Abstract

In several areas of optics and photonics the behavior of the electromagnetic waves has to be calculated with the scalar theory of diffraction by computational methods. Many of these high-speed diffraction algorithms based on a fast-Fourier-transformation are approximations of the Rayleigh-Sommerfeld-diffraction (RSD) theory. In this article a novel sampling condition for the well-sampling of the Riemann integral of the RSD is demonstrated, the fundamental restrictions due to this condition are discussed, it will be demonstrated that the restrictions are completely removed by a sampling below the Abbe resolution limit and a very general unified approach for applying the RSD outside its sampling domain is given.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. Sommerfeld, Optics, Lectures on Theoretical Physics, Vol. IV, New York (Academic, 1954).
  2. J. W. Goodman, Introduction to Fourier Optics, 4th Ed., (W. H. Freeman, 2017).
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    [PubMed]
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    [PubMed]
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    [PubMed]
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  19. E. Kreyszig, Introductory Functional Analysis with Applications (John Wiley & Sons, 1989).

2014 (1)

2010 (1)

2009 (1)

2006 (2)

2004 (1)

1990 (1)

1968 (1)

1964 (1)

1942 (1)

H. Scheffers, “Vereinfachte Ableitung der Formeln für die Fraunhoferschen Beugungserscheinungen,” Ann. Phys. 434, 211 (1942).

Engelberg, Y. M.

Farn, M. W.

Goodman, J. W.

Lalor, E.

Logofatu, P. C.

Marchand, E. W.

Matsushima, K.

Nascov, V.

Osten, W.

Pedrini, G.

Ritter, A.

Ruschin, S.

Scheffers, H.

H. Scheffers, “Vereinfachte Ableitung der Formeln für die Fraunhoferschen Beugungserscheinungen,” Ann. Phys. 434, 211 (1942).

Shen, F.

Wang, A.

Wolf, E.

Zhang, F.

Ann. Phys. (1)

H. Scheffers, “Vereinfachte Ableitung der Formeln für die Fraunhoferschen Beugungserscheinungen,” Ann. Phys. 434, 211 (1942).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Express (2)

Opt. Lett. (1)

Other (9)

Sommerfeld, Optics, Lectures on Theoretical Physics, Vol. IV, New York (Academic, 1954).

J. W. Goodman, Introduction to Fourier Optics, 4th Ed., (W. H. Freeman, 2017).

M. Born and E. Wolf, Principles of Optics, New York (Cambridge University, 2005).

T. C. Poon and J. P. Liu, Introduction to Modern Digital Holography with MATLAB (Cambridge University, 2014).

P. Picart and J. C. Li, Digital Holography (Wiley, 2012).

M.-K. Kim, Digital holographic Microscopy (Springer, 2011).

R. J. Marks II, The Joy of Fourier (Baylor University, 2006).

H. Gross, Handbook of Optical Systems, Volume 1: Fundamentals of Technical Optics (Wiley-VCH, 2005).

E. Kreyszig, Introductory Functional Analysis with Applications (John Wiley & Sons, 1989).

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Figures (9)

Fig. 1
Fig. 1

Coherent imaging of an object in the input plane (1). The second and third planes refer to the diffracted images of the object at the distance z 12 and z 13 from the object plane, respectively.

Fig. 2
Fig. 2

Input object plane, (a) magnitude and real part (b) of the complex amplitude. If not otherwise stated, the following parameters were used for the presented simulations: Wavelength λ=0.633μm, sampling spacing in the input plane δ x 1 =δ y 1 =0.76μm and in the output plane δ x 2 =δ y 2 =0.94μm. The pixel numbers in thex andy axis as well as in the input and output plane are the same N 1,2,x,y =265. The width of the computational domain in the input plane is P x 1 = P y 1 =201μm, whereas for the output plane it is P x 2 = P y 2 =250μm.

Fig. 3
Fig. 3

Reconstruction errors for z< z c (a) magnitude and (b) real part of the complex amplitude of the diffracted image in plane 2. (c) and (d) magnitude and real part of the reconstructed object in plane 1. N F =1000.

Fig. 4
Fig. 4

(a), (b) magnitude and real part of the image complex amplitude, respectively for z 13 > z c . (c) and (d) corresponding reconstructed amplitude and real part of the reconstructed object in the input plane. N F =27.

Fig. 5
Fig. 5

(a) and (b) magnitude and real part of the complex amplitude according to the RSD at the same propagation distance like Fig. 3 but with a sampling spacing of the object and image smaller than the Abbe limit. Figure 5. (c) and (d) reconstruction of the object. The structures in the object are larger than the Abbe limit. N F =1000.

Fig. 6
Fig. 6

(a), (b) magnitude and the real part of the complex amplitude for the rescaled input object. (c) and (d) magnitude and real part of the complex amplitude for the reconstructed object for a sampling spacing below the Abbe limit δ x 1 =δ y 1 =0.025μm < λ/2 = 0.32 μm. N F =10.

Fig. 7
Fig. 7

(a) and (b) magnitude and real part of the complex amplitude of the input object. (c) and (d) magnitude and real part of the complex amplitude for the corresponding output by a sampling below the Abbe limit δ x 1 =δ y 1 =0.013μmλ/2 = 0.32 μm. N F =2.5.

Fig. 8
Fig. 8

(a) and (b) magnitude and real part of the complex amplitude u 132 . (c) and (d) magnitude and real part of the complex amplitude u 12 by sampling below the Abbe limit, used as a reference. N F =1000.

Fig. 9
Fig. 9

(a), (b) amplitude and real part of the input object (c), (d) magnitude and real part of the amplitude for the reconstructed object.

Equations (11)

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u 2 ( R 2 )= Ω d 2 r 1 α u 1 ( r 1 )exp( ik| R 2 r 1 | ), R 2 = r 2 + z 12 e ^ z ;α= 1 2π ( ik 1 | R 2 r 1 | ) z 12 | R 2 r 1 | 2 ,
φ=k| R 2 r 1 |=k R 2 2 + r 1 2 2 R 2 r 1 =k x 2 2 + y 2 2 + z 12 2 + x 1 2 + y 1 2 2( x 1 x 2 + y 1 y 2 ) ,
f x = 1 2π | φ x 1 |= k| x 1 x 2 | 2π ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 + z 12 2 .
f x,max = 1 2π | φ x 1 | max = x 1fp + x 2fp λ ( x 1fp + x 2fp ) 2 + z 12 2 .
1 δ x 1 2 x 1fp + x 2fp λ ( x 1fp + x 2fp ) 2 + z 12 2 .
z 12 2 ( 4δ x 1 2 λ 2 1 ) ( x 1fp + x 2fp ) 2 .
z 1cx 2 ( 4δ x 1 2 λ 2 1 ) ( x 1fp + x 2fp ) 2 .
z 1cy 2 ( 4δ y 1 2 λ 2 1 ) ( y 1fp + y 2fp ) 2 .
z 1c =max( z 1cx , z 1cy ).
z 12 > z c .
R 231 R 132 =( R 31 R 23 )( R 32 R 13 )= R 31 R 23 R 32 R 13 = R 13 1 R 32 1 R 32 R 13 = R 13 1 I R 13 =I.