Abstract

Rough surfaces in translucent protective sheets are used in imaging systems, such as displays, to decrease specular reflections of external sources. However, they modify the quality of the images formed by transmission. Using a geometric approximation, we have modeled the behavior of rough surfaces in imaging systems. This model provides an analytical expression for the modulation transfer function of rough surfaces.

© 1999 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987).
  2. F. Perez-Quintian, M. A. Rebollo, N. G. Gaggioli, “Diffusion of light transmitted from rough surfaces,” J. Mod. Opt. 44, 447–460 (1997).
    [CrossRef]
  3. E. L. O’Neill, Introduction to Statistical Optics (Dover, New York, 1992).
  4. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, Bristol, 1991).
  5. F. Perez-Quintian, M. A. Rebollo, N. G. Gagglioli, C. A. Raffo, “The nonrefractive effect in translucent diffusers. A geometrical model,” J. Mod. Opt. 45, 441–448 (1998).
    [CrossRef]
  6. A. Papoulis, Probability, Random Variables and Stochastic Processes, International Student Edition, (McGraw-Hill Kogakusha, Tokyo, 1965).
  7. J. W. Coltman, “The specification of imaging properties by response to a sine wave input,” J. Opt. Soc. Am. 44, 468–471 (1954).
    [CrossRef]

1998

F. Perez-Quintian, M. A. Rebollo, N. G. Gagglioli, C. A. Raffo, “The nonrefractive effect in translucent diffusers. A geometrical model,” J. Mod. Opt. 45, 441–448 (1998).
[CrossRef]

1997

F. Perez-Quintian, M. A. Rebollo, N. G. Gaggioli, “Diffusion of light transmitted from rough surfaces,” J. Mod. Opt. 44, 447–460 (1997).
[CrossRef]

1954

Beckmann, P.

P. Beckmann, A. Spizzino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987).

Coltman, J. W.

Gaggioli, N. G.

F. Perez-Quintian, M. A. Rebollo, N. G. Gaggioli, “Diffusion of light transmitted from rough surfaces,” J. Mod. Opt. 44, 447–460 (1997).
[CrossRef]

Gagglioli, N. G.

F. Perez-Quintian, M. A. Rebollo, N. G. Gagglioli, C. A. Raffo, “The nonrefractive effect in translucent diffusers. A geometrical model,” J. Mod. Opt. 45, 441–448 (1998).
[CrossRef]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Dover, New York, 1992).

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, Bristol, 1991).

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, International Student Edition, (McGraw-Hill Kogakusha, Tokyo, 1965).

Perez-Quintian, F.

F. Perez-Quintian, M. A. Rebollo, N. G. Gagglioli, C. A. Raffo, “The nonrefractive effect in translucent diffusers. A geometrical model,” J. Mod. Opt. 45, 441–448 (1998).
[CrossRef]

F. Perez-Quintian, M. A. Rebollo, N. G. Gaggioli, “Diffusion of light transmitted from rough surfaces,” J. Mod. Opt. 44, 447–460 (1997).
[CrossRef]

Raffo, C. A.

F. Perez-Quintian, M. A. Rebollo, N. G. Gagglioli, C. A. Raffo, “The nonrefractive effect in translucent diffusers. A geometrical model,” J. Mod. Opt. 45, 441–448 (1998).
[CrossRef]

Rebollo, M. A.

F. Perez-Quintian, M. A. Rebollo, N. G. Gagglioli, C. A. Raffo, “The nonrefractive effect in translucent diffusers. A geometrical model,” J. Mod. Opt. 45, 441–448 (1998).
[CrossRef]

F. Perez-Quintian, M. A. Rebollo, N. G. Gaggioli, “Diffusion of light transmitted from rough surfaces,” J. Mod. Opt. 44, 447–460 (1997).
[CrossRef]

Spizzino, A.

P. Beckmann, A. Spizzino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987).

J. Mod. Opt.

F. Perez-Quintian, M. A. Rebollo, N. G. Gaggioli, “Diffusion of light transmitted from rough surfaces,” J. Mod. Opt. 44, 447–460 (1997).
[CrossRef]

F. Perez-Quintian, M. A. Rebollo, N. G. Gagglioli, C. A. Raffo, “The nonrefractive effect in translucent diffusers. A geometrical model,” J. Mod. Opt. 45, 441–448 (1998).
[CrossRef]

J. Opt. Soc. Am.

Other

P. Beckmann, A. Spizzino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, Mass., 1987).

A. Papoulis, Probability, Random Variables and Stochastic Processes, International Student Edition, (McGraw-Hill Kogakusha, Tokyo, 1965).

E. L. O’Neill, Introduction to Statistical Optics (Dover, New York, 1992).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, Bristol, 1991).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Experimental system for indirect MTF measurements.

Fig. 3
Fig. 3

Illuminance profile (arbitrary units) of the white-light beam used in the experiment.

Fig. 4
Fig. 4

Values deduced of CTF (continuous curves) compared with experimental results at two distances between the object and the rough sheet for (a) sample 1, (b) sample 2: ●, d 1 = 50 mm; *, d 2 = 100 mm.

Fig. 5
Fig. 5

Values deduced of MTF (continuous curves) compared with experimental results at two distances between the object and the rough sheet for (a) sample 1, (b) sample 2: ●, d 1 = 50 mm; *, d 2 = 100 mm.

Fig. 6
Fig. 6

MTF dependence on the distance between the object and the rough sheet d at five spatial frequencies (0.5, 1, 1.5, 2, 2.5) for sample 2.

Equations (13)

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Iθ2  n1 cosθ2-θ1-12n1 cos θ1-cos θ23×exp-sin θ2-n1 sin θ1n1 cos θ1-cos θ2τ2σ2,
CTFf=4πk=0-1kMTF2k+1f2k+1.
Tmax=12+k=0-1k2k+1πcos2k+12πyh,
Imax=- Iθ2yTmaxydy,
Imax=I1+I2,
I1=12n-1- exp-y2n-1dτσ2dy=πdστ,
I2=2n-1πk=0-1k2k+1- cos2k+12πyh×exp-y2n-1dτσdy=4ππdστk=0-1k2k+1×exp-2k+1πhσdn-1τ2.
Tmin=1-Tmax.
Imin=- Iθ2yTminydy,
Imin=I1-I2.
CTFh, d, σ/τ=I2I1=4πk=0-1k12k+1×exp-2k+1πhσdn-1τ2.
MTFh, d, σ/τ=exp-πhσdn-1τ2.
MTFf=π4CTFf+CTF3f3-CTF5f5+CTF7f7,

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