Abstract

A simple expression is obtained for the wavelength-dependent optical path length through a completely general N-prism sequence. Once the prisms’ rotational orientations are fixed, a one-time calculation yields analytic expressions for the group-delay dispersion and cubic dispersion. The technique can be generalized for higher dispersion orders. As an example, expressions for Proctor–Wise prism sequences [Opt. Lett. 17, 1295 (1992)] constructed from a variety of glasses are produced. The analytic expressions for this sequence permit an exhaustive search of relative prism translations. The arrangement that produces the smallest cubic dispersion for a given group-delay dispersion and total optical path is found.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. L. Fork, O. E. Martinez, and J. P. Gordon, Opt. Lett. 9, 150 (1984).
    [CrossRef] [PubMed]
  2. O. E. Martinez, J. P. Gordon, and R. L. Fork, J. Opt. Soc. Am. A 1, 1003 (1984).
    [CrossRef]
  3. B. E. Lemoff and C. P. J. Barty, Opt. Lett. 17, 1367 (1992).
    [CrossRef]
  4. B. Proctor and F. Wise, Opt. Lett. 17, 1295 (1992).
    [CrossRef] [PubMed]
  5. The required refractive-index derivatives were obtained by differentiation of the Sellmeir equation. The Sellmeir coefficients for all materials except fused silica were obtained from the manufacturer, Schott Glass Technologies, Duryea, Pa. The coefficients for fused silica were taken from the Optics and Coatings Catalog (1994), CVI Laser Corporation, 200 Dorado Place SE, P.O. Box 11308, Albuquerque, N.M. 87192.
  6. R. L. Fork, C. H. Brito Cruz, P. C. Becker, and C. V. Shank, Opt. Lett. 12, 483 (1987).
    [CrossRef] [PubMed]

1992

1987

1984

J. Opt. Soc. Am. A

Opt. Lett.

Other

The required refractive-index derivatives were obtained by differentiation of the Sellmeir equation. The Sellmeir coefficients for all materials except fused silica were obtained from the manufacturer, Schott Glass Technologies, Duryea, Pa. The coefficients for fused silica were taken from the Optics and Coatings Catalog (1994), CVI Laser Corporation, 200 Dorado Place SE, P.O. Box 11308, Albuquerque, N.M. 87192.

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 (5)

Fig. 1
Fig. 1

Two possible relative orientations of the principal vertices in a prism pair: (a) the uncrossed configuration, (b) the crossed configuration. A dashed line is the principal line for a pair. The dotted lines are principal phase fronts.

Fig. 2
Fig. 2

Two possible ways in which a ray can traverse a prism: (a) type I traversal, (b) type II traversal. For both, the phase front A is arbitrarily chosen somewhere along the incident ray, which is labeled 1. B is the principal phase front. Ray 2 is parallel to ray 1 and goes through the principal vertex.

Fig. 3
Fig. 3

Construction used in the proof that the optical path from PPFj to PPFj+1 is given by l cos β. The principal vertex of each prism is indicated by a filled circle, and the dashed line is the principal line for the pair. The solid line through point x is parallel to the transit ray of wavelength λ and highlights the specification of β. The dotted lines are phase fronts.

Fig. 4
Fig. 4

Transit angle nomenclature and sign conventions used in the text. (a) Nomenclature for a single prism. The arrows indicate the positive direction for each transit angle. The dotted line is the principal phase front. (b) Sign convention for the prism pair angle βj. The dashed line represents the principal line for the pair.

Fig. 5
Fig. 5

Plot of the optimal cubic dispersion and l2 cos β2 versus the total single-pass optical path through a PW-type prism sequence arranged to produce a single-pass GDD of -560 fs2. The minimum single-pass geometric path length inside a single prism is 0.4 cm.

Tables (1)

Tables Icon

Table 1 Coefficients of sin βj and cos βj in the Analytic Expressions for the Second and Third Derivatives of the Optical Path Length in a PW Prism Sequence a

Equations (40)

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

d2Φdω2ωc=λ32πc2 d2Pdλ2λc,
d3Φdω3ωc=-λ44π2c3 3 d2Pdλ2+λ d3Pdλ3λc.
P(λ)=j=1N-1lj cos βj(λ).
d2Pdλ2=j=1N-1lj(fj sin βj+gj cos βj),
d3Pdλ3=j=1N-1lj(hj sin βj+kj cosβj),
fj=-βj,
gj=-(βj)2,
hj=(βj)3-βj,
kj=-3(βj)(βj).
βj(m)(λ)=-ϕj(m)(λ),
ij(m)(λ)=sj-1ϕj-1(m)(λ),
sj=+1crossedpair-1uncrossedpair.
ϕ=1cos ϕ n sin ϕn-nr(sin α sin r+cos α cos r),
ϕ=(ϕ)2+nn-(r)2tan ϕ-(2nr+nr)cos ϕ (sin α sin r+cos α cos r),
ϕ=(ϕ)3+[3ϕϕn+n-3n(r)2-3nrr]×tanϕn-1cos ϕ [3nr+3nr+nr-n(r)3]×(sin α sin r+cos α cos r),
r=1n cos r (i cos i-n sin r),
r=1n cos r -2nr cos r+(r)2-nn-(i)2sin i+i cos i,
r=(r)3-3n (nr+nr)+tan r3(rr-ii)+1n [3n(r)2-n]+cos in cos r [i-(i)3].
ϕ=2n-i,
ϕ=2n-i+A(n)2+B[(i)2-2ni],
ϕ=2n-i+C(n)3+D(n)2i+En(i)2+F(i)3-3B(ni+ni-ii)+3Ann,
A=2n3 (2n4-1),
B=2n3 (n4-1),
C=2n6 (12n8+4n6-6n4+3n2+3),
D=-6n6 (6n8+2n6-5n4+2n2+3),
E=6n6 (4n8+n6-5n4+n2+3),
F=-6n6 (n8-2n4+1)
s1, s3=-1,s2=+1.
f3=-f1,g3=g1,h3=-h1,k3=k1.
t2=l1 sin(-β1)+t1,
t4=l3 sin(-β3)+t3,
l2 sin β2=t2.
l3 sin β3Δt,
l1 sin β1-Δt.
l2 cos β2=1g2 2πc2λ3 d2Φdω2-f2t2-l1(f1 sin β1+g1 cos β1)-l3(f3 sin β3+g3 cos β3).
d3Φdω3=-λ44π2c3 6πc2λ3 d2Φdω2+λl1ψ1(β1)+l3Ψ3(β3)+(tmin+Δt)h2-k2f2g2,
Ψ1(β)=h1-k2g2 f1sin β+k1-k2g2 g1cos βX sin β+Y cos β,
Ψ3(β)=h3-k2g2 f3sin β+k3-k2g2 g3cos β-X sin β+Y cos β.
d3Φdω3=-λ44π2c3 2πc2λ3 d2Φdω2 3+λ k2g2+λ(l1+l3)Y+h2-k2f2g2-2XΔt+tminh2-k2f2g2.
d3Φdω3=-929fs3+(0.86fs3/cm)(l1+l3)+(1.17fs3/cm)Δt.

Metrics