Abstract

We show that the cavity round-trip Gouy phase leads to pulse-to-pulse variation of the absolute phase and temporal profile of circulating few- or single-cycle pulses in empty resonators. This pulse-to-pulse variation can be eliminated by the proper insertion of a lens into the cavity. An application to terahertz resonators with phase-locked feedback is discussed.

© 2000 Optical Society of America

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References

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  1. M. van Exter, Ch. Fattinger, D. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
    [CrossRef]
  2. X.-C. Zhang, B. B. Hu, S. H. Xin, D. H. Auston, “Optically induced femtosecond electromagnetic pulses from GaSb/AlSb strained-layer superlattices,” Appl. Phys. Lett. 57, 753–755 (1990).
    [CrossRef]
  3. I. Brener, D. Dykaar, A. Frommer, L. N. Pfeiffer, J. Lopata, J. Wynn, K. West, M. C. Nuss, “Terahertz emission from electric field singularities in biased semiconductors,” Opt. Lett. 21, 1924–1926 (1996).
    [CrossRef] [PubMed]
  4. K. Victor, H. G. Roskos, C. Waschke, “Efficiency of submillimeter-wave generation and amplification by coherent wave-packet oscillations in semiconductor structures,” J. Opt. Soc. Am. B 11, 2470–2479 (1994).
    [CrossRef]
  5. U. Morgner, F. Kärtner, S. Cho, Y. Chen, H. Haus, J. Fujimoto, E. Ippen, V. Scheuer, G. Angelow, T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
    [CrossRef]
  6. D. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, T. Tschudi, “Semiconductor saturable-absorber mirror-assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett. 24, 631–633 (1999).
    [CrossRef]
  7. A. deBohan, P. Antoine, D. B. Milošević, B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. 81, 1837–1840 (1998).
    [CrossRef]
  8. S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
    [CrossRef]
  9. S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
    [CrossRef]
  10. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
    [CrossRef]
  11. S. Feng, H. G. Winful, “Spatiotemporal transformation of isodiffracting ultrashort pulses by nondispersive quadratic phase media,” J. Opt. Soc. Am. A 16, 2500–2509 (1999).
    [CrossRef]
  12. Ch. Fattinger, D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54, 490–492 (1989).
    [CrossRef]
  13. L. Xu, X.-C. Zhang, D. H. Auston, “Terahertz radiation from large aperture Si p-i-n diodes,” Appl. Phys. Lett. 59, 3357–3359 (1991).
    [CrossRef]
  14. P. Uhd Jepsen, S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. 20, 807–809 (1995).
    [CrossRef]
  15. D. Grischkowsky, S. Keiding, M. van Exter, Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
    [CrossRef]
  16. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley-Interscience, New York, 1984), p. 51.
  17. A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986), p. 601.

2000 (1)

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

1999 (4)

1998 (2)

A. deBohan, P. Antoine, D. B. Milošević, B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. 81, 1837–1840 (1998).
[CrossRef]

S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

1991 (1)

L. Xu, X.-C. Zhang, D. H. Auston, “Terahertz radiation from large aperture Si p-i-n diodes,” Appl. Phys. Lett. 59, 3357–3359 (1991).
[CrossRef]

1990 (2)

D. Grischkowsky, S. Keiding, M. van Exter, Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

X.-C. Zhang, B. B. Hu, S. H. Xin, D. H. Auston, “Optically induced femtosecond electromagnetic pulses from GaSb/AlSb strained-layer superlattices,” Appl. Phys. Lett. 57, 753–755 (1990).
[CrossRef]

1989 (2)

M. van Exter, Ch. Fattinger, D. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Ch. Fattinger, D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54, 490–492 (1989).
[CrossRef]

Angelow, G.

Antoine, P.

A. deBohan, P. Antoine, D. B. Milošević, B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. 81, 1837–1840 (1998).
[CrossRef]

Auston, D. H.

L. Xu, X.-C. Zhang, D. H. Auston, “Terahertz radiation from large aperture Si p-i-n diodes,” Appl. Phys. Lett. 59, 3357–3359 (1991).
[CrossRef]

X.-C. Zhang, B. B. Hu, S. H. Xin, D. H. Auston, “Optically induced femtosecond electromagnetic pulses from GaSb/AlSb strained-layer superlattices,” Appl. Phys. Lett. 57, 753–755 (1990).
[CrossRef]

Brener, I.

Chen, Y.

Cho, S.

deBohan, A.

A. deBohan, P. Antoine, D. B. Milošević, B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. 81, 1837–1840 (1998).
[CrossRef]

Dykaar, D.

Fattinger, Ch.

D. Grischkowsky, S. Keiding, M. van Exter, Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

Ch. Fattinger, D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54, 490–492 (1989).
[CrossRef]

M. van Exter, Ch. Fattinger, D. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Feng, S.

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

S. Feng, H. G. Winful, “Spatiotemporal transformation of isodiffracting ultrashort pulses by nondispersive quadratic phase media,” J. Opt. Soc. Am. A 16, 2500–2509 (1999).
[CrossRef]

S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
[CrossRef]

Frommer, A.

Fujimoto, J.

Gallmann, L.

Grischkowsky, D.

D. Grischkowsky, S. Keiding, M. van Exter, Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

Ch. Fattinger, D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54, 490–492 (1989).
[CrossRef]

M. van Exter, Ch. Fattinger, D. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Haus, H.

Hellwarth, R. W.

Hu, B. B.

X.-C. Zhang, B. B. Hu, S. H. Xin, D. H. Auston, “Optically induced femtosecond electromagnetic pulses from GaSb/AlSb strained-layer superlattices,” Appl. Phys. Lett. 57, 753–755 (1990).
[CrossRef]

Ippen, E.

Kärtner, F.

Keiding, S.

Keiding, S. R.

Keller, U.

Lopata, J.

Matuschek, N.

Miloševic, D. B.

A. deBohan, P. Antoine, D. B. Milošević, B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. 81, 1837–1840 (1998).
[CrossRef]

Morgner, U.

Morier-Genoud, F.

Nuss, M. C.

Pfeiffer, L. N.

Piraux, B.

A. deBohan, P. Antoine, D. B. Milošević, B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. 81, 1837–1840 (1998).
[CrossRef]

Roskos, H. G.

Rudd, J. V.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

Ruffin, A. B.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

Scheuer, V.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986), p. 601.

Steinmeyer, G.

Sutter, D.

Tschudi, T.

Uhd Jepsen, P.

van Exter, M.

D. Grischkowsky, S. Keiding, M. van Exter, Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

M. van Exter, Ch. Fattinger, D. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

Victor, K.

Waschke, C.

West, K.

Whitaker, J. F.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

Winful, H. G.

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

S. Feng, H. G. Winful, “Spatiotemporal transformation of isodiffracting ultrashort pulses by nondispersive quadratic phase media,” J. Opt. Soc. Am. A 16, 2500–2509 (1999).
[CrossRef]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
[CrossRef]

Wynn, J.

Xin, S. H.

X.-C. Zhang, B. B. Hu, S. H. Xin, D. H. Auston, “Optically induced femtosecond electromagnetic pulses from GaSb/AlSb strained-layer superlattices,” Appl. Phys. Lett. 57, 753–755 (1990).
[CrossRef]

Xu, L.

L. Xu, X.-C. Zhang, D. H. Auston, “Terahertz radiation from large aperture Si p-i-n diodes,” Appl. Phys. Lett. 59, 3357–3359 (1991).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley-Interscience, New York, 1984), p. 51.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley-Interscience, New York, 1984), p. 51.

Zhang, X.-C.

L. Xu, X.-C. Zhang, D. H. Auston, “Terahertz radiation from large aperture Si p-i-n diodes,” Appl. Phys. Lett. 59, 3357–3359 (1991).
[CrossRef]

X.-C. Zhang, B. B. Hu, S. H. Xin, D. H. Auston, “Optically induced femtosecond electromagnetic pulses from GaSb/AlSb strained-layer superlattices,” Appl. Phys. Lett. 57, 753–755 (1990).
[CrossRef]

Appl. Phys. Lett. (4)

M. van Exter, Ch. Fattinger, D. Grischkowsky, “High-brightness terahertz beams characterized with an ultrafast detector,” Appl. Phys. Lett. 55, 337–339 (1989).
[CrossRef]

X.-C. Zhang, B. B. Hu, S. H. Xin, D. H. Auston, “Optically induced femtosecond electromagnetic pulses from GaSb/AlSb strained-layer superlattices,” Appl. Phys. Lett. 57, 753–755 (1990).
[CrossRef]

Ch. Fattinger, D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54, 490–492 (1989).
[CrossRef]

L. Xu, X.-C. Zhang, D. H. Auston, “Terahertz radiation from large aperture Si p-i-n diodes,” Appl. Phys. Lett. 59, 3357–3359 (1991).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

Opt. Lett. (5)

Phys. Rev. E (1)

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle terahertz pulses,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

A. deBohan, P. Antoine, D. B. Milošević, B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. 81, 1837–1840 (1998).
[CrossRef]

Other (2)

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley-Interscience, New York, 1984), p. 51.

A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986), p. 601.

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Figures (4)

Fig. 1
Fig. 1

Evolution of the absolute phase after each round trip in a cavity with an irrational cavity winding number W#=0.4713 . This cavity is quasi periodic in the absolute phase. Here R=200 mm and L=182 mm.

Fig. 2
Fig. 2

Intensity envelope and underlying oscillation of a recirculating pulse after a number of round trips indicated on the plots for the same cavity as in Fig. 1. The quasi period is 17 round trips. The absolute phase and temporal waveform are almost periodic over a short time span (top row) but not in a long term (bottom row).

Fig. 3
Fig. 3

Possible setup of a modified terahertz resonator with a nondispersive thin lens at the center of a symmetric cavity. The left and the right parts of the cavity are equivalent. The emitters, which are represented by a small section of thick lines, can be placed at the focus in either or both sides of the cavity. If the phase difference of the two emitters is set to equal the Gouy shift between the two focal points, one feedback terahertz pulse can drive the two emitters.

Fig. 4
Fig. 4

Stable regime of the cavity [shaded area, from Eq. (4.1)] and the stabilization curves (solid curves), along which the absolute phase of the circulating pulse is π (L<R) or 3π (L>2R) after each round trip.

Equations (33)

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F˜(ω)=(ω-ω0)pexp[-(ω-ω0)τ0](ωω0),
E(0, z, t)=A0(z, T)exp[-iω0τ-iα(T)-iG(z)],
α(T)=(p+1)arctan(T),
A0(z, T)=1(1+z2/zR2)1/2(1+T2)(p+1)/2.
Grt=4 arctanL2zR=4 arctanL2R-L1/2.
Em(0, z, t)=A0(z, T)exp-iω0τ+α(T)+arctanzzR+mGrt,
-L2<z<L2,
W#2πarctanL2R-L1/2.
d2+zR2=2df.
R=L2-d+zR2[(L/2)-d].
d=L(2R-L)4(2 f+R-L),
zR=[L(2R-L)(4f-L)(4f+2R-L)]1/24|2 f+R-L|.
(2R-L)(4f-L)(4f+2R-L)>0.
4f+2R-L>0,
(2R-L)(4f-L)>0,
(2R-L)(2 f+R-L)>0,
(4f-L)(2 f+R-L)>0.
ϕ=arctanL-2d2zR+arctandzR.
tan(ϕ)=zRLd(4f-L).
tan(ϕ)=±L(4f+2R-L)(4f-L)(2R-L)1/2,
f=L(2R-L)4(R-L).
d=L(2R-L)(R-L)(2R-L)2+L2,
zR=RL|2R-L|(2R-L)2+L2.
zR2+d-R22=R24.
12RL+4fL1+8RfL2(f>0).
M1=10-2/R1×1L/201×10-1/f1×1L/201
=1-L2 fL-L24f-2R-1f+LRf1-2LR-L2 f+L22RfA1B1C1D1.
M=M1M1ABCD=A12+B1C1B1(A1+D1)C1(A1+D1)D12+B1C1.
A+D2=B1C1+12 (A12+D12)
=1-L4f-2LR-Lf+L2Rf+121-L2 f2+121-2LR-L2 f+L22Rf2.
A+D2=1-8(α+β-αβ)+8(α+β-αβ)2,
αβα+βαβ+1.
12RL+4fL1+8RfL2(f>0).

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