Abstract

The optical design and analysis of modern micro-optical elements with high index contrasts and large numerical apertures is still challenging, as fast and accurate wave-optical simulations beyond the thin-element-approximation are required. We introduce a modified formulation of the wave-propagation-method and assess its performance in comparison to different beam-propagation-methods with respect to accuracy, required sampling densities, and computational performance. For typical micro-optical components, the wave-propagation-method is found to be considerably faster and more accurate at even lower sampling densities compared to the different beam-propagation-methods. This enables realistic wave-optical simulations beyond the thin-element-approximation for micro-optical components. As an example, the modified wave-propagation-method is applied for in-line holographic measurements of strongly diffracting objects. From a direct comparison of experimental results and corresponding simulations, the geometric parameters of a test object could be retrieved with high accuracy.

© 2016 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).
  2. H. P. Herzig, Micro-Optics (Taylor & Francis, 1997).
  3. H. Gross, W. Singer, and M. Totzeck, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 1.
  4. D. Infante-Gómez and H. P. Herzig, “Design, simulation, and quality evaluation of micro-optical freeform beam shapers at different illumination conditions,” Appl. Opt. 55, 8340–8346 (2016).
    [Crossref] [PubMed]
  5. W. Harm, S. Bernet, M. Ritsch-Marte, I. Harder, and N. Lindlein, “Adjustable diffractive spiral phase plates,” Opt. Express 23, 413–421 (2015).
    [Crossref] [PubMed]
  6. W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 2.
  7. W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design of binary subwavelength multiphase level computer generated holograms,” Opt. Lett. 35, 676–678 (2010).
    [Crossref] [PubMed]
  8. W. Eckstein, E.-B. Kley, and A. Tünnermann, “Comparison of different simulation methods for effective medium computer generated holograms,” Opt. Express 21, 12424–12433 (2013).
    [Crossref] [PubMed]
  9. S. Thiele, A. Seifert, and A. M. Herkommer, “Wave-optical design of a combined refractive-diffractive varifocal lens,” Opt. Express 22, 13343–13350 (2014).
    [Crossref] [PubMed]
  10. T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nature Photon. 10, 554–560 (2016).
    [Crossref]
  11. T. Ando, T. Korenaga, M.-a. Suzuki, and J. Tanida, “Diffraction light analysis method for a diffraction grating imaging lens,” Appl. Opt. 53, 2532–2538 (2014).
    [Crossref] [PubMed]
  12. M.-S. Kim, T. Scharf, S. Mühlig, M. Fruhnert, C. Rockstuhl, R. Bitterli, W. Noell, R. Voelkel, and H. P. Herzig, “Refraction limit of miniaturized optical systems: a ball-lens example,” Opt. Express 24, 6996–7005 (2016).
    [Crossref] [PubMed]
  13. M. Kempkes, E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, M. Mazzotti, T. J. Naughton, and A. K. Asundi, “Three dimensional digital holographic profiling of micro-fibers,” Opt. Express 17, 2938–2943 (2009).
    [Crossref] [PubMed]
  14. M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
    [Crossref]
  15. Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
    [Crossref]
  16. S. Peterhänsel, C. Pruss, and W. Osten, “Limits of diffractometric reconstruction of line gratings when using scalar diffraction theory,” Opt. Lett. 39, 3764–3766 (2014).
    [Crossref] [PubMed]
  17. K.-H. Brenner and W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32, 4984–4988 (1993).
    [Crossref] [PubMed]
  18. G. R. Hadley, “Wide-angle beam propagation using pade approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
    [Crossref] [PubMed]
  19. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17, 1743–1745 (1992).
    [Crossref] [PubMed]
  20. D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quant. Electron. 26, 185–197 (1994).
    [Crossref]
  21. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tünnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express 11, 818–823 (2003).
    [Crossref] [PubMed]
  22. J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14, 2715–2720 (2006).
    [Crossref] [PubMed]
  23. M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990 (1978).
    [Crossref] [PubMed]
  24. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
    [Crossref]
  25. V. A. Soifer, Diffractive Nanophotonics (CRC Press, 2014).
    [Crossref]
  26. E. Bekker, P. Sewell, T. Benson, and A. Vukovic, “Wide-angle alternating-direction implicit finite-difference beam propagation method,” IEEE J. Lightwave Techn. 27, 2595–2604 (2009).
    [Crossref]
  27. R. Ratowsky and J. Fleck, “Accurate numerical solution of the helmholtz equation by iterative lanczos reduction,” Opt. Lett. 16, 787–789 (1991).
    [Crossref] [PubMed]
  28. J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photon. Technol. Lett. 18, 661–663 (2006).
    [Crossref]
  29. J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized douglas scheme,” IEEE Photon. Technol. Lett. 18, 2535–2537 (2006).
    [Crossref]
  30. C. Ma and E. Van Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express 14, 4668–4674 (2006).
    [Crossref] [PubMed]
  31. S. H. Wei and Y. Y. Lu, “Application of bi-cgstab to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
    [Crossref]
  32. M. Fertig and K.-H. Brenner, “Vector wave propagation method,” J. Opt. Soc. Am. A 27, 709–717 (2010).
    [Crossref]
  33. T. Op’t Root and C. Stolk, “One-way wave propagation with amplitude based on pseudo-differential operators,” Wave Motion 47, 67–84 (2010).
    [Crossref]
  34. L. Hörmander, The Analysis of Linear Partial Differential Operators III (Springer, 1980).
  35. M. V. de Hoop, J. H. Le Rousseau, and R.-S. Wu, “Generalization of the phase-screen approximation for the scattering of acoustic waves,” Wave Motion 31, 43–70 (2000).
    [Crossref]
  36. D. Yevick, “The application of complex pade approximants to vector field propagation,” IEEE Photon. Technol. Lett. 12, 1636–1638 (2000).
    [Crossref]
  37. K. Q. Le, “Complex padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282, 1252–1254 (2009).
    [Crossref]
  38. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
    [Crossref] [PubMed]

2016 (3)

2015 (1)

2014 (3)

2013 (2)

W. Eckstein, E.-B. Kley, and A. Tünnermann, “Comparison of different simulation methods for effective medium computer generated holograms,” Opt. Express 21, 12424–12433 (2013).
[Crossref] [PubMed]

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

2010 (3)

2009 (3)

M. Kempkes, E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, M. Mazzotti, T. J. Naughton, and A. K. Asundi, “Three dimensional digital holographic profiling of micro-fibers,” Opt. Express 17, 2938–2943 (2009).
[Crossref] [PubMed]

K. Q. Le, “Complex padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282, 1252–1254 (2009).
[Crossref]

E. Bekker, P. Sewell, T. Benson, and A. Vukovic, “Wide-angle alternating-direction implicit finite-difference beam propagation method,” IEEE J. Lightwave Techn. 27, 2595–2604 (2009).
[Crossref]

2006 (5)

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photon. Technol. Lett. 18, 661–663 (2006).
[Crossref]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized douglas scheme,” IEEE Photon. Technol. Lett. 18, 2535–2537 (2006).
[Crossref]

F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
[Crossref] [PubMed]

J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14, 2715–2720 (2006).
[Crossref] [PubMed]

C. Ma and E. Van Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express 14, 4668–4674 (2006).
[Crossref] [PubMed]

2004 (1)

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[Crossref]

2003 (2)

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[Crossref]

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tünnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express 11, 818–823 (2003).
[Crossref] [PubMed]

2002 (1)

S. H. Wei and Y. Y. Lu, “Application of bi-cgstab to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
[Crossref]

2000 (2)

M. V. de Hoop, J. H. Le Rousseau, and R.-S. Wu, “Generalization of the phase-screen approximation for the scattering of acoustic waves,” Wave Motion 31, 43–70 (2000).
[Crossref]

D. Yevick, “The application of complex pade approximants to vector field propagation,” IEEE Photon. Technol. Lett. 12, 1636–1638 (2000).
[Crossref]

1994 (1)

D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quant. Electron. 26, 185–197 (1994).
[Crossref]

1993 (1)

1992 (2)

1991 (1)

1978 (1)

Allano, D.

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[Crossref]

Ando, T.

Asundi, A. K.

Bekker, E.

E. Bekker, P. Sewell, T. Benson, and A. Vukovic, “Wide-angle alternating-direction implicit finite-difference beam propagation method,” IEEE J. Lightwave Techn. 27, 2595–2604 (2009).
[Crossref]

Benson, T.

E. Bekker, P. Sewell, T. Benson, and A. Vukovic, “Wide-angle alternating-direction implicit finite-difference beam propagation method,” IEEE J. Lightwave Techn. 27, 2595–2604 (2009).
[Crossref]

Bernet, S.

Bitterli, R.

Brenner, K.-H.

Broeng, J.

Cen, K.

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

Chen, L.

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

Coëtmellec, S.

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[Crossref]

Darakis, E.

de Hoop, M. V.

M. V. de Hoop, J. H. Le Rousseau, and R.-S. Wu, “Generalization of the phase-screen approximation for the scattering of acoustic waves,” Wave Motion 31, 43–70 (2000).
[Crossref]

Eckstein, W.

Ermeneux, S.

Feit, M. D.

Fertig, M.

Fleck, J.

Fleck, J. A.

Freese, W.

Fruhnert, M.

Gao, X.

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

Giessen, H.

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nature Photon. 10, 554–560 (2016).
[Crossref]

Gissibl, T.

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nature Photon. 10, 554–560 (2016).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

Gréhan, G.

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

Gross, H.

H. Gross, W. Singer, and M. Totzeck, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 1.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 2.

Hadley, G. R.

Harder, I.

Harm, W.

Herkommer, A.

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nature Photon. 10, 554–560 (2016).
[Crossref]

Herkommer, A. M.

Herzig, H. P.

Hörmander, L.

L. Hörmander, The Analysis of Linear Partial Differential Operators III (Springer, 1980).

Iliew, R.

Infante-Gómez, D.

Jakobsen, C.

Kämpfe, T.

Kariwala, V.

Kempkes, M.

Khanam, T.

Kim, M.-S.

Kley, E.-B.

Kolesik, M.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[Crossref]

Korenaga, T.

Le, K. Q.

K. Q. Le, “Complex padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282, 1252–1254 (2009).
[Crossref]

Le Rousseau, J. H.

M. V. de Hoop, J. H. Le Rousseau, and R.-S. Wu, “Generalization of the phase-screen approximation for the scattering of acoustic waves,” Wave Motion 31, 43–70 (2000).
[Crossref]

Lebrun, D.

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[Crossref]

Lederer, F.

Limpert, J.

Lindlein, N.

Lu, Y. Y.

S. H. Wei and Y. Y. Lu, “Application of bi-cgstab to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
[Crossref]

Ma, C.

Malek, M.

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[Crossref]

Mazzotti, M.

Meunier-Guttin-Cluzel, S.

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

Moloney, J. V.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[Crossref]

Mühlig, S.

Nakano, H.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized douglas scheme,” IEEE Photon. Technol. Lett. 18, 2535–2537 (2006).
[Crossref]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photon. Technol. Lett. 18, 661–663 (2006).
[Crossref]

Naughton, T. J.

Noell, W.

Nolte, S.

Op’t Root, T.

T. Op’t Root and C. Stolk, “One-way wave propagation with amplitude based on pseudo-differential operators,” Wave Motion 47, 67–84 (2010).
[Crossref]

Osten, W.

Peterhänsel, S.

Petersson, A.

Pruss, C.

Qiu, K.

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

Rajendran, A.

Ratowsky, R.

Ritsch-Marte, M.

Rockstuhl, C.

Röser, F.

Rothhardt, J.

Saengkaew, S.

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

Salin, F.

Scharf, T.

Schmidt, O.

Schreiber, T.

Seifert, A.

Sewell, P.

E. Bekker, P. Sewell, T. Benson, and A. Vukovic, “Wide-angle alternating-direction implicit finite-difference beam propagation method,” IEEE J. Lightwave Techn. 27, 2595–2604 (2009).
[Crossref]

Shen, F.

Shibayama, J.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized douglas scheme,” IEEE Photon. Technol. Lett. 18, 2535–2537 (2006).
[Crossref]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photon. Technol. Lett. 18, 661–663 (2006).
[Crossref]

Singer, W.

K.-H. Brenner and W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32, 4984–4988 (1993).
[Crossref] [PubMed]

H. Gross, W. Singer, and M. Totzeck, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 1.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 2.

Soifer, V. A.

V. A. Soifer, Diffractive Nanophotonics (CRC Press, 2014).
[Crossref]

Stolk, C.

T. Op’t Root and C. Stolk, “One-way wave propagation with amplitude based on pseudo-differential operators,” Wave Motion 47, 67–84 (2010).
[Crossref]

Suzuki, M.-a.

Takahashi, T.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized douglas scheme,” IEEE Photon. Technol. Lett. 18, 2535–2537 (2006).
[Crossref]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photon. Technol. Lett. 18, 661–663 (2006).
[Crossref]

Tanida, J.

Thiele, S.

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nature Photon. 10, 554–560 (2016).
[Crossref]

S. Thiele, A. Seifert, and A. M. Herkommer, “Wave-optical design of a combined refractive-diffractive varifocal lens,” Opt. Express 22, 13343–13350 (2014).
[Crossref] [PubMed]

Totzeck, M.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 2.

H. Gross, W. Singer, and M. Totzeck, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 1.

Tünnermann, A.

Tünnermann, T.

Van Keuren, E.

Vienne, G.

Voelkel, R.

Vukovic, A.

E. Bekker, P. Sewell, T. Benson, and A. Vukovic, “Wide-angle alternating-direction implicit finite-difference beam propagation method,” IEEE J. Lightwave Techn. 27, 2595–2604 (2009).
[Crossref]

Wang, A.

Wei, S. H.

S. H. Wei and Y. Y. Lu, “Application of bi-cgstab to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
[Crossref]

Wu, R.-S.

M. V. de Hoop, J. H. Le Rousseau, and R.-S. Wu, “Generalization of the phase-screen approximation for the scattering of acoustic waves,” Wave Motion 31, 43–70 (2000).
[Crossref]

Wu, X.

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

Wu, Y.

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

Yamauchi, J.

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photon. Technol. Lett. 18, 661–663 (2006).
[Crossref]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized douglas scheme,” IEEE Photon. Technol. Lett. 18, 2535–2537 (2006).
[Crossref]

Yevick, D.

D. Yevick, “The application of complex pade approximants to vector field propagation,” IEEE Photon. Technol. Lett. 12, 1636–1638 (2000).
[Crossref]

D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quant. Electron. 26, 185–197 (1994).
[Crossref]

Yvernault, P.

Zellmer, H.

Appl. Opt. (5)

IEEE J. Lightwave Techn. (1)

E. Bekker, P. Sewell, T. Benson, and A. Vukovic, “Wide-angle alternating-direction implicit finite-difference beam propagation method,” IEEE J. Lightwave Techn. 27, 2595–2604 (2009).
[Crossref]

IEEE Photon. Technol. Lett. (4)

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional horizontally wide-angle noniterative beam-propagation method based on the alternating-direction implicit scheme,” IEEE Photon. Technol. Lett. 18, 661–663 (2006).
[Crossref]

J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized douglas scheme,” IEEE Photon. Technol. Lett. 18, 2535–2537 (2006).
[Crossref]

D. Yevick, “The application of complex pade approximants to vector field propagation,” IEEE Photon. Technol. Lett. 12, 1636–1638 (2000).
[Crossref]

S. H. Wei and Y. Y. Lu, “Application of bi-cgstab to waveguide discontinuity problems,” IEEE Photon. Technol. Lett. 14, 645–647 (2002).
[Crossref]

J. Opt. Soc. Am. A (1)

Nature Photon. (1)

T. Gissibl, S. Thiele, A. Herkommer, and H. Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nature Photon. 10, 554–560 (2016).
[Crossref]

Opt. Commun. (3)

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[Crossref]

Y. Wu, X. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, L. Chen, K. Qiu, X. Gao, G. Gréhan, and K. Cen, “Digital Gabor and off-axis particle holography by shaped beams: A numerical investigation with glmt,” Opt. Commun. 305, 247–254 (2013).
[Crossref]

K. Q. Le, “Complex padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282, 1252–1254 (2009).
[Crossref]

Opt. Express (8)

C. Ma and E. Van Keuren, “A simple three dimensional wide-angle beam propagation method,” Opt. Express 14, 4668–4674 (2006).
[Crossref] [PubMed]

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tünnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express 11, 818–823 (2003).
[Crossref] [PubMed]

J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14, 2715–2720 (2006).
[Crossref] [PubMed]

M.-S. Kim, T. Scharf, S. Mühlig, M. Fruhnert, C. Rockstuhl, R. Bitterli, W. Noell, R. Voelkel, and H. P. Herzig, “Refraction limit of miniaturized optical systems: a ball-lens example,” Opt. Express 24, 6996–7005 (2016).
[Crossref] [PubMed]

M. Kempkes, E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, M. Mazzotti, T. J. Naughton, and A. K. Asundi, “Three dimensional digital holographic profiling of micro-fibers,” Opt. Express 17, 2938–2943 (2009).
[Crossref] [PubMed]

W. Eckstein, E.-B. Kley, and A. Tünnermann, “Comparison of different simulation methods for effective medium computer generated holograms,” Opt. Express 21, 12424–12433 (2013).
[Crossref] [PubMed]

S. Thiele, A. Seifert, and A. M. Herkommer, “Wave-optical design of a combined refractive-diffractive varifocal lens,” Opt. Express 22, 13343–13350 (2014).
[Crossref] [PubMed]

W. Harm, S. Bernet, M. Ritsch-Marte, I. Harder, and N. Lindlein, “Adjustable diffractive spiral phase plates,” Opt. Express 23, 413–421 (2015).
[Crossref] [PubMed]

Opt. Lett. (5)

Opt. Quant. Electron. (1)

D. Yevick, “A guide to electric field propagation techniques for guided-wave optics,” Opt. Quant. Electron. 26, 185–197 (1994).
[Crossref]

Phys. Rev. E (1)

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[Crossref]

Wave Motion (2)

M. V. de Hoop, J. H. Le Rousseau, and R.-S. Wu, “Generalization of the phase-screen approximation for the scattering of acoustic waves,” Wave Motion 31, 43–70 (2000).
[Crossref]

T. Op’t Root and C. Stolk, “One-way wave propagation with amplitude based on pseudo-differential operators,” Wave Motion 47, 67–84 (2010).
[Crossref]

Other (6)

L. Hörmander, The Analysis of Linear Partial Differential Operators III (Springer, 1980).

V. A. Soifer, Diffractive Nanophotonics (CRC Press, 2014).
[Crossref]

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 2.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

H. P. Herzig, Micro-Optics (Taylor & Francis, 1997).

H. Gross, W. Singer, and M. Totzeck, Handbook of Optical Systems (Wiley-VCH, 2005) Vol. 1.

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Figures (4)

Fig. 1
Fig. 1 Focal field distributions for a gaussian beam with wavelength λ = 632 nm and a waist of ω0 = 15 μm focused through a cylindrical rod lens made out of a glass described by a refractive index of n = 1.5 with a radius of r = 15 μm (NA ≈ 0.8). Respectively, in subfigures b) and c) the simulations with the WPM and the CN-BPM are compared to the rigorous simulation in subfigure a). In subfigure d) the normalized field-distributions on the optical axis are compared in between the various algorithms.
Fig. 2
Fig. 2 a): Comparison of the calculated back focal distances BFD of the WPM and the different BPMs in comparison to the rigorous simulation for varying micro-optical lens diameters. b): Relative back focal distance error | BFD Alg . BFD Rig . BFD Rig . | of the individual algorithms related to rigorous simulations. The errors are evaluated for a micro-optical lens with a radius of r = 15 μm for differing transverse discretization densities.
Fig. 3
Fig. 3 Illustrative sketch of the measurement configuration. An incoming beam is diffracted by an optical fiber, leading to a characteristic interference pattern on the detector specific for this investigated fiber. The distance between the source and fiber is specified by ds and the further distance to the detector by dd. Detected signals are evaluated perpendicular to the principal axis of symmetry.
Fig. 4
Fig. 4 Comparison of measured and simulated signals. The origin of the diagram corresponds to the center of symmetry, i.e. the optical axis: a) Comparison to the optimized cladding diameter DClad of DClad = 124.89 μm. Correspondingly, this belongs to a nominal deviation of ΔDClad = 0.11 μm in comparison to the nominally specified diameter as proposed by the vendor. b) Comparison between slightly mismatching parameters and the measurement. In particular, the simulation of the nominally specified fiber DClad = 125 μm and the equivalent bare No-Core fiber with DClad = 124.89 μm are shown.

Tables (1)

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Table 1 Computational runtime

Equations (17)

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E ( x , y , z + Δ z ) = 1 2 π E ˜ ( k x , k y , z ) e i k z ( k x , k y , x , y ) Δ z e i ( k x x + k y y ) d k x d k y ,
k z ( k x , k y , x , y ) = k 0 2 n 2 ( x , y , z + Δ z z ) k x 2 k y 2 , E ˜ ( k x , k y , z ) = x , y { E ( x , y , z ) } = 1 2 π E ( x , y , z ) e i ( k x x + k y y ) d x d y .
Δ k z 0 ( x , k x ) ~ i k 0 2 n z ( x ) k x x n z ( x ) k z ( k x , x ) 3 .
I m z ( x , y ) = { 1 n z ( x , y ) = n m , 0 n z ( x , y ) n m ,
E ( x , y , z + Δ z ) = m I m z ( x , y ) 1 { e i k z m ( k x , k y ) Δ z { E ( x , y , z ) } } , k z m ( k x , k y ) = k 0 2 n m 2 k x 2 k y 2 + κ ( k x , k y ) .
( Δ + k 0 2 n 2 ( x , y , z ) ) E = 0 , ( z 2 + Δ t + k 0 2 n 2 ( x , y , z ) T ) E = 0 , T = transverse Helmholtz operator ,
( i T + z ) ( i T z ) E = 0 .
z E = i T E ,
E ( x , y , z 0 + z ) = e i z T E ( x , y , z 0 ) .
E ( x , y , z + Δ z ) = 1 2 π E ˜ ( k x , k y , z ) e i k z ( k x , k y , x , y ) Δ z e i ( k x x + k y y ) d k x d k y , E ˜ ( k x , k y , z ) = 1 2 π E ( x , y , z ) e i ( k x x + k y y ) d x d y , k z ( k x , k y , x , y ) = k 0 2 n 2 ( x , y , z + Δ z 2 ) k x 2 k y 2 .
𝒫 { u ( x ) } = ( m c m ( x ) x m ) 1 2 π u ˜ ( k x ) e i ( k x x ) d k x = 1 2 π m c m ( x ) i m k x m u ˜ ( k x ) 1 2 π u ( x ˜ ) e i ( k x x ˜ ) d x ˜ e i ( k x x ) d k x = 1 2 π ( m c m ( x ) i m k x m u ( x ˜ ) e i ( k x ( x x ˜ ) ) ) d x ˜ d k x = 1 2 π ( p ( x , k x ) u ( x ˜ ) e i ( k x ( x x ˜ ) ) ) d x ˜ d k x
𝒫 { u } = 1 { p ( k x ) { u ( x ) } } .
𝒫 { 𝒫 { u ( x ) } } = 1 2 π p ( x , k x ) { 𝒫 { u ( x ) } e i ( k x ( x x ) ) d x d k x = 1 4 π 2 p ( x , k x ) ( p ( x , k x ) u ( x ˜ ) e i k x ( x x ˜ ) ) + e k x ( x x ) d x d k x d x ˜ d k x = 1 4 π 2 ( p ( x , k x ) p ( x , k x ) e i ( k x k x ) ( x x ) d x d k x ) u ( x ˜ ) e i k x ( x x ˜ ) d x ˜ d k x .
( p # p ) ( x , k x ) = p ( x , k x ) p ( x , k x ) e i ( k x k x ) ( x x ) d x d k x ! = h T ( x , k x ) .
( p # p ) ( x , k x ) = m = 0 1 i m m ! k x m p ( x , k x ) x m p ( x , k x ) = p ( x , k x ) 2 i x p ( x , k x ) k x p ( x , k x ) + m = 2 1 i m m ! k x m p ( x , k x ) x m p ( x , k x ) .
E ( x , z 0 + Δ z ) = 1 2 π e i k 0 2 n 0 2 k x 2 Δ z E ˜ ( k x , z 0 ) e i ( k x x ) d k x , E ˜ ( k x , k y ) = 1 2 π E ( x , z 0 ) e i ( k x x ) d x .
Δ k z ( x , k x ) = ( k z # k z ) ( x , k x ) h T k z ( x , k x ) = i x k z ( x , k x ) k x k z ( x , k x ) k z ( x , k y ) + 1 k z ( x , k x ) m = 2 1 i m m ! k x m k z ( x , k x ) x m k z ( x , k x ) = i k x k 0 2 n ( x ) x n ( x ) k z ( x , k x ) 3 + 1 k z ( x , k x ) m = 2 1 i m m ! k x m k z ( x , k x ) x m k z ( x , k x ) .

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