Abstract

We present a method to calculate wave propagation between arbitrary curved surfaces using a staircase approximation approach. The entire curved surface is divided into multiple subregions and each curved subregion is approximated by a piecewise flat subplane allowing the application of conventional diffraction theory. In addition, in order to reflect the local curvature of each subregion, we apply the phase compensation technique. Analytical expressions are derived based on the angular spectrum method and numerical studies are conducted to validate our method.

© 2014 Optical Society of America

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References

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2013

2012

2011

L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A 28(3), 290–295 (2011).
[CrossRef] [PubMed]

G. B. Esmer, L. Onural, H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284(24), 5537–5548 (2011).
[CrossRef]

2010

2009

2008

2006

2003

1998

1992

Ahrenberg, L.

Benzie, P.

Bianco, B.

Delen, N.

Esmer, G. B.

G. B. Esmer, L. Onural, H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284(24), 5537–5548 (2011).
[CrossRef]

Hao, P.

Hooker, B.

Ichihashi, Y.

Ito, T.

Kurita, T.

Magnor, M.

Masuda, N.

Matsushima, K.

Oi, R.

Onural, L.

Ozaktas, H. M.

G. B. Esmer, L. Onural, H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284(24), 5537–5548 (2011).
[CrossRef]

Sahin, E.

Schimmel, H.

Senoh, T.

Shen, F.

Shimobaba, T.

Tommasi, T.

Wang, A.

Watson, J.

Wei, W.

Wyrowski, F.

Xiahui, T.

Xiong, Q. Y.

Yamamoto, K.

Yu, X.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

G. B. Esmer, L. Onural, H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284(24), 5537–5548 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Other

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2004).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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Figures (7)

Fig. 1
Fig. 1

Schematics of the transversely uniform segmentation method for calculation of (a) propagation from an arbitrary curved surface to a flat plane and (b) the opposite case.

Fig. 2
Fig. 2

Schematics of the longitudinally uniform stratification method for calculation of (a) propagation from an arbitrary curved surface to a flat plane and (b) the opposite case.

Fig. 3
Fig. 3

Curved geometry and calculation window for the numerical simulation.

Fig. 4
Fig. 4

Calculated amplitude profiles of diffracted fields from the geometry shown in Fig. 3 using method 1 with R = 500μm and (a) N1 = 5, (b) N1 = 15, and (c) N1 = 35. The corresponding results for R = 400μm are shown in (d)–(f).

Fig. 5
Fig. 5

Calculated amplitude profiles of diffracted fields from the geometry shown in Fig. 3 using method 2 with R = 500μm and (a) N2 = 5, (b) N2 = 15, and (c) N2 = 35. The corresponding results for R = 400μm are shown in (d)–(f).

Fig. 6
Fig. 6

Normalized amplitude distributions calculated from the Rayleigh-Sommerfeld diffraction formula (solid black line), method 1 (dashed blue line), and method 2 (dotted red line) at z = R = 400μm.

Fig. 7
Fig. 7

Amplitude profiles around the focused spot when (a) Gaussian and (b) rectangular window functions are used, with method 1 and parameters of R = 400μm and N1 = 35. (c) Wave field without the phase compensation function.

Equations (20)

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A n ( f x ;z=g(nΔx))= u i (x) g p (nΔx,x)rect( xnΔx Δx )exp(i2π f x x)dx ,
g p (nΔx,x)=exp[ ik{ g(nΔx)g(x) } ],
rect(x)={ 1, | x |<1/2 1/2,| x |=1/2 0.otherwise
u o,n (x)= A n ( f x ;z=g(nΔx))exp[ ik{ z 0 g(nΔx) } 1 λ 2 f x 2 ]exp(i2π f x x)d f x .
u o (x)= n u o,n (x) .
u o (x)= n u o,n (x) g p (x,nΔx)rect( xnΔx Δx ) ,
u o,n (x)= A( f x ;z= z 0 )exp[ ik{ g(nΔx) z 0 } 1 λ 2 f x 2 ]exp(i2π f x x)d f x ,
A( f x ;z= z 0 )= u i (x)exp(i2π f x x)dx .
w n (x)=exp[ π ( xnΔx Δx ) 2 ].
u o (x)= A( f x ;z= z 0 )exp(i2π f x x)d f x ,
A( f x ;z= z 0 )= u i (x)exp[ ik{ z 0 g(x) } 1 λ 2 f x 2 ]exp(i2π f x x)dx ,
u o (x)= A( f x ;z= z 0 )exp[ ik{ g(x) z 0 } 1 λ 2 f x 2 ]exp(i2π f x x)d f x ,
A( f x ;z= z 0 )= u i (x)exp(i2π f x x)dx .
A n ( f x ;z=nΔz)= u i (x)exp[ ik{ nΔzg(x) } ]rect( nΔzg(x) Δz )exp(i2π f x x)dx .
u o (x;z= z 0 )= n u o,n (x;z= z 0 ) ,
u o,n (x;z= z 0 )= A n ( f x ;z=nΔz)exp[ ik{ z 0 nΔz } 1 λ 2 f x 2 ]exp(i2π f x x)d f x .
u o (x;z=g(x))= n u o,n (x;z=nΔz)exp[ ik{ g(x)nΔz } ]rect( nΔzg(x) Δz ) ,
u o,n (x;z=nΔz)= A n ( f x ;z= z 0 )exp[ ik{ nΔz z 0 } 1 λ 2 f x 2 ]exp(i2π f x x)d f x ,
A n ( f x ;z= z 0 )= u i (x)exp(i2π f x x)dx .
w n (x)=exp[ π ( nΔzg(x) Δz ) 2 ].

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