Abstract

In quantitative measurement of the speckle observed in laser projection displays, it is essential how accurately one simulates the human eye. Based on the eye model given by Westheimer, we succeeded in simulating the optical transfer function of the eye model using that for the circular aperture. The equivalent circular aperture diameter is dependent on the eye model used in the simulation and its viewing conditions, but particularly for the eye models given by Westheimer and Williams, they substantially agreed, yielding approximately 1.2 mm at green under a 3 mm pupil diameter viewing condition.

© 2014 Optical Society of America

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References

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  1. S. Kubota and J. W. Goodman, “Very efficient speckle contrast reduction realized by moving diffuser,” Appl. Opt. 49, 4385–4391 (2010).
    [CrossRef]
  2. J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).
  3. G. Westheimer, “The eye as an optical instrument,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, and J. P. Thomas, eds. (Wiley, 1986), Vol. 1, pp. 4/1–4/20.
  4. R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, 2000).
  5. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).
  6. B. A. Wandell, Foundations of Vision (Sinauer, 1995).
  7. D. Williams, D. Brainard, M. McMahon, and R. Navarro, “Double-pass and interferometric measures of the optical quality of the eye,” J. Opt. Soc. Am. A 11, 3123–3135 (1994).
    [CrossRef]
  8. P. Moon and D. Spencer, “Visual data applied to lighting design,” J. Opt. Soc. Am. 34, 605–614 (1944).
    [CrossRef]
  9. F. Campbell and R. Gubisch, “Optical quality of the human eye,” J. Physiol. 186, 558–578 (1966).
  10. R. Deeley, N. Drasdo, and W. Charman, “A simple parametric model of the human ocular modulation transfer function,” Ophthal. Physiol. Opt. 11, 91–93 (1991).
    [CrossRef]
  11. S. Moriguchi, K. Udagawa, and S. Hitotsumatsu, The Mathematical Formulae (Iwanami, 1956), in Japanese.
  12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

2010 (1)

1994 (1)

1991 (1)

R. Deeley, N. Drasdo, and W. Charman, “A simple parametric model of the human ocular modulation transfer function,” Ophthal. Physiol. Opt. 11, 91–93 (1991).
[CrossRef]

1966 (1)

F. Campbell and R. Gubisch, “Optical quality of the human eye,” J. Physiol. 186, 558–578 (1966).

1944 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, 2000).

Brainard, D.

Campbell, F.

F. Campbell and R. Gubisch, “Optical quality of the human eye,” J. Physiol. 186, 558–578 (1966).

Charman, W.

R. Deeley, N. Drasdo, and W. Charman, “A simple parametric model of the human ocular modulation transfer function,” Ophthal. Physiol. Opt. 11, 91–93 (1991).
[CrossRef]

Deeley, R.

R. Deeley, N. Drasdo, and W. Charman, “A simple parametric model of the human ocular modulation transfer function,” Ophthal. Physiol. Opt. 11, 91–93 (1991).
[CrossRef]

Drasdo, N.

R. Deeley, N. Drasdo, and W. Charman, “A simple parametric model of the human ocular modulation transfer function,” Ophthal. Physiol. Opt. 11, 91–93 (1991).
[CrossRef]

Goodman, J. W.

S. Kubota and J. W. Goodman, “Very efficient speckle contrast reduction realized by moving diffuser,” Appl. Opt. 49, 4385–4391 (2010).
[CrossRef]

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Gubisch, R.

F. Campbell and R. Gubisch, “Optical quality of the human eye,” J. Physiol. 186, 558–578 (1966).

Hitotsumatsu, S.

S. Moriguchi, K. Udagawa, and S. Hitotsumatsu, The Mathematical Formulae (Iwanami, 1956), in Japanese.

Kubota, S.

McMahon, M.

Moon, P.

Moriguchi, S.

S. Moriguchi, K. Udagawa, and S. Hitotsumatsu, The Mathematical Formulae (Iwanami, 1956), in Japanese.

Navarro, R.

Spencer, D.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Udagawa, K.

S. Moriguchi, K. Udagawa, and S. Hitotsumatsu, The Mathematical Formulae (Iwanami, 1956), in Japanese.

Wandell, B. A.

B. A. Wandell, Foundations of Vision (Sinauer, 1995).

Westheimer, G.

G. Westheimer, “The eye as an optical instrument,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, and J. P. Thomas, eds. (Wiley, 1986), Vol. 1, pp. 4/1–4/20.

Williams, D.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

J. Physiol. (1)

F. Campbell and R. Gubisch, “Optical quality of the human eye,” J. Physiol. 186, 558–578 (1966).

Ophthal. Physiol. Opt. (1)

R. Deeley, N. Drasdo, and W. Charman, “A simple parametric model of the human ocular modulation transfer function,” Ophthal. Physiol. Opt. 11, 91–93 (1991).
[CrossRef]

Other (7)

S. Moriguchi, K. Udagawa, and S. Hitotsumatsu, The Mathematical Formulae (Iwanami, 1956), in Japanese.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

G. Westheimer, “The eye as an optical instrument,” in Handbook of Perception and Human Performance, K. R. Boff, L. Kaufman, and J. P. Thomas, eds. (Wiley, 1986), Vol. 1, pp. 4/1–4/20.

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, 2000).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

B. A. Wandell, Foundations of Vision (Sinauer, 1995).

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Figures (7)

Fig. 1.
Fig. 1.

Line spread function of the eye after Westheimer [3].

Fig. 2.
Fig. 2.

OTF for the eye (solid) in comparison with the OTF for the 3 mm diffraction-limited circular aperture (dashed). Dots are the calculation by Wandell [6].

Fig. 3.
Fig. 3.

Plot of volume under the squared OTF for the diffraction-limited circular aperture as a function of cutoff frequency; dotted line shows the volume under the squared Westheimer OTF model of the eye.

Fig. 4.
Fig. 4.

Plots of the equalized (dotted) and Westheimer (solid) OTFs, respectively, dashed curve corresponds to the OTF for the 3 mm diffraction-limited circular aperture.

Fig. 5.
Fig. 5.

Plots of the equalized and Westheimer autocorrelation functions, respectively.

Fig. 6.
Fig. 6.

Plots of speckle contrast as a function of projection lens diameter, where the curves correspond to those for the simulated eye (dashed) and for the Westheimer (solid) eye model, respectively, and the dotted curve is calculated based on Goodman’s approximation.

Fig. 7.
Fig. 7.

Nonlinearity between the speckle contrasts for simulated eye and for the Westheimer eye model.

Tables (1)

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Table 1. Pupil and Pinhole Size in mm

Equations (31)

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le(α)=0.47e3.3α2+0.53e0.93|α|,
K=[1Ae2Pe(Δx,Δy)|μp(Δx,Δy)|2dΔxdΔy]1,
Le(ν)=0.458e2.99ν2+10.877+40.047ν2.
Pe(ρ)Le(ρ)=0.458e2.99ρ2+10.877+40.047ρ2,
e(ρ)Pe(ρ)/Pe(0)=0.2865e2.99ρ2+0.015620.14792+ρ2,
h(r)=[2J1(πDλzer)πDλzer]2,
Aa=h(x,y)dxdy,
H(ρ)h(x,y)exp[i2π(fxx+fyy)]dxdyh(x,y)dxdy,
H(ρ)=2π[arccos(ρρ0)(ρρ0)1(ρρ0)2],
ρ0=D/λze=(3×1030.543×106)×π180,
p˜e(s)=0.481e3.3008s2+0.1568K0(0.9298s).
pe(r)=0.481e3.9×107r2/ze2+0.1568K0(3.2×103r/ze),
K[Pe(0,0)Ae2|μp(Δx,Δy)|2dΔxdΔy]1.
μp(r)=[2J1(πDpλzpr)πDpλzpr]2
K4.59×107(D˜pλ)2,
KD(Δx,Δy)=h(x,y)h(xΔx,yΔy)dxdy.
F1(|H(ρ)|2)=h(x,y)h(xΔx,yΔy)dxdy(h(x,y)dxdy)2.
PE(Δx,Δy)=pe(x,y)pe(xΔx,yΔy)dxdy,
F1(|e(ρ)|2)=pe(x,y)pe(xΔx,yΔy)dxdy(pe(x,y)dxdy)2
Vρ=0|H(ρ)|2ρdρ=0|e(ρ)|2ρdρ,
Vr=0KD(r)Aa2rdr=0PE(r)Ae2rdr,
y=0.00686+1.0014x0.01637x20.6789x3+0.8051x4,
OTF(ρ)=eb2ρ2+1ρ2+a2,
f(ρ)=1(ρ2+a2)μ+1,g(r)=2π0ρf(ρ)J0(2πrρ)dρ.
0ρν+1f(ρ)J0(2πrρ)dρ=(2πr)μaνμKνμ(2πra)2μΓ(μ+1),limr0K0(2πra)=(μ=0,ν=0),limr02πra1K1(2πra)2Γ(2)=12a2(μ=1,ν=0).
0ρ|f(ρ)|dρ=01(ρ2+a2)μ+1ρdρ={[12ln(ρ2+a2)]0(μ=0)[121(ρ2+a2)μ]012a2(μ1).
g(r)=2π0ρf(ρ)J0(2πrρ)dρ.
0ρf(ρ)J0(2πrρ)dρ=12b2e(πr)2/b2
f(ρ)=eb2ρ21ρ2+a2,g(r)=2π0ρf(ρ)J0(2πrρ)dρ,=2π0ρeb2ρ21ρ2+a2J0(2πrρ)dρ.
t=ρ2,dt=2ρdρ,p=b2,
0eb2ρ2ρρ2+a2dρ=120ept1t+a2dt=12ea2pEi(a2p)<,

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