Abstract

A random phase plate is prepared by illuminating a photoresist plate with a fully developed speckle field and using the developed phase plate (DPP) as a diffuser. Wavefront sensing is implemented using phase retrieval based on the recording of speckle intensity patterns at various distances from the DPP and the wave propagation equation. The effects of the roughness height of the DPP on the phase retrieval are investigated. From simulations a roughness height of λ/10 results in a speckle field that yields good phase reconstruction for the spherical test wavefront incident on the DPP. From the experiments different portions of the DPP that received varying exposures are examined. A section of the phase plate with a characteristic roughness height facilitated the generation of a speckle field that is optimum for the phase retrieval algorithm. Thus a random phase plate with varying roughness height allows optimized measurements of wavefronts with different curvatures. Analytical expressions describing the second-order intensity statistics (fourth-order field statistics) for a field traversing a specific diffuser are presented. This DPP will not give rise to a fully developed speckle field, but knowing the statistics of the depth of the DPP will facilitate a rigorous treatment of the problem.

© 2008 Optical Society of America

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References

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2008 (1)

2007 (4)

2006 (1)

2005 (1)

2004 (1)

1999 (1)

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

1998 (1)

1987 (1)

1984 (1)

1973 (1)

Aksenov, V.

Almoro, P.

Anand, A.

Atuchin, V.

Bao, P.

Benzoni, J.

Chang, S.-I.

Danson, C.

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

DePalma, J.

Doyle, L.

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

Escamilla, H.

Garcia-Guerrero, E.

Goodman, J.

J. Goodman, Statistical Optics (Wiley, 2000).

J. Goodman, Speckle Phenomena in Optics (Roberts & Company, 2006), Eq. 4-41, p. 69.

Hanson, S.

Hoadley, H.

Izmailov, I.

Kanev, F.

Kim, H.

Kim, J.-J.

Kochemasov, G.

Kowalczyk, M.

Kulikov, S.

Kurtz, C.

Lee, B.-K.

Leskova, T.

Lewis, C.

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

Manachinsky, A.

Maradudin, A.

Maradudin, E.

Martin, G.

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

Maslov, N.

Mendez, E.

Morrow, T.

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

Munoz-Lopez, J.

Ogorodnikov, A.

Osten, W.

Pedrini, G.

Pepler, D.

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

Rose, B.

Ross, I.

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

Sarkar, S.

Sherrington, D.

Shin, D. H.

Soldatenkov, I.

Starikov, F.

Sukharev, S.

Weaver, I.

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

Yoon, J.-B.

Yura, H.

Zhang, F.

P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. 33, 309-311 (2008).
[CrossRef] [PubMed]

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Zhang, Y.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. A (1)

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

Rev. Sci. Instrum. (1)

C. Lewis, I. Weaver, L. Doyle, G. Martin, T. Morrow, D. Pepler, C. Danson, and I. Ross, “Use of random phase plate as a KrF laser beam homogenizer for thin film deposition applications,” Rev. Sci. Instrum. 70, 2116-2121 (1999).
[CrossRef]

Other (3)

J. Goodman, Statistical Optics (Wiley, 2000).

J. Goodman, Speckle Phenomena in Optics (Roberts & Company, 2006), Eq. 4-41, p. 69.

Mathematica 5.2.

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Figures (7)

Fig. 1
Fig. 1

Setup for the projection of speckles and the exposure of the photoresist plate. A coherent beam from a helium–cadmium laser incident on a ground glass diffuser projects a 3D speckle pattern in the photoresist. The mean transverse speckle size is proportional to λ and z and inversely to the aperture diameter . Upon exposure the average depth of indentations on the photoresist plate depends on the intensity distribution and exposure time.

Fig. 2
Fig. 2

Sequence for the photolithographic process. The photoresist is spun onto the glass plate and coated with an AR resin. The photoresist plate is preilluminated in UV light to achieve a linear response to intensity. The photoresist plate is then exposed to a speckle intensity distribution. The AR coating is peeled off and the photoresist plate is processed chemically. After rinsing and drying, the final DPP is obtained.

Fig. 3
Fig. 3

Schematic for wavefront sensing via phase retrieval and a volume speckle field using the DPP. The test wavefront is a diverging beam from the focus of a spherical lens that is illuminated by a collimated laser beam. The wavefront is then directed onto the DPP and generates a 3D speckle distribution. The speckle field is sampled sequentially at 20 planes using a CMOS camera and positioning stage. The intensity measurements are inputted to a computer algorithm to reconstruct the test wavefront.

Fig. 4
Fig. 4

Simulations of the effects of varying roughness height (h) of DPP on light scattering and phase reconstruction. (a)–(c) The wavefronts at the plane of the DPP with h = 0 , λ / 2 , and λ / 10 , respectively. (d)–(f) The corresponding sets of intensity measurements as plotted in the y z vertical plane showing varying degrees of light scattering. (g)–(i) After carrying out phase retrieval with the same number of iterations on each of the three sets of intensity measurements, successful phase reconstruction is observed for h = λ / 10 .

Fig. 5
Fig. 5

Experimental results on the effects of varying roughness height (due to varying exposure of the DPP) on the reconstruction of the test wavefront. Circles show portions of the DPP with different degrees of roughness approximated to be (a) shallow, (b) deep, and (c) medium. (d)–(f) The corresponding sets of intensity measurements as plotted in the y z vertical plane showing varying degrees of light scattering. (g)–(i) Successful phase retrieval is achieved when the portion of the DPP with medium roughness was used. (j) 3D stack of field intensities generated using the reconstructed wavefront. The wavefront is brought to a spot at the focal plane of the lens indicating phase retrieval. (k) Unwrapped phase distribution at the plane of the DPP exhibiting spherical symmetry.

Fig. 6
Fig. 6

Absolute value of the field correlation function for ρ coh = land : (a)  Q 2 I ¯ 2 = 1 , (b)  Q 2 I ¯ 2 = 4 , and (c) Q 2 I ¯ 2 = 9 .

Fig. 7
Fig. 7

Absolute value of the field correlation function for ρ coh = 1 : (a)  Q 2 I ¯ 2 = 9 and (b)  Q 2 I ¯ 2 = 1 . The lower curves are the approximations.

Equations (21)

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U ( x , y , d ) = C u ( f x , f y ) exp [ ( i 2 π d / y ) ( 1 λ 2 f x 2 λ 2 f y 2 ) 1 / 2 ] exp [ i 2 π ( f x x f y y ) ] d f x d f y ,
h ( r ) = η T I ( r ) .
φ ( r ) = k 0 ( n 1 ) h ( r ) = k 0 ( n 1 ) η T I ( r ) ,
φ ( r ) = Q I ( r ) .
U ( r ) = U 0 ( r ) exp [ i φ ( r ) ] .
U ( p ) = U 0 ( r ) exp [ i φ ( r ) ] G ( r , p ) d r ,
G ( r , p ) = i k 2 π B exp [ i k 2 B ( A r 2 2 r p + D ρ 2 ) ]
C I ( p 1 , p 2 ) I ( p 1 ) I ( p 2 ) I ( p 1 ) I ( p 2 ) ,
C I ( p 1 , p 2 ) = | U ( p 1 ) U * ( p 2 ) | 2 .
U ( p 1 ) U * ( p 2 ) = U 0 ( r 1 ) U 0 * ( r 2 ) exp [ i ( φ ( r 1 ) φ ( r 2 ) ) ] G ( r 1 , p 1 ) G * ( r 2 , p 2 ) d r 1 d r 2 .
p ( I 1 , I 2 ) = 1 I ¯ 2 ( 1 μ 2 ) exp [ I 1 + I 2 I ¯ ( 1 μ 2 ) ] I 0 [ 2 μ I 1 I 2 I ¯ ( 1 μ 2 ] .
μ ( p 1 , p 2 ) U ( p 1 ) U * ( p 2 ) I ( p 1 ) I ( p 2 ) ,
exp [ i ( φ ( r 1 ) φ ( r 2 ) ) ] = 0 0 p ( I 1 , I 2 ) exp [ i Q ( I 1 I 2 ) ] d I 1 d I 2 .
exp [ i ( φ ( r 1 ) φ ( r 2 ) ) ] = i 1 + I ¯ 2 Q 2 ( 1 μ 2 ) .
μ ( r 1 , r 2 ) = exp [ ( r 1 r 2 ) 2 ρ coh 2 ] ,
exp [ i ( φ ( r 1 ) φ ( r 2 ) ) ] = i 1 + I ¯ 2 Q 2 ( 1 exp [ 2 ( r 1 r 2 ) 2 ρ coh 2 ] ) .
exp [ i ( φ ( r 1 ) φ ( r 2 ) ) ] = i 1 + I ¯ 2 Q 2 ( 1 exp [ 2 Δ r 2 ρ coh 2 ] ) .
exp [ i ( φ ( r 1 ) φ ( r 2 ) ) ] = i 1 + I ¯ 2 Q 2 .
exp [ i ( φ ( r 1 ) φ ( r 2 ) ) ] = i ( 1 1 + I ¯ 2 Q 2 + I ¯ 2 Q 2 1 + I ¯ 2 Q 2 exp [ 2 ( 1 + I ¯ 2 Q 2 ) Δ x 2 ρ 2 ] ) .
ρ obs = [ 8 | B | 2 k 0 2 r s 2 + 4 k 0 Im [ B A * ] + r c 2 ( | A | 2 + 4 | B | 2 k 0 2 r s 4 + 4 Im [ B A * ] k 0 2 r s 2 ) ] 1 2 ,
ρ obs , free space = 8 L 2 k 0 2 r s 2 + ρ coh 2 1 + I ¯ 2 Q 2 ( 1 + 4 L 2 k 0 2 r s 4 ) .

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