Abstract

Simple polynomial formulas to calculate the FWHM and full width at 1/e 2 intensity diffraction spot size and the depth of focus at a Strehl ratio of 0.8 and 0.5 as a function of a Gaussian beam truncation ratio and a system f-number are presented. Formulas are obtained by use of the numerical integration of a Huygens-Fresnel diffraction integral and can be used to calculate the number of resolvable spots, the modulation transfer function, and the defocus tolerance of optical systems that employ laser beams. I also derived analytical formulas for the diffraction ring intensity as a function of the Gaussian beam truncation ratio and the system f-number. Such formulas can be used to estimate the diffraction-limited contrast of display and imaging systems.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Urey, D. Wine, T. Osborn, “Optical performance requirements for MEMS scanner based microdisplays,” in MOEMS and Miniaturized Systems, M. E. Motamedi, R. Goering, eds., Proc. SPIE4178, 176–185 (2000).
    [CrossRef]
  2. G. Marshall, ed., Optical Scanning (Marcel-Dekker, New York, 1991).
  3. H. Urey, F. B. Mccormick, “Storage limits of two-photon based three-dimensional memories,” in Optical Computing, Vol. 8 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1997), pp. 134–136.
  4. M. M. Wang, S. C. Esener, F. B. McCormick, I. Cokgor, A. S. Dvornikov, P. M. Rentzepis, “Experimental characterization of a two-photon memory,” Opt. Lett. 22, 558–560 (1997).
    [CrossRef] [PubMed]
  5. V. N. Mahajan, “Axial irradiance and optimum focusing of laser beams,” Appl. Opt. 22, 3042–3053 (1983).
    [CrossRef] [PubMed]
  6. V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of the diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A 3, 470–485 (1986).
    [CrossRef]
  7. Y. Li, “Degeneracy in the Fraunhofer diffraction of truncated Gaussian beam,” J. Opt. Soc. Am. A 4, 1237–1242 (1987).
    [CrossRef]
  8. H. T. Yura, “Optimum truncation of a Gaussian beam for propagation through atmospheric turbulence,” Appl. Opt. 34, 2774–2779 (1995).
    [CrossRef] [PubMed]
  9. H. T. Yura, T. S. Rose, “Gaussian beam transfer through hard-aperture optics,” Appl. Opt. 34, 6826–6828 (1995).
    [CrossRef] [PubMed]
  10. P. Belland, J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt. 21, 522–527 (1982).
    [CrossRef] [PubMed]
  11. K. Tanaka, N. Saga, H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt. 24, 1102–1106 (1985).
    [CrossRef] [PubMed]
  12. G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
    [CrossRef]
  13. V. Nourrit, J.-L. de Bougrenet de la Tocnaye, P. Chanclou, “Propagation and diffraction of truncated Gaussian beams,” J. Opt. Soc. Am. A 18, 546–556 (2001).
    [CrossRef]
  14. E. M. Drege, N. G. Skinner, D. M. Byrne, “Analytical far-field divergence angle of a truncated Gaussian beam,” Appl. Opt. 39, 4918–4925 (2000).
    [CrossRef]
  15. V. N. Mahajan, Optical Imaging and Aberration Part II, Wave Diffraction Optics (SPIE Press, Bellingham, Wash., 2001).
    [CrossRef]
  16. M. Abromovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  17. H. Urey, “Diffraction limited resolution and maximum contrast for scanning displays,” Proc. Soc. Inf. Disp. 31, 866–869 (2000).
  18. G. de Wit, “Contrast budget of head mounted displays,” Opt. Eng. 41, 2419–2426 (2002).
    [CrossRef]
  19. H. Urey, “Diffractive exit-pupil expander for display applications,” Appl. Opt. 40, 5840–5851 (2001).
    [CrossRef]

2002

G. de Wit, “Contrast budget of head mounted displays,” Opt. Eng. 41, 2419–2426 (2002).
[CrossRef]

2001

2000

E. M. Drege, N. G. Skinner, D. M. Byrne, “Analytical far-field divergence angle of a truncated Gaussian beam,” Appl. Opt. 39, 4918–4925 (2000).
[CrossRef]

H. Urey, “Diffraction limited resolution and maximum contrast for scanning displays,” Proc. Soc. Inf. Disp. 31, 866–869 (2000).

1997

1996

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[CrossRef]

1995

H. T. Yura, “Optimum truncation of a Gaussian beam for propagation through atmospheric turbulence,” Appl. Opt. 34, 2774–2779 (1995).
[CrossRef] [PubMed]

H. T. Yura, T. S. Rose, “Gaussian beam transfer through hard-aperture optics,” Appl. Opt. 34, 6826–6828 (1995).
[CrossRef] [PubMed]

1987

Y. Li, “Degeneracy in the Fraunhofer diffraction of truncated Gaussian beam,” J. Opt. Soc. Am. A 4, 1237–1242 (1987).
[CrossRef]

1986

1985

1983

1982

Abromovitz, M.

M. Abromovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Belland, P.

Byrne, D. M.

Chanclou, P.

Cokgor, I.

Crenn, J. P.

de Bougrenet de la Tocnaye, J.-L.

de Wit, G.

G. de Wit, “Contrast budget of head mounted displays,” Opt. Eng. 41, 2419–2426 (2002).
[CrossRef]

Drege, E. M.

Dvornikov, A. S.

Esener, S. C.

Lenz, G.

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[CrossRef]

Li, Y.

Y. Li, “Degeneracy in the Fraunhofer diffraction of truncated Gaussian beam,” J. Opt. Soc. Am. A 4, 1237–1242 (1987).
[CrossRef]

Mahajan, V. N.

McCormick, F. B.

M. M. Wang, S. C. Esener, F. B. McCormick, I. Cokgor, A. S. Dvornikov, P. M. Rentzepis, “Experimental characterization of a two-photon memory,” Opt. Lett. 22, 558–560 (1997).
[CrossRef] [PubMed]

H. Urey, F. B. Mccormick, “Storage limits of two-photon based three-dimensional memories,” in Optical Computing, Vol. 8 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1997), pp. 134–136.

Mizokami, H.

Nourrit, V.

Osborn, T.

H. Urey, D. Wine, T. Osborn, “Optical performance requirements for MEMS scanner based microdisplays,” in MOEMS and Miniaturized Systems, M. E. Motamedi, R. Goering, eds., Proc. SPIE4178, 176–185 (2000).
[CrossRef]

Rentzepis, P. M.

Rose, T. S.

Saga, N.

Skinner, N. G.

Stegun, I. A.

M. Abromovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Tanaka, K.

Urey, H.

H. Urey, “Diffractive exit-pupil expander for display applications,” Appl. Opt. 40, 5840–5851 (2001).
[CrossRef]

H. Urey, “Diffraction limited resolution and maximum contrast for scanning displays,” Proc. Soc. Inf. Disp. 31, 866–869 (2000).

H. Urey, D. Wine, T. Osborn, “Optical performance requirements for MEMS scanner based microdisplays,” in MOEMS and Miniaturized Systems, M. E. Motamedi, R. Goering, eds., Proc. SPIE4178, 176–185 (2000).
[CrossRef]

H. Urey, F. B. Mccormick, “Storage limits of two-photon based three-dimensional memories,” in Optical Computing, Vol. 8 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1997), pp. 134–136.

Wang, M. M.

Wine, D.

H. Urey, D. Wine, T. Osborn, “Optical performance requirements for MEMS scanner based microdisplays,” in MOEMS and Miniaturized Systems, M. E. Motamedi, R. Goering, eds., Proc. SPIE4178, 176–185 (2000).
[CrossRef]

Yura, H. T.

H. T. Yura, T. S. Rose, “Gaussian beam transfer through hard-aperture optics,” Appl. Opt. 34, 6826–6828 (1995).
[CrossRef] [PubMed]

H. T. Yura, “Optimum truncation of a Gaussian beam for propagation through atmospheric turbulence,” Appl. Opt. 34, 2774–2779 (1995).
[CrossRef] [PubMed]

Appl. Opt.

H. T. Yura, “Optimum truncation of a Gaussian beam for propagation through atmospheric turbulence,” Appl. Opt. 34, 2774–2779 (1995).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am. A

Y. Li, “Degeneracy in the Fraunhofer diffraction of truncated Gaussian beam,” J. Opt. Soc. Am. A 4, 1237–1242 (1987).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[CrossRef]

Opt. Eng.

G. de Wit, “Contrast budget of head mounted displays,” Opt. Eng. 41, 2419–2426 (2002).
[CrossRef]

Opt. Lett.

Proc. Soc. Inf. Disp.

H. Urey, “Diffraction limited resolution and maximum contrast for scanning displays,” Proc. Soc. Inf. Disp. 31, 866–869 (2000).

Other

V. N. Mahajan, Optical Imaging and Aberration Part II, Wave Diffraction Optics (SPIE Press, Bellingham, Wash., 2001).
[CrossRef]

M. Abromovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

H. Urey, D. Wine, T. Osborn, “Optical performance requirements for MEMS scanner based microdisplays,” in MOEMS and Miniaturized Systems, M. E. Motamedi, R. Goering, eds., Proc. SPIE4178, 176–185 (2000).
[CrossRef]

G. Marshall, ed., Optical Scanning (Marcel-Dekker, New York, 1991).

H. Urey, F. B. Mccormick, “Storage limits of two-photon based three-dimensional memories,” in Optical Computing, Vol. 8 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1997), pp. 134–136.

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 (4)

Fig. 1
Fig. 1

(a) Schematic of converging Gaussian beam truncated by a hard aperture; (b) Power loss at the aperture and peak focal plane irradiance as a function of T.

Fig. 2
Fig. 2

Focal plane irradiance cross sections for different Gaussian beam truncation ratios assuming unity total power beam. (a) Irradiance for r n < 2, (b) Irradiance in log scale for r n > 2. Solid curves are numerical solution of Eq. (5) normalized by the beam power transferred from the aperture, and dashed curves are approximate analytical solutions of Eqs. (12) and (13). Peak ring irradiances are also shown.

Fig. 3
Fig. 3

Spot-size constant and the depth-of-focus constants as a function of T by use of the formulas in Table 1.

Fig. 4
Fig. 4

(a) Integrand and the result of integration in Eq. (12) as a function of ρ for different values of r n , with dashed curves showing the exact formula and solid curves showing the approximate formula in Eq. (12); (b) the result of the integral for the integrand in (a) with integration limit from 0 to ρ (result for r n = 1 is too large and is not shown).

Tables (2)

Tables Icon

Table 1 Optical System Parameters for Three Exemplary Optical Imaging Systems for Different Applications Assuming T= 1a

Tables Icon

Table 2 Spot Size and Depth-of-Focus Formulas as a Function of Truncation Ratio T. K and K 2 for T < 0.5 Are Calculated by Use of Standard Gaussian Beam Formulas

Equations (15)

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

| U r ,   w m | = 2 π 1 w m exp -   r 2 w m 2 ,
P beam = 1 - exp - 2 a 2 / w m 2 = 1 - exp - 2 / T 2 ,
I T ,   r 2 ,   z = 0 a 2 π λ z   | U r ,   w m | exp i π r 2 λ 1 z -   1 R J o 2 π rr 2 λ z r d r 2 ,
I T ,   r n ,   z =   8 π a 2 λ 2 z 2 T 2 P beam   0 1 exp - ρ 2 / T 2 exp [ i π ρ 2 a 2 λ ( 1 z 1 R ) ] i π ρ 2 N Δ z z   J o π ρ r n R z ρ d ρ 2 ,
I focal T ,   r n =   2 π λ 2 f # 2 T 2 P beam   × 0 1 exp - ρ 2 / T 2 J o π ρ r n ρ d ρ 2 .
I 0 T =   π T 2 1 - exp - 1 / T 2 2 2 λ 2 f # 2 1 - exp - 2 / T 2 ,   I 0 T =   π 4 λ 2 f # 2 .
I axial T ,   z =   2 π a 2 T 2 coth 1 / T 2 - cos π N Δ z z sinh 1 / T 2 λ 2 z 2 1 +   π 2 T 4 N 2 Δ z 2 z 2 .
s = K λ f # ,
Δ z = K 2 z / 4 N K 2 λ f # 2 ,
K FWE 2 =   0.97 T e 1 - exp - 1 / T 2 - 1 1 / 2 .
J o x   2 / π x   cos π / 4 - x .
Integrand _ exact = 0 1 exp - ρ 2 / T 2 J o π ρ r n ρ d ρ ,   Integrand _ approx = exp - 1 / T 2 2 π 2 r n x 1 cos π / 4 - π ρ r n d ρ ,
I focal T ,   r n   2 π   exp - 2 / T 2 λ 2 f # 2 T 2 P beam 2 π 2 r n × x 1 cos π / 4 - π ρ r n d ρ 2   4   exp - 2 / T 2 λ 2 f # 2 π 3 r n 3 T 2 P beam sin 2 π r n - q ,
I focal T ,   r n   2   sin 2 π r n - π / 4 λ 2 f # 2 π 3 r n 3 .
I focal , avg T ,   r n =   2   exp - 2 / T 2 λ 2 f # 2 π 3 r n 3 T 2 1 - exp - 2 / T 2 ,   I focal , avg T ,   r n =   1 λ 2 f # 2 π 3 r n 3 .

Metrics