Abstract

An approach for designing purely refractive optical elements that generate engineered, multi-order-helix point spread functions (PSFs) with large peak separation for passive, optical depth measurement is presented. The influence of aberrations on the PSF’s rotation angle, which limits the depth retrieval accuracy, is studied numerically and analytically. It appears that only Zernike modes with an azimuthal index that is an integer multiple of the number of PSF peaks introduce PSF rotation, and hence depth estimation errors. This implies that high-order-helix designs have superior robustness with respect to aberrations. This is experimentally demonstrated by imaging an extended scene in the presence of severe system aberrations using novel, cost-efficient phase elements based on UV-replication on the wafer-scale.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [Crossref] [PubMed]
  2. A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474 (2004).
    [Crossref] [PubMed]
  3. Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
    [Crossref]
  4. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
    [Crossref] [PubMed]
  5. S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions”, Opt. Express 16, 3484–3489 (2008).
    [Crossref] [PubMed]
  6. S. Prasad, “Rotating point spread function via pupil-phase engineering,” Opt. Lett. 38, 585–587 (2013).
    [Crossref] [PubMed]
  7. S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
    [Crossref] [PubMed]
  8. S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
    [Crossref]
  9. G. Grover, S. Quirin, C. Fiedler, and R. Piestun, “Photon efficient double-helix PSF microscopy with application to 3D photo-activation localization imaging,” Biomed. Opt. Express 2, 3010–3020 (2011).
    [Crossref] [PubMed]
  10. A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
    [Crossref]
  11. C. Roider, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Axial super-localisation using rotating point spread functions shaped by polarisation-dependent phase modulation,” Opt. Express 22, 4029–4037 (2014).
    [Crossref] [PubMed]
  12. S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
    [Crossref] [PubMed]
  13. C. Roider, R. Heintzmann, R. Piestun, and A. Jesacher, “Deconvolution approach for 3D scanning microscopy with helical phase engineering,” Opt. Express 24, 15456 (2016).
    [Crossref] [PubMed]
  14. S. Quirin and R. Piestun, “Depth estimation and image recovery using broadband, incoherent illumination with engineered point spread functions [Invited],” Appl. Opt. 52, A367–A376 (2013).
    [Crossref] [PubMed]
  15. R. Berlich, A. Bräuer, and S. Stallinga, “Single shot three-dimensional imaging using an engineered point spread function,” Opt. Express 24, 5946 (2016).
    [Crossref] [PubMed]
  16. S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
    [Crossref] [PubMed]
  17. Z. Cao and K. Wang, “Effects of astigmatism and coma on rotating point spread function,” Appl. Opt. 53, 7325–7330 (2014).
    [Crossref] [PubMed]
  18. M. Baránek, P. Bouchal, M. Šiler, and Z. Bouchal, “Aberration resistant axial localization using a self-imaging of vortices,” Opt. Express 23, 15316 (2015).
    [Crossref] [PubMed]
  19. R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
    [Crossref]
  20. R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
    [Crossref] [PubMed]
  21. G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
    [Crossref]
  22. K. Itoh, “Analysis of the Phase Unwrapping Algorithm,” Appl. Opt. 21, 2470 (1982).
    [Crossref] [PubMed]
  23. D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
    [Crossref]
  24. A. N. Simonov and M. C. Rombach, “Passive ranging and three-dimensional imaging through chiral phase coding,” Opt. Lett. 36, 115–117 (2011).
    [Crossref] [PubMed]
  25. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).
  26. V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A2 (1985).
    [Crossref]
  27. H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
    [Crossref]
  28. K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
    [Crossref]

2016 (3)

2015 (1)

2014 (3)

2013 (4)

S. Quirin and R. Piestun, “Depth estimation and image recovery using broadband, incoherent illumination with engineered point spread functions [Invited],” Appl. Opt. 52, A367–A376 (2013).
[Crossref] [PubMed]

S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
[Crossref] [PubMed]

S. Prasad, “Rotating point spread function via pupil-phase engineering,” Opt. Lett. 38, 585–587 (2013).
[Crossref] [PubMed]

G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
[Crossref]

2012 (1)

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
[Crossref] [PubMed]

2011 (3)

2009 (1)

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

2008 (2)

2007 (1)

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

2006 (1)

2004 (1)

2000 (1)

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

1999 (1)

K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
[Crossref]

1998 (1)

1995 (1)

1982 (1)

Baddeley, D.

D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
[Crossref]

Baránek, M.

Barsic, A.

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
[Crossref]

Berlich, R.

Bernet, S.

Born, M.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Bouchal, P.

Bouchal, Z.

Bräuer, A.

R. Berlich, A. Bräuer, and S. Stallinga, “Single shot three-dimensional imaging using an engineered point spread function,” Opt. Express 24, 5946 (2016).
[Crossref] [PubMed]

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Cannell, M. B.

D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
[Crossref]

Cao, Z.

Castro, A.

Cathey, W. T.

Dowski, E. R.

Eckstein, H.-C.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Fiedler, C.

Ghosh, S.

S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
[Crossref] [PubMed]

Greengard, A.

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
[Crossref] [PubMed]

Grover, G.

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
[Crossref]

G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
[Crossref]

G. Grover, S. Quirin, C. Fiedler, and R. Piestun, “Photon efficient double-helix PSF microscopy with application to 3D photo-activation localization imaging,” Biomed. Opt. Express 2, 3010–3020 (2011).
[Crossref] [PubMed]

Haas, K.-H.

K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
[Crossref]

Heintzmann, R.

Itoh, K.

Jesacher, A.

Leitel, R.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Liu, L.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

Mahajan, V. N.

V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A2 (1985).
[Crossref]

Ojeda-Castañeda, J.

Pavani, S. R. P.

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
[Crossref] [PubMed]

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions”, Opt. Express 16, 3484–3489 (2008).
[Crossref] [PubMed]

S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
[Crossref] [PubMed]

Piestun, R.

C. Roider, R. Heintzmann, R. Piestun, and A. Jesacher, “Deconvolution approach for 3D scanning microscopy with helical phase engineering,” Opt. Express 24, 15456 (2016).
[Crossref] [PubMed]

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
[Crossref]

G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
[Crossref]

S. Quirin and R. Piestun, “Depth estimation and image recovery using broadband, incoherent illumination with engineered point spread functions [Invited],” Appl. Opt. 52, A367–A376 (2013).
[Crossref] [PubMed]

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
[Crossref] [PubMed]

G. Grover, S. Quirin, C. Fiedler, and R. Piestun, “Photon efficient double-helix PSF microscopy with application to 3D photo-activation localization imaging,” Biomed. Opt. Express 2, 3010–3020 (2011).
[Crossref] [PubMed]

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

S. R. P. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
[Crossref] [PubMed]

S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions”, Opt. Express 16, 3484–3489 (2008).
[Crossref] [PubMed]

A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
[Crossref] [PubMed]

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[Crossref]

Prasad, S.

Preza, C.

S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
[Crossref] [PubMed]

Quirin, S.

Ritsch-Marte, M.

Roider, C.

Rombach, M. C.

Schechner, Y.

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

Schechner, Y. Y.

Schleicher, P.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Shamir, J.

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[Crossref]

Šiler, M.

Simonov, A. N.

Soeller, C.

D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
[Crossref]

Stallinga, S.

Stumpf, M.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Sun, J.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

Tünnermann, A.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Wang, K.

Wolf, E.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Wolter, H.

K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
[Crossref]

Yang, Q.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

Zeitner, U. D.

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[Crossref]

Biomed. Opt. Express (1)

Curr. Opin. Solid State Mater. Sci. (1)

K.-H. Haas and H. Wolter, “Synthesis, properties and applications of inorganic–organic copolymers (ORMOCERs),” Curr. Opin. Solid State Mater. Sci. 4, 571–580 (1999).
[Crossref]

J. Biomed. Opt. (1)

S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration,” J. Biomed. Opt. 18, 036010 (2013).
[Crossref] [PubMed]

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

JOSA A (1)

R. Piestun, Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” JOSA A 17, 294–303 (2000).
[Crossref] [PubMed]

Nano Res. (1)

D. Baddeley, M. B. Cannell, and C. Soeller, “Three-dimensional sub-100 nm super-resolution imaging of biological samples using a phase ramp in the objective pupil,” Nano Res. 4, 589–598 (2011).
[Crossref]

Opt. Commun. (1)

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Proc. Natl. Acad. Sci. (1)

S. Quirin, S. R. P. Pavani, and R. Piestun, “Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions,” Proc. Natl. Acad. Sci. 109, 675–679 (2012).
[Crossref] [PubMed]

Proc. SPIE (2)

G. Grover and R. Piestun, “New approach to double-helix point spread function design for 3D super-resolution microscopy,” Proc. SPIE 8590, 85900M (2013).
[Crossref]

H.-C. Eckstein, U. D. Zeitner, R. Leitel, M. Stumpf, P. Schleicher, A. Bräuer, and A. Tünnermann, “High dynamic grayscale lithography with an LED-based micro-image stepper,” Proc. SPIE 9780, 97800T (2016).
[Crossref]

Sci. Reports (1)

A. Barsic, G. Grover, and R. Piestun, “Three-dimensional super-resolution and localization of dense clusters of single molecules,” Sci. Reports 4, 5388 (2014).
[Crossref]

Other (2)

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A2 (1985).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 The individual steps of the proposed design process for multi-order-helix PSFs are exemplary illustrated for a double- (left column) and a triple- (right column) helix design. The phase distributions Φp as well as the corresponding PSFs are shown for the individual design steps. The Fresnel zone start designs (a,b) are initially truncated to an angular section, dedicated to a single PSF peak (c,d). The respective section is optimized for a high peak confinement (e,f) and subsequently used to fill the overall phase element with rotated copies (g,h). A purely refractive design is finally obtained by phase unwrapping in an optional step (i,j).
Fig. 2
Fig. 2 Top: Simulated, axial PSF dependency H′[v, w; z] of five considered pupil functions as indicated in the left part: A, nominal PSF for a clear aperture; B, Fresnel zone double-helix PSF; C, optimized double-helix PSF; D, optimized triple-helix PSF; E, refractive double-helix PSF. A broadband illumination spectrum ranging from 450 – 650 nm and a nominal object distance of z = 1000 mm are assumed. Note that the individual PSF plots are normalized to their respective maximum irradiance level for illustration purposes. Bottom: Corresponding axial dependencies of the standard deviation of the lower limit to measurement precision σCRLB.
Fig. 3
Fig. 3 Influence of primary Zernike aberrations on the shape of an engineered (a) double- and (b) triple-helix PSF design including the effect of defocus ( Z 2 0) as well as first order astigmatism ( Z 2 2, Z 2 2), coma ( Z 3 1, Z 3 1) and spherical aberration ( Z 4 0).
Fig. 4
Fig. 4 Comparison of the rotation angle dependency α of multi-helix-PSFs on the rms phase error for the first 21 Zernike aberrations (excluding piston, tip and tilt). Aberrations that cause a significant rotation, i.e. |α(π/2)| > 1°, on a particular multi-helix design are highlighted by rectangular boxes with the respective plot color.
Fig. 5
Fig. 5 Wavefront distribution of (a) phase element part Wp and (b) defocus aberration part W Z 2 0 (b) as well as respective gradient fields ∇⃗Wp and ∇⃗Wz4. The two highlighted subareas S1 and S2 directly correspond to the two peaks of the double-helix PSF. (c) PSF distribution associated with Wp. The direction of the centroid vectors 〈p⃗〉 and 〈z⃗k〉 are indicated for both subareas S1 and S2, respectively.
Fig. 6
Fig. 6 Top: Measured surface profile height h(x, y) of (a) the tool and (b) the replication of the double-helix CGH design. The dashed white circle indicates the aperture size of 10 mm used in the optical setup. Bottom: horizontal cross-section of the profile height h(x) at y = 1 mm as indicated by dashed blue lines in the respective upper plots.
Fig. 7
Fig. 7 Top: Measured on-axis distributions of the engineered double-, triple- and tetra-helix PSFs at multiple object distances z (λ = 540 nm). Bottom: Corresponding dependency of the rotation angle α on the object distance z.
Fig. 8
Fig. 8 (a) Nominal distribution of the imaged test screen if no CGH is implemented in the demonstration system. The subimage shown in (b) correspond to the inset indicated in (a). The subimages (c)–(e) show the same part of the image after the double-, triple- and tetra-helix CGHs are implemented in the demonstration system, respectively.
Fig. 9
Fig. 9 Processed Ccepstrum distributions of the subimages shown in Fig. 8(c)–8(e).
Fig. 10
Fig. 10 Measured rotation angle distribution α of the imaged demonstration scene using a (a) double- and (b) triple-helix PSF. (c) and (d) show the residual rotation angle distributions αr(x, y) for the two cases after eliminating the effect of field curvature by fitting and subtracting a parabolic function from α(x, y).

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

Φ l ( ϕ ) = [ ( l 1 ) N + 1 ] ϕ
Δ z max = 2 λ d 0 2 L R 2 + 2 λ l d 0 L ,
σ CRLB 2 ( z ) = 1 σ N 2 v = 1 V w = 1 W ( z H [ v , w ; z ] ) 2
H [ v , w ; z ] = λ 1 λ 2 η ( λ ) H [ v , w ; z ; λ ] d λ
Φ ( u , v ) = Φ p ( u , v ) + n , m A n m Z n m ( u , v ) ,
Z n m ( ρ , ϕ ) = R n m ( ρ ) cos m ϕ
Z n m ( ρ , ϕ ) = R n m ( ρ ) sin m ϕ
α = 1 N j = 1 N Δ α j ,
c = D E EP W ( u , v ) d u d v
W ( u , v ) = λ 2 π Φ ( u , v ) = λ 2 π Φ p ( u , v ) W p + λ 2 π A n m Z n m ( u , v ) W n m
2 π ( j 1 ) N ( ϕ u ϕ 0 ) < 2 π j N | j = 1 , , N ,
c j = D E S j W ( u , v ) d u d v = D E S j W p ( u , v ) d u d v D E S j W n m ( u , v ) d u d v = p j + z n m j
M n m = z ^ ( j = 1 N p j × z n m j ) ,
m = κ N | κ = 0 , 1 , 2 ,
p j = p ( cos ϕ j sin ϕ j ) ,
ϕ j = ϕ 0 + π 2 π N + j 2 π N
p j = p cos ( ϕ ϕ j ) ρ ^ p sin ( ϕ ϕ j ) ϕ ^ ,
z n m j = W n m j = W n m ρ ρ ^ 1 ρ W n m ϕ ϕ ^ j
M n m = p j = 1 N 1 ρ cos ( ϕ ϕ j ) W n m ϕ sin ( ϕ ϕ j ) W n m ρ j
0 ψ 2 π N
M n m = p D E ρ = 0 1 ψ = 0 2 π / N j = 1 N [ 1 ρ sin ( π N ψ ) W n m ( ρ , 2 π j N + ϕ 0 ψ ) ψ + cos ( π N ψ ) W n m ( ρ , 2 π j N + ϕ 0 ψ ) ρ ] ρ d ρ d ψ
M n m = 2 π p D A n m λ E [ m F s s ρ = 0 1 R n m ( ρ ) d ρ + F c c ρ = 0 1 ρ R n m ( ρ ) ρ d ρ ]
F s s = ψ = 0 2 π / N [ sin ( π N ψ ) j = 1 N sin ( 2 π j m N + m ϕ 0 m ψ ) ] d ψ
F c c = ψ = 0 2 π / N [ cos ( π N ψ ) j = 1 N cos ( 2 π j m N + m ϕ 0 m ψ ) ] d ψ
m = κ N | κ = 0 , 1 , 2 ,
F s s = N κ 2 N 2 1 [ 2 κ N sin ( π N ) cos ( κ N ϕ 0 ) ]
F c c = N κ 2 N 2 1 [ 2 sin ( π N ) cos ( κ N ϕ 0 ) ]
ρ = 0 1 ρ R n m ( ρ ) ρ d ρ = 1 ρ = 0 1 R n m ( ρ ) d ρ
ρ = 0 1 R n m ( ρ ) d ρ = ( 1 ) ( n m ) / 2 ( n + 1 )
M n m = 2 π p D A n m λ E [ F c c + ( m F s s F c c ) ( ( 1 ) ( n m ) / 2 1 ( n + 1 ) ) ]
M n κ N = 4 π p D A n κ N N λ E ( κ 2 N 2 1 ) sin ( π N ) cos ( κ N ϕ 0 ) [ 1 + ( ( 1 ) ( n κ N ) / 2 ( κ 2 N 2 1 ) ( n + 1 ) ) ]
F s c = N κ 2 N 2 1 [ 2 κ N ( 1 ) κ sin ( π N ) sin ( κ N ϕ 0 ) ]
F c s = N κ 2 N 2 1 [ 2 ( 1 ) κ sin ( π N ) sin ( κ N ϕ 0 ) ]
M n κ N = ( 1 ) κ 4 π p D A n κ N N λ E ( κ 2 N 2 1 ) sin ( π N ) sin ( κ N ϕ 0 ) [ 1 ( ( 1 ) ( n κ N ) / 2 ( κ 2 N 2 + 1 ) ( n + 1 ) ) ]

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