Abstract

In the framework of geometrical optics, we show that the general ray-tracing formula for finding the direction of the diffracted rays for thin phase holographic optical elements located on any arbitrary continuous surface can be derived in a form similar to Fermat's principle.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. D. Brorson, H. A. Haus, “Diffraction gratings and geometrical optics,” J. Opt. Soc. Am. B 5, 247–248 (1988).
    [CrossRef]
  2. N. Abramson, “Principle of least wave change,” J. Opt. Soc. Am. A 6, 627–629 (1989).
    [CrossRef] [PubMed]
  3. R. Collier, C. Burkhardt, L. Lin, Optical Holography (Academic, New York, 1971), Chap. 8, pp. 224–225.
  4. W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322–323 (1975).
    [CrossRef]
  5. R. N. Smith, “A note on practical formulae for finite ray-tracing through holograms and diffractive optical elements,” Opt. Commun. 55, 11–12 (1985).
    [CrossRef]

1989 (1)

1988 (1)

1985 (1)

R. N. Smith, “A note on practical formulae for finite ray-tracing through holograms and diffractive optical elements,” Opt. Commun. 55, 11–12 (1985).
[CrossRef]

1975 (1)

W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322–323 (1975).
[CrossRef]

Abramson, N.

Brorson, S. D.

Burkhardt, C.

R. Collier, C. Burkhardt, L. Lin, Optical Holography (Academic, New York, 1971), Chap. 8, pp. 224–225.

Collier, R.

R. Collier, C. Burkhardt, L. Lin, Optical Holography (Academic, New York, 1971), Chap. 8, pp. 224–225.

Haus, H. A.

Lin, L.

R. Collier, C. Burkhardt, L. Lin, Optical Holography (Academic, New York, 1971), Chap. 8, pp. 224–225.

Smith, R. N.

R. N. Smith, “A note on practical formulae for finite ray-tracing through holograms and diffractive optical elements,” Opt. Commun. 55, 11–12 (1985).
[CrossRef]

Welford, W. T.

W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322–323 (1975).
[CrossRef]

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

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

Opt. Commun. (2)

W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322–323 (1975).
[CrossRef]

R. N. Smith, “A note on practical formulae for finite ray-tracing through holograms and diffractive optical elements,” Opt. Commun. 55, 11–12 (1985).
[CrossRef]

Other (1)

R. Collier, C. Burkhardt, L. Lin, Optical Holography (Academic, New York, 1971), Chap. 8, pp. 224–225.

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 (2)

Fig. 1
Fig. 1

Example of curved modulation lines for a flat phase HOE; the refractive index of the modulated medium is constant along a modulation line. In the real hologram pattern, the density of the lines is much higher (on the figure, only a few out of a thousand lines is retained).

Fig. 2
Fig. 2

Readout hologram ray-tracing geometry for a given order of diffraction in the case of a transmission hologram (a) and a reflection hologram (b). (Characters with arrows represent boldface characters without arrows in text.)

Equations (41)

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

U(B)=pathsU(A) exp[ikCl(AB)]l(AB),
U(B)=SU(A) exp{ikC[l(AH)+l(HB)]}[l(AH)+l(HB)]× T(H)dS,
T(H)=exp[+iΦ(H)]
Φ(H)=ΦO(H)-ΦR(H),
Φ(H)=k0 p(H),
θ=p(H)p0,0,0+p1,0,0s+p0,1,0t+p0,0,1u,
p0,0,0=p[x0, y0, z0=s(x0, y0)]=p(H0),
p1,0,0=px|H=H0,p0,1,0=py|H=H0,
p0,0,1=pz|H=H0,
s=(x-x0),t=(y-y0),u=(z-z0).
T(k0θ)=m=-+am exp(+imk0θ),
am=k02π 02πT(k0θ)exp(-imk0θ)dθ.
U(B)=m=-+amSU(A) ×expikCl(AH)+l(HB)+m k0kC θ[l(AH)+l(HB)] dS.
Wx=0,Wy=0,Wz=0,
W=l(AH)+l(HB)+m k0kC θ,
rC=AHAH=rCXeX+rCYeY+rCZeZ,
rI=HBHB=rIXeX+rIYeY+rIZeZ,
l(AH)=nCAH,l(HB)=nIHB,
kC(nCrCX-nIrIX)+k0mp1,0,0=0,
kC(nCrCY-nIrIY)+k0mp0,1,0=0,
kC(nCrCZ-nIrIZ)+k0mp0,0,1=0.
N(x, y, z)=sx2+sy2+1-1/2×-sx eX-sy eY+eZ,
N(x, y, z)=NXeX+NYeY+NZeZ.
q(x, y)=p[x, y, z=s(x, y)].
px=qx-pz sx,
py=qy-pz sy.
Np=0.
pz=F2(x, y)G(x, y),
G(x, y)=qx sx+qy sy,
F(x, y)=sx2+sy2+1-1/2.
kC(nIrIX-nCrCX)=k0mq1,0+k0mF(H0)G(H0)NX,
kC(nIrIY-nCrCY)=k0mq0,1+k0mF(H0)G(H0)NY,
kC(nIrIZ-nCrCZ)=k0mF(H0)G(H0)NZ,
q1,0=qx|H=H0,q0,1=qy|H=H0.
nIrI-nCrC=λCλ0 mq(x, y)+λCλ0 mF(H0)G(H0)N,
nIrI-nCrC=λCλ0 mp[x, y, z=s(x, y)].
N×(nIrI-nCrC)=N×λCλ0 mq(x, y),
N×(nIrI-nCrC)=N×λCλ0 mp[x, y,z=s(x, y)],
N×q(x, y)=N×p[x, y, z=s(x, y)],
p(H)=nOrO-nRrR,
N×(nIrI-nCrC)=N×λCλ0 m(nOrO-nRrR),

Metrics