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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 6 — Mar. 15, 2010
  • pp: 5776–5784
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Jones matrix analysis of dichroic phase retarders realized in soft matter composite materials.

Roberto Caputo, Ivan Trebisacce, Luciano De Sio, and Cesare Umeton  »View Author Affiliations


Optics Express, Vol. 18, Issue 6, pp. 5776-5784 (2010)
http://dx.doi.org/10.1364/OE.18.005776


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Abstract

Materials showing birefringence and polarization selective absorption (dichroism) affect the polarization state of incoming light in a peculiar way, quite different from the one exhibited by phase retarders like waveplates. In this paper, we report on the characterization of a Polymer LIquid CRYstal Polymer Slices (POLICRYPS) diffraction grating used as a dichroic phase retarder; the dichroic behaviour of the grating is due to the polarization-dependent diffraction efficiency. Experimental data are validated with a theoretical model based on the Jones matrix formalism, while the grating behavior is modeled by means of the dichroic matrix. In this way, the birefringence of the analyzed structure is easily obtained. For comparison purposes, also two systems different from POLICRYPS have been fabricated and tested.

© 2010 OSA

1. Introduction

2. The POLICRYPS grating as a dichroic absorber

The grating used for this experiment has a thickness L=3.03μm and a fringe spacing Λ=1.22μm. From Fig. 1, it can be noticed that the diffraction efficiency assumes a very high value for a p-polarization of the impinging radiation (η≈94%), while it goes down to a very low value when the probe beam is s-polarized (η≈8%).

3. Phase retardation behavior of a birefringent dichroic material

We derive, now, the equations for calculating the complex electric field E˜out(β) and hence the intensity Iout(β) of light transmitted by the analyzer A in our experimental geometry. We define β as the angle between directions of analyzer axis and incident polarization (therefore β=0 when the axis of the analyzer A is parallel to that of the polarizer P).

The Jones Matrix of an analyzer put at a generic angle β can be written as:
A(β)=(cosβsinβsinβcosβ)(0001)(cosβsinβsinβcosβ)=(sin2βsinβcosβsinβcosβcos2β)
(2)
In Eq. (2), in order to take into account any angular position the analyzer can assume during the experiment, we have used the unitary transformation A(β) = R(-β)AR(β) for the Jones Matrix A=(0001)of an analyzer oriented along the y axis of the reference system; R(β)=(cosβsinβsinβcosβ) is the β-angle rotation matrix. As for the sample, the Jones Matrix of a retardation plate can be written as:
M=(eiδ200eiδ2)
(3)
Where δ is the unknown phase retardation (introduced by the plate) between the two orthogonal components E|| and E (with respect to the optical axis of the birefringent material) which the impinging wave is decomposed into. If the optical axis of the sample plate is oriented parallel to the first polarizer (θ=0), we have no retardation (δ=0). When the plate is, instead, oriented at an angle θ=π/4 the field components have the same amplitude (E||=E) and the plate introduces the maximum retardation. In the Jones matrix sequence representing our system, this choice can be accounted for by applying to M a rotation matrix R of angle θ=π/4. Unfortunately, substituting M with R(-π/4)MR(π/4) would yield an over complication of calculations; a more straightforward procedure is that of leaving the matrix M as it is in Eq. (3) and rotate, instead, the direction of the incoming polarization by the same θ = π/4 angle. This corresponds to express the impinging electric field, in terms of Jones Matrices, as E˜inc=22Iinc(11). By taking into account this consideration and using matrices A(β), M and L, the complex electric field of the radiation coming out from the analyzer in the experimental geometry of Fig. 2 is written as:
E˜out=(sin2βsinβcosβsinβcosβcos2β)[(H00V)(eiδ200eiδ2)]22Iinc(11)==22Iinc(Heiδ2sin2βVeiδ2sinβcosβHeiδ2sinβcosβ+Veiδ2cos2β)
(4)
From Eq. (4) we derive the intensity of the light transmitted by the analyzer:
Iout(β)=E˜out(β)E˜out*(β)=Iinc2[H2sin2β+V2cos2β+HVsin2βcosδ]
(5)
which depends on the angle β between the analyzer A and the first polarizer P. In order to evaluate H, V and δ from Eq. (5), we notice that when the analyzer is parallel (β=0) or perpendicular (β=π/2) to the polarizer, the output intensity holds:
Iout(β=π2)=IincH22
or
Iout(β=0)=IincV22
which yield:
H=2Iout(β=π2)Iinc
(6)
and
V=2Iout(β=0)Iinc
(7)
respectively. Then, we can calculate the phase retardation δ introduced by the sample as:
cosδ=1HV[2Iout(β=π/4)IincH2+V22]
(8)
Hence, by using the measured values of the transmitted intensity Iout when the analyzer is put at particular angles (β = 0, π/2, π/4), it is possible to obtain the values of all parameters appearing in Eq. (5), which are characteristic of the system under investigation. Finally, it is worth noting that, by combining Eq. (6) and Eq. (7) we obtain a relationship between H and V parameters and the measured intensity values utilized for their evaluation:
H2V2=Iout(β=π2)Iout(β=0)
(9)
As we are going to show in the following, Eq. (9) gives an easy way to deduce the dichroism of the investigated sample directly from the plot of experimental results.

4. Experiment

Knowledge of parameters H, V and δ of a physical system gives information about the system itself; on the other hand, features of different systems can be compared when these parameters are known for each of them. In the following, we report results of the above described experiment, performed on a POLICRYPS grating. In order to compare this system with other birefringent ones which can, eventually, show a dichroic behavior, we have analyzed the characteristic of both a HPDLC grating and a thin film of NLC with planar alignment. For each sample, parameters H and V and the phase retardation δ are calculated by means of Eqs. (6-8), while experimental curves of the intensity Iout, transmitted by the analyzer for different values of β in the interval 0≤β≤2π, are compared with theoretical predictions obtained by means of Eq. (5). Results show a good agreement in all considered cases and confirm the validity of our assumptions. In the following we describe in detail the features of investigated systems and results obtained from experiments.

a) POLICRYPS grating

In Fig. 4, segments of different length put into evidence output intensity values for β=0 and β=π/2 respectively. By means of Eq. (9), these values enable to calculate how large is the dichroism of the birefringent sample.

b) Comparison with other soft matter systems

i) HPDLC grating

Where the HPDLC is concerned, we have fabricated a sample with the same fringe spacing of the POLICRYPS (Λ=1.22μm) but a slightly different thickness (L=1.9μm instead of 3.03μm). The main difference between HPDLC and POLICRYPS gratings is in their morphology (Fig. 5a
Fig. 5 Scanning Electron Microscope comparison between (a) a typical HPDLC and (b) a POLICRYPS morphology.
and Fig. 5b). In a typical HPDLC, the liquid crystal nucleates into droplets where the configuration of the LC director is, in general, the bipolar one. Unless particular actions are made on the sample, (e.g. stretching in a particular direction [9

9. V. P. Tondiglia, R. L. Sutherland, L. V. Natarajan, P. F. Lloyd, and T. J. Bunning, “Droplet deformation and alignment for high-efficiency polarization-dependent holographic polymer-dispersed liquid-crystal reflection gratings,” Opt. Lett. 33(16), 1890–1892 (2008). [CrossRef] [PubMed]

]), orientation of the main axis of different droplets is completely random. The HPDLC exhibits, therefore, a quite low value of birefringence, as shown in Fig. 6
Fig. 6 Behavior of the intensity transmitted by the analyzer put after a HPDLC grating as a function of the angle β between the electric field of the impinging wave and the axis of the analyzer itself. Two segments in the graph evidence output intensity values for the analyzer positions β=0 and β=π/2 respectively. Experimental error is of the order of the dimension of crosses.
.

Calculations made by using Eqs. (6-8) give H=0.630, V=0.699 and δ=0.198rad, which corresponds, for a probe wavelength λ=632.8nm (He-Ne Laser), to a very low birefringence of the structure: Δn=δλ/2πL=0.010. This result confirms above considerations on the average orientation of the birefringent material. Also in this case, by comparing the output intensity values measured for β=0 and β=π/2, we get that the dichroism is very limited.

ii) Thin film of liquid crystals

A thin film of planarly aligned NLC represented, for a long time, the basic system for realizing a tuneable phase retarder. Its simplicity yields, however, its main drawback: the anchoring forces which induce the director alignment are due to the glass substrates, prepared to give a planar orientation of molecules; thus, the NLC alignment can be easily perturbed by external means (e.g. a high power probe beam). A NLC cell can be referred to, therefore, as a standard reference system, to be compared with structures where the NLC material is stabilized by a polymer (HPDLC and POLICRYPS). The sample has been fabricated by filling in with E7 NLC a standard cell (L=2.1 μm, produced by EHC Corporation) whose glass substrates have been treated to give a planar alignment of the NLC director. A check of the sample at the optical microscope, between crossed polarizers, evidences a good alignment of the NLC material. Results for the transmitted intensity, obtained by using our technique, are plotted in Fig. 7
Fig. 7 Behavior of the intensity transmitted by the analyzer put after a thin film of NLC as a function of the angle β between the electric field of the impinging wave and the axis of the analyzer itself. Two segments in the graph evidence output intensity values for the analyzer positions β=0 and β=π/2 respectively. Experimental error is of the order of the dimension of crosses.
. Calculation of H and V coefficients provides in this case the values H=0.719, V=0.693, with a retardation δ=1.76rad, which corresponds, for a thickness L=2.1μm and a probe wavelength λ=632.8nm (He-Ne Laser), to a birefringence Δnavg=0.147.

This value is slightly lower than the ideal one ΔnLC=0.214 (at λ=632.8nm) expected for a perfectly aligned NLC film, and confirms that our technique can be efficiently used also to obtain information about the molecular organization of the birefringent material under investigation.

5. Discussion and conclusions

In this paper we have developed an implementation of the Jones Matrix formalism which enables to describe the behavior of light transmitted by structures containing birefringent materials and exhibiting polarization selective absorption of light (dichroism). The idea came from the need of describing the behavior of a POLICRYPS diffraction grating when used as a phase retarder. We realized soon that the technique can have a more general validity, since it can be used to describe any birefringent material which behaves as a dichroic absorber too. In order to check the model, we have chosen, at first, a POLICRYPS diffraction grating and we have designed an experiment devoted to measure the intensity of light transmitted by an analyzer put after the sample. The same experiment has been repeated with other, different, physical systems (HPDLC diffraction grating, thin film of aligned NLC) in order to confirm and compare obtained results. In all considered cases, results show a good agreement with theoretical predictions. Moreover, the evaluation of the birefringence of the different structures has been used for carrying out a comparison of their properties. In particular, we have observed that the degree of order of the birefringent material is very high in the sample containing only an aligned NLC film, while it is lower in geometries where the liquid crystal material is confined and stabilized by a polymer structure (HPDLC and POLICRYPS gratings). It is worth noting that the birefringence value drastically increases (about five times) when we move from the HPDLC grating, where a stochastic distribution of the nematic droplets is present, to the POLICRYPS structure, which contains layers of nematic material homogeneously aligned. This result puts into evidence, once more, the high quality of POLICRYPS and the possibility to exploit these systems in technological applications as switchable phase retarders.

Acknowledgments

This research has been supported by PRIN 2006 - Umeton - prot. 2006022132_001.

References

1.

E. Bartholinus, Experimenta crystalli islandici disdiaclastici quibus mira & insolita refractio detegitur, (Hafniæ, Denmark 1669).

2.

L. De Sio, N. V. Tabyrian, R. Caputo, A. Veltri, and C. Umeton, “POLICRYPS Structures as Switchable Optical Phase Modulators,” Opt. Express 16(11), 7619 (2008). [CrossRef] [PubMed]

3.

R. Caputo, L. De Sio, A. Veltri, C. Umeton, and A. V. Sukhov, “Development of a new kind of switchable holographic grating made of liquid-crystal films separated by slices of polymeric material,” Opt. Lett. 29(11), 1261–1263 (2004). [CrossRef] [PubMed]

4.

M. Born, and E. Wolf, Principles of Optics (Pergamon, New York, 1980).

5.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).

6.

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811 (1981). [CrossRef]

7.

R. Caputo, A. V. Sukhov, C. Umeton, and R. F. Ushakov, “Formation of a Grating of Submicron Nematic Layers by Photopolymerization of Nematic-Containing Mixtures,” J. Exp. Theor. Phys. 91(6), 1190–1197 (2000). [CrossRef]

8.

R. C. Jones, “New calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31(7), 488–493 (1941). [CrossRef]

9.

V. P. Tondiglia, R. L. Sutherland, L. V. Natarajan, P. F. Lloyd, and T. J. Bunning, “Droplet deformation and alignment for high-efficiency polarization-dependent holographic polymer-dispersed liquid-crystal reflection gratings,” Opt. Lett. 33(16), 1890–1892 (2008). [CrossRef] [PubMed]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(160.3710) Materials : Liquid crystals
(230.1950) Optical devices : Diffraction gratings
(230.5440) Optical devices : Polarization-selective devices

ToC Category:
Optical Devices

History
Original Manuscript: December 3, 2009
Revised Manuscript: February 2, 2010
Manuscript Accepted: February 8, 2010
Published: March 8, 2010

Citation
Roberto Caputo, Ivan Trebisacce, Luciano De Sio, and Cesare Umeton, "Jones matrix analysis of dichroic phase retarders realized in soft matter composite materials.," Opt. Express 18, 5776-5784 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-5776


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References

  1. E. Bartholinus, Experimenta crystalli islandici disdiaclastici quibus mira & insolita refractio detegitur, (Hafniæ, Denmark 1669).
  2. L. De Sio, N. V. Tabyrian, R. Caputo, A. Veltri, and C. Umeton, “POLICRYPS Structures as Switchable Optical Phase Modulators,” Opt. Express 16(11), 7619 (2008). [CrossRef] [PubMed]
  3. R. Caputo, L. De Sio, A. Veltri, C. Umeton, and A. V. Sukhov, “Development of a new kind of switchable holographic grating made of liquid-crystal films separated by slices of polymeric material,” Opt. Lett. 29(11), 1261–1263 (2004). [CrossRef] [PubMed]
  4. M. Born, and E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  5. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
  6. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811 (1981). [CrossRef]
  7. R. Caputo, A. V. Sukhov, C. Umeton, and R. F. Ushakov, “Formation of a Grating of Submicron Nematic Layers by Photopolymerization of Nematic-Containing Mixtures,” J. Exp. Theor. Phys. 91(6), 1190–1197 (2000). [CrossRef]
  8. R. C. Jones, “New calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31(7), 488–493 (1941). [CrossRef]
  9. V. P. Tondiglia, R. L. Sutherland, L. V. Natarajan, P. F. Lloyd, and T. J. Bunning, “Droplet deformation and alignment for high-efficiency polarization-dependent holographic polymer-dispersed liquid-crystal reflection gratings,” Opt. Lett. 33(16), 1890–1892 (2008). [CrossRef] [PubMed]

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