φ_{2 (1)} (x_{1})]} 〉 = exp (- σ^{2}) .

Equations (A8) and (A9) are the required results to obtain Eq. (11) in the main text.

Acknowledgements

The authors gratefully acknowledge the financial support from the Autonomous University of Puebla under project OSA-EXT-10-G.

References and links

1. A. S. Ostrovsky, P. Martínez-Vara, M. Á. Olvera-Santamaría, and G. Martínez-Niconoff, Vector coherence theory: An overview of basic concepts and definitions, in Recent Research Developments in Optics, S.G. Pandalai, ed. (Research Signpost, Kerala, India, 2009) Chap. 5.

2. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screen and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004). [CrossRef]

3. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005). [CrossRef]

4. A. S. Ostrovsky, G. Martínez-Niconoff, V. Arrizón, P. Martínez-Vara, M. Á. Olvera-Santamaría, and C. Rickenstorff-Parrao, “Modulation of coherence and polarization using liquid crystal spatial light modulators,” Opt. Express 17(7), 5257–5264 (2009). [CrossRef] [PubMed]

5. G. Piquero, F. Gori, P. Romanini, M. Satarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002). [CrossRef]

6. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]

7. H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226(1-6), 57–60 (2003). [CrossRef]

8. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, UK, 2007).

9. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

10. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21(11), 2205–2215 (2004). [CrossRef]

11. M. A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281(9), 2393–2396 (2008). [CrossRef]

12. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef] [PubMed]

13. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24(4), 1063–1068 (2007). [CrossRef]

14. P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13(16), 6051–6060 (2005). [CrossRef] [PubMed]

15. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20(1), 78–84 (2003). [CrossRef]

16. A. S. Ostrovsky, Coherent-Mode Representations in Optics (SPIE Press, Bellingham, WA, USA, 2006).

Experimental generating the partially coherent and partially polarized electromagnetic source

Abstract

1. Introduction

2. Background

3. Technique for generation

4. Technique for characterization

5. Experiments and results

6. Conclusions

Appendix: To derivation of Eq. (11)

Acknowledgements

References and links

Cited By

Figures (4)

Equations (29)

Optics Express