APPLICATION OF THE MATRIX FORMALISM IN A MUELLER MATRIX IMAGING POLARIMETRY

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1 Roanian Reports in Physics, Vol. 60, No. 4, P , 2008 APPLICATION OF THE MATRIX FORMALISM IN A MUELLER MATRIX IMAGING POLARIMETRY O. TOMA, E. DINESCU Faculty of Physics, University of Bucharest, P.O.Box MG-11, Bucharest Magurele, Roania E-ail: thtoa72@yahoo.co (Received Septeber 1, 2008) Abstract. We present the practical realization and the theoretical analysis in the frae of the Mueller atrix foralis of a Mueller iaging polarieter, functioning in transission. The Mueller atrices of both the polarization state generator and the polarization state analyzer of the Mueller atrix iaging polarieter will be theoretically calculated. Key words: light polarization, anisotropy, Mueller atrix polarietry. 1. INTRODUCTION Ellipsoetry is a odern technique with lot of applications in various fields of physics and science, in general [1]. Soe of the up to date directions of the odern ellipsoetry are: the dynaic ellipsoetry [2], the spectroscopic ellipsoetry [3] and the iaging polarietry [4]. The description of various ellipsoeters can be handled in various atrix (Jones and Mueller) [5 8], or pure operatorial [9 12] foraliss. This work is focused on the theoretical calculation, based on a Mueller atrix approach of the Mueller atrices for PSG (polarization state generator) and PSA (polarization state analyzer) systes. As it is known, Mueller atrix polarietry iaging [13] is a powerful iaging technique used to provide high precision easureents for the Mueller atrices at every pixel of an iage captured with a CCD detector. We reind two classical configurations Fig. 1 for a Mueller atrix iaging polarieter. The first configuration uses a focused bea: the saple (which ust posses a polarization inhoogeneous structure depending on its internal structure and on the different interfaces) that ight be a biological tissue, or a thin fil, is illuinated Paper presented at the Annual Scientific Conference, June 6, 2008, Faculty of Physics, Bucharest University, Roania.

2 1066 O. Toa, E. Dinescu 2 Fig. 1 Classical configurations for a Mueller atrix iaging polarieter, in the focused bea (above) or expanded bea (below) setup: a) light polarizer; b) quarter wave plate. with a focused bea and a second lens (or icroscope objective) collects the transitted light. This second lens is iaged onto the CCD, so that each pixel will correspond to a different incidence angle on the saple. Thus, this configuration is used for deterining how the polarization properties of a plan saple vary with the incident direction of the light. The second configuration is in the expanded bea and it is used to confir the level of polarization aberrations in the bea expanding optics (this kind of polarieter have increased costs because the polarization eleents costs increase exponentially with the clear aperture). Both these systes perit the acquisition of 16 Mueller atrix iages and therefore the possibility of extracting a relatively reduced quantity of inforation regarding the polarization properties of the investigated saple. 2. EXPERIMENTAL SET-UP In Fig. 2 it is presented the optical schee [14] used in the acquisition of the Mueller atrix iages. We have build up this polarieter using a blue laser diode (with the wavelength of 457 n) and with all the optical polarization coponents adapted to this wavelength.

3 3 Matrix foralis in a Mueller atrix iaging polarietry 1067 Fig. 2 Optical schee of the Mueller polarization iaging syste. A colliated laser bea (1), enters into the PSG syste fored by two quarter-wave plates (3) and (5) and a polarizer (4). Thus, this syste provides, for different rotation angles of its obile parts, the corresponding Stokes vectors of the incident bea of light on the saple (6). The Mueller atrix polarization iages are projected onto the plane of a light sensitive CCD caera (10) with the iniu resolution pixels, using a icroscope objective (7) with approx. focal length 15 c. This experiental set-up perits the recording of 24 polarization iages, thanks to the suppleentary quarter-wave plate (5) and therefore the possibility of extracting ore inforation regarding the investigated object. The PSA syste is fored by a quarter-wave plate (8) and another polarizer (9), so this syste generates the enseble of the Stokes vectors for the light arriving at the CCD. 3. THEORETICAL CONSIDERATIONS AND RESULTS We will obtain soe analytical expressions for the Mueller atrix eleents for both systes involved in our experiental set-up: the polarization state generator syste (PSG) and the polarization state analyzer syste (PSA). The first quarter-wave plate (3) (an ideal, hoogeneous, linear retarder with ρ the aziuth of its fast axis, equal to 0), has the following associated atrix: M = (1)

4 1068 O. Toa, E. Dinescu 4 The rotating polarizer (4) (an ideal, hoogeneous, linear polarizer, with θ the aziuth of its transission axis) has this corresponding atrix: M 1 cos2θ sin2θ 0 cos2 cos2 1 θ 2θ cos2θsin 2θ 0 =. 2 sin2 θ cos2 θ sin2 θ sin 2 θ The second quarter-wave plate (5) (an ideal, hoogeneous, linear retarder with ρ # 0) has its corresponding atrix as it follows: M cos2 2ρ cos2ρsin2ρ sin2ρ =. 0 cos2 ρ sin2 ρ sin 2 ρ cos2 ρ 0 sin2ρ cos2ρ Thus, the polarization state generator atrix will be: MG 3 2 1, (2) (3) = M M M (4) and the Mueller atrix eleents are calculated below: 11 = 1 31 = sin2ρcos2( ρ θ) 12 = cos2θ 32 = sin 2ρcos2θcos2( ρ θ) 13 = 0 33 = 0 14 = sin2θ 34 = sin2ρsin2θcos2( ρ θ) 21 = cos2ρcos2( δ θ ) 41 = sin 2( ρ θ) 22 = cos2ρcos2θcos2( ρ θ ) 42 = cos2θsin 2( ρ θ) 23 = 0 43 = 0 24 = cos2ρsin 2θcos2( ρ θ ) 44 = sin2θsin2( ρ θ). In the sae anner we will calculate the eleents corresponding to the PSA syste. So, the rotating quarter-wave plate (8) (an ideal, hoogeneous, linear retarder with ρ # 0) and the rotating polarizer (9) (an ideal, hoogeneous, linear polarizer, with θ the aziuth of its transission axis) have their associated atrices M 3 and, respectively, M 2, described by the relations (3) and (2). Therefore, the polarization state analyzer atrix will be: MA = M M (5) 2 3 and the Mueller atrix eleents are also calculated below:

5 5 Matrix foralis in a Mueller atrix iaging polarietry = 1 = sin 2θ = cos2ρcos2( ρ θ ) = sin2θcos2ρcos2( ρ θ) = sin2ρcos2( ρ θ ) = sin2ρsin 2θcos2( ρ θ) = sin2( θ ρ ) 34 = sin2θsin 2( θ ρ) = cos2 θ = 0 = cos2ρcos2θcos2( ρ θ ) = 0 = sin2ρcos2θcos2( ρ θ) 43 = 0 = cos2θsin2( θ ρ ) = Each rotating polarization coponent of the experiental set-up perit us the obtaining of different values for the angles θ and ρ. Thus, different values for the Mueller atrix eleents corresponding to the PSG and PSA systes Mueller atrices can be obtained, and fro these atrices an iportant quantity of inforation regarding the polarization properties of the investigated saple can be extracted. 4. CONCLUSIONS The calculus of the eleents corresponding to the Mueller atrices for the polarization state generator and the polarization state analyzer systes involved in this Mueller atrix iaging polarieter, was perfored. In coparison with the classical acquisition of 16 Mueller atrix iages of the saple, the experiental set-up used perits the recording of 24 polarization iages and therefore the possibility of extracting ore inforation regarding the investigated object. REFERENCES 1. R. M. Azza, N. M. Bashara, Ellipsoetry and Polarized Light, Elsevier, Asterda, E. Bernabeu, J. J. Gil, An experiental device for the dynaic deterination of Mueller atrices, J. Opt., 16, 3, (1985). 3. H. Fujiwara, Spectroscopic ellipsoetry, Principles and applications, John Wiley and Sons Ltd., J. S.Tyo, D. L. Goldstein, D. B. Chenault, J. A. Shaw, Review of passing iage polarietry for reote sensing applications, Appl. Opt., 45, (2006). 5. D. H. Goldstein, E. Collett, Polarized Light, CRS Press, NewYork, S. Huard, Polarisation de la luière, Masson, Paris, S. Y. Lu, R. A. Chipan, Mueller atrices and the degree of polarization, Opt. Coun., 146, (1998). 8. S. N. Savenkov, Optiization and structuring of the instruent atrix for polarietric easureents, Opt. Eng., 41, (2002).

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