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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 8116–8124
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Point diffraction interferometer with a liquid crystal monopixel

Claudio Ramírez, Eva Otón, Claudio Iemmi, Ignacio Moreno, Noureddine Bennis, José Manuel Otón, and Juan Campos  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 8116-8124 (2013)
http://dx.doi.org/10.1364/OE.21.008116


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Abstract

In this work a novel point diffraction interferometer based on a variable liquid crystal wave plate (LCWP) has been implemented. The LCWP consists of a 3x3 cm2 monopixel cell with parallel alignment. The monopixel cell was manufactured such that the electrode covers the entire surface except in a centered circular area of 50 μm of diameter. This circle acts as a point perturbation which diffracts the incident wave front giving rise to a spherical reference wave. By applying a voltage to the LCWP we can change the phase of the wave front that passes through the monopixel, except at the center. Phase shifting techniques are used in order to calculate the amplitude and phase distribution of the object wave front. The system allows a digital hologram to be obtained, and by using the Fresnel diffraction integral it is possible to digitally reconstruct the different planes that constitute the three dimensional object.

© 2013 OSA

1. Introduction

Several other works have made proposals of point diffraction interferometers by using polarization optics. Some of them [7

7. G. Wang, Y. Zheng, A. Sun, S. Wu, and Z. Wang, “Polarization pinhole interferometer,” Opt. Lett. 16(17), 1352–1354 (1991). [CrossRef] [PubMed]

] and more recently [8

8. H. Kihm and Y. Lee, “Polarization point diffraction interferometer for fringe stabilization,” Proc. SPIE 7790, 779013 (2010). [CrossRef]

] use pinholes in polarizers and waveplates to allow the interferences to form. These proposals cannot perform phase shifting between the object and reference beams, making thus more difficult the fringe analysis. Neal and Wyant have proposed [9

9. R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt. 45(15), 3463–3476 (2006). [CrossRef] [PubMed]

] a system able to perform the phase shifting. First a linear polarization state is generated and with an electro-optical modulator the relative phase between the s and p components can be varied. The point diffraction is edged in a half wave-plate at 45 degrees. By means of a polarizer the s and p components are projected to allow the interference to be formed. In [10

10. M. Paturzo, F. Pignatiello, S. Grilli, S. De Nicola, and P. Ferraro, “Phase-shifting point-diffraction interferometer developed by using the electro-optic effect in ferroelectric crystals,” Opt. Lett. 31(24), 3597–3599 (2006). [CrossRef] [PubMed]

] a simpler configuration is proposed by using a z-cut lithium niobate crystal. A thin aluminum layer is deposited on both sides and a pinhole is edged in one side. By applying a voltage the relative phase between the two beams can be changed. This device has the advantage that can be operated a very fast speed, and as the authors said the main drawback is the high voltage necessary to use in comparison with devices based on liquid crystals.

In this work we propose the fabrication and use of a liquid crystal monopixel cell with parallel alignment. The electrode covers the whole liquid crystal cell, except a circular area of 50 μm at center. When a voltage is applied the phase of the wave front that impinges on the liquid crystal is modified, except in this circular region. Therefore, this monopixel allows efficiently implementing a PDI of common path with capability of interferometry by phase shifting, without any additional artifact produced by the pixelated structure of a microdisplay.

In section 2 we explain the monopixel fabrication process. In section 3 we describe the proposed PDI and the method for calculating the amplitude and phase distribution of the object wave front. Finally, in section 4 we show the obtained experimental results.

2. Liquid crystal monopixel cell

The monopixel device is based on a liquid crystal cell having an active area of 3 × 3 cm2 approximately. The cell is a sandwich-like flat structure, formed by two ultra-flat glasses of 0.7 mm of thickness between which a thin layer of liquid crystal is trapped. All the process is performed in a clean room.

The inner surfaces of the glasses are coated with a transparent semiconductor (Indium Tin Oxide, ITO) which acts as electrode (about 20 Ω/☐). The coating on one of the glasses is left intact. In the opposite glass, a circular area of 50 μm diameter is removed by means of a photolithographic process, as is shown in Fig. 1
Fig. 1 Monopixel of liquid crystal wave plate (LCWP), with parallel alignment, made by Universidad Politécnica de Madrid. On the right, a magnified view of the central part to see the circular area without electrode on one of the glass plates of the monopixel.
.

Next, a polyimide layer (PIA2000, Chisso) is spin-coated onto the ITO electrode of both glasses. The polyimide (actually, the monomer) is cured in an oven to promote polymerization, and eventually is mechanically rubbed with a special lint-free cloth mounted on a cylinder onto which the glasses circulate under controlled conditions. The buffing process induces a preferential direction on the polyimide layer of the glasses, which eventually will determine the orientation of the liquid crystal layer.

Once the surface orientation is conditioned, the cell is mounted by placing the active surfaces in front to each other. Calibrated silica microspheres of 4 μm diameter are deposited on one of the surfaces, with aid of an aerograph. The microspheres act as spacers, allowing assembling the cell with a predetermined separation between the glasses. The cell edges are sealed with a serigraphed adhesive (gasket), leaving just a slit in the center of one of the edges. The glasses do not overlap fully to each other. A small lip is intentionally left on opposite sides, so that the inner surface of each glass is accessible from the outside for electrical connections.

The result of the above processes is ultimately a prismatic bottle of 3 cm × 3 cm × 4 μm. This bottle is filled, inside a vacuum chamber, with liquid crystal MDA-98 (Merck). The slit is then sealed with a photopolymerizable adhesive (NOA-65). Indium-metal contacts are soldered to the ITO electrodes of the inner surfaces, in order to apply the desired electric signals to the liquid crystal.

Surface conditioning induces the liquid crystal to orient parallel to the glass surfaces, along the predefined rubbing direction (so called homogeneous alignment). On the other hand, MDA-98 is a liquid crystal mixture having positive dielectric anisotropy (i.e., the dielectric constant along the molecular axis is higher than the dielectric constant of its perpendicular plane) and high birefringence (Δn = 0.266 @ 589 nm). The positive dielectric anisotropy will induce the liquid crystal molecules to orient their axes parallel to the electric field. The birefringence will then induce phase delays between the switched and unswitched regions of the liquid crystal cell.

Therefore, when an electric signal is applied to the electrodes, the liquid crystal molecules tend to be oriented, aligning its director axis parallel to the applied electric field (i.e., perpendicular to the glass plates). As the initial position –induced by the conditioning surface– is parallel to the glass plates, a reorientation of the liquid crystal molecules occurs, modifying the refractive index that “sees” the light with linear polarization parallel to the orientation direction. Thus, a variable delay controlled by the external voltage is produced. The liquid crystal located inside the central circle does not switch (the electrode has been removed), so that a variable phase difference is generated between this circular area and the remaining area of the monopixel. The aspect ratio of the circle (50:4 μm diameter-thickness) is flat enough, so that the deformation effects of the electric field in the circle edges (fringing) can be considered negligible.

3. Point diffraction interferometer

The transfer function of the (LCWP) can be described as

H(u)=exp(i2πnN)+δ(u)[exp(i2πnN)+1],
(1)
δ(u)={1u=00u0.

We can rewrite Eq. (1) as
H(u)=exp(iϕn)+δ(u)[exp(iϕn)+1],
(2)
where ϕn=2πnN is the phase delay introduced at each of the N steps employed to apply the phase shifting interferometry technique.

Let us define the complex amplitude of the object wave front as

O(x)=o(x)exp[iΨ(x)].
(3)

In this way the complex amplitude on the CCD plane can be expressed as the convolution of the object wave front with the impulse response of the (LCWP):

An(x)=O(x)H(x).
(4)

The convolution theorem lets us rewrite Eq. (4) as

An(x)=F1{O(u)H(u)},An(x)=F1{O(u)[exp(iϕn)+δ(u)(exp(iϕn)+1)]},An(x)=o(x)exp[i(Ψ(x)+ϕn)][exp(iϕn)1]F1{O(u)δ(u)}.
(5)

If the complex constant K=|K|exp(iσ) is defined as the average value of O(x) then

An(x)=o(x)exp[i(Ψ(x)+ϕn)][exp(iϕn)1]K.
(6)

Keeping in mind that the CCD detects intensity, i.e., the square modulus of the amplitude An(x), and doing the substitution o^=o(x)exp[i(Ψ(x)+ϕn)] yields

|An|2=|o^[exp(iϕn)1]K|2,
(7a)
|An|2=|o^|2o^[exp(iϕn)1]Ko^[exp(iϕn)1]K+[exp(iϕn)1][exp(iϕn)1]|K|2,|An|2=|o^|2+2|K|2+[2|K|2cos(ϕn)+|K|2cos(σΨ(x))o(x)]cos(ϕn)|K|2cos[σΨ(x)]o(x)+|K|2sin[σΨ(x)]sin(ϕn)o(x).
(7b)

Let us multiply the detected intensities bycos(ϕn) and sin(ϕn) in order to obtain the terms

C=n=0N1|An|2cos(ϕn)S=n=0N1|An|2sin(ϕn).
(8)

Then, substituting Eq. (7b) into Eq. (8) and using the orthogonality properties of sinusoidal functions, the following relations can be obtained:

C=N2[2|K|2+|K|2cos(σΨ(x))o(x)],
S=N2[|K|2sin(σΨ(x))o(x)],
C=N|K|2+N|K|cos[σΨ(x)]o(x),
(9a)
S=N|K|sin[σΨ(x)]o(x).
(9b)

The images captured by the CCD camera contain regions that belong to the image of the aperture placed behind the lens L1. In these regions only the constant term proportional to |K|2 appears, so it can be determined.

By dividing Eq. (9)a) by Eq. (9b), one obtains

CS=N|K|2+N|K|cos(σΨ(x))o(x)N|K|sin(σΨ(x))o(x)=|K|sin(σΨ(x))o(x)+cos(σΨ(x))sin(σΨ(x)),cos(σΨ(x))sin(σΨ(x))=CS+|K|sin(σΨ(x))o(x)=CS+N|K|2S=C+N|K|2S,tan(σΨ(x))=sin(σΨ(x))cos(σΨ(x))=SC+N|K|2.
(10)

Finally the unknown phase of the object wave front can be recovered by solving Ψ(x) from Eq. (10):
Ψ(x)σ=arctan(SC+N|K|2),
(11)
where σ is a constant phase that does not affect the complex amplitude reconstruction of the object. The magnitude of the wave-front can be obtained directly from Eq. (7a) by taking the root square of the intensity registered by the CCD camera when ϕn = 0, i.e. when no voltage is applied to the cell.

4. Focusing objects placed at different planes

Once the complex amplitude is known at a given plane, it can be digitally evaluated at other parallel planes by using the Fresnel diffraction integral [11

11. J. W. Goodman, Chapter IV “Fresnel and Fraunhofer diffraction” in Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

13

13. U. Schnars and W. Jueptner, “Digital holography, numerical reconstruction” in Digital Holography (Springer Verlag, 2005), Chap. 3.2.

]:

O(x,y)=exp(ikd)idλ{O(x,y)exp[ik2d((xx)2+(yy)2)]dxdy}.
(12)

Let us group terms and define h(x′, y′) as the optical transfer function at plane (x′, y′), after propagating a distance d:

h(xx,yy)=exp(ikd)idλexp[ik2d((xx)2+(yy)2)].
(13)

The Fresnel integral (12) can be rewritten as the convolution of the complex amplitude of the object wave front O(x, y) with the optical transfer function h(x′, y′):

O(x,y)={O(x,y)h(xx,yy)dxdy}.
(14)

By applying the convolution theorem in Eq. (14), one obtains
F{O(x,y)}=O(fx,fy)H(fx,fy),
(15)
where

O(fx,fy)=F{O(x,y)},
(16)
H(fx,fy)=F{exp(ikd)idλexp[ik2d(x2+y2)]},H(fx,fy)=exp(ikd)exp[iπλd(fx2+fy2)].
(17)

We use the inverse Fourier transform in order to obtain a more convenient expression to the near field propagation, due to its dependency with distance d in the phase factor. The result is

O(x,y)=F1{O(fx,fy)H(fx,fy)},
(18)
O(x,y)=exp(ikd)F1{F[O(x,y)]exp[iπλd(fx2+fy2)]}.
(19)

5. Experimental results

Before using the LC device in the interferometer, we have experimentally determined the axis orientation of the liquid crystal (α=90.77º with respect to the laboratory vertical) and the phase delay as a function of the voltage applied between the electrodes (see Fig. 3
Fig. 3 LCWP phase retardation versus applied voltage. Dots indicate the experimental data, and the continuous line the corresponding polynomial fit. Negligible phase modulation is observed for values voltages below of 1 V o above of 3.4 V.
) [14

14. I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express 16(21), 16711–16722 (2008). [CrossRef] [PubMed]

,15

15. C. Ramirez, B. Karakus, A. Lizana, and J. Campos, “Polarimetric method for liquid crystal displays characterization in presence of phase fluctuations,” Opt. Express (to be published).

].

To know the voltage necessary to obtain the required phase shift, a look up table (LUT) was obtained by means of a polynomial fit of the measured phases. The electrodes of the LCWP were connected to a voltage source that applies a square bipolar signal between –V and +V at a frequency of 1 Khz. The applied voltage V can be varied by a digital to analog converter controlled by a PC.

The experimental set up is shown in Fig. 4
Fig. 4 Point diffraction interferometer implemented in the laboratory.
. We used a convergent beam from a HeNe laser (λ = 632.8 nm) as light source. This beam impinges a rotating diffuser to decrease the coherence and the associated noise. A circular aperture was employed to limit the wave front extension of the laser beam. Lens L2 in Fig. 2 forms the image of this stop on the CCD camera. In this way we can calculate the constant K of Eq. (6). To test the ability of the proposed technique to focus different planes of an extended object, we placed two thin plates of glass (microscope slides denoted as Object plane 1 and 2) in separated parallel planes with printed letters, and a reticule in diagonal (Object 3D in Fig. 4). Lenses L1 and L2 have focal length of 20 cm. Object planes 1 and 2 are separated approximately 35 cm each other. Object plane 1 is approximately 10 cm apart from Lens L1.

It should be pointed out that when there is no object in the set-up, the convergent wave front is focused on the LCWP in the circular area without electrode. In this case, the point spread function is about the same size as that of the circular area.

The four step phase shift algorithm was employed to obtain the complex distribution of the object wave front. The interferograms acquired with the CCD, for phase delays of 0, π/2, π and 3π/2 between object and reference wave fronts, are shown in Fig. 5
Fig. 5 Fringe patterns with phase retardance between the object and diffracted wave fronts of: (a) ϕn = 0, (b) ϕn = 0.5π, (c) ϕn = π and (d) ϕn = 1.5π.
. The wrapped phase of the original wave front was evaluated by using Eqs. (8) and (11).

Figure 6
Fig. 6 (a) Magnitude of the object wave front o(x), (b) calculated wrapped phase Ψ(x).
represents the magnitude and phase of the wave-front corresponding to the focused plane at the CCD camera, which coincides with the circular aperture plane of Fig. 4. One can notice (Fig. 6(a)) that the characters and the rule are defocused, while the edges of the diaphragm are on focus.

By applying the Fresnel propagation Eq. (19) one can digitally focus other planes that were defocused on the original image, as is illustrated in Fig. 7
Fig. 7 Focusing different image planes using Fresnel diffraction equation: (a) d = −18.7 mm, (b) d ≈-35 mm y (c) d = −56.5 mm. (Media 1).
. At d = 0 we obtain the result of Fig. 6(a). By selecting different planes along the z axis we can see how the successive planes are focused (first the letter J, then the reticule and the letter O). In the movie more images are taken around these planes. Therefore, these results show evidences of the proposed optical system.

6. Conclusions

We have implemented a novel point diffraction interferometer (PDI) based in a liquid crystal wave plate (LCWP). In the LCWP the electrode covers whole the surface, except by a circular area of 50 μm of diameter. This area acts as a point that diffracts a wave front. The phase of the wave front that impinges on the monopixel, in contrast to the central point, is changed by applying different voltages to the LCWP. A phase shifting technique was used to obtain the amplitude and phase distribution of the object wave front. We have applied this system to obtain a digital hologram and, by digitally focusing in several planes, have used this digital hologram to reconstruct a three-dimensional object using the Fresnel diffraction integral.

Acknowledgments

This work has been funded by Ministerio de Ciencia e Innovación and FEDER funds (FIS2009-13955-C02-01 and 02). C. Iemmi appreciates the support from the Universidad de Buenos Aires and CONICET (Argentina).

References and links

1.

U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33(2), 179–181 (1994). [CrossRef] [PubMed]

2.

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]

3.

V. Linnik, “Simple interferometer for the investigation of optical systems,” Proc. Acad. Sci. USSR 1, 208–210 (1933).

4.

C. R. Mercer and K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett. 19(12), 916–918 (1994). [CrossRef] [PubMed]

5.

C. Iemmi, A. Moreno, J. Nicolás, and J. Campos, “Evaluation and correction of aberrations in an optical correlator by phase-shifting interferometry,” Opt. Lett. 28(13), 1117–1119 (2003). [CrossRef] [PubMed]

6.

C. Iemmi, A. Moreno, and J. Campos, “Digital holography with a point diffraction interferometer,” Opt. Express 13(6), 1885–1891 (2005). [CrossRef] [PubMed]

7.

G. Wang, Y. Zheng, A. Sun, S. Wu, and Z. Wang, “Polarization pinhole interferometer,” Opt. Lett. 16(17), 1352–1354 (1991). [CrossRef] [PubMed]

8.

H. Kihm and Y. Lee, “Polarization point diffraction interferometer for fringe stabilization,” Proc. SPIE 7790, 779013 (2010). [CrossRef]

9.

R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt. 45(15), 3463–3476 (2006). [CrossRef] [PubMed]

10.

M. Paturzo, F. Pignatiello, S. Grilli, S. De Nicola, and P. Ferraro, “Phase-shifting point-diffraction interferometer developed by using the electro-optic effect in ferroelectric crystals,” Opt. Lett. 31(24), 3597–3599 (2006). [CrossRef] [PubMed]

11.

J. W. Goodman, Chapter IV “Fresnel and Fraunhofer diffraction” in Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

12.

M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1(1), 018005 (2010). [CrossRef]

13.

U. Schnars and W. Jueptner, “Digital holography, numerical reconstruction” in Digital Holography (Springer Verlag, 2005), Chap. 3.2.

14.

I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express 16(21), 16711–16722 (2008). [CrossRef] [PubMed]

15.

C. Ramirez, B. Karakus, A. Lizana, and J. Campos, “Polarimetric method for liquid crystal displays characterization in presence of phase fluctuations,” Opt. Express (to be published).

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(230.3720) Optical devices : Liquid-crystal devices
(090.1995) Holography : Digital holography

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: October 18, 2012
Revised Manuscript: January 5, 2013
Manuscript Accepted: January 6, 2013
Published: March 27, 2013

Citation
Claudio Ramírez, Eva Otón, Claudio Iemmi, Ignacio Moreno, Noureddine Bennis, José Manuel Otón, and Juan Campos, "Point diffraction interferometer with a liquid crystal monopixel," Opt. Express 21, 8116-8124 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8116


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References

  1. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt.33(2), 179–181 (1994). [CrossRef] [PubMed]
  2. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997). [CrossRef] [PubMed]
  3. V. Linnik, “Simple interferometer for the investigation of optical systems,” Proc. Acad. Sci. USSR1, 208–210 (1933).
  4. C. R. Mercer and K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett.19(12), 916–918 (1994). [CrossRef] [PubMed]
  5. C. Iemmi, A. Moreno, J. Nicolás, and J. Campos, “Evaluation and correction of aberrations in an optical correlator by phase-shifting interferometry,” Opt. Lett.28(13), 1117–1119 (2003). [CrossRef] [PubMed]
  6. C. Iemmi, A. Moreno, and J. Campos, “Digital holography with a point diffraction interferometer,” Opt. Express13(6), 1885–1891 (2005). [CrossRef] [PubMed]
  7. G. Wang, Y. Zheng, A. Sun, S. Wu, and Z. Wang, “Polarization pinhole interferometer,” Opt. Lett.16(17), 1352–1354 (1991). [CrossRef] [PubMed]
  8. H. Kihm and Y. Lee, “Polarization point diffraction interferometer for fringe stabilization,” Proc. SPIE7790, 779013 (2010). [CrossRef]
  9. R. M. Neal and J. C. Wyant, “Polarization phase-shifting point-diffraction interferometer,” Appl. Opt.45(15), 3463–3476 (2006). [CrossRef] [PubMed]
  10. M. Paturzo, F. Pignatiello, S. Grilli, S. De Nicola, and P. Ferraro, “Phase-shifting point-diffraction interferometer developed by using the electro-optic effect in ferroelectric crystals,” Opt. Lett.31(24), 3597–3599 (2006). [CrossRef] [PubMed]
  11. J. W. Goodman, Chapter IV “Fresnel and Fraunhofer diffraction” in Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  12. M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev.1(1), 018005 (2010). [CrossRef]
  13. U. Schnars and W. Jueptner, “Digital holography, numerical reconstruction” in Digital Holography (Springer Verlag, 2005), Chap. 3.2.
  14. I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express16(21), 16711–16722 (2008). [CrossRef] [PubMed]
  15. C. Ramirez, B. Karakus, A. Lizana, and J. Campos, “Polarimetric method for liquid crystal displays characterization in presence of phase fluctuations,” Opt. Express (to be published).

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