Abstract

Grating resonators are not characterized by a single cavity length, and thus the cavity-mode spacing cannot simply be obtained from the standing-wave pattern. This problem is studied in a Littrow grating cavity with geometric ray tracing and a result of the scalar diffraction theory for the phase of a plane wave diffracted at a grating. The round-trip phase in the cavity is considered, and it is shown that a grating cavity may be modeled by a tilted-mirror Fabry–Perot cavity. The tilt magnitude depends linearly on the wavelength deviation from the resonant Littrow wavelength, and the mirror separation, LC, is equal to the grating-cavity length at the center of the aperture. The model shows that the cavity axial mode separation may be determined from the standard expression, c/2LC. An effect of finesse decrease and mode broadening, which are linearly dependent on wavelength deviation from the central Littrow wavelength, are predicted. A passive grating cavity was experimentally studied with an interferometric method and a tunable laser to demonstrate the discussed hypotheses.

© 1994 Optical Society of America

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References

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  1. F. P. Schafer, ed., Dye Lasers, 3rd ed., Vol. 1 of Topics in Applied Physics (Springer-Verlag, Berlin, 1990).
  2. F. J. Duarte, L. W. Hillman, Dye Laser Principles, 1st ed. (Academic, San Diego, Calif., 1990), Chap. 4, pp. 134–144.
  3. J. E. Bjorkholm, T. C. Damen, J. Shah, “Improved use of grating in tunable lasers,” Opt. Commun.4, 283–284 (1971).
  4. G. J. Ernst, W. J. Witteman, “Transition selection with adjustable outcoupling for a laser device applied to CO2,” IEEE J. Quantum Electron. QE-7, 484–488 (1971).
    [CrossRef]
  5. P. McNicholl, H. J. Metcalf, “Synchronous cavity mode and feedback wavelength scanning in dye laser oscillators with gratings,” Appl. Opt. 24, 2757–2761 (1985).
    [CrossRef] [PubMed]
  6. W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Chap. 8, pp. 108–130.
  7. R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecommun. Technol. 3, 5–12 (1985).
  8. D. Basting, B. Burghardt, P. Lokai, W. Muckenheim, Zs. Bor, “Single-frequency dye laser with 50 ns pulse duration,” in Pulse Single-Frequency Lasers: Technology and Applications, W. K. Bischell, L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 87–94 (1988).
  9. M. de Labachelerie, G. Passedat, “Mode-hop suppression of Littrow grating-tuned lasers,” Appl. Opt. 32, 269–274 (1993).
    [CrossRef] [PubMed]
  10. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 11, pp. 432–435.
  11. M. G. Littman, H. J. Metcalf, “Spectrally narrow pulsed dye laser without beam expander,” Appl. Opt. 17, 2224–2227 (1984).
    [CrossRef]
  12. G. J. Sloggett, “Fringe broadening in Fabry–Perot interferometer,” Appl. Opt. 23, 2427–2432 (1984).
    [CrossRef] [PubMed]

1993 (1)

1985 (2)

P. McNicholl, H. J. Metcalf, “Synchronous cavity mode and feedback wavelength scanning in dye laser oscillators with gratings,” Appl. Opt. 24, 2757–2761 (1985).
[CrossRef] [PubMed]

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecommun. Technol. 3, 5–12 (1985).

1984 (2)

1971 (2)

J. E. Bjorkholm, T. C. Damen, J. Shah, “Improved use of grating in tunable lasers,” Opt. Commun.4, 283–284 (1971).

G. J. Ernst, W. J. Witteman, “Transition selection with adjustable outcoupling for a laser device applied to CO2,” IEEE J. Quantum Electron. QE-7, 484–488 (1971).
[CrossRef]

Basting, D.

D. Basting, B. Burghardt, P. Lokai, W. Muckenheim, Zs. Bor, “Single-frequency dye laser with 50 ns pulse duration,” in Pulse Single-Frequency Lasers: Technology and Applications, W. K. Bischell, L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 87–94 (1988).

Bjorkholm, J. E.

J. E. Bjorkholm, T. C. Damen, J. Shah, “Improved use of grating in tunable lasers,” Opt. Commun.4, 283–284 (1971).

Bor, Zs.

D. Basting, B. Burghardt, P. Lokai, W. Muckenheim, Zs. Bor, “Single-frequency dye laser with 50 ns pulse duration,” in Pulse Single-Frequency Lasers: Technology and Applications, W. K. Bischell, L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 87–94 (1988).

Burghardt, B.

D. Basting, B. Burghardt, P. Lokai, W. Muckenheim, Zs. Bor, “Single-frequency dye laser with 50 ns pulse duration,” in Pulse Single-Frequency Lasers: Technology and Applications, W. K. Bischell, L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 87–94 (1988).

Cameron, K. H.

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecommun. Technol. 3, 5–12 (1985).

Damen, T. C.

J. E. Bjorkholm, T. C. Damen, J. Shah, “Improved use of grating in tunable lasers,” Opt. Commun.4, 283–284 (1971).

de Labachelerie, M.

Duarte, F. J.

F. J. Duarte, L. W. Hillman, Dye Laser Principles, 1st ed. (Academic, San Diego, Calif., 1990), Chap. 4, pp. 134–144.

Ernst, G. J.

G. J. Ernst, W. J. Witteman, “Transition selection with adjustable outcoupling for a laser device applied to CO2,” IEEE J. Quantum Electron. QE-7, 484–488 (1971).
[CrossRef]

Hillman, L. W.

F. J. Duarte, L. W. Hillman, Dye Laser Principles, 1st ed. (Academic, San Diego, Calif., 1990), Chap. 4, pp. 134–144.

Littman, M. G.

Lokai, P.

D. Basting, B. Burghardt, P. Lokai, W. Muckenheim, Zs. Bor, “Single-frequency dye laser with 50 ns pulse duration,” in Pulse Single-Frequency Lasers: Technology and Applications, W. K. Bischell, L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 87–94 (1988).

Matthews, M. R.

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecommun. Technol. 3, 5–12 (1985).

McNicholl, P.

Metcalf, H. J.

Muckenheim, W.

D. Basting, B. Burghardt, P. Lokai, W. Muckenheim, Zs. Bor, “Single-frequency dye laser with 50 ns pulse duration,” in Pulse Single-Frequency Lasers: Technology and Applications, W. K. Bischell, L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 87–94 (1988).

Passedat, G.

Shah, J.

J. E. Bjorkholm, T. C. Damen, J. Shah, “Improved use of grating in tunable lasers,” Opt. Commun.4, 283–284 (1971).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 11, pp. 432–435.

Sloggett, G. J.

Steel, W. H.

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Chap. 8, pp. 108–130.

Witteman, W. J.

G. J. Ernst, W. J. Witteman, “Transition selection with adjustable outcoupling for a laser device applied to CO2,” IEEE J. Quantum Electron. QE-7, 484–488 (1971).
[CrossRef]

Wyatt, R.

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecommun. Technol. 3, 5–12 (1985).

Appl. Opt. (4)

Br. Telecommun. Technol. (1)

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecommun. Technol. 3, 5–12 (1985).

IEEE J. Quantum Electron. (1)

G. J. Ernst, W. J. Witteman, “Transition selection with adjustable outcoupling for a laser device applied to CO2,” IEEE J. Quantum Electron. QE-7, 484–488 (1971).
[CrossRef]

Opt. Commun. (1)

J. E. Bjorkholm, T. C. Damen, J. Shah, “Improved use of grating in tunable lasers,” Opt. Commun.4, 283–284 (1971).

Other (5)

F. P. Schafer, ed., Dye Lasers, 3rd ed., Vol. 1 of Topics in Applied Physics (Springer-Verlag, Berlin, 1990).

F. J. Duarte, L. W. Hillman, Dye Laser Principles, 1st ed. (Academic, San Diego, Calif., 1990), Chap. 4, pp. 134–144.

D. Basting, B. Burghardt, P. Lokai, W. Muckenheim, Zs. Bor, “Single-frequency dye laser with 50 ns pulse duration,” in Pulse Single-Frequency Lasers: Technology and Applications, W. K. Bischell, L. A. Rahn, eds., Proc. Soc. Photo-Opt. Instrum. Eng.912, 87–94 (1988).

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), Chap. 8, pp. 108–130.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 11, pp. 432–435.

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Figures (4)

Fig. 1
Fig. 1

Schematic description of the compensating-plate experiment (a) in a plane-parallel Fabry–Perot and (b) in a Littrow grating cavity. R is a partially transparent plane mirror. A Zygo interferometer (at 633 nm) combined with the examined cavity is used to observe a reflection inteferogram from the cavity (c). A glass slide is used as a compensating plate to tune the round-trip phase in the upper part of the cavity aperture. As the slide is tilted the fringe pattern always moves in the direction corresponding to cavity elongation.

Fig. 2
Fig. 2

Round-trip phase shift versus plate tilt angle in a Fabry–Perot cavity and a grating cavity based on Eq. (1), solid curve, and based on geometric considerations [Eq. (2)], dashed line. The plate parameters are T = 1.05 mm and n = 1.5; the grating is 1200 lines/mm, operated at M = 1 and λ L = 633 nm. The observed phase in the grating cavity of Fig. 1(b) versus plate angle, α, is indicated by filled circles and crosses for plate tilt axis parallel and normal to the grating lines, respectively.

Fig. 3
Fig. 3

Schematic description of the resonance-frequency-dependence experiment in a Littrow grating cavity. A beam from a tunable cw dye laser probes the cavity resonance. As the probe frequency is scanned, the first-order-diffracted beam from the grating is deflected by Γ, and the fringe pattern on the screen (by the zero-order reflection) is shifted. The amount of shift depends on the location within the cavity aperture. The grating cavity in (a) may be modeled by a plane–plane tilted-mirror cavity with mirror separation, LC , as shown in (b). The described beams in the schematic model cavity correspond to a case where λ < λ L . The tilted-mirror model is useful to explain broadening effects and the CMS in the cavity.

Fig. 4
Fig. 4

Phase shift in cycles versus cavity length, L, corresponding to 5.4 GHz frequency change of the probe laser in Fig. 3(a), as calculated from Eq. (6), solid line, and as experimentally obtained (solid circles). The dashed curve corresponds to the theoretical value of the CMS, whereas the squares present the measured CMS data.

Equations (14)

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P h S ( α ) = 2 L C λ + 2 P ( α ) λ ,
P ( α ) = 2 T { 1 n + [ n cos ( α β ) cos ( β ) ] } ,
P h ( α ) = P h S ( α ) + 2 T [ sin ( α β ) tan ( θ L ) ] [ λ cos ( β ) ] .
d [ sin ( θ INC ) + sin ( θ REF ) ] = M λ ,
G M ( U ) = G M ( p ) exp [ i Φ 0 ( p ) ] exp ( i 2 π M U / d ) ,
P h ( λ , L ) = 2 L λ { L M d sin ( θ L ) Φ 0 ( p ) } .
P h ( λ r , L C ) P h ( λ L , L C ) = ± 1 cycles ,
CMS ( L C ) = | c / λ r c / λ L | = c / ( 2 L C ) .
δ CMS ( D ) / CMS ( L C ) 1 ,
L C / [ Dtg ( θ L ) ] 1 .
Γ ( λ ) / 2 = M ( λ λ L ) / [ 2 d cos ( θ L ) ] .
δ Φ D ( λ ) = D Γ ( λ ) / λ 1 ,
Tilt Finesse = ½ δ Φ D ( λ ) 1 ,
N = [ L C + Dtg ( θ L ) ] / Dtg ( θ L ) .

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