Abstract

The difference-frequency terahertz (THz) pulse generation from Fourier components of a single infrared laser pulse in waveguiding heterostructures is considered. It is shown that a proper choice of the waveguide parameters allows for the significant reduction of the input pulse group-velocity dispersion (GVD). As a result, it is possible to noticeably increase the length of such devices and, consequently, the pump-THz pulse conversion efficiency by employing low absorbing materials with high intrinsic GVD coefficients for their fabrication. A particular example of such a scheme—a CGaAsC waveguiding heterostructure—is analyzed, a corresponding conversion rate is found, and ways for its further improvement are considered.

© 2006 Optical Society of America

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2005 (4)

S. Dhillon, J. Alton, S. Barbieri, C. Sirtori, A. de Rossi, M. Calligaro, H. E. Beere, and D. Ritchie, 'Ultralow threshold current terahertz quantum cascade lasers based on double-metal buried strip waveguides,' Appl. Phys. Lett. 87, 071101 (2005).
[CrossRef]

A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, 'High-intensity terahertz radiation from a microstructured large-area photoconductor,' Appl. Phys. Lett. 86, 121114 (2005).
[CrossRef]

A. G. Stepanov, J. Kuhl, I. Z. Kozma, E. Riedle, G. Almasi, and J. Hebling, 'Scaling up the energy of THz pulses created by optical rectification,' Opt. Express 13, 5762-5768 (2005).
[CrossRef] [PubMed]

R. A. Akhmedzhanov, A. I. Korytin, A. G. Litvak, A. M. Sergeev, and E. V. Suvorov, 'Generation and detection of ultrashort electromagnetic pulses in terahertz region and their application for spectroscopy,' Radiophys. Quantum Electron. 48, 837-843 (2005).
[CrossRef]

2004 (5)

Y. J. Ding, 'Quasi-single-cycle terahertz pulses based on broad-phase-matched difference-frequency generation in second-order nonlinear medium: high output powers and conversion efficiencies,' IEEE J. Sel. Top. Quantum Electron. 10, 1171-1179 (2004).
[CrossRef]

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, 'In vivo study of human skin using pulsed terahertz radiation,' Phys. Med. Biol. 49, 1595-1607 (2004).
[CrossRef] [PubMed]

T. Taniuchi, S. Okada, and H. Nakanishi, 'Widely tunable terahertz-wave generation in an organic crystal and its spectroscopic application,' J. Appl. Phys. 95, 5984-5988 (2004).
[CrossRef]

B. H. Wu and J. C. Cao, 'Nonlinear dynamics in multisubband semiconductor quantum wells under terahertz radiation,' Physica B 349, 322-326 (2004).
[CrossRef]

Y. J. Ding, 'Efficient generation of high-power quasi-single-cycle terahertz pulses from a single infrared beam in a second-order nonlinear medium,' Opt. Lett. 29, 2650-2652 (2004).
[CrossRef] [PubMed]

2003 (2)

W. Shi and Y. J. Ding, 'Designs of terahertz waveguides for efficient parametric terahertz generation,' Appl. Phys. Lett. 82, 4435-4437 (2003).
[CrossRef]

B. S. Williams, S. Kumar, H. Callebaut, and Q. Hu, 'Terahertz quantum-cascade laser at 100 μm using metal waveguide for mode confinement,' Appl. Phys. Lett. 83, 2124-2126 (2003).
[CrossRef]

2002 (3)

R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, 'Terahertz semiconductor-heterostructure laser,' Nature 417, 156-159 (2002).
[CrossRef] [PubMed]

A. G. Davies, E. H. Linfield, and M. B. Johnston, 'The development of terahertz sources and their applications,' Phys. Med. Biol. 47, 3679-3689 (2002).
[CrossRef] [PubMed]

J.-P. Caumes, L. Videau, C. Rouyer, and E. Freysz, 'Kerr-like nonlinearity induced via terahertz generation and the electro-optical effect in zinc blende crystals,' Phys. Rev. Lett. 89, 047401 (2002).
[CrossRef] [PubMed]

2001 (1)

L. Duvillaret, F.-F. Garet, J.-F. Roux, and J.-L. Coutaz, 'Analytical modeling and optimization of terahertz time-domain spectroscopy experiments, using photoswitches as antennas,' IEEE J. Sel. Top. Quantum Electron. 7, 615-623 (2001).
[CrossRef]

1999 (2)

R. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, 'Broadband phase-matched difference frequency mixing of femtosecond pulses in GaSe: experiment and theory,' Appl. Phys. Lett. 75, 1060-1062 (1999).
[CrossRef]

G. Gallot, J. Zhang, R. W. McGowan, T. Jeon, and D. Grischkowsky, 'Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,' Appl. Phys. Lett. 74, 3450-3452 (1999).
[CrossRef]

1998 (1)

1996 (2)

A. Nahata, A. S. Weling, and T. F. Heinz, 'A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,' Appl. Phys. Lett. 69, 2321-2323 (1996).
[CrossRef]

M. Hangyo, S. Tomozawa, Y. Murakami, M. Tonouchi, M. Tani, Z. Wang, K. Sakai, and S. Nakashima, 'Terahertz radiation from superconducting YBa2Cu3O7-thin films excited by femtosecond optical pulses,' Appl. Phys. Lett. 69, 2122-2124 (1996).
[CrossRef]

1995 (4)

T. J. Carrig, G. Rodriguez, T. S. Clement, A. J. Taylor, and K. R. Stewart, 'Scaling of terahertz radiation via optical rectification in electro-optic crystals,' Appl. Phys. Lett. 66, 121-123 (1995).
[CrossRef]

T. Carrig, I. Rodriguez, T. Clement, A. Taylor, and K. Stewart, 'Generation of terahertz radiation using electro-optic crystal mosaics,' Appl. Phys. Lett. 66, 10-12 (1995).
[CrossRef]

A. Nahata, D. Auston, C. Wu, and J. Yardley, 'Generation of terahertz radiation from a poled polymer,' Appl. Phys. Lett. 67, 1358-1360 (1995).
[CrossRef]

A. Bonvalet, M. Joffre, J. Martin, and A. Migus, 'Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,' Appl. Phys. Lett. 67, 2907-2909 (1995).
[CrossRef]

1994 (2)

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, 'Terahertz optical rectification from ‹110› zinc-blende crystals,' Appl. Phys. Lett. 64, 1324-1326 (1994).
[CrossRef]

J. U. Kang, A. Villeneuve, M. Sheik-Bahae, and G. I. Stegeman, 'Limitation due to three-photon absorption on the useful spectral range for nonlinear optics in AlGaAs below half band gap,' Appl. Phys. Lett. 65, 147-149 (1994).
[CrossRef]

1992 (6)

S. Chuang, S. Schmitt-Rink, B. Green, P. Saeta, and A. Levi, 'Optical rectification at semiconductor surfaces,' Phys. Rev. Lett. 68, 102-105 (1992).
[CrossRef] [PubMed]

B. Greene, P. Saeta, D. Dykaar, S. Schmitt-Rink, and S. Chuang, 'Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,' IEEE J. Quantum Electron. 28, 2302-2312 (1992).
[CrossRef]

X.-C. Zhang and D. Auston, 'Optoelectronic measurement of semiconductor surfaces and interfaces with femtosecond optics,' J. Appl. Phys. 71, 326-338 (1992).
[CrossRef]

L. Xu, X.-C. Zhang, and D. Auston, 'Terahertz beam generation by femtosecond optical pulses in electro-optic materials,' Appl. Phys. Lett. 61, 1784-1786 (1992).
[CrossRef]

H. Roskos, M. Nuss, J. Shah, K. Leo, D. Miller, A. Fox, S. Schmitt-Rink, and K. Kohler, 'Coherent submillimeter-wave emission from charge oscillations in a double-well potential,' Phys. Rev. Lett. 68, 2216-2219 (1992).
[CrossRef] [PubMed]

P. Planken, M. Nuss, I. Brener, K. Goossen, M. Luo, S. Chuang, and L. Pfeiffer, 'Terahertz emission in single quantum wells after coherent optical excitation of light hole and heavy hole excitons,' Phys. Rev. Lett. 69, 3800-3803 (1992).
[CrossRef] [PubMed]

1990 (1)

B. Hu, X.-C. Zhang, D. Auston, and P. Smith, 'Free-space radiation from electro-optic crystals,' Appl. Phys. Lett. 56, 506-508 (1990).
[CrossRef]

1984 (1)

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, 'Cherenkov radiation from femtosecond optical pulses in electro-optic media,' Phys. Rev. Lett. 53, 1555-1558 (1984).
[CrossRef]

1978 (1)

A. N. Pikhtin and A. D. Yas'kov, 'Refraction index dispersion in semiconductors with diamond and zinc blende structures,' Sov. Phys. Semicond. 12, 622-626 (1978).

1972 (1)

C. Flytzanis, 'Infrared dispersion of second-order electric susceptibilities in semiconducting compounds,' Phys. Rev. B 6, 1264-1290 (1972).
[CrossRef]

1971 (2)

F. P. Kapron and D. B. Keck, 'Pulse transmission through a dielectric optical waveguide,' Appl. Opt. 10, 1519-1523 (1971).
[CrossRef] [PubMed]

K. H. Yang, P. L. Richards, and Y. R. Shen, 'Generation of far-infrared radiation by picosecond light pulses in LiNbO3,' Appl. Phys. Lett. 19, 320-323 (1971).
[CrossRef]

1970 (1)

T. Yajima and N. Takeuchi, 'Far-infrared difference-frequency generation by picosecond laser pulses,' Jpn. J. Appl. Phys. 9, 1361-1371 (1970).
[CrossRef]

1959 (1)

R. H. Hall, 'Recombination processes in semiconductors,' Proc. IEE Suppl. 106 B, 923-931 (1959).

Agrawal, A.

A. Agrawal, H. Cao, and A. Nahata, 'Excitation and scattering of surface plasmon-polaritons on structured metal films and their application to pulse shaping and enhanced transmission,' New J. Phys. 7, doi: 10.1088/1367-2630/7/1/249 (2005).
[CrossRef]

Akhmedzhanov, R. A.

R. A. Akhmedzhanov, A. I. Korytin, A. G. Litvak, A. M. Sergeev, and E. V. Suvorov, 'Generation and detection of ultrashort electromagnetic pulses in terahertz region and their application for spectroscopy,' Radiophys. Quantum Electron. 48, 837-843 (2005).
[CrossRef]

Alexander, M.

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, 'Terahertz optical rectification from ‹110› zinc-blende crystals,' Appl. Phys. Lett. 64, 1324-1326 (1994).
[CrossRef]

Almasi, G.

Alton, J.

S. Dhillon, J. Alton, S. Barbieri, C. Sirtori, A. de Rossi, M. Calligaro, H. E. Beere, and D. Ritchie, 'Ultralow threshold current terahertz quantum cascade lasers based on double-metal buried strip waveguides,' Appl. Phys. Lett. 87, 071101 (2005).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart, and Winston, 1976).

Auston, D.

A. Nahata, D. Auston, C. Wu, and J. Yardley, 'Generation of terahertz radiation from a poled polymer,' Appl. Phys. Lett. 67, 1358-1360 (1995).
[CrossRef]

X.-C. Zhang and D. Auston, 'Optoelectronic measurement of semiconductor surfaces and interfaces with femtosecond optics,' J. Appl. Phys. 71, 326-338 (1992).
[CrossRef]

L. Xu, X.-C. Zhang, and D. Auston, 'Terahertz beam generation by femtosecond optical pulses in electro-optic materials,' Appl. Phys. Lett. 61, 1784-1786 (1992).
[CrossRef]

B. Hu, X.-C. Zhang, D. Auston, and P. Smith, 'Free-space radiation from electro-optic crystals,' Appl. Phys. Lett. 56, 506-508 (1990).
[CrossRef]

Auston, D. H.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, 'Cherenkov radiation from femtosecond optical pulses in electro-optic media,' Phys. Rev. Lett. 53, 1555-1558 (1984).
[CrossRef]

Barbieri, S.

S. Dhillon, J. Alton, S. Barbieri, C. Sirtori, A. de Rossi, M. Calligaro, H. E. Beere, and D. Ritchie, 'Ultralow threshold current terahertz quantum cascade lasers based on double-metal buried strip waveguides,' Appl. Phys. Lett. 87, 071101 (2005).
[CrossRef]

Beere, H. E.

S. Dhillon, J. Alton, S. Barbieri, C. Sirtori, A. de Rossi, M. Calligaro, H. E. Beere, and D. Ritchie, 'Ultralow threshold current terahertz quantum cascade lasers based on double-metal buried strip waveguides,' Appl. Phys. Lett. 87, 071101 (2005).
[CrossRef]

R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, 'Terahertz semiconductor-heterostructure laser,' Nature 417, 156-159 (2002).
[CrossRef] [PubMed]

Beltram, F.

R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, 'Terahertz semiconductor-heterostructure laser,' Nature 417, 156-159 (2002).
[CrossRef] [PubMed]

Bliss, D.

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, 'Terahertz optical rectification from ‹110› zinc-blende crystals,' Appl. Phys. Lett. 64, 1324-1326 (1994).
[CrossRef]

Bonvalet, A.

A. Bonvalet, M. Joffre, J. Martin, and A. Migus, 'Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,' Appl. Phys. Lett. 67, 2907-2909 (1995).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Brener, I.

P. Planken, M. Nuss, I. Brener, K. Goossen, M. Luo, S. Chuang, and L. Pfeiffer, 'Terahertz emission in single quantum wells after coherent optical excitation of light hole and heavy hole excitons,' Phys. Rev. Lett. 69, 3800-3803 (1992).
[CrossRef] [PubMed]

Callebaut, H.

B. S. Williams, S. Kumar, H. Callebaut, and Q. Hu, 'Terahertz quantum-cascade laser at 100 μm using metal waveguide for mode confinement,' Appl. Phys. Lett. 83, 2124-2126 (2003).
[CrossRef]

Calligaro, M.

S. Dhillon, J. Alton, S. Barbieri, C. Sirtori, A. de Rossi, M. Calligaro, H. E. Beere, and D. Ritchie, 'Ultralow threshold current terahertz quantum cascade lasers based on double-metal buried strip waveguides,' Appl. Phys. Lett. 87, 071101 (2005).
[CrossRef]

Cao, H.

A. Agrawal, H. Cao, and A. Nahata, 'Excitation and scattering of surface plasmon-polaritons on structured metal films and their application to pulse shaping and enhanced transmission,' New J. Phys. 7, doi: 10.1088/1367-2630/7/1/249 (2005).
[CrossRef]

Cao, J. C.

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M. Hangyo, S. Tomozawa, Y. Murakami, M. Tonouchi, M. Tani, Z. Wang, K. Sakai, and S. Nakashima, 'Terahertz radiation from superconducting YBa2Cu3O7-thin films excited by femtosecond optical pulses,' Appl. Phys. Lett. 69, 2122-2124 (1996).
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[CrossRef]

Roskos, H.

H. Roskos, M. Nuss, J. Shah, K. Leo, D. Miller, A. Fox, S. Schmitt-Rink, and K. Kohler, 'Coherent submillimeter-wave emission from charge oscillations in a double-well potential,' Phys. Rev. Lett. 68, 2216-2219 (1992).
[CrossRef] [PubMed]

Rossi, F.

R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, 'Terahertz semiconductor-heterostructure laser,' Nature 417, 156-159 (2002).
[CrossRef] [PubMed]

Roux, J.-F.

L. Duvillaret, F.-F. Garet, J.-F. Roux, and J.-L. Coutaz, 'Analytical modeling and optimization of terahertz time-domain spectroscopy experiments, using photoswitches as antennas,' IEEE J. Sel. Top. Quantum Electron. 7, 615-623 (2001).
[CrossRef]

Rouyer, C.

J.-P. Caumes, L. Videau, C. Rouyer, and E. Freysz, 'Kerr-like nonlinearity induced via terahertz generation and the electro-optical effect in zinc blende crystals,' Phys. Rev. Lett. 89, 047401 (2002).
[CrossRef] [PubMed]

Saeta, P.

B. Greene, P. Saeta, D. Dykaar, S. Schmitt-Rink, and S. Chuang, 'Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,' IEEE J. Quantum Electron. 28, 2302-2312 (1992).
[CrossRef]

S. Chuang, S. Schmitt-Rink, B. Green, P. Saeta, and A. Levi, 'Optical rectification at semiconductor surfaces,' Phys. Rev. Lett. 68, 102-105 (1992).
[CrossRef] [PubMed]

Sakai, K.

M. Hangyo, S. Tomozawa, Y. Murakami, M. Tonouchi, M. Tani, Z. Wang, K. Sakai, and S. Nakashima, 'Terahertz radiation from superconducting YBa2Cu3O7-thin films excited by femtosecond optical pulses,' Appl. Phys. Lett. 69, 2122-2124 (1996).
[CrossRef]

Schmitt-Rink, S.

B. Greene, P. Saeta, D. Dykaar, S. Schmitt-Rink, and S. Chuang, 'Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,' IEEE J. Quantum Electron. 28, 2302-2312 (1992).
[CrossRef]

S. Chuang, S. Schmitt-Rink, B. Green, P. Saeta, and A. Levi, 'Optical rectification at semiconductor surfaces,' Phys. Rev. Lett. 68, 102-105 (1992).
[CrossRef] [PubMed]

H. Roskos, M. Nuss, J. Shah, K. Leo, D. Miller, A. Fox, S. Schmitt-Rink, and K. Kohler, 'Coherent submillimeter-wave emission from charge oscillations in a double-well potential,' Phys. Rev. Lett. 68, 2216-2219 (1992).
[CrossRef] [PubMed]

Schroeder, C. B.

C. Toth, J. van Tilborg, C. G. Geddes, G. Fubiani, C. B. Schroeder, E. Esarey, J. Faure, G. Dugan, and W. P. Leemans, 'Powerful pulsed THz radiation from laser-accelerated relativistic electron bunches,' in Signal Processing, Sensor Fusion, and Target Recognition XIII, I.Kadar, ed., Proc. SPIE 5448, 491-504 (2004).

Sergeev, A. M.

R. A. Akhmedzhanov, A. I. Korytin, A. G. Litvak, A. M. Sergeev, and E. V. Suvorov, 'Generation and detection of ultrashort electromagnetic pulses in terahertz region and their application for spectroscopy,' Radiophys. Quantum Electron. 48, 837-843 (2005).
[CrossRef]

Shah, J.

H. Roskos, M. Nuss, J. Shah, K. Leo, D. Miller, A. Fox, S. Schmitt-Rink, and K. Kohler, 'Coherent submillimeter-wave emission from charge oscillations in a double-well potential,' Phys. Rev. Lett. 68, 2216-2219 (1992).
[CrossRef] [PubMed]

Sheik-Bahae, M.

J. U. Kang, A. Villeneuve, M. Sheik-Bahae, and G. I. Stegeman, 'Limitation due to three-photon absorption on the useful spectral range for nonlinear optics in AlGaAs below half band gap,' Appl. Phys. Lett. 65, 147-149 (1994).
[CrossRef]

Shen, Y. R.

K. H. Yang, P. L. Richards, and Y. R. Shen, 'Generation of far-infrared radiation by picosecond light pulses in LiNbO3,' Appl. Phys. Lett. 19, 320-323 (1971).
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

Shi, W.

W. Shi and Y. J. Ding, 'Designs of terahertz waveguides for efficient parametric terahertz generation,' Appl. Phys. Lett. 82, 4435-4437 (2003).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Sirtori, C.

S. Dhillon, J. Alton, S. Barbieri, C. Sirtori, A. de Rossi, M. Calligaro, H. E. Beere, and D. Ritchie, 'Ultralow threshold current terahertz quantum cascade lasers based on double-metal buried strip waveguides,' Appl. Phys. Lett. 87, 071101 (2005).
[CrossRef]

Smith, D.

Smith, P.

B. Hu, X.-C. Zhang, D. Auston, and P. Smith, 'Free-space radiation from electro-optic crystals,' Appl. Phys. Lett. 56, 506-508 (1990).
[CrossRef]

Stegeman, G. I.

J. U. Kang, A. Villeneuve, M. Sheik-Bahae, and G. I. Stegeman, 'Limitation due to three-photon absorption on the useful spectral range for nonlinear optics in AlGaAs below half band gap,' Appl. Phys. Lett. 65, 147-149 (1994).
[CrossRef]

Stepanov, A. G.

Stewart, K.

T. Carrig, I. Rodriguez, T. Clement, A. Taylor, and K. Stewart, 'Generation of terahertz radiation using electro-optic crystal mosaics,' Appl. Phys. Lett. 66, 10-12 (1995).
[CrossRef]

Stewart, K. R.

T. J. Carrig, G. Rodriguez, T. S. Clement, A. J. Taylor, and K. R. Stewart, 'Scaling of terahertz radiation via optical rectification in electro-optic crystals,' Appl. Phys. Lett. 66, 121-123 (1995).
[CrossRef]

Stuk, G.

D. Zimdars, J. A. Valdmanis, J. S. White, G. Stuk, S. Williamson, W. P. Winfree, and E. I. Madaras, 'Technology and applications of terahertz imaging non-destructive examination: inspection of space shuttle sprayed on foam insulation,' in Review of Progress in Quantitative Nondestructive Evaluation, D.O.Thompson and D.E.Chimenti, eds., AIP Conf. Proc. 760, 570-577 (2005).

Suvorov, E. V.

R. A. Akhmedzhanov, A. I. Korytin, A. G. Litvak, A. M. Sergeev, and E. V. Suvorov, 'Generation and detection of ultrashort electromagnetic pulses in terahertz region and their application for spectroscopy,' Radiophys. Quantum Electron. 48, 837-843 (2005).
[CrossRef]

Svelto, O.

O. Svelto, Principles of Lasers (Plenum, 1989).

Takeuchi, N.

T. Yajima and N. Takeuchi, 'Far-infrared difference-frequency generation by picosecond laser pulses,' Jpn. J. Appl. Phys. 9, 1361-1371 (1970).
[CrossRef]

Tani, M.

M. Hangyo, S. Tomozawa, Y. Murakami, M. Tonouchi, M. Tani, Z. Wang, K. Sakai, and S. Nakashima, 'Terahertz radiation from superconducting YBa2Cu3O7-thin films excited by femtosecond optical pulses,' Appl. Phys. Lett. 69, 2122-2124 (1996).
[CrossRef]

Taniuchi, T.

T. Taniuchi, S. Okada, and H. Nakanishi, 'Widely tunable terahertz-wave generation in an organic crystal and its spectroscopic application,' J. Appl. Phys. 95, 5984-5988 (2004).
[CrossRef]

Taylor, A.

T. Carrig, I. Rodriguez, T. Clement, A. Taylor, and K. Stewart, 'Generation of terahertz radiation using electro-optic crystal mosaics,' Appl. Phys. Lett. 66, 10-12 (1995).
[CrossRef]

Taylor, A. J.

T. J. Carrig, G. Rodriguez, T. S. Clement, A. J. Taylor, and K. R. Stewart, 'Scaling of terahertz radiation via optical rectification in electro-optic crystals,' Appl. Phys. Lett. 66, 121-123 (1995).
[CrossRef]

Tomozawa, S.

M. Hangyo, S. Tomozawa, Y. Murakami, M. Tonouchi, M. Tani, Z. Wang, K. Sakai, and S. Nakashima, 'Terahertz radiation from superconducting YBa2Cu3O7-thin films excited by femtosecond optical pulses,' Appl. Phys. Lett. 69, 2122-2124 (1996).
[CrossRef]

Tonouchi, M.

M. Hangyo, S. Tomozawa, Y. Murakami, M. Tonouchi, M. Tani, Z. Wang, K. Sakai, and S. Nakashima, 'Terahertz radiation from superconducting YBa2Cu3O7-thin films excited by femtosecond optical pulses,' Appl. Phys. Lett. 69, 2122-2124 (1996).
[CrossRef]

Toth, C.

C. Toth, J. van Tilborg, C. G. Geddes, G. Fubiani, C. B. Schroeder, E. Esarey, J. Faure, G. Dugan, and W. P. Leemans, 'Powerful pulsed THz radiation from laser-accelerated relativistic electron bunches,' in Signal Processing, Sensor Fusion, and Target Recognition XIII, I.Kadar, ed., Proc. SPIE 5448, 491-504 (2004).

Tredicucci, A.

R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, 'Terahertz semiconductor-heterostructure laser,' Nature 417, 156-159 (2002).
[CrossRef] [PubMed]

Vainshtein, L. A.

L. A. Vainshtein, Electromagnetic Waves (Sov. Radio, 1988) (in Russian).

Valdmanis, J. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, 'Cherenkov radiation from femtosecond optical pulses in electro-optic media,' Phys. Rev. Lett. 53, 1555-1558 (1984).
[CrossRef]

D. Zimdars, J. A. Valdmanis, J. S. White, G. Stuk, S. Williamson, W. P. Winfree, and E. I. Madaras, 'Technology and applications of terahertz imaging non-destructive examination: inspection of space shuttle sprayed on foam insulation,' in Review of Progress in Quantitative Nondestructive Evaluation, D.O.Thompson and D.E.Chimenti, eds., AIP Conf. Proc. 760, 570-577 (2005).

van Tilborg, J.

C. Toth, J. van Tilborg, C. G. Geddes, G. Fubiani, C. B. Schroeder, E. Esarey, J. Faure, G. Dugan, and W. P. Leemans, 'Powerful pulsed THz radiation from laser-accelerated relativistic electron bunches,' in Signal Processing, Sensor Fusion, and Target Recognition XIII, I.Kadar, ed., Proc. SPIE 5448, 491-504 (2004).

Videau, L.

J.-P. Caumes, L. Videau, C. Rouyer, and E. Freysz, 'Kerr-like nonlinearity induced via terahertz generation and the electro-optical effect in zinc blende crystals,' Phys. Rev. Lett. 89, 047401 (2002).
[CrossRef] [PubMed]

Villeneuve, A.

J. U. Kang, A. Villeneuve, M. Sheik-Bahae, and G. I. Stegeman, 'Limitation due to three-photon absorption on the useful spectral range for nonlinear optics in AlGaAs below half band gap,' Appl. Phys. Lett. 65, 147-149 (1994).
[CrossRef]

Wallace, V. P.

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, 'In vivo study of human skin using pulsed terahertz radiation,' Phys. Med. Biol. 49, 1595-1607 (2004).
[CrossRef] [PubMed]

Wang, Z.

M. Hangyo, S. Tomozawa, Y. Murakami, M. Tonouchi, M. Tani, Z. Wang, K. Sakai, and S. Nakashima, 'Terahertz radiation from superconducting YBa2Cu3O7-thin films excited by femtosecond optical pulses,' Appl. Phys. Lett. 69, 2122-2124 (1996).
[CrossRef]

Weber, M. J.

M. J. Weber, Handbook of Laser Wavelengths (CRC, 1999).

Weling, A. S.

A. Nahata, A. S. Weling, and T. F. Heinz, 'A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,' Appl. Phys. Lett. 69, 2321-2323 (1996).
[CrossRef]

White, J. S.

D. Zimdars, J. A. Valdmanis, J. S. White, G. Stuk, S. Williamson, W. P. Winfree, and E. I. Madaras, 'Technology and applications of terahertz imaging non-destructive examination: inspection of space shuttle sprayed on foam insulation,' in Review of Progress in Quantitative Nondestructive Evaluation, D.O.Thompson and D.E.Chimenti, eds., AIP Conf. Proc. 760, 570-577 (2005).

Williams, B. S.

B. S. Williams, S. Kumar, H. Callebaut, and Q. Hu, 'Terahertz quantum-cascade laser at 100 μm using metal waveguide for mode confinement,' Appl. Phys. Lett. 83, 2124-2126 (2003).
[CrossRef]

Williamson, S.

D. Zimdars, J. A. Valdmanis, J. S. White, G. Stuk, S. Williamson, W. P. Winfree, and E. I. Madaras, 'Technology and applications of terahertz imaging non-destructive examination: inspection of space shuttle sprayed on foam insulation,' in Review of Progress in Quantitative Nondestructive Evaluation, D.O.Thompson and D.E.Chimenti, eds., AIP Conf. Proc. 760, 570-577 (2005).

Winfree, W. P.

D. Zimdars, J. A. Valdmanis, J. S. White, G. Stuk, S. Williamson, W. P. Winfree, and E. I. Madaras, 'Technology and applications of terahertz imaging non-destructive examination: inspection of space shuttle sprayed on foam insulation,' in Review of Progress in Quantitative Nondestructive Evaluation, D.O.Thompson and D.E.Chimenti, eds., AIP Conf. Proc. 760, 570-577 (2005).

Winnerl, S.

A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, 'High-intensity terahertz radiation from a microstructured large-area photoconductor,' Appl. Phys. Lett. 86, 121114 (2005).
[CrossRef]

Woerner, M.

R. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, 'Broadband phase-matched difference frequency mixing of femtosecond pulses in GaSe: experiment and theory,' Appl. Phys. Lett. 75, 1060-1062 (1999).
[CrossRef]

R. Kaindl, D. Smith, M. Joschko, M. Hasselbeck, M. Woerner, and T. Elsaesser, 'Femtosecond infrared pulses tunable from 9to18 μm at an 88-MHz repetition rate,' Opt. Lett. 23, 861-863 (1998).
[CrossRef]

Wu, B. H.

B. H. Wu and J. C. Cao, 'Nonlinear dynamics in multisubband semiconductor quantum wells under terahertz radiation,' Physica B 349, 322-326 (2004).
[CrossRef]

Wu, C.

A. Nahata, D. Auston, C. Wu, and J. Yardley, 'Generation of terahertz radiation from a poled polymer,' Appl. Phys. Lett. 67, 1358-1360 (1995).
[CrossRef]

Xu, L.

L. Xu, X.-C. Zhang, and D. Auston, 'Terahertz beam generation by femtosecond optical pulses in electro-optic materials,' Appl. Phys. Lett. 61, 1784-1786 (1992).
[CrossRef]

Yajima, T.

T. Yajima and N. Takeuchi, 'Far-infrared difference-frequency generation by picosecond laser pulses,' Jpn. J. Appl. Phys. 9, 1361-1371 (1970).
[CrossRef]

Yang, K. H.

K. H. Yang, P. L. Richards, and Y. R. Shen, 'Generation of far-infrared radiation by picosecond light pulses in LiNbO3,' Appl. Phys. Lett. 19, 320-323 (1971).
[CrossRef]

Yardley, J.

A. Nahata, D. Auston, C. Wu, and J. Yardley, 'Generation of terahertz radiation from a poled polymer,' Appl. Phys. Lett. 67, 1358-1360 (1995).
[CrossRef]

Yas'kov, A. D.

A. N. Pikhtin and A. D. Yas'kov, 'Refraction index dispersion in semiconductors with diamond and zinc blende structures,' Sov. Phys. Semicond. 12, 622-626 (1978).

Zhang, J.

G. Gallot, J. Zhang, R. W. McGowan, T. Jeon, and D. Grischkowsky, 'Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,' Appl. Phys. Lett. 74, 3450-3452 (1999).
[CrossRef]

Zhang, X.-C.

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, 'Terahertz optical rectification from ‹110› zinc-blende crystals,' Appl. Phys. Lett. 64, 1324-1326 (1994).
[CrossRef]

X.-C. Zhang and D. Auston, 'Optoelectronic measurement of semiconductor surfaces and interfaces with femtosecond optics,' J. Appl. Phys. 71, 326-338 (1992).
[CrossRef]

L. Xu, X.-C. Zhang, and D. Auston, 'Terahertz beam generation by femtosecond optical pulses in electro-optic materials,' Appl. Phys. Lett. 61, 1784-1786 (1992).
[CrossRef]

B. Hu, X.-C. Zhang, D. Auston, and P. Smith, 'Free-space radiation from electro-optic crystals,' Appl. Phys. Lett. 56, 506-508 (1990).
[CrossRef]

Zimdars, D.

D. Zimdars, J. A. Valdmanis, J. S. White, G. Stuk, S. Williamson, W. P. Winfree, and E. I. Madaras, 'Technology and applications of terahertz imaging non-destructive examination: inspection of space shuttle sprayed on foam insulation,' in Review of Progress in Quantitative Nondestructive Evaluation, D.O.Thompson and D.E.Chimenti, eds., AIP Conf. Proc. 760, 570-577 (2005).

Appl. Opt. (1)

Appl. Phys. Lett. (17)

G. Gallot, J. Zhang, R. W. McGowan, T. Jeon, and D. Grischkowsky, 'Measurements of the THz absorption and dispersion of ZnTe and their relevance to the electro-optic detection of THz radiation,' Appl. Phys. Lett. 74, 3450-3452 (1999).
[CrossRef]

J. U. Kang, A. Villeneuve, M. Sheik-Bahae, and G. I. Stegeman, 'Limitation due to three-photon absorption on the useful spectral range for nonlinear optics in AlGaAs below half band gap,' Appl. Phys. Lett. 65, 147-149 (1994).
[CrossRef]

W. Shi and Y. J. Ding, 'Designs of terahertz waveguides for efficient parametric terahertz generation,' Appl. Phys. Lett. 82, 4435-4437 (2003).
[CrossRef]

B. S. Williams, S. Kumar, H. Callebaut, and Q. Hu, 'Terahertz quantum-cascade laser at 100 μm using metal waveguide for mode confinement,' Appl. Phys. Lett. 83, 2124-2126 (2003).
[CrossRef]

S. Dhillon, J. Alton, S. Barbieri, C. Sirtori, A. de Rossi, M. Calligaro, H. E. Beere, and D. Ritchie, 'Ultralow threshold current terahertz quantum cascade lasers based on double-metal buried strip waveguides,' Appl. Phys. Lett. 87, 071101 (2005).
[CrossRef]

A. Dreyhaupt, S. Winnerl, T. Dekorsy, and M. Helm, 'High-intensity terahertz radiation from a microstructured large-area photoconductor,' Appl. Phys. Lett. 86, 121114 (2005).
[CrossRef]

M. Hangyo, S. Tomozawa, Y. Murakami, M. Tonouchi, M. Tani, Z. Wang, K. Sakai, and S. Nakashima, 'Terahertz radiation from superconducting YBa2Cu3O7-thin films excited by femtosecond optical pulses,' Appl. Phys. Lett. 69, 2122-2124 (1996).
[CrossRef]

T. J. Carrig, G. Rodriguez, T. S. Clement, A. J. Taylor, and K. R. Stewart, 'Scaling of terahertz radiation via optical rectification in electro-optic crystals,' Appl. Phys. Lett. 66, 121-123 (1995).
[CrossRef]

B. Hu, X.-C. Zhang, D. Auston, and P. Smith, 'Free-space radiation from electro-optic crystals,' Appl. Phys. Lett. 56, 506-508 (1990).
[CrossRef]

L. Xu, X.-C. Zhang, and D. Auston, 'Terahertz beam generation by femtosecond optical pulses in electro-optic materials,' Appl. Phys. Lett. 61, 1784-1786 (1992).
[CrossRef]

A. Bonvalet, M. Joffre, J. Martin, and A. Migus, 'Generation of ultrabroadband femtosecond pulses in the mid-infrared by optical rectification of 15 fs light pulses at 100 MHz repetition rate,' Appl. Phys. Lett. 67, 2907-2909 (1995).
[CrossRef]

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, 'Terahertz optical rectification from ‹110› zinc-blende crystals,' Appl. Phys. Lett. 64, 1324-1326 (1994).
[CrossRef]

T. Carrig, I. Rodriguez, T. Clement, A. Taylor, and K. Stewart, 'Generation of terahertz radiation using electro-optic crystal mosaics,' Appl. Phys. Lett. 66, 10-12 (1995).
[CrossRef]

A. Nahata, D. Auston, C. Wu, and J. Yardley, 'Generation of terahertz radiation from a poled polymer,' Appl. Phys. Lett. 67, 1358-1360 (1995).
[CrossRef]

K. H. Yang, P. L. Richards, and Y. R. Shen, 'Generation of far-infrared radiation by picosecond light pulses in LiNbO3,' Appl. Phys. Lett. 19, 320-323 (1971).
[CrossRef]

A. Nahata, A. S. Weling, and T. F. Heinz, 'A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,' Appl. Phys. Lett. 69, 2321-2323 (1996).
[CrossRef]

R. Kaindl, F. Eickemeyer, M. Woerner, and T. Elsaesser, 'Broadband phase-matched difference frequency mixing of femtosecond pulses in GaSe: experiment and theory,' Appl. Phys. Lett. 75, 1060-1062 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. Greene, P. Saeta, D. Dykaar, S. Schmitt-Rink, and S. Chuang, 'Far-infrared light generation at semiconductor surfaces and its spectroscopic applications,' IEEE J. Quantum Electron. 28, 2302-2312 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

Y. J. Ding, 'Quasi-single-cycle terahertz pulses based on broad-phase-matched difference-frequency generation in second-order nonlinear medium: high output powers and conversion efficiencies,' IEEE J. Sel. Top. Quantum Electron. 10, 1171-1179 (2004).
[CrossRef]

L. Duvillaret, F.-F. Garet, J.-F. Roux, and J.-L. Coutaz, 'Analytical modeling and optimization of terahertz time-domain spectroscopy experiments, using photoswitches as antennas,' IEEE J. Sel. Top. Quantum Electron. 7, 615-623 (2001).
[CrossRef]

J. Appl. Phys. (2)

T. Taniuchi, S. Okada, and H. Nakanishi, 'Widely tunable terahertz-wave generation in an organic crystal and its spectroscopic application,' J. Appl. Phys. 95, 5984-5988 (2004).
[CrossRef]

X.-C. Zhang and D. Auston, 'Optoelectronic measurement of semiconductor surfaces and interfaces with femtosecond optics,' J. Appl. Phys. 71, 326-338 (1992).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Yajima and N. Takeuchi, 'Far-infrared difference-frequency generation by picosecond laser pulses,' Jpn. J. Appl. Phys. 9, 1361-1371 (1970).
[CrossRef]

Nature (1)

R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, 'Terahertz semiconductor-heterostructure laser,' Nature 417, 156-159 (2002).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Med. Biol. (2)

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, 'In vivo study of human skin using pulsed terahertz radiation,' Phys. Med. Biol. 49, 1595-1607 (2004).
[CrossRef] [PubMed]

A. G. Davies, E. H. Linfield, and M. B. Johnston, 'The development of terahertz sources and their applications,' Phys. Med. Biol. 47, 3679-3689 (2002).
[CrossRef] [PubMed]

Phys. Rev. B (1)

C. Flytzanis, 'Infrared dispersion of second-order electric susceptibilities in semiconducting compounds,' Phys. Rev. B 6, 1264-1290 (1972).
[CrossRef]

Phys. Rev. Lett. (5)

H. Roskos, M. Nuss, J. Shah, K. Leo, D. Miller, A. Fox, S. Schmitt-Rink, and K. Kohler, 'Coherent submillimeter-wave emission from charge oscillations in a double-well potential,' Phys. Rev. Lett. 68, 2216-2219 (1992).
[CrossRef] [PubMed]

P. Planken, M. Nuss, I. Brener, K. Goossen, M. Luo, S. Chuang, and L. Pfeiffer, 'Terahertz emission in single quantum wells after coherent optical excitation of light hole and heavy hole excitons,' Phys. Rev. Lett. 69, 3800-3803 (1992).
[CrossRef] [PubMed]

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, 'Cherenkov radiation from femtosecond optical pulses in electro-optic media,' Phys. Rev. Lett. 53, 1555-1558 (1984).
[CrossRef]

J.-P. Caumes, L. Videau, C. Rouyer, and E. Freysz, 'Kerr-like nonlinearity induced via terahertz generation and the electro-optical effect in zinc blende crystals,' Phys. Rev. Lett. 89, 047401 (2002).
[CrossRef] [PubMed]

S. Chuang, S. Schmitt-Rink, B. Green, P. Saeta, and A. Levi, 'Optical rectification at semiconductor surfaces,' Phys. Rev. Lett. 68, 102-105 (1992).
[CrossRef] [PubMed]

Physica B (1)

B. H. Wu and J. C. Cao, 'Nonlinear dynamics in multisubband semiconductor quantum wells under terahertz radiation,' Physica B 349, 322-326 (2004).
[CrossRef]

Proc. IEE Suppl. (1)

R. H. Hall, 'Recombination processes in semiconductors,' Proc. IEE Suppl. 106 B, 923-931 (1959).

Radiophys. Quantum Electron. (1)

R. A. Akhmedzhanov, A. I. Korytin, A. G. Litvak, A. M. Sergeev, and E. V. Suvorov, 'Generation and detection of ultrashort electromagnetic pulses in terahertz region and their application for spectroscopy,' Radiophys. Quantum Electron. 48, 837-843 (2005).
[CrossRef]

Sov. Phys. Semicond. (1)

A. N. Pikhtin and A. D. Yas'kov, 'Refraction index dispersion in semiconductors with diamond and zinc blende structures,' Sov. Phys. Semicond. 12, 622-626 (1978).

Other (14)

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

R. W. Boyd, Nonlinear Optics (Academic, 1992).

M.J.Weber, ed., Handbook of Optical Materials (CRC, 2003).

D.R.Lide ed., CRC Handbook of Chemistry and Physics (CRC, 2005).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart, and Winston, 1976).

M.K.Barnoski, ed., Introduction to Integrated Optics (Plenum, 1974).

M. J. Weber, Handbook of Laser Wavelengths (CRC, 1999).

L. A. Vainshtein, Electromagnetic Waves (Sov. Radio, 1988) (in Russian).

O. Svelto, Principles of Lasers (Plenum, 1989).

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (4)

Fig. 1
Fig. 1

Schematic of (a) 2PA and (b) 3PA processes in the semiconductors. E v and E c are the top energy of the valence band and the bottom energy of the conductivity band, respectively, h ω p 2 π is the photon energy of the pump field.

Fig. 2
Fig. 2

C Ga As C slab waveguide layout and the coordinate system used in the text.

Fig. 3
Fig. 3

(a) Difference between the waveguide pump group index, n ¯ pg , and the refraction index at the central frequency of the THz pulse, n ¯ THz , and (b) derivative ( n ¯ pg ω ) ω = ω p c ( 2 h ω 2 ) ω = ω p as functions of the pump wavelength, λ p . The thickness of the GaAs layer is 2 a = 74 μ m .

Fig. 4
Fig. 4

Module of the spectral electric field of the THz pulse, A THz , normalized by the maximal value of the same quantity for a Gaussian pump pulse, A p max π τ 2 I p max ( c n ¯ p ) [please see Eq. (14)], as a function of the dimensionless frequency ω ω THz . It is also assumed that z = 2.39 cm , τ = 225 fs , I p max = 6 × 10 10 W cm 2 , and λ p = 6 μ m .

Equations (23)

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L c π c [ ω n pg n ( ω ) ] ,
n pg = n ( ω THz ) n THz ,
d I p d z = β 2 I p 2 β 3 I p 3 ,
σ ( ω ) = σ 0 e ( 1 + i ω τ e ) + σ 0 h ( 1 + i ω τ h ) ,
ε ( ω ) n 2 ( ω ) + 4 π i σ ( ω ) ω ,
e y = cos ( g x ) exp ( i h z ) ,
b x = n ¯ cos ( g x ) exp ( i h z ) ,
b z = i ( g c ω ) sin ( g x ) exp ( i h z ) ,
e y = cos ( g a ) exp [ p ( x a ) + i h z ] ,
b x = n ¯ cos ( g a ) exp [ p ( x a ) + i h z ] ,
b z = ( i g c ω ) sin ( g x a x ) exp [ p ( x a ) + i h z ] ,
g a = arctan ( p g ) ,
h 2 + g 2 = ( ω c ) 2 ε Ga As ,
h 2 p 2 = ( ω c ) 2 ε C ,
P ̃ i ( ω ) = 2 cos 2 ( g p x ) exp ( i ω n ¯ pg z c ) χ i 22 + G ( ω ) G * ( ω ω ) d ω ( 2 π ) ,
A THz ( ω , z ) 2 b a a 0 z j ̃ y e THz y * d x d z N THz = i ω d eff M z ( 2 π N THz ) 1 sinc { ω [ n ¯ pg n ¯ ( ω ) ] z 2 c } × exp { i ω [ n ¯ pg n ¯ ( ω ) ] z 2 c } + G ( ω ) G * ( ω ω ) d ω ,
M 2 b a a e py 2 ( z = 0 ) e THz y ( z = 0 ) d x = 2 b × { sin ( g THz a ) g THz + [ 2 g p sin ( 2 g p a ) cos ( g THz a ) g THz cos ( 2 g p a ) sin ( g THz a ) ] ( 4 g p 2 g THz 2 ) }
N THz ( c 2 π ) 2 b + [ e THz , b THz ] z = 0 z 0 d x = ( b c n ¯ THz π ) [ a + cot ( g THz a ) g THz ]
η = N THz + A THz ( ω ) 2 d ω N p + A p ( ω ) 2 d ω ,
G ( ω ) = π τ 2 I p max ( c n ¯ p ) exp ( τ 2 ω 2 4 ) ,
+ A p ( ω ) 2 d ω = 4 2 π 5 2 I p max τ ( c n ¯ p ) ,
+ G ( ω ) G * ( ω ω ) d ω = 2 2 π 5 2 I p max τ exp ( τ 2 ω 2 8 ) ( c n ¯ p ) .
η = 2 π 3 2 d eff 2 M 2 z I p max τ ω THz 2 exp ( τ 2 ω THz 2 4 ) n ¯ p ( n ¯ THz g n ¯ pg ) N p N THz ,

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