Technical Field

The invention refers to optics, in particular to development of a new method for optical measurements and it can be used in polarimetry, spectropolarimetry and ellipsometry.

Background Art

The registration method of state of polarization vector of monochromatic light radiation using liquid crystal (LC) polarization diffraction gtating (PDG) is known [1-3], according to which the light beam is directed to PDG, at the output of which the diffracted beams in +/-1 orders with corresponding right- and left-circular polarizations are formed, registered by separate photodetectors. The non-diffracted beam in zero-order is directed to the common diffraction grating, from the output of which the diffracted beams in +/-1 orders are registered by separate photodetectors, previously passing through the 0° and 45° oriented linear polarizers. Thus, having values of light intensities registered by four different photodetectors the four ¹ Stokes' parameters can be restored, allowing to determine the beam polarization vector.

The method of monochromatic beam polarization vector state determining is chosen as the closest prototype, based on the Stokes' parameters calculation, using the values of light intensity registered on the output of liquid crystal polarizatsion diffraction gratings [4]. To obtain Stokes' parameters it is necessary to make three different measurements of the diffracted fields intensities: in the first case - polarization vector of the linearly polarized beam, directed to the grating, coincides with x axis of the analizer, placed before PDG; in the second case - the angle between this vector and x axis is 45°; and in the third case - there is an arbitrary angle. According to the prototype, the state of the beam polarization vector is determined as follows. The beam with arbitrary polarization is directed to PDG, at the output of which diffraction beams I ₀ , 1 ₊ i, Li are formed correspondingly at zero and +/-1 orders.

In the second case the beam is transmitted through the analizer, the rotation angle of which is zero, and I ₀ (0) value is registered at the PDG output. During the third measurement the beam is transmitted through the analizer, rotated to 45° and I ₀ (45) value is registered at the PDG output. The Stokes' paremeters are calculated on the base of registered intensities values according to the following

s ₀ = 8(I ₁ + i_ ₁ ), s ₁ = 8[i ₀ (0) - I _l - I_ ₁ ],

s ₂ - l_' ₁ ) So, as it is seen from the above-stated, to determine the state of polarization vector it is necessary to realize three consistent measurements. However, there are some problems in ellipsometry, connected with the change of the state of polarization vector of monochromatic beam, reflected from time-varying medium. For example, the reflection coefficient of semiconductor conditioned by life time of the free carriers, generated in the semiconductor under the influence of laser beam is time depended, and consequently, in this case the rotation of the polarization vector of monochromatic beam, conditioned by photogenerated carriers, is a function from time as well. So, in similar problems to register the state of polarization vector it is necessary to have another measurement method, allowing to determine Stokes' parameters by a single measurement.

The objective of this invention is to simplify the registration of the state of beam polarization vector.

Summary of Invention

What is claimed is:

To propose a method for registration the change of polarization vector of monochromatic light beam based on using axial symmetric liquid crystal retarder and polarization diffraction grating. In contrast to the prototype, in the proposed method, besides PDG, an axial symmetric liquid crystal retarder is used.

Brief Description of Drawings

In Fig.1 an optical scheme for writing of axialy symmetric liquid crystal retarder is presented. The linearly polarized beam of CW laser (1) at 325 nm is directed to rotating half-wave plate (2), at the output of which there is a slit (3) at 45° angle relative to polarization vector of incident beam. Passing through the slit, the beam falls to nematic liquid crystal layer (4) coated by oriented polymer ROP- 103/2 CP layer. Under the influence of incident light the radial symmetric orientation of polymer molecules takes place. The described radial symmetric phase retarder allows to determine the unique correspondence between the state of incident beam polarization vector and intensity distribution in radial symmetric beam passed through the analyzer placed after the retarder.

In Fig.2 an optical scheme of analyzer of beam polarization state is given. The above described radial symmetric phase retarder, at the output of which there is analyzer (4) crossed with input polarizer (1), is used in the considered polarisation state analyzer. The investigated monochromatic beam after passing through the polarizer is directed to the clockwise rotating half-wave plate (2), at the output of which the radial symmetric phase retarder (3) is placed. Thus, a phase modulated beam is formed at the retarder output. The modulation of phase of beam, passed throught the analyzer, which is placed after the retarder, is transformed to the radial symmetric amplidute modulation.

In Fig.3-11 the radial symmetric images of experimentally registered intensities at the analyzer output are presented, which correspond to different states of polarization vector of the investigated monochromatic beam (shown at the left top angle of each figure).

The components with right- and left-circular polarization are overlapped in the depicted radial symmetric intensities distributions. As it is seen from the figures, radial symmetric distribution with certain intensity corresponds to each state of polarization vector. For example, the four-lobe radial symmetric intensity distribution corresponds to the beam with right-circular polarization. In the Fig.12 the optical scheme of polarization state analyzer is presented, were PDG (4) is used instead of analyzer at the output of radial symmetric phase retarder, the left- and right-circularly polarized components, i.e. +/-1 order diffracted beams, can be spatially separated. The (1), (2), (3) elements in Fig.12 are: input polarizer - (1), the clockwise rotating half-wave plate - (2), and radial symmetric phase retarder (3). So, the beams with horizontal and vertical linear polarization correspond to the same radial symmetric eight-lobe intensity distribution (Fig.3 and Fig.11), where the right- and left-circular polarized components are at 45° relative to each other, so after PDG inserting the four-lobe distributions, corresponding to the right- and left-circularly polarized components, are spatially separated (Fig.12). It should be noted, at the PDG output, besides +/-1 diffraction orders corresponding to the right- and left-cilcularly polarized beams, a non-diffracted beam, corresponding to 0 order, is also formed.

In Fig.13, Fig.14, Fig.15 and Fig.16 intensities distributions for the investigated beams with right- and left-circular polarization are presented. Here the homogeneous circular distributions of intensity either in +1 and 0 or -1 and 0 orders are available. It should be noted, in case of the beam right (left) circular polarization, besided the seen +1 (-1) diffracted orders, a certain weak intensity distribution is observed in -1 (+1) orders as well. It is conditioned by imperfection of the grating spatial periodicity.

In Fig.17 and Fig.18 the numerically simulated distributions of the right- and left-circular polarized diffracted beams intensities, corresponding to the investigated horizontal-linearly polarized beam at the output of PDG are presented.

In Fig.19 and Fig.20 numerically simulated Radon distributions, which correspond to right- and left-circular polarized beams, shown in Fig.17 and Fig.18 are presented.

In Fig.21 and Fig.22 ID Radon distributions along the x ¹ = 0 axis, which correspond to right- and left-circular polarized beams (Fig.19, Fig.20) are presented. In Fig.23 and Fig.24 numerically obtained intensities distributions of the right- and left-circular diffracted polarized beams for -45° linearly polarized investigated beam are presented.

In Fig.25 and Fig.26 the 2D Radon distributions of the right- and left-circular polarized beams diffracted on the PDG are presented.

In Fig.27 and Fig.28 ID Radon distributions along the x ¹ = 0 axis, corresponding to right- and left-circular polarized beams (Fig.25, Fig.26) are presented.

In Fig.29 the angles χ and ψ are presented.

In Fig. 30, 31, 32, 33 the signs of SQ ,SI , S2 , S^ values, allowing to restore the polarization vector of the investigated beam (to replace V by S3 _, Q - by Si, U - by ¾ ) are presented.

Modes for Carrying out the Invention

So, there is unique correspondence between the state of polarization vector of the investigated beam and the radial symmetric intensity distributions corresponding to right- and left-circularly polarized beams.

For quantitative description of the mentioned correspondence it is proposed to use Radon transformation relative to intensities distributions [5]. Radon transformation is a symmetric image projection on the given axis, which in the ID case is a single-valued function of the axis rotation angle, and its maximum values are periodic. This corresponds to the transformation of intensity distribution along the circle length to the intensity distribution along the horizontal angle axis. That is, rotation of radial symmetric image will correspond to the shifting of maximum values of ID Radon transformation distribution along the angular axis.

Algorithm of restoration of monochromatic beam polarization vector state is based on Radon transformation. For the considered geometry of experiment the maximum values corresponds to symmetric axes of the registered images and are at 45° angle relative to each other.

Using Radon transformation it is possible to converse the intensity distribution along the circle length into the intensity distribution along the line. In this case the deviation of Radon transform image along the angular axis will correspond to rotation of radial symmetric image. The optical scheme shown in Fig.12 can be described by the transfer function

H = HpH _QW p{e)H _HWP {e)H _PDG where H p , H Q _WP {9) , H _HWP {e) and Hpp _Q - are Miiller matrices, corresponding to polarizer, rotating quarter- and half-wave plates, and PDG, Θ- rotation angle.

Thus, in the proposed method the state of polarization vector is determined by values of Stokes' parameters, obtained by a single measurement. The diffracted beams in orders correspond to the right- and left-circular polarized beams, respectively. In contrast to experimental results, in numerical model, where the ideal PDG is considered, the beam at zero order is absent. The mutual perpendicular components of the investigated beam can be expressed as

E _x = E _0x cos(cot + δ _χ ),E _y = Eoy + S _y )

E _0x = E ₀ cos(a),E _0y = E ₀ sin(a)

and the phase difference is δ = d _y - δ _χ .

For the fixed value δ the change of angle results to rotation of radial symmetric intensities, corresponding to right- and left-circular polarized beams, relative to the center of symmetry. It leads to ID Radon distribution shift along the axis x ¹ by half of a angle, herewith the sign of a is determined by direction of the shifting: right is positive and left is negative. The change of δ angle for fixed a leads only to change of ID Radon distribution contrast. So, the Stokes' parameters can be expressed by mutual orthogonal components of the investigated beam and registered intensities:

I = S ₀ = I+](RCP) + ^I -1(LCP) ^{= E} 0x ^{+ E} 0y

U = = -2im(plE _R )= 2E _0x E _0y cos^ ) ~ \ ^E R \ ² = ^2E 0x ^E 0y ^sin ( ⁵ ) where E _L = ith left circular polarization, and

E _R = ~={E _x - jEy ) is the beam field with right circular polarization.

S _Q and ¾ are registered experimettaly. If S3 > 0 , the polarization vector of investigated radiation is placed in Poincare lower half-sphere, and if S _j < 0 - in upper half-sphere.

The angle a is determined from ID Radon distribution, tangent of which is defined by tan( a) = Eo _y jE _QX , allowing to express E ₀ and EQ _X by measured value SQ : Then Sj is determined by E _0x ,Eo _y values according to S _j = Ε _χ - Eg _y , taking into account the sign of angle a , and sin(s) is determined as sin(S ) = S 2E _0x E ₀ y .

When S ₃ > 0 , then (2E _0x E _0y sin(S )) /(E ₀ ² _x + E¾ _y ) > 0 , and if S ₃ < 0 , then ( 2Eo _x E _0y sin( δ )) /( EQ _x + E^ _y ) < 0 , and by this way the sign of sin( S) \s determined. ^ is calculated according to the following expression ^ = ±2E _Qx E _oy j J - sin (δ) . In the last expression the sign ambiguity can be ignored, if the investigated beam is linearly polarized, which correspond to the interaction of linearly polarized radiation with medium. This correspond to the case with δ = 0 . In other words, it is considered the case, when the polarization vector of radiation is on the sphere equator. Generally, when ^≠ 0 , the sign of expression ± ^l - sin (S) is determined according to

ίαη(2ψ) tan(2a)cos(s)

In order to increase the determination accuracy of angle a it is proposed to use the average values of angles, at which maxima of ID Radon transformation take place. The number of summands, used for averaging depends on the number of lobes of radial-symmetric intensity distribution. In turn, the number of lobes can be varied by changing of slit rotation angle during the radial-symmetric transparent preparation. Namely, in this case to determine the estimating value of the rotation angle the following expression is used:

where N = 4.

In order to reduce random components of a determining errors, it is proposed to use a phase transparency with higher order of radial symmetry, recorded at the increased angular speed of slit rotation, leading to increasing of number of lobes. As a result, the number of maxima ΓΝ is number of summands) in ID Radon distribution increases, which in its turn reduced the random errors in determining the rotation angle of polarization vector ( a ) of linearly polarized beam [6]. To determine the linear dichroism it is proposed to combine the diffracted beams with mutually orthogonal circular polarizations by rotating one of them at angle a . After, it is necessary to photometrical measure in logarithmic scale the difference of combined intensities distributions pixel by pixel.

The proposed method allows determine the linear dichroism of medium. In the case of elliptically polarized continuous radiation, when contrast of ID Radon distribution is reduced, the proposed method may be used also for measuring the rotation angle of polarization vector, phase difference of orthogonal polarized components, as well as circular dichroism.

The sign of Si is determined according S _} = EQ _X - EQ _Y to

The sign of sin(2x )is determined by the sign of S3, and the sign of tan{2a) is determined from ID Radon distribution.

Thus, if the above given relation is presented as tgS = , where k =—— .

cos5 S _j tg(2 )

Using the sign of coefficient k it is possible to determine: if the sign of tg5 and cos is the same or not. So the sign of cos δ is determined this way, allowing to find the value of ¾ .

References

1. T. Todorov, L. Likolova. Spectrophotopolarimeter: fast simultaneous real-time measurement of light parameters. Optics letters, Vol.17, No.5, 1992

2. T. Todorov, L. Nikolova, G. Stoilov, B. Hristov. Spectral Stokesmeter. 1. Implementation of the device. Applied Optics, Vol.46, No.27, 2007

3. C. Provenzano, G. Cipparrone, A. Mazzulla. Photopolarimeter based on two gratings recorded in thin organic films. Applied Optics, Vol.45, No.17, 2006

4. F. Gori. Measuring Stokes parameters by Means of a Polarization Grating. Optics Letters, Vol.24, No.9, 1999.

5. Deans, Stanley R. The Radon Transform and Some of Its Applications, New York: John Wiley & Sons, 1983.

6. Vincenzo D'Ambrosio, Nicolo'Spagnolo, Lorenzo Del Re, Sergei Slussarenko, Ying Li, Leong Chuan Kwek, Lorenzo Marrucci, Stephen P. Walborn, Leandro Aolita, Fabio Sciarrino. Photonic polarization gears for ultra-sensitive angular measurements. Nature Communications. DOI: 10.1038/ncomms3432.