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.

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References

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    [CrossRef]
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2010 (2)

H. Kihm and Y. Lee, “Polarization point diffraction interferometer for fringe stabilization,” Proc. SPIE7790, 779013 (2010).
[CrossRef]

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

2008 (1)

2006 (2)

2005 (1)

2003 (1)

1997 (1)

1994 (2)

1991 (1)

1933 (1)

V. Linnik, “Simple interferometer for the investigation of optical systems,” Proc. Acad. Sci. USSR1, 208–210 (1933).

Campos, J.

Creath, K.

De Nicola, S.

Fernández, E.

Ferraro, P.

Grilli, S.

Iemmi, C.

Jüptner, W.

Karakus, B.

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).

Kihm, H.

H. Kihm and Y. Lee, “Polarization point diffraction interferometer for fringe stabilization,” Proc. SPIE7790, 779013 (2010).
[CrossRef]

Kim, M. K.

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

Lee, Y.

H. Kihm and Y. Lee, “Polarization point diffraction interferometer for fringe stabilization,” Proc. SPIE7790, 779013 (2010).
[CrossRef]

Linnik, V.

V. Linnik, “Simple interferometer for the investigation of optical systems,” Proc. Acad. Sci. USSR1, 208–210 (1933).

Lizana, A.

Márquez, A.

Mercer, C. R.

Moreno, A.

Moreno, I.

Neal, R. M.

Nicolás, J.

Paturzo, M.

Pignatiello, F.

Ramirez, C.

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).

Schnars, U.

Sun, A.

Wang, G.

Wang, Z.

Wu, S.

Wyant, J. C.

Yamaguchi, I.

Yzuel, M. J.

Zhang, T.

Zheng, Y.

Appl. Opt. (2)

Opt. Express (3)

Opt. Lett. (5)

Proc. Acad. Sci. USSR (1)

V. Linnik, “Simple interferometer for the investigation of optical systems,” Proc. Acad. Sci. USSR1, 208–210 (1933).

Proc. SPIE (1)

H. Kihm and Y. Lee, “Polarization point diffraction interferometer for fringe stabilization,” Proc. SPIE7790, 779013 (2010).
[CrossRef]

SPIE Rev. (1)

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

Other (2)

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

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

Supplementary Material (1)

» Media 1: AVI (2594 KB)     

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Figures (7)

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.

Fig. 2
Fig. 2

Experimental setup of the Point Diffraction Interferometer. The laser beam is expanded and filtered with a spatial filter. LP1 is a linear polarizer with its transmission axis parallel to the optical axis of the LCWP. Next of the liquid crystal, a diffracted wave front is generated. Lens L2 is used to focusing one object plane (object wave) and diffracted wave (reference wave) on CCD.

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.

Fig. 4
Fig. 4

Point diffraction interferometer implemented in the laboratory.

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.

Fig. 6
Fig. 6

(a) Magnitude of the object wave front o(x), (b) calculated wrapped phase Ψ(x).

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).

Equations (24)

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H( u )=exp( i 2πn N )+δ( u )[ exp( i 2πn N )+1 ],
δ( u )={ 1 u=0 0 u0 .
H( u )=exp( i ϕ n )+δ( u )[ exp( i ϕ n )+1 ],
O( x )=o( x )exp[ iΨ( x ) ].
A n ( x )=O( x )H( x ).
A n ( x )= F 1 { O( u )H( u ) }, A n ( x )= F 1 { O( u )[ exp( i ϕ n )+δ( u )( exp( i ϕ n )+1 ) ] }, A n ( x )=o( x )exp[ i( Ψ( x )+ ϕ n ) ][ exp( i ϕ n )1 ] F 1 { O( u )δ( u ) }.
A n ( x )=o( x )exp[ i( Ψ( x )+ ϕ n ) ][ exp( i ϕ n )1 ]K.
| A n | 2 = | o ^ [ exp( i ϕ n )1 ]K | 2 ,
| A n | 2 = | o ^ | 2 o ^ [ exp( i ϕ n )1 ] K o ^ [ exp( i ϕ n )1 ]K+[ exp( i ϕ n )1 ][ exp( i ϕ n )1 ] | K | 2 , | A n | 2 = | o ^ | 2 +2 | K | 2 +[ 2 | K | 2 cos( ϕ n )+| K |2cos( σΨ( x ) )o( x ) ]cos( ϕ n ) | K |2cos[ σ Ψ( x ) ]o( x )+| K |2sin[ σ Ψ( x ) ]sin( ϕ n )o( x ) .
C= n=0 N1 | A n | 2 cos( ϕ n ) S= n=0 N1 | A n | 2 sin( ϕ n ) .
C= N 2 [ 2 | K | 2 +| K |2cos( σ Ψ( x ) )o( x ) ],
S= N 2 [ | K |2sin( σ Ψ( x ) )o( x ) ],
C=N | K | 2 +N| K |cos[ σΨ( x ) ]o( x ),
S=N| K |sin[ σΨ( x ) ]o( x ).
C S = 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 ) ) = C S + | K | sin( σΨ( x ) )o( x ) = C S + N | K | 2 S = C+N | K | 2 S , tan( σΨ( x ) )= sin( σΨ( x ) ) cos( σΨ( x ) ) = S C+N | K | 2 .
Ψ( x )σ=arctan( S C+N | K | 2 ),
O( x , y )= exp( ikd ) idλ { O( x,y )exp[ ik 2d ( ( x x ) 2 + ( y y ) 2 ) ]dxdy } .
h( x x, y y )= exp( ikd ) idλ exp[ ik 2d ( ( x x ) 2 + ( y y ) 2 ) ].
O( x , y )= { O( x,y )h( x x, y y )dxdy } .
F{ O( x , y ) }=O( f x , f y )H( f x , f y ),
O( f x , f y )=F{ O( x,y ) },
H( f x , f y )=F{ exp( ikd ) idλ exp[ ik 2d ( x 2 + y 2 ) ] }, H( f x , f y )=exp( ikd )exp[ iπλd( f x 2 + f y 2 ) ].
O( x , y )= F 1 { O( f x , f y )H( f x , f y ) },
O( x , y )=exp( ikd ) F 1 { F[ O( x,y ) ]exp[ iπλd( f x 2 + f y 2 ) ] }.

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