Abstract

Noncollinear optical parametric amplifiers (NOPAs) pumped with ultrashort subpicosecond pulses often suffer from pulse-front tilting, resulting in angular dispersion and noncompressibility of the amplified signal pulses. We show that pulse-front matching (PFM) with a prism-telescope setup corrects for pulse-front tilts in a near-IR NOPA. We discuss the conditions that lead to pulse-front tilt and angular dispersion in NOPA-amplified signal pulses, thus requiring pulse-front mismatch correction. We review the method of PFM and describe the application of PFM to an 800 nm pumped near-IR NOPA based on a 2-mm-thick bulk potassium-titanyl phosphate (KTP) crystal. The introduction of pulse-front matching into the KTP-NOPA reduces the signal pulse angular dispersion significantly over the >35  THz bandwidth, thus nearly removing the pulse-front tilt. Compression of 1200 nm pulses to 25  fs can be readily achieved with a fused-silica prism pair compressor.

© 2009 Optical Society of America

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2008

2007

D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108 (2007).
[CrossRef] [PubMed]

G. Cerullo, C. Manzoni, L. Luer, and D. Polli, “Time-resolved methods in biophysics. 4. Broadband pump-probe spectroscopy system with sub-20 fs temporal resolution for the study of energy transfer processes in photosynthesis,” Photochem. Photobiol. Sci. 6, 135-144 (2007).
[CrossRef] [PubMed]

S. Park, K. Kwak, and M. D. Fayer, “Ultrafast 2D-IR vibrational echo spectroscopy: a probe of molecular dynamics,” Laser Phys. Lett. 4, 704-718 (2007).
[CrossRef]

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[CrossRef] [PubMed]

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

D. Kraemer, M. L. Cowan, R. Z. Hua, K. Franjic, and R. D. Miller, “High-power femtosecond infrared laser source based on noncollinear optical parametric chirped pulse amplification,” J. Opt. Soc. Am. B 24, 813-818 (2007).
[CrossRef]

F. Tavella, Y. Nomura, L. Veisz, V. Pervak, A. Marcinkevicius, and F. Krausz, “Dispersion management for a sub-10-fs, 10 TW optical parametric chirped-pulse amplifier,” Opt. Lett. 32, 2227-2229 (2007).
[CrossRef] [PubMed]

G. Cirmi, D. Brida, C. Manzoni, M. Marangoni, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses in the near-infrared from a noncollinear optical parametric amplifier,” Opt. Lett. 32, 2396-2398 (2007).
[CrossRef] [PubMed]

S. Adachi, H. Ishii, T. Kanai, N. Ishii, A. Kosuge, and S. Watanabe, “1.5 mJ, 6.4 fs parametric chirped-pulse amplification system at 1 kHz,” Opt. Lett. 32, 2487-2489 (2007).
[CrossRef] [PubMed]

2006

2005

2004

2003

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1-18 (2003).
[CrossRef]

2002

T. Kobayashi and A. Baltuska, “Sub-5 fs pulse generation from a noncollinear optical parametric amplifier,” Meas. Sci. Technol. 13, 1671-1682 (2002).
[CrossRef]

2001

T. D. Chinh, W. Seibt, and K. Siegbahn, “Dot patterns from second-harmonic and sum-frequency generation in polycrystalline ZnSe,” J. Appl. Phys. 90, 2612-2614 (2001).
[CrossRef]

2000

T. Kobayashi and A. Shirakawa, “Tunable visible and near-infrared pulse generator in a 5 fs regime,” Appl. Phys. B 70, S239-S246 (2000).
[CrossRef]

D. Bodlaki and E. Borguet, “Picosecond infrared optical parametric amplifier for nonlinear interface spectroscopy,” Rev. Sci. Instrum. 71, 4050-4056 (2000).
[CrossRef]

G. Pretzler, A. Kasper, and K. J. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1-9 (2000).
[CrossRef]

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

1998

1997

1996

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum Electron. 28, 1759-1763 (1996).
[CrossRef]

1995

1985

Z. Bor and B. Racz, “Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation,” Opt. Commun. 54, 165-170 (1985).
[CrossRef]

1984

O. E. Martinez, J. P. Gordon, and R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 1, 1003-1006 (1984).
[CrossRef]

R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150-152 (1984).
[CrossRef] [PubMed]

Adachi, S.

Agostini, P.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[CrossRef] [PubMed]

Akturk, S.

Andreoni, A.

Apolonski, A.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Arisholm, G.

Auguste, T.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[CrossRef] [PubMed]

Bakker, H. J.

Baltuska, A.

Beitel, D.

D. Beitel, L. Carrion, L. R. Chen, and R. Maciejko, “Development of broadband sources based on semiconductor optical amplifiers and erbium-doped fiber amplifiers for optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 243-250 (2008).
[CrossRef]

Benedetti, E.

Beutter, M.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Biegert, J.

Bodlaki, D.

D. Bodlaki and E. Borguet, “Picosecond infrared optical parametric amplifier for nonlinear interface spectroscopy,” Rev. Sci. Instrum. 71, 4050-4056 (2000).
[CrossRef]

Bor, Z.

Z. Bor and B. Racz, “Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation,” Opt. Commun. 54, 165-170 (1985).
[CrossRef]

Borguet, E.

O. Isaienko and E. Borguet, “Generation of ultra-broadband pulses in the near-IR by non-collinear optical parametric amplification in potassium titanyl phosphate,” Opt. Express 16, 3949-3954 (2008).
[CrossRef] [PubMed]

D. Bodlaki and E. Borguet, “Picosecond infrared optical parametric amplifier for nonlinear interface spectroscopy,” Rev. Sci. Instrum. 71, 4050-4056 (2000).
[CrossRef]

Brida, D.

Butkus, R.

Calegari, F.

Carrion, L.

D. Beitel, L. Carrion, L. R. Chen, and R. Maciejko, “Development of broadband sources based on semiconductor optical amplifiers and erbium-doped fiber amplifiers for optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 243-250 (2008).
[CrossRef]

Cerullo, G.

C. Vozzi, C. Manzoni, F. Calegari, E. Benedetti, G. Sansone, G. Cerullo, M. Nisoli, S. De Silvestri, and S. Stagira, “Characterization of a high-energy self-phase-stabilized near-infrared parametric source,” J. Opt. Soc. Am. B 25, B112-B117 (2008).
[CrossRef]

D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108 (2007).
[CrossRef] [PubMed]

G. Cirmi, D. Brida, C. Manzoni, M. Marangoni, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses in the near-infrared from a noncollinear optical parametric amplifier,” Opt. Lett. 32, 2396-2398 (2007).
[CrossRef] [PubMed]

G. Cerullo, C. Manzoni, L. Luer, and D. Polli, “Time-resolved methods in biophysics. 4. Broadband pump-probe spectroscopy system with sub-20 fs temporal resolution for the study of energy transfer processes in photosynthesis,” Photochem. Photobiol. Sci. 6, 135-144 (2007).
[CrossRef] [PubMed]

P. Tzankov, J. Zheng, M. Mero, D. Polli, C. Manzoni, and G. Cerullo, “300 μJ noncollinear optical parametric amplifier in the visible at 1 kHz repetition rate,” Opt. Lett. 31, 3629-3631 (2006).
[CrossRef] [PubMed]

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1-18 (2003).
[CrossRef]

G. Cerullo, M. Nisoli, S. Stagira, and S. De Silvestri, “Sub-8-fs pulses from an ultrabroadband optical parametric amplifier in the visible,” Opt. Lett. 23, 1283-1285 (1998).
[CrossRef]

Cha, M.

Chen, L. R.

D. Beitel, L. Carrion, L. R. Chen, and R. Maciejko, “Development of broadband sources based on semiconductor optical amplifiers and erbium-doped fiber amplifiers for optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 243-250 (2008).
[CrossRef]

Chinh, T. D.

T. D. Chinh, W. Seibt, and K. Siegbahn, “Dot patterns from second-harmonic and sum-frequency generation in polycrystalline ZnSe,” J. Appl. Phys. 90, 2612-2614 (2001).
[CrossRef]

Cirmi, G.

Clausnitzer, T.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Cowan, M. L.

Csatari, M.

K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10, 213-220 (2004).
[CrossRef]

Danielius, R.

De Silvestri, S.

Di Trapani, P.

DiMauro, L. F.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[CrossRef] [PubMed]

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, Berlin, 1999).

Dubietis, A.

Eikema, K. S. E.

Fayer, M. D.

S. Park, K. Kwak, and M. D. Fayer, “Ultrafast 2D-IR vibrational echo spectroscopy: a probe of molecular dynamics,” Laser Phys. Lett. 4, 704-718 (2007).
[CrossRef]

Foggi, P.

Fork, R. L.

R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150-152 (1984).
[CrossRef] [PubMed]

O. E. Martinez, J. P. Gordon, and R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 1, 1003-1006 (1984).
[CrossRef]

Franjic, K.

Fuji, T.

Fujimoto, Y.

Y. S. Seo, Y. Fujimoto, and M. Nakatsuka, “Optical amplification in a bismuth-doped silica glass at 1300 nm telecommunications window,” Opt. Commun. 266, 169-171 (2006).
[CrossRef]

Fulop, J. A.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Gordon, J. P.

R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150-152 (1984).
[CrossRef] [PubMed]

O. E. Martinez, J. P. Gordon, and R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 1, 1003-1006 (1984).
[CrossRef]

Gu, X.

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, Berlin, 1999).

Harris, T. J.

W. J. Tropf, M. E. Thomas, and T. J. Harris, in Handbook of Optics, edited by M.Bass (McGraw-Hill, New York, 1995), pp. 33.3-33.83.

Hauri, C. P.

Hebling, J.

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum Electron. 28, 1759-1763 (1996).
[CrossRef]

Heiner, Z.

K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10, 213-220 (2004).
[CrossRef]

Henig, A.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Hogervorst, W.

Horlein, R.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Hua, R.

Hua, R. Z.

Isaienko, O.

Ishii, H.

Ishii, N.

Itatani, J.

Jeon, O. Y.

Jin, M. J.

Kanai, T.

Karsch, S.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Kasper, A.

G. Pretzler, A. Kasper, and K. J. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1-9 (2000).
[CrossRef]

Keller, U.

Kim, B. J.

Klebniczki, J.

K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10, 213-220 (2004).
[CrossRef]

Kley, E. B.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Kobayashi, T.

T. Kobayashi and Z. Wang, “Correlations of instantaneous transition energy and intensity of absorption peaks during molecular vibration: toward potential hyper-surface,” New J. Phys. 10, 065015 (2008).
[CrossRef]

T. Kobayashi and A. Baltuska, “Sub-5 fs pulse generation from a noncollinear optical parametric amplifier,” Meas. Sci. Technol. 13, 1671-1682 (2002).
[CrossRef]

T. Kobayashi and A. Shirakawa, “Tunable visible and near-infrared pulse generator in a 5 fs regime,” Appl. Phys. B 70, S239-S246 (2000).
[CrossRef]

A. Shirakawa, I. Sakane, and T. Kobayashi, “Pulse-front-matched optical parametric amplification for sub-10-fs pulse generation tunable in the visible and near infrared,” Opt. Lett. 23, 1292-1294 (1998).
[CrossRef]

A. Shirakawa and T. Kobayashi, “Noncollinearly phase-matched femtosecond optical parametric amplification with a 2000 cm−1bandwidth,” Appl. Phys. Lett. 72, 147-149 (1998).
[CrossRef]

T. Kobayashi, in Femtosecond Optical Frequency Comb: Principle, Operation and Applications, edited by J.Ye and S.T.Cundiff (Springer, Berlin, 2005), pp. 133-175.
[CrossRef]

Kobayashi, Y.

Kosuge, A.

Kovacs, A. P.

K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10, 213-220 (2004).
[CrossRef]

Kraemer, D.

Krausz, F.

Kruber, S.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Kurdi, G.

K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10, 213-220 (2004).
[CrossRef]

Kwak, K.

S. Park, K. Kwak, and M. D. Fayer, “Ultrafast 2D-IR vibrational echo spectroscopy: a probe of molecular dynamics,” Laser Phys. Lett. 4, 704-718 (2007).
[CrossRef]

Lim, H. H.

Lochbrunner, S.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Luer, L.

G. Cerullo, C. Manzoni, L. Luer, and D. Polli, “Time-resolved methods in biophysics. 4. Broadband pump-probe spectroscopy system with sub-20 fs temporal resolution for the study of energy transfer processes in photosynthesis,” Photochem. Photobiol. Sci. 6, 135-144 (2007).
[CrossRef] [PubMed]

D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108 (2007).
[CrossRef] [PubMed]

Maciejko, R.

D. Beitel, L. Carrion, L. R. Chen, and R. Maciejko, “Development of broadband sources based on semiconductor optical amplifiers and erbium-doped fiber amplifiers for optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 243-250 (2008).
[CrossRef]

Major, Z.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Manzoni, C.

Marangoni, M.

Marcinkevicius, A.

Martinez, O. E.

R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150-152 (1984).
[CrossRef] [PubMed]

O. E. Martinez, J. P. Gordon, and R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 1, 1003-1006 (1984).
[CrossRef]

Mero, M.

Miller, R. D.

Miller, R. J. D.

Muller, H. G.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[CrossRef] [PubMed]

Nakatsuka, M.

Y. S. Seo, Y. Fujimoto, and M. Nakatsuka, “Optical amplification in a bismuth-doped silica glass at 1300 nm telecommunications window,” Opt. Commun. 266, 169-171 (2006).
[CrossRef]

Nienhuys, H. K.

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, Berlin, 1999).

Nisoli, M.

Nomura, Y.

Osterhoff, J.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Osvay, K.

K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10, 213-220 (2004).
[CrossRef]

Park, S.

S. Park, K. Kwak, and M. D. Fayer, “Ultrafast 2D-IR vibrational echo spectroscopy: a probe of molecular dynamics,” Laser Phys. Lett. 4, 704-718 (2007).
[CrossRef]

Pervak, V.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

F. Tavella, Y. Nomura, L. Veisz, V. Pervak, A. Marcinkevicius, and F. Krausz, “Dispersion management for a sub-10-fs, 10 TW optical parametric chirped-pulse amplifier,” Opt. Lett. 32, 2227-2229 (2007).
[CrossRef] [PubMed]

Piel, J.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

T. Wilhelm, J. Piel, and E. Riedle, “Sub-20-fs pulses tunable across the visible from a blue-pumped single-pass noncollinear parametric converter,” Opt. Lett. 22, 1494-1496 (1997).
[CrossRef]

Piskarskas, A.

Polli, D.

G. Cerullo, C. Manzoni, L. Luer, and D. Polli, “Time-resolved methods in biophysics. 4. Broadband pump-probe spectroscopy system with sub-20 fs temporal resolution for the study of energy transfer processes in photosynthesis,” Photochem. Photobiol. Sci. 6, 135-144 (2007).
[CrossRef] [PubMed]

D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108 (2007).
[CrossRef] [PubMed]

P. Tzankov, J. Zheng, M. Mero, D. Polli, C. Manzoni, and G. Cerullo, “300 μJ noncollinear optical parametric amplifier in the visible at 1 kHz repetition rate,” Opt. Lett. 31, 3629-3631 (2006).
[CrossRef] [PubMed]

Pretzler, G.

G. Pretzler, A. Kasper, and K. J. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1-9 (2000).
[CrossRef]

Racz, B.

Z. Bor and B. Racz, “Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation,” Opt. Commun. 54, 165-170 (1985).
[CrossRef]

Riedle, E.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

T. Wilhelm, J. Piel, and E. Riedle, “Sub-20-fs pulses tunable across the visible from a blue-pumped single-pass noncollinear parametric converter,” Opt. Lett. 22, 1494-1496 (1997).
[CrossRef]

Sakane, I.

Salieres, P.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[CrossRef] [PubMed]

Sansone, G.

Schenkl, S.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Schlup, P.

Seibt, W.

T. D. Chinh, W. Seibt, and K. Siegbahn, “Dot patterns from second-harmonic and sum-frequency generation in polycrystalline ZnSe,” J. Appl. Phys. 90, 2612-2614 (2001).
[CrossRef]

Seo, Y. S.

Y. S. Seo, Y. Fujimoto, and M. Nakatsuka, “Optical amplification in a bismuth-doped silica glass at 1300 nm telecommunications window,” Opt. Commun. 266, 169-171 (2006).
[CrossRef]

Shirakawa, A.

T. Kobayashi and A. Shirakawa, “Tunable visible and near-infrared pulse generator in a 5 fs regime,” Appl. Phys. B 70, S239-S246 (2000).
[CrossRef]

A. Shirakawa and T. Kobayashi, “Noncollinearly phase-matched femtosecond optical parametric amplification with a 2000 cm−1bandwidth,” Appl. Phys. Lett. 72, 147-149 (1998).
[CrossRef]

A. Shirakawa, I. Sakane, and T. Kobayashi, “Pulse-front-matched optical parametric amplification for sub-10-fs pulse generation tunable in the visible and near infrared,” Opt. Lett. 23, 1292-1294 (1998).
[CrossRef]

Siegbahn, K.

T. D. Chinh, W. Seibt, and K. Siegbahn, “Dot patterns from second-harmonic and sum-frequency generation in polycrystalline ZnSe,” J. Appl. Phys. 90, 2612-2614 (2001).
[CrossRef]

Smilgevicius, V.

Smith, A.

A. Smith, “SNLO software package,” http://www.as-photonics.com/?q=SNLO.

Solcia, C.

Sporlein, S.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Stagira, S.

Tate, J.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[CrossRef] [PubMed]

Tavella, F.

Thomas, M. E.

W. J. Tropf, M. E. Thomas, and T. J. Harris, in Handbook of Optics, edited by M.Bass (McGraw-Hill, New York, 1995), pp. 33.3-33.83.

Torizuka, K.

Trebino, R.

Tropf, W. J.

W. J. Tropf, M. E. Thomas, and T. J. Harris, in Handbook of Optics, edited by M.Bass (McGraw-Hill, New York, 1995), pp. 33.3-33.83.

Tunnermann, A.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Turi, L.

Tzankov, P.

Ubachs, W.

Veisz, L.

Veitas, G.

Vozzi, C.

Wang, Z.

T. Kobayashi and Z. Wang, “Correlations of instantaneous transition energy and intensity of absorption peaks during molecular vibration: toward potential hyper-surface,” New J. Phys. 10, 065015 (2008).
[CrossRef]

Watanabe, S.

Weingartner, R.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

Wilhelm, T.

Witte, K. J.

G. Pretzler, A. Kasper, and K. J. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1-9 (2000).
[CrossRef]

Witte, S.

Wolf, A. L.

Yakovlev, V. S.

Yoshitomi, D.

Zeek, E.

Zheng, J.

Zinkstok, R. T.

Zinth, W.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Appl. Opt.

Appl. Phys. B

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, “Generation of 10 to 50 fs pulses tunable through all of the visible and the NIR,” Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

T. Kobayashi and A. Shirakawa, “Tunable visible and near-infrared pulse generator in a 5 fs regime,” Appl. Phys. B 70, S239-S246 (2000).
[CrossRef]

G. Pretzler, A. Kasper, and K. J. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1-9 (2000).
[CrossRef]

Appl. Phys. Lett.

A. Shirakawa and T. Kobayashi, “Noncollinearly phase-matched femtosecond optical parametric amplification with a 2000 cm−1bandwidth,” Appl. Phys. Lett. 72, 147-149 (1998).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10, 213-220 (2004).
[CrossRef]

D. Beitel, L. Carrion, L. R. Chen, and R. Maciejko, “Development of broadband sources based on semiconductor optical amplifiers and erbium-doped fiber amplifiers for optical coherence tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 243-250 (2008).
[CrossRef]

J. Appl. Phys.

T. D. Chinh, W. Seibt, and K. Siegbahn, “Dot patterns from second-harmonic and sum-frequency generation in polycrystalline ZnSe,” J. Appl. Phys. 90, 2612-2614 (2001).
[CrossRef]

J. Opt. Soc. Am. A Opt. Image Sci. Vis

O. E. Martinez, J. P. Gordon, and R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 1, 1003-1006 (1984).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys. Lett.

S. Park, K. Kwak, and M. D. Fayer, “Ultrafast 2D-IR vibrational echo spectroscopy: a probe of molecular dynamics,” Laser Phys. Lett. 4, 704-718 (2007).
[CrossRef]

Meas. Sci. Technol.

T. Kobayashi and A. Baltuska, “Sub-5 fs pulse generation from a noncollinear optical parametric amplifier,” Meas. Sci. Technol. 13, 1671-1682 (2002).
[CrossRef]

New J. Phys.

J. A. Fulop, Z. Major, A. Henig, S. Kruber, R. Weingartner, T. Clausnitzer, E. B. Kley, A. Tunnermann, V. Pervak, A. Apolonski, J. Osterhoff, R. Horlein, F. Krausz, and S. Karsch, “Short-pulse optical parametric chirped-pulse amplification for the generation of high-power few-cycle pulses,” New J. Phys. 9, 438 (2007).
[CrossRef]

T. Kobayashi and Z. Wang, “Correlations of instantaneous transition energy and intensity of absorption peaks during molecular vibration: toward potential hyper-surface,” New J. Phys. 10, 065015 (2008).
[CrossRef]

Opt. Commun.

Y. S. Seo, Y. Fujimoto, and M. Nakatsuka, “Optical amplification in a bismuth-doped silica glass at 1300 nm telecommunications window,” Opt. Commun. 266, 169-171 (2006).
[CrossRef]

Z. Bor and B. Racz, “Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation,” Opt. Commun. 54, 165-170 (1985).
[CrossRef]

Opt. Express

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399-4410 (2004).
[CrossRef] [PubMed]

S. Witte, R. T. Zinkstok, W. Hogervorst, and K. S. E. Eikema, “Generation of few-cycle terawatt light pulses using optical parametric chirped pulse amplification,” Opt. Express 13, 4903-4908 (2005).
[CrossRef] [PubMed]

O. Y. Jeon, M. J. Jin, H. H. Lim, B. J. Kim, and M. Cha, “Broadband optical parametric amplification at the communication band with periodically poled lithium niobate,” Opt. Express 14, 7210-7215 (2006).
[CrossRef] [PubMed]

S. Witte, R. T. Zinkstok, A. L. Wolf, W. Hogervorst, W. Ubachs, and K. S. E. Eikema, “A source of 2 terawatt, 2.7 cycle laser pulses based on noncollinear optical parametric chirped pulse amplification,” Opt. Express 14, 8168-8177 (2006).
[CrossRef] [PubMed]

F. Tavella, A. Marcinkevicius, and F. Krausz, “90 mJ parametric chirped pulse amplification of 10 fs pulses,” Opt. Express 14, 12822-12827 (2006).
[CrossRef] [PubMed]

O. Isaienko and E. Borguet, “Generation of ultra-broadband pulses in the near-IR by non-collinear optical parametric amplification in potassium titanyl phosphate,” Opt. Express 16, 3949-3954 (2008).
[CrossRef] [PubMed]

S. Adachi, N. Ishii, T. Kanai, A. Kosuge, J. Itatani, Y. Kobayashi, D. Yoshitomi, K. Torizuka, and S. Watanabe, “5-fs, multi-mJ, CEP-locked parametric chirped-pulse amplifier pumped by a 450-nm source at 1 kHz,” Opt. Express 16, 14341-14352 (2008).
[CrossRef] [PubMed]

Opt. Lett.

P. Tzankov, J. Zheng, M. Mero, D. Polli, C. Manzoni, and G. Cerullo, “300 μJ noncollinear optical parametric amplifier in the visible at 1 kHz repetition rate,” Opt. Lett. 31, 3629-3631 (2006).
[CrossRef] [PubMed]

F. Tavella, Y. Nomura, L. Veisz, V. Pervak, A. Marcinkevicius, and F. Krausz, “Dispersion management for a sub-10-fs, 10 TW optical parametric chirped-pulse amplifier,” Opt. Lett. 32, 2227-2229 (2007).
[CrossRef] [PubMed]

G. Cirmi, D. Brida, C. Manzoni, M. Marangoni, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses in the near-infrared from a noncollinear optical parametric amplifier,” Opt. Lett. 32, 2396-2398 (2007).
[CrossRef] [PubMed]

S. Adachi, H. Ishii, T. Kanai, N. Ishii, A. Kosuge, and S. Watanabe, “1.5 mJ, 6.4 fs parametric chirped-pulse amplification system at 1 kHz,” Opt. Lett. 32, 2487-2489 (2007).
[CrossRef] [PubMed]

D. Kraemer, R. Hua, M. L. Cowan, K. Franjic, and R. J. D. Miller, “Ultrafast noncollinear optical parametric chirped pulse amplification in KTiOAsO4,” Opt. Lett. 31, 981-983 (2006).
[CrossRef] [PubMed]

N. Ishii, L. Turi, V. S. Yakovlev, T. Fuji, F. Krausz, A. Baltuska, R. Butkus, G. Veitas, V. Smilgevicius, R. Danielius, and A. Piskarskas, “Multimillijoule chirped parametric amplification of few-cycle pulses,” Opt. Lett. 30, 567-569 (2005).
[CrossRef] [PubMed]

T. Wilhelm, J. Piel, and E. Riedle, “Sub-20-fs pulses tunable across the visible from a blue-pumped single-pass noncollinear parametric converter,” Opt. Lett. 22, 1494-1496 (1997).
[CrossRef]

G. Cerullo, M. Nisoli, S. Stagira, and S. De Silvestri, “Sub-8-fs pulses from an ultrabroadband optical parametric amplifier in the visible,” Opt. Lett. 23, 1283-1285 (1998).
[CrossRef]

A. Shirakawa, I. Sakane, and T. Kobayashi, “Pulse-front-matched optical parametric amplification for sub-10-fs pulse generation tunable in the visible and near infrared,” Opt. Lett. 23, 1292-1294 (1998).
[CrossRef]

C. P. Hauri, P. Schlup, G. Arisholm, J. Biegert, and U. Keller, “Phase-preserving chirped-pulse optical parametric amplification to 17.3 fs directly from a Ti: sapphire oscillator,” Opt. Lett. 29, 1369-1371 (2004).
[CrossRef] [PubMed]

R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150-152 (1984).
[CrossRef] [PubMed]

Opt. Quantum Electron.

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum Electron. 28, 1759-1763 (1996).
[CrossRef]

Photochem. Photobiol. Sci.

G. Cerullo, C. Manzoni, L. Luer, and D. Polli, “Time-resolved methods in biophysics. 4. Broadband pump-probe spectroscopy system with sub-20 fs temporal resolution for the study of energy transfer processes in photosynthesis,” Photochem. Photobiol. Sci. 6, 135-144 (2007).
[CrossRef] [PubMed]

Phys. Rev. Lett.

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108 (2007).
[CrossRef] [PubMed]

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1-18 (2003).
[CrossRef]

D. Bodlaki and E. Borguet, “Picosecond infrared optical parametric amplifier for nonlinear interface spectroscopy,” Rev. Sci. Instrum. 71, 4050-4056 (2000).
[CrossRef]

Other

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, Berlin, 1999).

“Newport Corporation, Application Note 29: Prism Compressor for Ultrashort Laser Pulses,”http://www.newport.com/file_store/Optics_and_Mechanics/AppsNote29.pdf.

A. Smith, “SNLO software package,” http://www.as-photonics.com/?q=SNLO.

W. J. Tropf, M. E. Thomas, and T. J. Harris, in Handbook of Optics, edited by M.Bass (McGraw-Hill, New York, 1995), pp. 33.3-33.83.

T. Kobayashi, in Femtosecond Optical Frequency Comb: Principle, Operation and Applications, edited by J.Ye and S.T.Cundiff (Springer, Berlin, 2005), pp. 133-175.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Noncollinear geometry between femtosecond signal and pump beams induces a pulse-front tilt into signal pulses [24]. The pulse front of amplified signal pulses (gray bars) follows the pulse front of the pump pulses (black bars) resulting in the signal front tilted by angle α inside the nonlinear crystal (or η ext after exiting the nonlinear crystal). The signal front tilt causes a nonzero angular dispersion d ε / d λ , where ε is the exit angle of a wavelength component λ.

Fig. 2
Fig. 2

Schematic representation of the electric field profile of an undistorted pulsed beam (a) and a beam with a pulse front tilted by angle γ (b). The x ( z ) axis is normal (along) the beam propagation direction. Phase fronts, as well as the pulse front in each case, are shown. k = k ( λ ) is the wave vector corresponding to a frequency component ω = 2 π c / λ , and ε ( λ ) is the exit angle of the same frequency component (adapted from [26]).

Fig. 3
Fig. 3

Dependence of the signal gain coefficient on wavelength for the following parameters: crystal thickness L = 2   mm , λ pump = 800   nm , pump pulse intensity I pump = 110   GW / cm 2 , θ pump = 48 ° , and d eff = 2.955   pm / V (type-II phase matching in x z plane [40, 43]), α int = 4 ° . Inset: geometry of pump, signal, and idler beam wave vectors interacting inside the nonlinear optical crystal; the phase-matching angle, θ, between the pump wave vector and the z axis is defined, as well as the internal signal-pump noncollinear angle, α.

Fig. 4
Fig. 4

Prism-telescope setup for pump-signal pulse-front matching [21]. γ prism and γ ext indicate the tilt angle of the 800 nm pulses after passing through the prism and the f 1 / f 2 telescope, respectively. γ int is the pump pulse-front tilt angle inside the nonlinear crystal, which is matched with signal-pump internal noncollinear angle α.

Fig. 5
Fig. 5

Pulse-front tilt angle of the 800 nm pump pulses inside the KTP crystal induced by a CaF 2 prism with θ apex = 60 ° (a), a CaF 2 prism with θ apex = 45 ° (b), and a Brewster-angle fused-silica prism with θ apex = 69.0 ° and a 200–50 mm telescope, as a function of the incidence angle ϕ 1 . (d) Pulse-front tilt angle of the 395 nm pump pulses as a function of the incidence angle ϕ 1 inside a BBO crystal calculated for the PFM-NOPA setup in [21].

Fig. 6
Fig. 6

Experimental setup of the pulse-front matched KTP-NOPA: BS, beam splitter; L, BK7 lenses; SM, spherical mirrors; CaF 2 , equilateral prism (2.5 cm face size); sapphire, 2-mm-thick plate; KTP, 2-mm-thick crystal, cut at θ = 42 ° for phase matching in x z plane (axes are shown); FS, Brewster-angle fused-silica prisms; and AC, autocorrelator. Double arrows and dotted circle indicate the polarization of 800 nm beam in and perpendicular to the plane of the drawing, respectively. γ prism and γ ext indicate the tilt angle of the 800 nm pulses after passing through the CaF 2 prism and the L2-L3 telescope, respectively.

Fig. 7
Fig. 7

Left: optical setup of KTP-NOPA without pulse-front matching: SF18, equilateral prisms; L4, BK7 lens. Right: angular dispersion of the pump pulses after passing through the two SF18 prisms (the solid line is a guide for the eyes).

Fig. 8
Fig. 8

Second-harmonic spectra of the signal pulses from the KTP-NOPA without (a) and with (b) pulse-front matching of the pump and seed pulses measured at different exit angles.

Fig. 9
Fig. 9

Center-of-mass wavelengths of the signal pulses vs the exit angle for the non-PFM (open circles) and PFM (triangles) signal outputs; dashed line corresponds to the calculated angular dispersion of the signal pulses arising from noncollinear geometry between the seed and nontilted pump pulses for internal noncollinear angle α int = η int = 4.0 ° ; Eqs. (2, 3) were used. The solid line is a guide for the eyes.

Fig. 10
Fig. 10

(a) A typical SH spectrum of the amplified signal pulses from the PFM-NOPA (solid curve). For comparison, the gain spectrum from Fig. 3 is shown (dashed curve). (b) Autocorrelation of compressed signal pulses on a 300 - μ m -thick BBO crystal (triangles) and sech 2 fit (dotted curve).

Equations (6)

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tan   γ = n n λ d n d λ λ d ε d λ = ν group ν phase λ d ε d λ ,
d ε ( λ ) d λ = tan   η ext λ s ,
tan   η int = ν s c tan   η ext ,
tan   γ prism = sin   θ apex cos   ϕ 1   cos   ϕ 2 λ p d n d λ p ,
tan   γ ext = f 1 f 2 tan   γ prism .
tan   γ int = v p c f 1 f 2 tan   γ prism .

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