Abstract

In imaging systems and especially in confocal microscopy systems there is a trade-off between the lateral resolution and the obtained depth of focus. The use of complex pupils to improve the lateral resolution by engineering the point spread function is a common approach; however the lateral improvement reduces the effective depth of focus and therefore the fluorescence efficiency. In this work we analytically develop an optimized approach for obtaining a complex pupil with an extended depth of focus. The proposed solution is numerically applied and tested in designing an improved focal depth in confocal microscope configuration.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. S. Dam, I. R. Perch-Nielsen, D. Palima, and J. Glückstad, “Three-dimensional imaging in three-dimensional optical multi-beam micromanipulation,” Opt. Express 16, 7244–7250 (2008).
    [CrossRef] [PubMed]
  2. S. Delica and C. M. Blanca, “Wide-field depth-sectioning fluorescence microscopy using projector-generated patterned illumination,” Appl. Opt. 46, 7237–7243 (2007).
    [CrossRef] [PubMed]
  3. B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
    [CrossRef]
  4. R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34, 293–300 (2003).
    [CrossRef] [PubMed]
  5. A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. G. Martinez, “Superresolving optical system with time multiplexing and computer decoding,” Appl. Opt. 38, 7245–7251 (1999).
    [CrossRef]
  6. I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. 72, 1287–1291 (1982).
    [CrossRef]
  7. D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11, 2747–2752 (2003).
    [CrossRef] [PubMed]
  8. Z. S. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [CrossRef]
  9. P. Dufour, M. Piché, Y. De Koninck, and N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. 45, 9246–9252 (2006).
    [CrossRef] [PubMed]
  10. S. A. Rolfe and J. D. Scholes, “Extended depth-of-focus imaging of chlorophyll fluorescence from intact leaves,” Photosynth. Res. 72, 107–115 (2002).
    [CrossRef]
  11. W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc. Am. 50, 749–752 (1960).
    [CrossRef]
  12. C. J. R. Sheppard and T. Wilson, “Imaging properties of annular lenses,” Appl. Opt. 18, 3764–3769 (1979).
    [CrossRef] [PubMed]
  13. E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
    [CrossRef]
  14. G. Shabtay, Z. Zalevsky, U. Levy, and D. Mendlovic, “Optimal synthesis of 3-D complex amplitude distributions,” Opt. Lett. 25, 363–365 (2000).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  16. U. Levy, G. Shabtay, D. Mendlovic, Z. Zalevsky, and E. Marom, “Iterative algorithm for determining optimal beam profiles in a 3-D space,” Appl. Opt. 38, 6732–6736 (1999).
    [CrossRef]
  17. D. Sazbon, Z. Zalevsky, and E. Rivlin, “Qualitative real-time range extraction for preplanned scene partitioning using laser beam coding,” Pattern Recogn. Lett. 26, 1772–1781 (2005).
    [CrossRef]
  18. D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–413 (1997).
    [CrossRef]
  19. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer–Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 1–12 (2009).
    [CrossRef]
  20. J. W. Y. Lit and R. Tremblay, “Focal depth of a transmitting axicon,” J. Opt. Soc. Am. 63, 445–449 (1973).
    [CrossRef]

2009

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer–Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 1–12 (2009).
[CrossRef]

2008

2007

S. Delica and C. M. Blanca, “Wide-field depth-sectioning fluorescence microscopy using projector-generated patterned illumination,” Appl. Opt. 46, 7237–7243 (2007).
[CrossRef] [PubMed]

B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
[CrossRef]

2006

E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[CrossRef]

P. Dufour, M. Piché, Y. De Koninck, and N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. 45, 9246–9252 (2006).
[CrossRef] [PubMed]

2005

D. Sazbon, Z. Zalevsky, and E. Rivlin, “Qualitative real-time range extraction for preplanned scene partitioning using laser beam coding,” Pattern Recogn. Lett. 26, 1772–1781 (2005).
[CrossRef]

2003

D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11, 2747–2752 (2003).
[CrossRef] [PubMed]

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34, 293–300 (2003).
[CrossRef] [PubMed]

2002

S. A. Rolfe and J. D. Scholes, “Extended depth-of-focus imaging of chlorophyll fluorescence from intact leaves,” Photosynth. Res. 72, 107–115 (2002).
[CrossRef]

2000

1999

1997

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–413 (1997).
[CrossRef]

1996

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

1986

1982

1979

1973

1960

Blanca, C. M.

Botcherby, E. J.

E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[CrossRef]

Braat, J. J. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer–Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 1–12 (2009).
[CrossRef]

Cox, I. J.

Dam, J. S.

De Koninck, Y.

Delica, S.

Dufour, P.

Gan, X.

Ganic, D.

Garcia, J.

Glückstad, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gu, M.

Hanley, Q. S.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34, 293–300 (2003).
[CrossRef] [PubMed]

Heckenberga, N.

B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
[CrossRef]

Hegedus, Z. S.

Heintzmann, R.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34, 293–300 (2003).
[CrossRef] [PubMed]

Janssen, A. J. E. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer–Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 1–12 (2009).
[CrossRef]

Jovin, T. M.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34, 293–300 (2003).
[CrossRef] [PubMed]

Juskaitis, R.

E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[CrossRef]

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–413 (1997).
[CrossRef]

Laib, K.

B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
[CrossRef]

Levy, U.

Lit, J. W. Y.

Littleton, B.

B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
[CrossRef]

Longstaffb, D.

B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
[CrossRef]

Marom, E.

Martinez, P. G.

McCarthy, N.

Mendlovic, D.

Munroe, P.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34, 293–300 (2003).
[CrossRef] [PubMed]

Munroee, P.

B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
[CrossRef]

Nailon, J.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34, 293–300 (2003).
[CrossRef] [PubMed]

Palima, D.

Perch-Nielsen, I. R.

Pereira, S. F.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer–Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 1–12 (2009).
[CrossRef]

Piché, M.

Rivlin, E.

D. Sazbon, Z. Zalevsky, and E. Rivlin, “Qualitative real-time range extraction for preplanned scene partitioning using laser beam coding,” Pattern Recogn. Lett. 26, 1772–1781 (2005).
[CrossRef]

Rolfe, S. A.

S. A. Rolfe and J. D. Scholes, “Extended depth-of-focus imaging of chlorophyll fluorescence from intact leaves,” Photosynth. Res. 72, 107–115 (2002).
[CrossRef]

Rubinsztein-Dunlopa, H.

B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
[CrossRef]

Sarafis, V.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34, 293–300 (2003).
[CrossRef] [PubMed]

Z. S. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
[CrossRef]

Sarafisa, V.

B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
[CrossRef]

Sazbon, D.

D. Sazbon, Z. Zalevsky, and E. Rivlin, “Qualitative real-time range extraction for preplanned scene partitioning using laser beam coding,” Pattern Recogn. Lett. 26, 1772–1781 (2005).
[CrossRef]

Scholes, J. D.

S. A. Rolfe and J. D. Scholes, “Extended depth-of-focus imaging of chlorophyll fluorescence from intact leaves,” Photosynth. Res. 72, 107–115 (2002).
[CrossRef]

Shabtay, G.

Shemer, A.

Sheppard, C. J. R.

Tremblay, R.

van Haver, S.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer–Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 1–12 (2009).
[CrossRef]

Welford, W. T.

Wilson, T.

Zalevsky, Z.

Appl. Opt.

J. Eur. Opt. Soc. Rapid Publ.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer–Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. 4, 09048 1–12 (2009).
[CrossRef]

J. Mod. Opt.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–413 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Micron

B. Littleton, K. Laib, D. Longstaffb, V. Sarafisa, P. Munroee, N. Heckenberga, and H. Rubinsztein-Dunlopa, “Coherent super-resolution microscopy via laterally structured illumination,” Micron 38, 150–157 (2007).
[CrossRef]

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34, 293–300 (2003).
[CrossRef] [PubMed]

Opt. Commun.

E. J. Botcherby, R. Juskaitis, and T. Wilson, “Scanning two photon fluorescence microscopy with extended depth of field,” Opt. Commun. 268, 253–260 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Pattern Recogn. Lett.

D. Sazbon, Z. Zalevsky, and E. Rivlin, “Qualitative real-time range extraction for preplanned scene partitioning using laser beam coding,” Pattern Recogn. Lett. 26, 1772–1781 (2005).
[CrossRef]

Photosynth. Res.

S. A. Rolfe and J. D. Scholes, “Extended depth-of-focus imaging of chlorophyll fluorescence from intact leaves,” Photosynth. Res. 72, 107–115 (2002).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Optical setup for confocal microscope scanning system.

Fig. 2
Fig. 2

Diffraction of a Gaussian beam ( X - Z cross section). (a) Without phase element. (b) With positive axicon. (c) With the proposed phase element.

Fig. 3
Fig. 3

Desired illumination PSF in scanning plane ( X - Y cross section).

Fig. 4
Fig. 4

The designed phase element.

Fig. 5
Fig. 5

Diffraction of a desired pattern ( X - Z cross section) of beam profile in Fig. 3. (a) Without phase element. (b) With the element. Zoomed image of the edge of the DOF region. (c) Without phase element. (d) With the element.

Fig. 6
Fig. 6

Comparison of intensity patterns at different axial positions ( X - Y cross section). Images (a)–(d) present the intensities without the phase element at axial distances of 1.8, 3.6, 5.4, and 7.2 μ m , respectively. Images (e)–(h) present the intensities with the phase element at axial distances of 1.8, 3.6, 5.4, and 7.2 μ m , respectively.

Equations (25)

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

ε = x z [ | u d ( x ) exp ( i ϕ ( x ) ) h ( x , z ) | | u d ( x ) | ] 2 d x d z ,
h ( x , z ) = exp ( π i x 2 λ z ) .
ε = x z | u d ( x ) exp ( i ϕ ( x ) ) h ( x , z ) | 2 d x d z + x z | u d ( x ) | 2 d x d z 2 x z | u d ( x ) exp ( i ϕ ( x ) ) h ( x , z ) | | u d ( x ) | d x d z .
δ ε = ε ϕ δ ϕ .
ϕ [ x z | u d ( x ) exp ( i ϕ ( x ) ) h ( x , z ) | 2 d x d z ] = 0.
ϕ [ x z | u d ( x ) | 2 d x d z ] = 0.
ε + δ ε = 2 x z | u d ( x ) | { x 1 x 2 u d ( x 1 ) u d ( x 2 ) exp [ i ( ϕ ( x 1 ) + δ ϕ ( x 1 ) ) ] exp [ i ( ϕ ( x 2 ) + δ ϕ ( x 2 ) ) ] exp [ π i λ z ( x 1 2 x 2 2 ) ] exp [ 2 π i x λ z ( x 1 x 2 ) ] d x 2 d x 1 } 0.5 d z d x .
ε + δ ε = 2 x z | u d ( x ) | { x 1 x 2 u d ( x 1 ) u d ( x 2 ) exp [ i ( ϕ ( x 1 ) ϕ ( x 2 ) ) ] exp [ π i λ z ( x 1 2 x 2 2 ) ] exp [ 2 π i x λ z ( x 1 x 2 ) ] [ 1 + i ( δ ϕ ( x 1 ) δ ϕ ( x 2 ) ) ] d x 1 d x 2 } 0.5 d x d z .
1 + 2 = 1 1 + 2 / 1 1 ( 1 + 2 2 1 ) = 1 + 2 2 1 ,
ε + δ ε = 2 x z | u d ( x ) | [ { x 1 x 2 u d ( x 1 ) u d ( x 2 ) exp [ i ( ϕ ( x 1 ) ϕ ( x 2 ) ) ] exp [ π i λ z ( x 1 2 x 2 2 ) ] exp [ 2 π i x λ z ( x 1 x 2 ) ] d x 1 d x 2 } 0.5 + 0.5 { x 1 x 2 u d ( x 1 ) u d ( x 2 ) exp [ i ( ϕ ( x 1 ) ϕ ( x 2 ) ) + π i λ z ( x 1 2 x 2 2 ) + 2 π i x λ z ( x 1 x 2 ) ] i [ δ ϕ ( x 1 ) δ ϕ ( x 2 ) ] d x 1 d x 2 } { x 1 x 2 u d ( x 1 ) u d ( x 2 ) exp [ i ( ϕ ( x 1 ) ϕ ( x 2 ) ) + π i λ z ( x 1 2 x 2 2 ) + 2 π i x λ z ( x 1 x 2 ) ] d x 1 d x 2 } 0.5 ] d x d z .
δ ε = x z | u d ( x ) | x 1 x 2 A z ( x , x 1 ) A z ( x , x 1 ) [ δ ϕ ( x 1 ) δ ϕ ( x 2 ) ] d x 1 d x 2 x 1 x 2 A z ( x , x 1 ) A z ( x , x 1 ) d x 1 d x 2 d x d z ,
A z ( x , x 1 ) u d ( x 1 ) exp [ π i x 1 2 λ z ] exp [ 2 π i x x 1 λ z ] exp [ i ϕ ( x 1 ) ] .
x z | u d ( x ) | ( x 1 A z ( x , x 1 ) δ ϕ ( x 1 ) d x 1 ) ( x 1 A z ( x , x 1 ) d x 1 ) ( x 1 A z ( x , x 1 ) d x 1 ) ( x 1 A z ( x , x 1 ) d x 1 ) d x d z = x z | u d ( x ) | ( x 1 A z ( x , x 1 ) d x 1 ) ( x 1 A z ( x , x 1 ) δ ϕ ( x 1 ) d x 1 ) ( x 1 A z ( x , x 1 ) d x 1 ) ( x 1 A z ( x , x 1 ) d x 1 ) d x d z .
x z | u d ( x ) | ( x 1 A z ( x , x 1 ) δ ϕ ( x 1 ) d x 1 ) ( x 1 A z ( x , x 1 ) d x 1 ) | x 1 A z ( x , x 1 ) d x 1 | d x d z = x z | u d ( x ) | ( x 1 A z ( x , x 1 ) d x 1 ) ( x 1 A z ( x , x 1 ) δ ϕ ( x 1 ) d x 1 ) | x 1 A z ( x , x 1 ) d x 1 | d x d z ,
Im { x z | u d ( x ) | ( x 1 A z ( x , x 1 ) δ ϕ ( x 1 ) d x 1 ) ( x 1 A z ( x , x 1 ) d x 1 ) | x 1 A z ( x , x 1 ) d x 1 | d x d z } = 0.
x 1 A z ( x , x 1 ) d x 1 β z ( x ) exp [ i ψ z ( x ) ] ,
Im { x z | u d ( x ) | exp [ i ψ z ( x ) ] ( x 1 A z ( x , x 1 ) δ ϕ ( x 1 ) d x 1 ) d x d z } = 0.
Im { x 1 δ ϕ ( x 1 ) exp ( i ϕ ( x 1 ) ) u d ( x 1 ) [ x z | u d ( x ) | exp [ i ψ z ( x ) + π i x 1 2 λ z + 2 π i x x 1 λ z ] d x d z ] d x 1 } = 0.
Im { x 1 δ ϕ ( x 1 ) u d ( x 1 ) [ cos ( ϕ ( x 1 ) ) + i   sin ( ϕ ( x 1 ) ) ] [ x z | u d ( x ) | ( cos ( π x 1 2 λ z 2 π x x 1 λ z ψ z ( x ) ) + i   sin ( π x 1 2 λ z 2 π x x 1 λ z ψ z ( x ) ) ) d x d z ] d x 1 } = 0.
cos ( ϕ ( x 1 ) ) x z | u d ( x ) | sin ( π x 1 2 λ z 2 π x x 1 λ z ψ z ( x ) ) d x d z = sin ( ϕ ( x 1 ) ) x z | u d ( x ) | cos ( π x 1 2 λ z 2 π x x 1 λ z ψ z ( x ) ) d x d z .
tan ( ϕ ( x 1 ) ) = x z | u d ( x ) | sin ( π x 1 2 λ z 2 π x x 1 λ z ψ z ( x ) ) d x d z x z | u d ( x ) | cos ( π x 1 2 λ z 2 π x x 1 λ z ψ z ( x ) ) d x d z ,
i ψ z ( x ) = ln [ x 1 u d ( x 1 ) exp ( π i x 1 2 λ z ) exp ( 2 π i x x 1 λ z ) exp ( i ϕ ( x 1 ) ) d x 1 | x 1 u d ( x 1 ) exp ( π i x 1 2 λ z ) exp ( 2 π i x x 1 λ z ) exp ( i ϕ ( x 1 ) ) d x 1 | ] .
ϕ ( x 1 ) = angle [ x z | u d ( x ) | exp ( i ψ z ( x ) ) exp ( i π x 1 2 λ z + 2 π i x x 1 λ z ) d x d z ] ,
ψ z ( x ) = angle [ x 1 u d ( x 1 ) exp ( i ϕ ( x 1 ) ) exp ( π i x 1 2 λ z ) exp ( 2 π i x x 1 λ z ) d x 1 ] .
ϕ ( x 1 ) = angle [ z Fresnel 1 { u d ( x ) exp ( i   angle [ Fresnel { u d ( x 1 ) exp ( i ϕ ( x 1 ) ) } ] ) } d z ] ,

Metrics