2410 OPTICS LETTERS / Vol. 33, No. 20 / October 15, 2008

Experimental determination of electriccross-spectral density matrix and

generalized Stokes parameters for a laser beam

Bhaskar Kanseri and Hem Chandra Kandpal*Optical Radiation Standards, National Physical Laboratory, New Delhi 110012, India

*Corresponding author: hckandpal@mail.nplindia.ernet.in

Received May 29, 2008; revised July 13, 2008; accepted August 6, 2008;posted September 18, 2008 (Doc. ID 96820); published October 15, 2008

We report an experimental method to determine the elements of the electric cross-spectral density matrix forlaser light. For this purpose an additional setup consisting of mirrors and reflecting prisms is utilized withthe conventional Young’s interferometer to overcome existing experimental limitations. The generalizedStokes parameters, which are the characteristics of two spatial points of the electromagnetic field, are alsoobtained for a pair of points. The knowledge of these two quantities might be useful in determining thechange in polarization properties of light in propagation and their effects in optical measurements. © 2008Optical Society of America

OCIS codes: 260.3160, 030.1640.

The recently developed unified theory of coherenceand polarization by Wolf [1] suits well with the ob-served theoretical and experimental observations[2–4] that could not fully be explained by the Stokesparameters [5] or 2�2 coherence matrix [6]. The uni-fied theory elucidates an intimate relationship exist-ing between the coherence and the polarization ofelectromagnetic beams. The central quantity in thistheory is a 2�2 electric cross-spectral density matrixWJ �r1 ,r2 ,��, which characterizes the correlations be-tween two mutually orthogonal spectral componentsof the electric field at a pair of points in a plane per-pendicular to the direction of propagation of thebeam. Later, using the newly presented matrix, a lotof theoretical and experimental work has been donetreating the light field as a vector quantity [7–9]. Therecently introduced generalized Stokes parameters[10] also depend on two spatial variables. In a nut-shell we can say that the generalized Stokes param-eters, like the cross-spectral density matrix, couldalso be used to study both the coherence and the po-larization properties of random electromagneticbeams.

After the invention of the laser, most of the opticalexperiments are conducted using a laser source. Thusit becomes of the utmost importance to have informa-tion about the polarization properties of the laserbeam. In the present Letter, we have experimentallydetermined the cross-spectral density matrix of anexpanded laser beam at a pair of points in the double-slit plane. An expanded laser beam was used so thatthe secondary source (the pinhole of the beam ex-pander) could approximately simulate the pointsource with reduced spatial coherence.

The method used here is an experimental realiza-tion of the theoretical approach presented in [11] toobtain the components of the cross-spectral densitymatrix using polarizers and rotators in front of thepinholes and taking spectral measurements at theobservation plane. To our knowledge this approach

[11] has yet not been realized experimentally prob-

0146-9592/08/202410-3/$15.00 ©

ably owing to the complexities involved in existingspatial coherence based interferometers for placingpolarizers and rotators before the individual pin-holes, which were having very narrow separation(typically few hundreds of micrometers). Introducinga combination of prisms and mirrors after the doubleslit, the two beams were separated by a few centime-ters to overcome the prevailing complexities. Thefour generalized Stokes parameters that involve thesame pair of points were also determined.

Let us consider a random electromagnetic beampropagating in the z direction. The fluctuations of thebeam could be considered statistically stationary atleast in a wide sense. Let E�r ,��= �Ei�r ,���, �i=x ,y�be the statistical ensemble of frequency � of the fluc-tuating electric field E�r ,�� at a point r in space. Thesecond-order coherence properties of the beam maybe characterized by a 2�2 electric cross-spectral den-sity matrix [1]:

WJ �r1,r2,�� = Wij�r1,r2,�� = �Ei*�r1,��Ej�r2,���,

i = x,y, j = x,y, �1�

where the asterisk denotes the complex conjugate.Let an opaque screen be placed across the plane z

=0 with two narrow slits symmetrically positionedabout the line perpendicular to the plane (inset ofFig. 1). Let Q1 and Q2 be any two points having posi-

Fig. 1. (Color online) Schematics of the experimentalsetup. The inset illustrates the notations. Abbreviations

are defined in the text.

2008 Optical Society of America

October 15, 2008 / Vol. 33, No. 20 / OPTICS LETTERS 2411

tion vectors r1 and r2, respectively, in the differentslits. The spectral degree of coherence of the electricfield of the beam is given by [4]

��r1,r2,�� =Tr WJ �r1,r2,��

�Tr WJ �r1,r1,���Tr WJ �r2,r2,��. �2�

Equation (2) can be expressed in the form [11]

Tr WJ �r1,r2,�� = �S�r1,���S�r2,����r1,r2,��. �3�

According to [11], by using different polarizing ele-ments before the separate slits, the components ofthe cross-spectral density matrix WJ �r1 ,r2 ,�� will begiven by

WJ �r1,r2,�� = Wij�r1,r2,��

= �Si�r1,���Sj�r2,���ij�r1,r2,��,

i = x,y, j = x,y. �4�

The generalized Stokes parameters, which dependupon two spatial points, are different than the usualStokes parameters, which are one-point quantities.The first parameter S0�r1 ,r2 ,�� gives the total inten-sity of the field at two points, while the second pa-rameter S1�r1 ,r2 ,�� is the measure of the linear po-larization in orthogonal x and y directions of thepoints. The third parameter S2�r1 ,r2 ,�� provides in-formation about the linear polarization at 45° of the xand the y axis at the two points, and the fourth pa-rameter S3�r1 ,r2 ,�� is a measure of the circular po-larization for the points. All the four generalizedStokes parameters could be expressed as [10]

Si�r1,r2,�� = Tr�WJ �r1,r2,�� · �Ji �i = 0, 1, 2, 3�,

�5�

where �Ji �i=1,2,3� are the Pauli spin matrices [6]and �J0 is the unit matrix.

A randomly polarized helium-neon laser (MellesGriot, wavelength 632.8 nm, power 2 mW) was usedas a source (Fig. 1). The laser beam was expanded us-ing a spatial filter beam expander assembly (BE) andwas passed through a double slit (D, slit width150 �m and slit separation 200 �m) put in the beampath at 30 cm away from the expander and was sym-metric about the line passing through the pinhole ofthe BE, as shown in Fig. 1. Two right-angle front-coated prisms Q1 and Q2 were introduced after D, oneof which �Q1� separated the two beams approxi-mately by 8 cm, and the other �Q2� recombined themagain at a distance 10 cm (along the z axis) from thefirst prism. Mirrors (M) were used with prisms toguide the beam through required path. The straight-line interference fringes were obtained in the obser-vation plane R at a distance 135 cm from the secondprism Q2 and were photographed as shown in Fig. 2.A fiber (F)-coupled spectrometer (SP, Photon ControlModel No. SPM-002) mounted with a computer (DP)-controlled motorized micropositioner was used to

measure the spectra at the observation plane.

To calculate the Wxx�r1 ,r2 ,�� component of theelectric cross-spectral density matrix, two identicaldichroic sheet polarizers (Melles Griot, wavelengthrange 380–780 nm) having direction of polarizationalong the x axis �P1,P2� were inserted in the separatebeam paths (Fig. 1). This allowed the x components ofthe beams only to pass through, and good visibilityinterference fringes were obtained at R as shown inFig. 2(a). The maximum and minimum values of thespectral density were recorded by the spectrometertracing the fiber tip horizontally over the central in-terference fringe. The data output of the spectrom-eter (saved in excel format) looks like Fig. 2(e). Thespectral densities at an axial point P over the centralfringe due to the individual slits (points Q1 and Q2)were obtained by making measurements when onlyone slit was open and the other closed. To computethe Wyy�r1 ,r2 ,�� component, both polarizers were ro-tated clockwise by 90° (this made the direction of po-larization y axis), and similar measurements weremade for both the slits which open simultaneouslyand alternately [fringes are shown in Fig. 2(b)].

The Wxy�r1 ,r2 ,�� component of the matrix was ob-tained when the direction of polarization of the polar-izer placed in one beam path was along the x axis�P1� while the other was along the y axis �P2�. Tobring the vibrations in the same plane so that inter-ference could be possible, a half-wave plate (H, mul-tiorder, �=632.8 nm), having optics axis at 45° withthe incident polarization of light was placed after thepolarizer P2, which worked as a 90° polarization ro-tator. Another half-wave plate C with identical speci-fications having optic axis along the incident polar-ization of light was placed in another arm for pathcompensation (Fig. 1). The maximum and minimumvalues of the spectral density for the same pair ofpoints were recorded by the spectrometer at the ob-servation plane for the central fringe [Fig. 2(c)]. Withone slit closed and the other open, the spectral den-sities due to the individual slits were also recorded atthe same point P. The Wyx �r1 ,r2 ,�� component wasobtained by rotating both the polarizers by 90° andtaking similar spectral measurements for both theslits as well as the individual slits [fringes in Fig.2(d)]. It is evident from the photographs that thefringe visibility due to xy and yx components is muchless than the xx and yy components.

Following the procedure as stated earlier, the val-

Fig. 2. (Color online) (a–d) Interference fringes observedat the observation plane R for different direction of polar-izations (DOPs) of the polarizers. (e) Data recorded by thespectrometer shows the spectral profile of laser beam.

ues of spectral densities about the central bright

2412 OPTICS LETTERS / Vol. 33, No. 20 / October 15, 2008

fringe were measured at the observation plane R.The modulus of spectral degree of coherence of light(also called spectral visibility) for the two points wascalculated using the formula [6]

Smax − Smin

Smax + Smin, �6�

where Smax and Smin are the respective maximumand the minimum values of spectral density aroundthe central fringe when both the slits are open.

The values of the spectral degree of coherence ofthe electric field ��r1 ,r2 ,�� for all the four cases aredetermined from the experimental data using thewell-known methods [11] and are given in Table 1. Ina similar manner, the spectral densities of the ex-panded laser beam due to the individual slits weremeasured at the axial point P. Using Eq. (4), we havecalculated the elements of the electric cross-spectraldensity matrix as shown in Table 1. As the experi-ment was conducted eight times, the statistical errorassociated with it is shown as the standard deviationin the calculated values.

Using Eq. (4) the matrix would be given as

WJ �r1,r2,�� = 410.6 ± 4.0 118.2 ± 1.6

119.7 ± 1.8 366.4 ± 3.4� . �7�

Using the elements of the electric cross-spectral den-sity matrix, the generalized Stokes parameters couldbe obtained using Eq. (5). The results are presentedin Table 2.

Equation (7) shows that the spectral degree of co-herence, i.e., visibility of interference fringes forcross-polarized cases (xy and yx) is less than that forthe copolarized cases (xx and yy). It is due to the factthat there is no correlation (practically very small)between the two orthogonal components of the ran-domly polarized light [6]. Hence the condition for in-terference is satisfied only for the electric field, whichis very little correlated [see Figs. 2(c) and 2(d)].

Table 1. Components of the Spectral Degreeof Coherence and the Cross-Spectral Density Matrix

Obtained for the Laser Light

�xx�r1 ,r2 ,�� 0.84±0.09 Wxx�r1 ,r2 ,�� 410.6±4.0�xy�r1 ,r2 ,�� 0.32±0.06 Wxy�r1 ,r2 ,�� 118.2±1.6�yx�r1 ,r2 ,�� 0.24±0.04 Wyx�r1 ,r2 ,�� 119.7±1.8�yy�r1 ,r2 ,�� 0.71±0.08 Wyy�r1 ,r2 ,�� 366.4±3.4

Table 2. Generalized Stokes Parameters for a Pairof Points in the Laser Beam

S0�r1 ,r2 ,�� 777.0±5.1S1�r1 ,r2 ,�� 44.2±2.5S2�r1 ,r2 ,�� 237.9±2.4S3�r1 ,r2 ,�� �1.5±0.81�i

It is evident from the positive value of the secondStokes parameter that for the two points, the laserbeam has the x-axis polarization stronger than they-axis polarization [Figs. 2(a) and 2(b)]. The same re-sult could be observed (Fig. 3) by using a linear po-larizer in front of the random laser beam and record-ing the output intensity as a function of the angularposition of the transmission axis of the polarizer (0°to 360°). Determining the Stokes parameters for dif-ferent pairs of points in the propagating beam, thechange in polarization owing to propagation could bemeasured.

In conclusion, in this study we have determinedthe four elements of the electric cross-spectral den-sity matrix for a laser beam using a modified inter-ferometer. The generalized Stokes parameters for apair of points in the double-slit plane were also ob-tained using the density matrix. This simple methodcould be beneficial to study the properties of coher-ence and polarization of the random electromagneticbeams in propagation.

The authors thank the Director, National PhysicalLaboratory New Delhi for permission to publish thepaper. B. Kanseri is grateful to the Council of Scien-tific and Industrial Research (CSIR) (India) for finan-cial support as a Junior Research Fellowship (JRF).

References

1. E. Wolf, Phys. Lett. A 312, 263 (2003).2. D. F. V. James, J. Opt. Soc. Am. A 11, 1641 (1994).3. G. P. Agrawal and E. Wolf, J. Opt. Soc. Am. A 17, 2019

(2000).4. E. Wolf, Opt. Lett. 28, 1078 (2003).5. G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399

(1852).6. L. Mandel and E. Wolf, Optical Coherence and

Quantum Optics (Cambridge U. Press, 1995).7. H. Roychowdhury and E. Wolf, Opt. Commun. 252, 268

(2005).8. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, Opt.

Lett. 31, 688 (2006).9. E. Wolf, Introduction to Theory of Coherence and

Polarization of Light (Cambridge U. Press, 2007).10. O. Korotkova and E. Wolf, Opt. Lett. 30, 198 (2005).11. H. Roychowdhury and E. Wolf, Opt. Commun. 226, 57

(2003).

Fig. 3. Variation in the output intensity of the laser beamwith different DOPs of the linear polarizer. 0° and 180° ofthe transmission axis correspond to the x axis polarization.