Abstract

It is shown that a negative contribution to the group-velocity dispersion always accompanies angular dispersion. We describe structures, such as slabs and prisms, that use this contribution to provide adjustable group-velocity dispersion. We discuss, in particular, considerations regarding incorporation of these structures with a laser resonator. A description of possible submillimeter semiconductor devices is also given.

© 1984 Optical Society of America

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References

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  1. E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
    [CrossRef]
  2. R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, IEEE J. Quantum Electron. QE-19, 500 (1983).
    [CrossRef]
  3. C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
    [CrossRef]
  4. O. E. Martinez, R. L. Fork, J. P. Gordon, Opt. Lett. 9, 156 (1984).
    [CrossRef] [PubMed]
  5. R. L. Fork, O. E. Martinez, J. P. Gordon, Opt. Lett. 9, 150 (1984).
    [CrossRef] [PubMed]
  6. R. E. Fern, A. Onton, J. Appl. Phys. 42, 3499 (1971).
    [CrossRef]
  7. J. P. Gordon, R. L. Fork, Opt. Lett. 9, 153 (1984).
    [CrossRef] [PubMed]
  8. W. Dietel, J. J. Fontaine, J.-C. Diels, Opt. Lett. 8, 4 (1983).
    [CrossRef] [PubMed]
  9. K. Furuya, B. I. Miller, L. A. Coldren, R. Howard, IEEE J. Quantum Electron. QE-18, 1783 (1982).
    [CrossRef]
  10. J. P. Van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983).
    [CrossRef]

1984 (3)

1983 (3)

J. P. Van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983).
[CrossRef]

W. Dietel, J. J. Fontaine, J.-C. Diels, Opt. Lett. 8, 4 (1983).
[CrossRef] [PubMed]

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, IEEE J. Quantum Electron. QE-19, 500 (1983).
[CrossRef]

1982 (2)

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

K. Furuya, B. I. Miller, L. A. Coldren, R. Howard, IEEE J. Quantum Electron. QE-18, 1783 (1982).
[CrossRef]

1971 (1)

R. E. Fern, A. Onton, J. Appl. Phys. 42, 3499 (1971).
[CrossRef]

1969 (1)

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

Coldren, L. A.

K. Furuya, B. I. Miller, L. A. Coldren, R. Howard, IEEE J. Quantum Electron. QE-18, 1783 (1982).
[CrossRef]

Diels, J.-C.

Dietel, W.

Fern, R. E.

R. E. Fern, A. Onton, J. Appl. Phys. 42, 3499 (1971).
[CrossRef]

Fontaine, J. J.

Fork, R. L.

O. E. Martinez, R. L. Fork, J. P. Gordon, Opt. Lett. 9, 156 (1984).
[CrossRef] [PubMed]

R. L. Fork, O. E. Martinez, J. P. Gordon, Opt. Lett. 9, 150 (1984).
[CrossRef] [PubMed]

J. P. Gordon, R. L. Fork, Opt. Lett. 9, 153 (1984).
[CrossRef] [PubMed]

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, IEEE J. Quantum Electron. QE-19, 500 (1983).
[CrossRef]

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

Furuya, K.

K. Furuya, B. I. Miller, L. A. Coldren, R. Howard, IEEE J. Quantum Electron. QE-18, 1783 (1982).
[CrossRef]

Gordon, J. P.

Hirlimann, C. A.

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, IEEE J. Quantum Electron. QE-19, 500 (1983).
[CrossRef]

Howard, R.

K. Furuya, B. I. Miller, L. A. Coldren, R. Howard, IEEE J. Quantum Electron. QE-18, 1783 (1982).
[CrossRef]

Logan, R. A.

J. P. Van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983).
[CrossRef]

Martinez, O. E.

Miller, B. I.

K. Furuya, B. I. Miller, L. A. Coldren, R. Howard, IEEE J. Quantum Electron. QE-18, 1783 (1982).
[CrossRef]

Onton, A.

R. E. Fern, A. Onton, J. Appl. Phys. 42, 3499 (1971).
[CrossRef]

Shank, C. V.

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, IEEE J. Quantum Electron. QE-19, 500 (1983).
[CrossRef]

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

Stolen, R. H.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

Tomlinson, W. J.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

Treacy, E. B.

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

Van der Ziel, J. P.

J. P. Van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983).
[CrossRef]

Yen, R.

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, IEEE J. Quantum Electron. QE-19, 500 (1983).
[CrossRef]

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

Appl. Phys. Lett. (1)

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, W. J. Tomlinson, Appl. Phys. Lett. 40, 761 (1982).
[CrossRef]

IEEE J. Quantum Electron. (4)

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

R. L. Fork, C. V. Shank, R. Yen, C. A. Hirlimann, IEEE J. Quantum Electron. QE-19, 500 (1983).
[CrossRef]

K. Furuya, B. I. Miller, L. A. Coldren, R. Howard, IEEE J. Quantum Electron. QE-18, 1783 (1982).
[CrossRef]

J. P. Van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983).
[CrossRef]

J. Appl. Phys. (1)

R. E. Fern, A. Onton, J. Appl. Phys. 42, 3499 (1971).
[CrossRef]

Opt. Lett. (4)

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Figures (5)

Fig. 1
Fig. 1

(a) Slab structure. The phase shift between A and B for a plane wave refracted in the direction of k is given by the scalar product between k and AB. (b) Two-prism structure. This is equivalent to the structure shown in (a) with the difference that a slab-shaped region of space has been inserted within the original slab. The phase shift can be calculated in a manner similar to that used for (a). The path for the center of the beam is shown by the dashed line. The distance AB is denoted by l.

Fig. 2
Fig. 2

Effect of a telescope. Here, f1 and f2 are the focal distances of the lenses. The distance between O and O′ (focal planes) must be subtracted from P, because the optical paths for waves propagating at different angles are identical. The increase in the angular dispersion can, however, usually overcome this disadvantage.

Fig. 3
Fig. 3

Basic period of the equivalent waveguide for the laser cavity. Only the slabs are shown, and all other optical components are eliminated for simplicity and generality. The paths for two different wavelengths are shown. In regions II and IV, dispersion takes place. Regions I and III are equivalent, and transverse displacement can occur in either region depending on the other optical elements of the cavity.

Fig. 4
Fig. 4

Different configurations with fewer than four prisms. The solid line represents the path for one wavelength, and the dashed line represents the path for a different wavelength. (a) Standing-wave cavity. An aperture forces the modes to overlap in region I. (b) Ring cavity folded in the nondispersive region. An odd number of inversions of the mode in the plane containing the dispersed rays is required per transit of the resonator, and the full period occurs after two transits. (c) Ring cavity folded in the dispersion region. The two prisms may now be fused into a single one, and again an odd number of inversions is required per transit. The regions are labeled according to the nomenclature described in Fig. 3.

Fig. 5
Fig. 5

A small semiconductor structure. Region II is a dielectric waveguide, and regions I and III are a semiconductor waveguide. The required gain could be any of these regions.

Equations (15)

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P = ϕ c / w = n 2 l cos θ ,
d 2 ϕ d w 2 = d 2 w P / c d w 2 = λ 3 2 π c 2 d 2 P d λ 2 .
D = d d λ ( 1 v g ) = - w λ l d 2 ϕ d w 2 .
d 2 P d λ 2 = l cos θ d 2 n 2 d λ 2 - 2 l sin θ d n 2 d λ d θ d λ - n 2 l cos θ ( d θ d λ ) 2 - n 2 l sin θ d 2 θ d λ 2 .
d 2 P d λ 2 = [ d 2 n 2 d λ 2 - n 2 ( d θ d λ ) 2 ] l .
d 2 P d λ 2 = l [ - cos θ ( d θ d λ ) 2 - sin θ d 2 θ d λ 2 ] .
d 2 P d λ 2 = 0.52 μ m - 2 mm - l ( 3.75 × 10 - 3 μ m - 2 )
l eff = [ l - 2 ( f 1 + f 2 ) ] ( f 1 / f 2 ) 2 .
d 3 ϕ d w 3 = - λ 4 ( 2 π ) 2 c 3 ( 3 d 2 P d λ 2 + λ d 3 P d λ 3 ) .
3 ( Δ w / w ) 1.
d 3 P d λ 3 = l sin θ ( d θ d λ ) 3 - 3 l cos θ d θ d λ d 2 θ d λ 2 - l sin θ d 3 θ d λ 3 ,
d θ d λ = d θ d n d n d λ ,
d 2 θ d λ 2 = d 2 θ d n 2 ( d n d λ ) 2 + d θ d λ d n 2 d λ 2 ,
d 3 θ d λ 3 = d 3 θ d n 3 ( d n d λ ) 3 + 3 d 2 θ d n 2 d n d λ d 2 n d λ 2 + d θ d n d 3 n d λ 3 .
d 3 P d λ 3 - 3 l cos θ ( d θ d n ) 2 d n d λ d 2 n d λ 2 - l sin θ d θ d n d 3 n d λ 3 .

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