Abstract

In several optical systems, a specific Point Spread Function (PSF) needs to be generated. This can be achieved by shaping the complex field at the pupil. The Extended Nijboer-Zernike (ENZ) theory relates complex Zernike modes on the pupil directly to functions in the focal region. In this paper, we introduce a method to engineer a PSF using the ENZ theory. In particular, we present an optimization algorithm to design an extended depth of focus with high lateral resolution, while keeping the transmission of light high (over 60%). We also have demonstrated three outcomes of the algorithm using a Spatial Light Modulator (SLM).

© 2014 Optical Society of America

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  1. G. B. Airy, “On the diffraction of an annular aperture,” Philos. Mag. 18, 1–10 (1841).
  2. J. Durnin, J. J. Miceli, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. Y. Xu, J. Singh, C. Sheppard, N. Chen, “Ultra long high resolution beam by multi-zone rotationally symmetrical complex pupil filter,” Opt. Express 15, 6409–6413 (2007).
    [CrossRef] [PubMed]
  4. H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photonics Rev. 6, 354–392 (2012).
    [CrossRef]
  5. B. J. Thompson, “Diffraction by semitransparent and phase annuli,” J. Opt. Soc. Am. 55, 145–148 (1965).
    [CrossRef]
  6. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
    [CrossRef]
  7. H. Wang, F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. 40, 5658–5662 (2001).
    [CrossRef]
  8. C. J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Opt. 43, 525–536 (1996).
    [CrossRef]
  9. J. Ojeda-Castañeda, L. R. Berriel-Valdos, “Arbitrarily high focal depth with finite apertures,” Opt. Lett. 13, 183–185 (1988).
    [CrossRef] [PubMed]
  10. H. Wang, G. Yuan, W. Tan, L. Shi, T. Chong, “Spot size and depth of focus in optical data storage system,” Opt. Eng. 46, 065201 (2007).
    [CrossRef]
  11. L. Wei, “Evaluation of Extended Nijboer-Zernike theory as a tool for complex point spread function calculation,” Master thesis (Delft University of Technology, 2012).
  12. J. J. M. Braat, A. J. E. M. Janssen, P. Dirksen, S. van Haver, “Assessment of optical systems by means of point spread functions,” Prog. Opt. 51, 349–468 (2008).
  13. C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5, 81–101 (2011).
    [CrossRef]
  14. A. J. E. M. Janssen, “ENZ approach for the computation of optical point spread function,” J. Opt. Soc. Am. A 19, 849–857 (2002).
    [CrossRef]
  15. A. J. E. M. Janssen, J. J. M. Braat, P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
    [CrossRef]
  16. S. van Haver, A. J. E. M. Janssen, “Advanced analytic treatment and efficient computation of the diffraction integrals in the Extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rap. Publ. 8, 13044 (2013).
    [CrossRef]
  17. Holoeye, “PLUTO Phase Only Spatial Light Modulator (Reflective),” http://holoeye.com/spatial-light-modulators/slm-pluto-phase-only/ , June 24, 2013.
  18. O. El Gawhary, A. Wiegmann, N. Kumar, S. F. Pereira, H. P. Urbach, “Through-focus phase retrieval and its connection to the spatial correlation for propagating fields,” Opt. Express 21, 5550–5560 (2013).
    [CrossRef] [PubMed]
  19. H. Zhang, J. Xie, J. Liu, Y. Wang, “Elimination of a zero-order beam induced by a pixelated spatial light modulator for holographic projection,” Appl. Opt. 48, 5834–5841 (2009).
    [CrossRef] [PubMed]

2013

S. van Haver, A. J. E. M. Janssen, “Advanced analytic treatment and efficient computation of the diffraction integrals in the Extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rap. Publ. 8, 13044 (2013).
[CrossRef]

O. El Gawhary, A. Wiegmann, N. Kumar, S. F. Pereira, H. P. Urbach, “Through-focus phase retrieval and its connection to the spatial correlation for propagating fields,” Opt. Express 21, 5550–5560 (2013).
[CrossRef] [PubMed]

2012

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photonics Rev. 6, 354–392 (2012).
[CrossRef]

2011

C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5, 81–101 (2011).
[CrossRef]

2009

2008

J. J. M. Braat, A. J. E. M. Janssen, P. Dirksen, S. van Haver, “Assessment of optical systems by means of point spread functions,” Prog. Opt. 51, 349–468 (2008).

2007

Y. Xu, J. Singh, C. Sheppard, N. Chen, “Ultra long high resolution beam by multi-zone rotationally symmetrical complex pupil filter,” Opt. Express 15, 6409–6413 (2007).
[CrossRef] [PubMed]

H. Wang, G. Yuan, W. Tan, L. Shi, T. Chong, “Spot size and depth of focus in optical data storage system,” Opt. Eng. 46, 065201 (2007).
[CrossRef]

2004

A. J. E. M. Janssen, J. J. M. Braat, P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
[CrossRef]

2002

2001

1996

C. J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Opt. 43, 525–536 (1996).
[CrossRef]

1988

1987

J. Durnin, J. J. Miceli, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1965

1952

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[CrossRef]

1841

G. B. Airy, “On the diffraction of an annular aperture,” Philos. Mag. 18, 1–10 (1841).

Airy, G. B.

G. B. Airy, “On the diffraction of an annular aperture,” Philos. Mag. 18, 1–10 (1841).

Bernet, S.

C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5, 81–101 (2011).
[CrossRef]

Berriel-Valdos, L. R.

Braat, J. J. M.

J. J. M. Braat, A. J. E. M. Janssen, P. Dirksen, S. van Haver, “Assessment of optical systems by means of point spread functions,” Prog. Opt. 51, 349–468 (2008).

A. J. E. M. Janssen, J. J. M. Braat, P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
[CrossRef]

Chen, N.

Chong, T.

H. Wang, G. Yuan, W. Tan, L. Shi, T. Chong, “Spot size and depth of focus in optical data storage system,” Opt. Eng. 46, 065201 (2007).
[CrossRef]

Dirksen, P.

J. J. M. Braat, A. J. E. M. Janssen, P. Dirksen, S. van Haver, “Assessment of optical systems by means of point spread functions,” Prog. Opt. 51, 349–468 (2008).

A. J. E. M. Janssen, J. J. M. Braat, P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

El Gawhary, O.

Gan, F.

Ho, S. T.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photonics Rev. 6, 354–392 (2012).
[CrossRef]

Janssen, A. J. E. M.

S. van Haver, A. J. E. M. Janssen, “Advanced analytic treatment and efficient computation of the diffraction integrals in the Extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rap. Publ. 8, 13044 (2013).
[CrossRef]

J. J. M. Braat, A. J. E. M. Janssen, P. Dirksen, S. van Haver, “Assessment of optical systems by means of point spread functions,” Prog. Opt. 51, 349–468 (2008).

A. J. E. M. Janssen, J. J. M. Braat, P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
[CrossRef]

A. J. E. M. Janssen, “ENZ approach for the computation of optical point spread function,” J. Opt. Soc. Am. A 19, 849–857 (2002).
[CrossRef]

Jesacher, A.

C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5, 81–101 (2011).
[CrossRef]

Kumar, N.

Liu, J.

Maurer, C.

C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5, 81–101 (2011).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Ojeda-Castañeda, J.

Pereira, S. F.

Ravi, K.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photonics Rev. 6, 354–392 (2012).
[CrossRef]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5, 81–101 (2011).
[CrossRef]

Sheppard, C.

Sheppard, C. J. R.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photonics Rev. 6, 354–392 (2012).
[CrossRef]

C. J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Opt. 43, 525–536 (1996).
[CrossRef]

Shi, L.

H. Wang, G. Yuan, W. Tan, L. Shi, T. Chong, “Spot size and depth of focus in optical data storage system,” Opt. Eng. 46, 065201 (2007).
[CrossRef]

Singh, J.

Tan, W.

H. Wang, G. Yuan, W. Tan, L. Shi, T. Chong, “Spot size and depth of focus in optical data storage system,” Opt. Eng. 46, 065201 (2007).
[CrossRef]

Thompson, B. J.

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[CrossRef]

Urbach, H. P.

van Haver, S.

S. van Haver, A. J. E. M. Janssen, “Advanced analytic treatment and efficient computation of the diffraction integrals in the Extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rap. Publ. 8, 13044 (2013).
[CrossRef]

J. J. M. Braat, A. J. E. M. Janssen, P. Dirksen, S. van Haver, “Assessment of optical systems by means of point spread functions,” Prog. Opt. 51, 349–468 (2008).

Vienne, G.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photonics Rev. 6, 354–392 (2012).
[CrossRef]

Wang, H.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photonics Rev. 6, 354–392 (2012).
[CrossRef]

H. Wang, G. Yuan, W. Tan, L. Shi, T. Chong, “Spot size and depth of focus in optical data storage system,” Opt. Eng. 46, 065201 (2007).
[CrossRef]

H. Wang, F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. 40, 5658–5662 (2001).
[CrossRef]

Wang, Y.

Wei, L.

L. Wei, “Evaluation of Extended Nijboer-Zernike theory as a tool for complex point spread function calculation,” Master thesis (Delft University of Technology, 2012).

Wiegmann, A.

Xie, J.

Xu, Y.

Yuan, G.

H. Wang, G. Yuan, W. Tan, L. Shi, T. Chong, “Spot size and depth of focus in optical data storage system,” Opt. Eng. 46, 065201 (2007).
[CrossRef]

Zhang, H.

Appl. Opt.

J. Eur. Opt. Soc. Rap. Publ.

S. van Haver, A. J. E. M. Janssen, “Advanced analytic treatment and efficient computation of the diffraction integrals in the Extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rap. Publ. 8, 13044 (2013).
[CrossRef]

J. Mod. Opt.

A. J. E. M. Janssen, J. J. M. Braat, P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
[CrossRef]

C. J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Opt. 43, 525–536 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Laser Photonics Rev.

C. Maurer, A. Jesacher, S. Bernet, M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5, 81–101 (2011).
[CrossRef]

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photonics Rev. 6, 354–392 (2012).
[CrossRef]

Nuovo Cimento

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[CrossRef]

Opt. Eng.

H. Wang, G. Yuan, W. Tan, L. Shi, T. Chong, “Spot size and depth of focus in optical data storage system,” Opt. Eng. 46, 065201 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Philos. Mag.

G. B. Airy, “On the diffraction of an annular aperture,” Philos. Mag. 18, 1–10 (1841).

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Prog. Opt.

J. J. M. Braat, A. J. E. M. Janssen, P. Dirksen, S. van Haver, “Assessment of optical systems by means of point spread functions,” Prog. Opt. 51, 349–468 (2008).

Other

L. Wei, “Evaluation of Extended Nijboer-Zernike theory as a tool for complex point spread function calculation,” Master thesis (Delft University of Technology, 2012).

Holoeye, “PLUTO Phase Only Spatial Light Modulator (Reflective),” http://holoeye.com/spatial-light-modulators/slm-pluto-phase-only/ , June 24, 2013.

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Figures (9)

Fig. 1
Fig. 1

Example of Itarget(x, 0) and Itarget(0, z) which may be used to find an initial solution β⃗1. Here R A = 0.61 λ NA and R E = n λ NA 2. The vertical line in the figure of Itarget(x, 0) indicates where x = RA. These target functions are approximated using a least-squares method.

Fig. 2
Fig. 2

Figures a and b show the cross sections of the amplitude and phase respectively of one of the pupil functions found with the algorithm. The pupil function is circularly symmetric, so only the cross sections along the radius (which is in this case normalized to the radius a of the aperture) are needed. This pupil function produces an elongated focal spot along the optical axis and a high lateral resolution in the focal plane, as is shown in Figures c and d respectively. The Zernike coefficients are given in Table 1 in the row C2.

Fig. 3
Fig. 3

Figures a, b and c show the pupil masks according to Table 1 which are assigned to the SLM. In the black regions the phase is 0. In the grey regions the phase is π. In the regions with a phase ramp the light is tilted away, which corresponds to the amplitude being modulated to 0. Figures d, e and f show the theoretical intensity distributions in the focal region and the corresponding profiles along the optical axis for the three pupil functions. For comparison, the aberration-free case (uniform illumination) is included.

Fig. 4
Fig. 4

Schematic of the experimental setup used to modulate the phase of the laser beam and scan through the focal field of the 0.4 NA microscope objective.

Fig. 5
Fig. 5

Result of the measurement performed to determine how the Zernike defocus coefficient α 2 0 relates to the Rayleigh unit RE. By seeing for what value of α 2 0 the intensity has dropped by a factor 1/e ≈ 0.37, we know which α 2 0 corresponds to 1 RE (as indicated by the red dashed lines). In this case it turns out to be α 2 0 1.69.

Fig. 6
Fig. 6

The experimental results of intensity distributions of the focal fields for the three pupil functions (from top to bottom) ‘Phase’, ‘Complex 1’ and ‘Complex 2’. The left column shows the on-axis intensity distributions. The right column shows the through-focus intensity distributions in the xz-plane.

Fig. 7
Fig. 7

Comparison of the widths of the focal spots for the aberration free case and for the pupil function ‘Complex 2’.

Fig. 8
Fig. 8

Comparison of the two scanning methods (adding a lens phase and moving the microscope objective). The graphs have been normalized to the intensity at z = 0, since that is the point where no lens phase is added and thus the measurements are the same.

Fig. 9
Fig. 9

Flowchart of the algorithm

Tables (2)

Tables Icon

Table 1 The complex Zernike coefficients used to create the pupil functions ‘Phase’ (P), ‘Complex1’ (C1) and ‘Complex’ (C2) which are tested experimentally.

Tables Icon

Table 2 Details of the setup used to measure the focal field of a wavefront modulated by the PLUTO SLM and focused by a microscope objective. The schematic of the setup is shown in Fig. 4.

Equations (25)

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f = n | m | β n m Z n m ,
Z n m ( ρ , ϕ ) = R n | m | ( ρ ) e i m ϕ ,
R n m ( ρ ) = s = 0 p ( 1 ) s ( n s ) ! s ! ( q s ) ! ( p s ) ! ρ n 2 s , p = 1 2 ( n | m | ) , q = 1 2 ( n + | m | ) .
E 0 ( ρ , θ ) = n | m | β n m Z n m ( ρ , θ ) E ( r , ϕ , f ) n | m | β n m V n | m | ( r , f ) 2 i | m | e i m ϕ ,
I = | E | 2 | k = 0 N β 2 k 0 V 2 k 0 | 2 ,
E 0 ( ρ , ϕ ) = k = 0 N β 2 k 0 Z 2 k 0 .
E ˜ 0 ( ρ , θ ) = k = 0 N β 2 k 0 Z 2 k 0 ( ρ , θ ) ,
E 0 , phase ( ρ ) = sgn ( E ˜ 0 ( ρ ) ) .
E 0 , complex ( ρ ) = { 0 if | E ˜ 0 ( ρ ) | t sgn ( E ˜ 0 ( ρ ) ) if | E ˜ 0 ( ρ ) | > t ,
V n m ( r , f ) = ε m exp ( 1 2 i f ) k = 0 ( 2 k + 1 ) i k j k ( 1 2 f ) l min k + p ( 1 ) l w k l J | m | + 2 l + 1 ( 2 π r ) 2 π r ,
l min = { p k , if 0 < k < p 0 , if p k q k q , if q < k w k l = s = 0 p t = 0 min ( k , s ) f p s | m | b k s t g k + s 2 t , l | m | ,
w k , k + p 2 j = b k p j ; j = 0 , 1 , , min ( k , p ) .
f p s m = ( 1 ) p s [ 2 s + 1 p + s + 1 ( m + p s 1 p s ) ( m + p + s s ) / ( p + s s ) ] , s = 0 , , p
g u l m = m + 2 l + 1 m + u + l + 1 [ ( m u l ) ( u + l u ) / ( m + l + u u ) ] = m + 2 l + 1 m + u + l + 1 ( m u l ) i = 1 u l + i l + m + i , u = l , , l + m ,
b s 1 s 2 t = 2 s 1 + 2 s 2 4 t + 1 2 s 1 + 2 s 2 2 t + 1 ( A s 1 t A t A s 2 t A s 1 + s 2 t ) , t = 0 , , min ( s 1 , s 2 ) ,
f p s 0 = δ p s , g u l 0 = δ u l ,
T = 2 0 1 ρ | E 0 ( ρ ) | 2 d ρ .
I = | E | 2 ( | k = 0 N β 2 k 0 V 2 k 0 | ) 2 ,
E 0 ( ρ , ϕ ) = k = 0 N β 2 k 0 Z 2 k 0 .
x : = ( x 1 x max ) , z : = ( z 1 z max ) , ρ : = ( ρ 1 ρ max ) .
V x , k ( x ) = ( V 2 k 0 ( x 1 , 0 ) V 2 k 0 ( x max , 0 ) ) , V z , k ( z ) = ( V 2 k 0 ( 0 , z 1 ) V 2 k 0 ( 0 , z max ) ) , Z ρ , k ( ρ ) = ( Z 2 k 0 ( ρ 1 ) Z 2 k 0 ( ρ max ) ) .
X = [ V x , 0 ( x ) V x , 1 ( x ) V x , N ( x ) ] , Z = [ V z , 0 ( z ) V z , 1 ( z ) V z , N ( z ) ] , R = [ Z ρ , 0 ( ρ ) Z ρ , 1 ( ρ ) Z ρ , N ( ρ ) ] ,
I ( x , 0 ) = | X β | 2 max | X β | 2 , I ( 0 , z ) = | Z β | 2 max | Z β | 2 , | E 0 ( ρ ) | 2 = | R β | 2 max | R β | 2 .
f = | [ I target ( x , 0 ) I target ( 0 , z ) ] [ I ( x , 0 ) I ( 0 , z ) ] | 2
M T ( β ) = { 1 T ( β ) if x 0 ( β ) x 0 * , z 0 ( β ) z 0 * and M < 0.2 otherwise

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