Abstract

We computationally investigate polarized second-harmonic generation (SHG) of a spectrally broad femtosecond pulse following transmission through traditional quarter wave plates (QWPs). Because the sideband modes of a broadband pulse can interact through sum-frequency generation processes, the SHG responses for several experimentally relevant cases exhibit asymmetries between individual sideband modes, spectral peak shifts, and, critically, artificial chiral signatures. Remarkably, errors in the various sum-frequency sidebands are found to compensate for each other so that the total SHG response approaches the ideal narrowband response. This occurs in the absence of significant axis misalignment in a compound QWP. Hence, our results suggest that polarized femtosecond SHG can be remarkably tolerant against the broad bandwidth of ultrashort pulses.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  34. L. D. Acquisto, G. Petrucci, and B. Zuccarello, 'Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,' Opt. Lasers Eng. 37, 389-400 (2002).
    [CrossRef]
  35. S. Cattaneo and M. Kauranen, 'Application of second-harmonic generation to retardation measurements,' J. Opt. Soc. Am. B 20, 520-528 (2003).
    [CrossRef]
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    [CrossRef]
  37. M. Kauranen, T. Verbiest, S. V. Elshocht, and A. Persoons, 'Chirality in surface nonlinear optics,' Opt. Mater. 9, 286-294 (1998).
    [CrossRef]
  38. B. K. Canfield, S. Kujala, K. Laiho, K. Jefimovs, J. Turunen, and M. Kauranen, 'Chirality arising from small defects in gold nanoparticle arrays,' Opt. Express 14, 950-955 (2006).
    [CrossRef] [PubMed]
  39. M. Born and E. Wolf, Principles of Optics, Seventh ed. (Cambridge U. Press, 1999).
  40. D. Goldstein, Polarized Light, 2nd ed. (Dekker, 2003).
    [CrossRef]
  41. B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, 278-284 (2006).
    [CrossRef]
  42. W. H. Glenn, 'Second-harmonic generation by picosecond optical pulses,' IEEE J. Quantum Electron. 5, 284-290 (1969).
    [CrossRef]
  43. Y. Nabekawa and K. Midorikawa, 'Broadband sum frequency mixing using noncollinear angularly dispersed geometry for indirect phase control of sub-20-femtosecond UV pulses,' Opt. Express 11, 324-338 (2003).
    [CrossRef] [PubMed]
  44. The dispersion of crystal quartz was obtained from http://www.cvilaser.com/Common/PDFs/Dispersion_Equations.pdf. Note: Because this URL is too long for the printed column, the reader will need to type it into his or her browser directly.
  45. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
    [CrossRef]
  46. N. H. Burnett, C. Kan, and P. B. Corkum, 'Ellipticity and polarization effects in harmonic generation in ionizing neon,' Phys. Rev. A 51, 3418-3121 (1995).
    [CrossRef]
  47. K. D. Singer, M. G. Kuzyk, and J. E. Sohn, 'Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,' J. Opt. Soc. Am. B 4, 968-976 (1987).
    [CrossRef]

2006

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, 278-284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, K. Laiho, K. Jefimovs, J. Turunen, and M. Kauranen, 'Chirality arising from small defects in gold nanoparticle arrays,' Opt. Express 14, 950-955 (2006).
[CrossRef] [PubMed]

2005

W. Huang, W. Qian, and M. A. El-Sayed, 'Photothermal reshaping of prismatic Au nanoparticles in periodic monolayer arrays by femtosecond laser pulses,' J. Appl. Phys. 98, 114301 (2005).
[CrossRef]

B. K. Canfield, S. Kujala, M. Kauranen, K. Jefimovs, T. Vallius, and J. Turunen, 'Remarkable polarization sensitivity of gold nanoparticle arrays,' Appl. Phys. Lett. 86, 183109 (2005).
[CrossRef]

X. Xiao, C. Yang, S. Gao, and H. Miao, 'Analysis of ultrashort-pulse second-harmonic generation in both phase- and group-velocity-matched structures,' IEEE J. Quantum Electron. 41, 85-93 (2005).
[CrossRef]

D. Hovhannisyan, K. Stepanyan, and R. Avagyan, 'Sum and difference frequency generation by femtosecond laser pulses in a nonlinear crystal,' Opt. Commun. 245, 443-456 (2005).
[CrossRef]

2004

M. H. Smith, 'Polarization metrology moves beyond 'home-brewed' solutions,' Laser Focus World 40, 123-129 (2004).

T. Atay, J.-H. Song, and A. V. Nurmikko, 'Strongly interacting plasmon nanoparticle pairs: from dipole-dipole interaction to conductively coupled regime,' Nano Lett. 4, 1627-1631 (2004).
[CrossRef]

H. Zhu, T. Wang, W. Zheng, P. Yuan, L. Qian, and D. Fan, 'Efficient second harmonic generation of femtosecond laser at 1μm,' Opt. Express 12, 2150-2155 (2004).
[CrossRef] [PubMed]

2003

2002

A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, 'Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,' Phys. Rev. Lett. 89, 087401 (2002).
[CrossRef] [PubMed]

B. Boulbry, B. L. Jeune, F. Pellen, J. Cariou, and J. Lotrian, 'Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source,' J. Phys. D 35, 2508-2515 (2002).
[CrossRef]

L. D. Acquisto, G. Petrucci, and B. Zuccarello, 'Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,' Opt. Lasers Eng. 37, 389-400 (2002).
[CrossRef]

V. Pasiskevicius, S. J. Holmgren, S. Wang, and F. Laurell, 'Simultaneous second-harmonic generation with two orthogonal polarization states in periodically poled KTP,' Opt. Lett. 27, 1628-1630 (2002).
[CrossRef]

1999

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, 'Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,' Phys. Rev. Lett. 83, 2934-2937 (1999).
[CrossRef]

B. Lamprecht, A. Leitner, and F. R. Aussenegg, 'SHG studies of plasmon dephasing in nanoparticles,' Appl. Phys. B 68, 419-423 (1999).
[CrossRef]

1998

M. Kauranen, T. Verbiest, S. V. Elshocht, and A. Persoons, 'Chirality in surface nonlinear optics,' Opt. Mater. 9, 286-294 (1998).
[CrossRef]

1996

1995

E. Sidick, A. Knoesen, and A. Dienes, 'Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses,' J. Opt. Soc. Am. B 12, 1704-1712 (1995).
[CrossRef]

E. Sidick, A. Dienes, and A. Knoesen, 'Ultrashort-pulse second-harmonic generation. II. Non-transform-limited fundamental pulses,' J. Opt. Soc. Am. B 12, 1713-1722 (1995).
[CrossRef]

J. J. Maki, M. Kauranen, and A. Persoons, 'Surface second-harmonic generation from chiral materials,' Phys. Rev. B 51, 1425-1434 (1995).
[CrossRef]

N. H. Burnett, C. Kan, and P. B. Corkum, 'Ellipticity and polarization effects in harmonic generation in ionizing neon,' Phys. Rev. A 51, 3418-3121 (1995).
[CrossRef]

E. A. West and M. H. Smith, 'Polarization errors associated with birefringent waveplates,' Opt. Eng. 34, 1574-1580 (1995).
[CrossRef]

1994

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

1993

See, for instance, the special issue on femtochemistry, J. Phys. Chem. 97, 12460-12465 (1993). Includes the major paper by A. Zewail, who won the 1999 Nobel in chemistry for time-resolved studies of molecular interactions.
[CrossRef]

J. J. Macklin, J. D. Kmetec, and C. L. I. Gordon, 'High-order harmonic generation using intense femtosecond pulses,' Phys. Rev. Lett. 70, 766-769 (1993).
[CrossRef] [PubMed]

C. Radzewicz, J. S. Krasinski, and Y. B Band, 'Increased efficiency for sumfrequency generation for broadband input fields,' Opt. Lett. 18, 331-333 (1993).
[CrossRef] [PubMed]

1992

1990

Y. B. Band, D. F. Heller, J. R. Ackerhalt, and J. S. Krasinski, 'Spectrum of second-harmonic generation for multimode fields,' Phys. Rev. A 42, 1515--1521 (1990).
[CrossRef] [PubMed]

T. R. Zhang, H. R. Choo, and M. C. Downer, 'Phase and group velocity matching for second harmonic generation of femtosecond pulses,' Appl. Opt. 29, 3927-3933 (1990).
[CrossRef] [PubMed]

1988

P. D. Hale and G. W. Day, 'Stability of birefringent linear retarders (wave plates),' Appl. Opt. 27, 5146-5153 (1988).
[CrossRef] [PubMed]

C. J. Sun and J. T. Lue, 'Second harmonic generation with focused broad-band and high-order transverse mode lasers,' IEEE J. Quantum Electron. 24, 113-117 (1988).
[CrossRef]

1987

1982

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, 'Up-Conversion of subpicosecond light pulses,' IEEE J. Quantum Electron. 18, 2048-2056 (1982).
[CrossRef]

1969

W. H. Glenn, 'Second-harmonic generation by picosecond optical pulses,' IEEE J. Quantum Electron. 5, 284-290 (1969).
[CrossRef]

Ackerhalt, J. R.

Y. B. Band, D. F. Heller, J. R. Ackerhalt, and J. S. Krasinski, 'Spectrum of second-harmonic generation for multimode fields,' Phys. Rev. A 42, 1515--1521 (1990).
[CrossRef] [PubMed]

Acquisto, L. D.

L. D. Acquisto, G. Petrucci, and B. Zuccarello, 'Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,' Opt. Lasers Eng. 37, 389-400 (2002).
[CrossRef]

Allione, M.

A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, 'Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,' Phys. Rev. Lett. 89, 087401 (2002).
[CrossRef] [PubMed]

Anceau, C.

Antonetti, A.

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

Atay, T.

T. Atay, J.-H. Song, and A. V. Nurmikko, 'Strongly interacting plasmon nanoparticle pairs: from dipole-dipole interaction to conductively coupled regime,' Nano Lett. 4, 1627-1631 (2004).
[CrossRef]

Aussenegg, F. R.

B. Lamprecht, A. Leitner, and F. R. Aussenegg, 'SHG studies of plasmon dephasing in nanoparticles,' Appl. Phys. B 68, 419-423 (1999).
[CrossRef]

Avagyan, R.

D. Hovhannisyan, K. Stepanyan, and R. Avagyan, 'Sum and difference frequency generation by femtosecond laser pulses in a nonlinear crystal,' Opt. Commun. 245, 443-456 (2005).
[CrossRef]

Band, Y. B

Band, Y. B.

Y. B. Band, D. F. Heller, J. R. Ackerhalt, and J. S. Krasinski, 'Spectrum of second-harmonic generation for multimode fields,' Phys. Rev. A 42, 1515--1521 (1990).
[CrossRef] [PubMed]

Bloembergen, N.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, 'Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,' Phys. Rev. Lett. 83, 2934-2937 (1999).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, Seventh ed. (Cambridge U. Press, 1999).

Boulbry, B.

B. Boulbry, B. L. Jeune, F. Pellen, J. Cariou, and J. Lotrian, 'Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source,' J. Phys. D 35, 2508-2515 (2002).
[CrossRef]

Boyd, R. W.

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

Brabec, T.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

Brasselet, S.

Burnett, N. H.

N. H. Burnett, C. Kan, and P. B. Corkum, 'Ellipticity and polarization effects in harmonic generation in ionizing neon,' Phys. Rev. A 51, 3418-3121 (1995).
[CrossRef]

Canfield, B. K.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, 278-284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, K. Laiho, K. Jefimovs, J. Turunen, and M. Kauranen, 'Chirality arising from small defects in gold nanoparticle arrays,' Opt. Express 14, 950-955 (2006).
[CrossRef] [PubMed]

B. K. Canfield, S. Kujala, M. Kauranen, K. Jefimovs, T. Vallius, and J. Turunen, 'Remarkable polarization sensitivity of gold nanoparticle arrays,' Appl. Phys. Lett. 86, 183109 (2005).
[CrossRef]

Cariou, J.

B. Boulbry, B. L. Jeune, F. Pellen, J. Cariou, and J. Lotrian, 'Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source,' J. Phys. D 35, 2508-2515 (2002).
[CrossRef]

Cattaneo, S.

Cheville, R. A.

Cheyssac, P.

A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, 'Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,' Phys. Rev. Lett. 89, 087401 (2002).
[CrossRef] [PubMed]

Choo, H. R.

Corkum, P. B.

N. H. Burnett, C. Kan, and P. B. Corkum, 'Ellipticity and polarization effects in harmonic generation in ionizing neon,' Phys. Rev. A 51, 3418-3121 (1995).
[CrossRef]

Curley, P. F.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

Day, G. W.

Dienes, A.

Downer, M. C.

El-Sayed, M. A.

W. Huang, W. Qian, and M. A. El-Sayed, 'Photothermal reshaping of prismatic Au nanoparticles in periodic monolayer arrays by femtosecond laser pulses,' J. Appl. Phys. 98, 114301 (2005).
[CrossRef]

Elshocht, S. V.

M. Kauranen, T. Verbiest, S. V. Elshocht, and A. Persoons, 'Chirality in surface nonlinear optics,' Opt. Mater. 9, 286-294 (1998).
[CrossRef]

Fan, D.

Fedosejevs, R.

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, 'Up-Conversion of subpicosecond light pulses,' IEEE J. Quantum Electron. 18, 2048-2056 (1982).
[CrossRef]

Fermann, M. E.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

Gadenne, P.

Ganeev, R. A.

R. A. Ganeev, A. Ishizawa, T. Kanai, T. Ozaki, and H. Kuroda, 'Polarization peculiarities of femtosecond laser induced harmonic generation from solid surface plasma,' Opt. Commun. 227, 175-182 (2003).
[CrossRef]

Gao, S.

X. Xiao, C. Yang, S. Gao, and H. Miao, 'Analysis of ultrashort-pulse second-harmonic generation in both phase- and group-velocity-matched structures,' IEEE J. Quantum Electron. 41, 85-93 (2005).
[CrossRef]

Glenn, W. H.

W. H. Glenn, 'Second-harmonic generation by picosecond optical pulses,' IEEE J. Quantum Electron. 5, 284-290 (1969).
[CrossRef]

Goldstein, D.

D. Goldstein, Polarized Light, 2nd ed. (Dekker, 2003).
[CrossRef]

Gordon, C. L. I.

J. J. Macklin, J. D. Kmetec, and C. L. I. Gordon, 'High-order harmonic generation using intense femtosecond pulses,' Phys. Rev. Lett. 70, 766-769 (1993).
[CrossRef] [PubMed]

Halas, N. J.

Hale, P. D.

Heller, D. F.

Y. B. Band, D. F. Heller, J. R. Ackerhalt, and J. S. Krasinski, 'Spectrum of second-harmonic generation for multimode fields,' Phys. Rev. A 42, 1515--1521 (1990).
[CrossRef] [PubMed]

Hierle, R.

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

Hofer, M.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

Hofmann, T.

Holmgren, S. J.

Hovhannisyan, D.

D. Hovhannisyan, K. Stepanyan, and R. Avagyan, 'Sum and difference frequency generation by femtosecond laser pulses in a nonlinear crystal,' Opt. Commun. 245, 443-456 (2005).
[CrossRef]

Huang, W.

W. Huang, W. Qian, and M. A. El-Sayed, 'Photothermal reshaping of prismatic Au nanoparticles in periodic monolayer arrays by femtosecond laser pulses,' J. Appl. Phys. 98, 114301 (2005).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Hulin, D.

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

Ishizawa, A.

R. A. Ganeev, A. Ishizawa, T. Kanai, T. Ozaki, and H. Kuroda, 'Polarization peculiarities of femtosecond laser induced harmonic generation from solid surface plasma,' Opt. Commun. 227, 175-182 (2003).
[CrossRef]

Jefimovs, K.

B. K. Canfield, S. Kujala, K. Laiho, K. Jefimovs, J. Turunen, and M. Kauranen, 'Chirality arising from small defects in gold nanoparticle arrays,' Opt. Express 14, 950-955 (2006).
[CrossRef] [PubMed]

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, 278-284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, M. Kauranen, K. Jefimovs, T. Vallius, and J. Turunen, 'Remarkable polarization sensitivity of gold nanoparticle arrays,' Appl. Phys. Lett. 86, 183109 (2005).
[CrossRef]

Jeune, B. L.

B. Boulbry, B. L. Jeune, F. Pellen, J. Cariou, and J. Lotrian, 'Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source,' J. Phys. D 35, 2508-2515 (2002).
[CrossRef]

Joffre, M.

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

Kan, C.

N. H. Burnett, C. Kan, and P. B. Corkum, 'Ellipticity and polarization effects in harmonic generation in ionizing neon,' Phys. Rev. A 51, 3418-3121 (1995).
[CrossRef]

Kanai, T.

R. A. Ganeev, A. Ishizawa, T. Kanai, T. Ozaki, and H. Kuroda, 'Polarization peculiarities of femtosecond laser induced harmonic generation from solid surface plasma,' Opt. Commun. 227, 175-182 (2003).
[CrossRef]

Kauranen, M.

B. K. Canfield, S. Kujala, K. Laiho, K. Jefimovs, J. Turunen, and M. Kauranen, 'Chirality arising from small defects in gold nanoparticle arrays,' Opt. Express 14, 950-955 (2006).
[CrossRef] [PubMed]

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, 278-284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, M. Kauranen, K. Jefimovs, T. Vallius, and J. Turunen, 'Remarkable polarization sensitivity of gold nanoparticle arrays,' Appl. Phys. Lett. 86, 183109 (2005).
[CrossRef]

S. Cattaneo and M. Kauranen, 'Application of second-harmonic generation to retardation measurements,' J. Opt. Soc. Am. B 20, 520-528 (2003).
[CrossRef]

M. Kauranen, T. Verbiest, S. V. Elshocht, and A. Persoons, 'Chirality in surface nonlinear optics,' Opt. Mater. 9, 286-294 (1998).
[CrossRef]

J. J. Maki, M. Kauranen, and A. Persoons, 'Surface second-harmonic generation from chiral materials,' Phys. Rev. B 51, 1425-1434 (1995).
[CrossRef]

Kmetec, J. D.

J. J. Macklin, J. D. Kmetec, and C. L. I. Gordon, 'High-order harmonic generation using intense femtosecond pulses,' Phys. Rev. Lett. 70, 766-769 (1993).
[CrossRef] [PubMed]

Knoesen, A.

Kofman, R.

A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, 'Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,' Phys. Rev. Lett. 89, 087401 (2002).
[CrossRef] [PubMed]

Krasinski, J. S.

C. Radzewicz, J. S. Krasinski, and Y. B Band, 'Increased efficiency for sumfrequency generation for broadband input fields,' Opt. Lett. 18, 331-333 (1993).
[CrossRef] [PubMed]

Y. B. Band, D. F. Heller, J. R. Ackerhalt, and J. S. Krasinski, 'Spectrum of second-harmonic generation for multimode fields,' Phys. Rev. A 42, 1515--1521 (1990).
[CrossRef] [PubMed]

Krausz, F.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

Kujala, S.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, 278-284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, K. Laiho, K. Jefimovs, J. Turunen, and M. Kauranen, 'Chirality arising from small defects in gold nanoparticle arrays,' Opt. Express 14, 950-955 (2006).
[CrossRef] [PubMed]

B. K. Canfield, S. Kujala, M. Kauranen, K. Jefimovs, T. Vallius, and J. Turunen, 'Remarkable polarization sensitivity of gold nanoparticle arrays,' Appl. Phys. Lett. 86, 183109 (2005).
[CrossRef]

Kuroda, H.

R. A. Ganeev, A. Ishizawa, T. Kanai, T. Ozaki, and H. Kuroda, 'Polarization peculiarities of femtosecond laser induced harmonic generation from solid surface plasma,' Opt. Commun. 227, 175-182 (2003).
[CrossRef]

Kuzyk, M. G.

Laiho, K.

Lamprecht, B.

B. Lamprecht, A. Leitner, and F. R. Aussenegg, 'SHG studies of plasmon dephasing in nanoparticles,' Appl. Phys. B 68, 419-423 (1999).
[CrossRef]

Laurell, F.

Leitner, A.

B. Lamprecht, A. Leitner, and F. R. Aussenegg, 'SHG studies of plasmon dephasing in nanoparticles,' Appl. Phys. B 68, 419-423 (1999).
[CrossRef]

Lotrian, J.

B. Boulbry, B. L. Jeune, F. Pellen, J. Cariou, and J. Lotrian, 'Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source,' J. Phys. D 35, 2508-2515 (2002).
[CrossRef]

Lue, J. T.

C. J. Sun and J. T. Lue, 'Second harmonic generation with focused broad-band and high-order transverse mode lasers,' IEEE J. Quantum Electron. 24, 113-117 (1988).
[CrossRef]

Macklin, J. J.

J. J. Macklin, J. D. Kmetec, and C. L. I. Gordon, 'High-order harmonic generation using intense femtosecond pulses,' Phys. Rev. Lett. 70, 766-769 (1993).
[CrossRef] [PubMed]

Maki, J. J.

J. J. Maki, M. Kauranen, and A. Persoons, 'Surface second-harmonic generation from chiral materials,' Phys. Rev. B 51, 1425-1434 (1995).
[CrossRef]

Malvezzi, A. M.

A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, 'Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,' Phys. Rev. Lett. 89, 087401 (2002).
[CrossRef] [PubMed]

Miao, H.

X. Xiao, C. Yang, S. Gao, and H. Miao, 'Analysis of ultrashort-pulse second-harmonic generation in both phase- and group-velocity-matched structures,' IEEE J. Quantum Electron. 41, 85-93 (2005).
[CrossRef]

Midorikawa, K.

Migus, A.

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

Mlejnek, M.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, 'Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,' Phys. Rev. Lett. 83, 2934-2937 (1999).
[CrossRef]

Moloney, J. V.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, 'Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,' Phys. Rev. Lett. 83, 2934-2937 (1999).
[CrossRef]

Mossavi, K.

Nabekawa, Y.

Nurmikko, A. V.

T. Atay, J.-H. Song, and A. V. Nurmikko, 'Strongly interacting plasmon nanoparticle pairs: from dipole-dipole interaction to conductively coupled regime,' Nano Lett. 4, 1627-1631 (2004).
[CrossRef]

Ober, M. H.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

Offenberger, A. A.

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, 'Up-Conversion of subpicosecond light pulses,' IEEE J. Quantum Electron. 18, 2048-2056 (1982).
[CrossRef]

Osvay, K.

Ozaki, T.

R. A. Ganeev, A. Ishizawa, T. Kanai, T. Ozaki, and H. Kuroda, 'Polarization peculiarities of femtosecond laser induced harmonic generation from solid surface plasma,' Opt. Commun. 227, 175-182 (2003).
[CrossRef]

Pasiskevicius, V.

Patrini, M.

A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, 'Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,' Phys. Rev. Lett. 89, 087401 (2002).
[CrossRef] [PubMed]

Pellen, F.

B. Boulbry, B. L. Jeune, F. Pellen, J. Cariou, and J. Lotrian, 'Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source,' J. Phys. D 35, 2508-2515 (2002).
[CrossRef]

Persoons, A.

M. Kauranen, T. Verbiest, S. V. Elshocht, and A. Persoons, 'Chirality in surface nonlinear optics,' Opt. Mater. 9, 286-294 (1998).
[CrossRef]

J. J. Maki, M. Kauranen, and A. Persoons, 'Surface second-harmonic generation from chiral materials,' Phys. Rev. B 51, 1425-1434 (1995).
[CrossRef]

Petrucci, G.

L. D. Acquisto, G. Petrucci, and B. Zuccarello, 'Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,' Opt. Lasers Eng. 37, 389-400 (2002).
[CrossRef]

Qian, L.

Qian, W.

W. Huang, W. Qian, and M. A. El-Sayed, 'Photothermal reshaping of prismatic Au nanoparticles in periodic monolayer arrays by femtosecond laser pulses,' J. Appl. Phys. 98, 114301 (2005).
[CrossRef]

Radzewicz, C.

Reiten, M. T.

Ross, I. N.

Schmidt, A. J.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

Sidick, E.

Siegman, A. E.

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

Singer, K. D.

Smith, M. H.

M. H. Smith, 'Polarization metrology moves beyond 'home-brewed' solutions,' Laser Focus World 40, 123-129 (2004).

E. A. West and M. H. Smith, 'Polarization errors associated with birefringent waveplates,' Opt. Eng. 34, 1574-1580 (1995).
[CrossRef]

Sohn, J. E.

Song, J.-H.

T. Atay, J.-H. Song, and A. V. Nurmikko, 'Strongly interacting plasmon nanoparticle pairs: from dipole-dipole interaction to conductively coupled regime,' Nano Lett. 4, 1627-1631 (2004).
[CrossRef]

Spielmann, C.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

Stella, A.

A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, 'Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,' Phys. Rev. Lett. 89, 087401 (2002).
[CrossRef] [PubMed]

Stepanyan, K.

D. Hovhannisyan, K. Stepanyan, and R. Avagyan, 'Sum and difference frequency generation by femtosecond laser pulses in a nonlinear crystal,' Opt. Commun. 245, 443-456 (2005).
[CrossRef]

Sun, C. J.

C. J. Sun and J. T. Lue, 'Second harmonic generation with focused broad-band and high-order transverse mode lasers,' IEEE J. Quantum Electron. 24, 113-117 (1988).
[CrossRef]

Svirko, Y.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, 278-284 (2006).
[CrossRef]

Szabó, G.

Tittel, F. K.

Tomov, I. V.

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, 'Up-Conversion of subpicosecond light pulses,' IEEE J. Quantum Electron. 18, 2048-2056 (1982).
[CrossRef]

Toussaere, E.

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

Turunen, J.

B. K. Canfield, S. Kujala, K. Laiho, K. Jefimovs, J. Turunen, and M. Kauranen, 'Chirality arising from small defects in gold nanoparticle arrays,' Opt. Express 14, 950-955 (2006).
[CrossRef] [PubMed]

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, 278-284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, M. Kauranen, K. Jefimovs, T. Vallius, and J. Turunen, 'Remarkable polarization sensitivity of gold nanoparticle arrays,' Appl. Phys. Lett. 86, 183109 (2005).
[CrossRef]

Vallius, T.

B. K. Canfield, S. Kujala, M. Kauranen, K. Jefimovs, T. Vallius, and J. Turunen, 'Remarkable polarization sensitivity of gold nanoparticle arrays,' Appl. Phys. Lett. 86, 183109 (2005).
[CrossRef]

Verbiest, T.

M. Kauranen, T. Verbiest, S. V. Elshocht, and A. Persoons, 'Chirality in surface nonlinear optics,' Opt. Mater. 9, 286-294 (1998).
[CrossRef]

Wang, S.

Wang, T.

West, E. A.

E. A. West and M. H. Smith, 'Polarization errors associated with birefringent waveplates,' Opt. Eng. 34, 1574-1580 (1995).
[CrossRef]

White, J. O.

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

Wintner, E.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, Seventh ed. (Cambridge U. Press, 1999).

Wright, E. M.

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, 'Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,' Phys. Rev. Lett. 83, 2934-2937 (1999).
[CrossRef]

Xiao, X.

X. Xiao, C. Yang, S. Gao, and H. Miao, 'Analysis of ultrashort-pulse second-harmonic generation in both phase- and group-velocity-matched structures,' IEEE J. Quantum Electron. 41, 85-93 (2005).
[CrossRef]

Yang, C.

X. Xiao, C. Yang, S. Gao, and H. Miao, 'Analysis of ultrashort-pulse second-harmonic generation in both phase- and group-velocity-matched structures,' IEEE J. Quantum Electron. 41, 85-93 (2005).
[CrossRef]

Yuan, P.

Zhang, T. R.

Zheng, W.

Zhu, H.

Zuccarello, B.

L. D. Acquisto, G. Petrucci, and B. Zuccarello, 'Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,' Opt. Lasers Eng. 37, 389-400 (2002).
[CrossRef]

Zyss, J.

C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, 'Local second-harmonic generation enhancement on gold nanostructures probed by two-photon microscopy,' Opt. Lett. 28, 713-715 (2003).
[CrossRef] [PubMed]

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

Appl. Opt.

Appl. Phys. B

B. Lamprecht, A. Leitner, and F. R. Aussenegg, 'SHG studies of plasmon dephasing in nanoparticles,' Appl. Phys. B 68, 419-423 (1999).
[CrossRef]

Appl. Phys. Lett.

B. K. Canfield, S. Kujala, M. Kauranen, K. Jefimovs, T. Vallius, and J. Turunen, 'Remarkable polarization sensitivity of gold nanoparticle arrays,' Appl. Phys. Lett. 86, 183109 (2005).
[CrossRef]

J. O. White, D. Hulin, M. Joffre, A. Migus, A. Antonetti, E. Toussaere, R. Hierle, and J. Zyss, 'Ultrabroadband second-harmonic generation in organic and inorganic thin crystals,' Appl. Phys. Lett. 64, 264-266 (1994).
[CrossRef]

IEEE J. Quantum Electron.

I. V. Tomov, R. Fedosejevs, and A. A. Offenberger, 'Up-Conversion of subpicosecond light pulses,' IEEE J. Quantum Electron. 18, 2048-2056 (1982).
[CrossRef]

X. Xiao, C. Yang, S. Gao, and H. Miao, 'Analysis of ultrashort-pulse second-harmonic generation in both phase- and group-velocity-matched structures,' IEEE J. Quantum Electron. 41, 85-93 (2005).
[CrossRef]

C. J. Sun and J. T. Lue, 'Second harmonic generation with focused broad-band and high-order transverse mode lasers,' IEEE J. Quantum Electron. 24, 113-117 (1988).
[CrossRef]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, 'Femtosecond solid-state lasers,' IEEE J. Quantum Electron. 28, 2097-2122 (1992).
[CrossRef]

W. H. Glenn, 'Second-harmonic generation by picosecond optical pulses,' IEEE J. Quantum Electron. 5, 284-290 (1969).
[CrossRef]

J. Appl. Phys.

W. Huang, W. Qian, and M. A. El-Sayed, 'Photothermal reshaping of prismatic Au nanoparticles in periodic monolayer arrays by femtosecond laser pulses,' J. Appl. Phys. 98, 114301 (2005).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, 278-284 (2006).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem.

See, for instance, the special issue on femtochemistry, J. Phys. Chem. 97, 12460-12465 (1993). Includes the major paper by A. Zewail, who won the 1999 Nobel in chemistry for time-resolved studies of molecular interactions.
[CrossRef]

J. Phys. D

B. Boulbry, B. L. Jeune, F. Pellen, J. Cariou, and J. Lotrian, 'Identification of error parameters and calibration of a double-crystal birefringent wave plate with a broadband spectral light source,' J. Phys. D 35, 2508-2515 (2002).
[CrossRef]

Laser Focus World

M. H. Smith, 'Polarization metrology moves beyond 'home-brewed' solutions,' Laser Focus World 40, 123-129 (2004).

Nano Lett.

T. Atay, J.-H. Song, and A. V. Nurmikko, 'Strongly interacting plasmon nanoparticle pairs: from dipole-dipole interaction to conductively coupled regime,' Nano Lett. 4, 1627-1631 (2004).
[CrossRef]

Opt. Commun.

R. A. Ganeev, A. Ishizawa, T. Kanai, T. Ozaki, and H. Kuroda, 'Polarization peculiarities of femtosecond laser induced harmonic generation from solid surface plasma,' Opt. Commun. 227, 175-182 (2003).
[CrossRef]

D. Hovhannisyan, K. Stepanyan, and R. Avagyan, 'Sum and difference frequency generation by femtosecond laser pulses in a nonlinear crystal,' Opt. Commun. 245, 443-456 (2005).
[CrossRef]

Opt. Eng.

E. A. West and M. H. Smith, 'Polarization errors associated with birefringent waveplates,' Opt. Eng. 34, 1574-1580 (1995).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

L. D. Acquisto, G. Petrucci, and B. Zuccarello, 'Full field automated evaluation of the quarter wave plate retardation by phase stepping technique,' Opt. Lasers Eng. 37, 389-400 (2002).
[CrossRef]

Opt. Lett.

Opt. Mater.

M. Kauranen, T. Verbiest, S. V. Elshocht, and A. Persoons, 'Chirality in surface nonlinear optics,' Opt. Mater. 9, 286-294 (1998).
[CrossRef]

Phys. Rev. A

N. H. Burnett, C. Kan, and P. B. Corkum, 'Ellipticity and polarization effects in harmonic generation in ionizing neon,' Phys. Rev. A 51, 3418-3121 (1995).
[CrossRef]

Y. B. Band, D. F. Heller, J. R. Ackerhalt, and J. S. Krasinski, 'Spectrum of second-harmonic generation for multimode fields,' Phys. Rev. A 42, 1515--1521 (1990).
[CrossRef] [PubMed]

Phys. Rev. B

J. J. Maki, M. Kauranen, and A. Persoons, 'Surface second-harmonic generation from chiral materials,' Phys. Rev. B 51, 1425-1434 (1995).
[CrossRef]

Phys. Rev. Lett.

J. J. Macklin, J. D. Kmetec, and C. L. I. Gordon, 'High-order harmonic generation using intense femtosecond pulses,' Phys. Rev. Lett. 70, 766-769 (1993).
[CrossRef] [PubMed]

A. M. Malvezzi, M. Allione, M. Patrini, A. Stella, P. Cheyssac, and R. Kofman, 'Melting-induced enhancement of the second-harmonic generation from metal nanoparticles,' Phys. Rev. Lett. 89, 087401 (2002).
[CrossRef] [PubMed]

M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, 'Second harmonic generation of femtosecond pulses at the boundary of a nonlinear dielectric,' Phys. Rev. Lett. 83, 2934-2937 (1999).
[CrossRef]

Other

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

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

Selected manufacturers' wave-plate specifications can be found at Casix (http://www.casix.com), CVI (http://www.cvilaser.com), Ekspla (http://www.ekspla.com), and Elan (http://www.elan.spb.ru).

M. Born and E. Wolf, Principles of Optics, Seventh ed. (Cambridge U. Press, 1999).

D. Goldstein, Polarized Light, 2nd ed. (Dekker, 2003).
[CrossRef]

The dispersion of crystal quartz was obtained from http://www.cvilaser.com/Common/PDFs/Dispersion_Equations.pdf. Note: Because this URL is too long for the printed column, the reader will need to type it into his or her browser directly.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Structure of a compound WP with a misalignment α between the axes of the two crystals. Distances along z ̂ are greatly exaggerated for illustrative purposes.

Fig. 2
Fig. 2

Core configuration of a BB-SHG experiment.

Fig. 3
Fig. 3

Phase deviations from CP ( 90 ° ) for QWPT and 3 ° -misaligned QWPC for thicknesses of 1 and 2 mm . The shaded region indicates the 5 ° tolerance condition.

Fig. 4
Fig. 4

Signed eccentricity of the polarization for QWPT and 1 mm thick QWPC with misalignments of 0.1 ° , 1 ° , and 3 ° . The sign of ϵ indicates whether the projection of a is greater along p ̂ (positive) or s ̂ (negative) in Fig. 5.

Fig. 5
Fig. 5

Polarization ellipse orientation for the QWP configurations in Fig. 4.

Fig. 6
Fig. 6

Normalized SHG responses at 2 ω 0 and in the sidebands generated from coefficient set S3 with QWPM, BW3.

Fig. 7
Fig. 7

Normalized total SHG responses generated from coefficient set S3 with QWPT, BW1 and QWPM, BW3, and BW4.

Fig. 8
Fig. 8

Normalized SHG responses at 2 ω 0 and in the sidebands generated from coefficient set S2 with QWPT, BW4.

Fig. 9
Fig. 9

Spectral shift of the SHG peak generated from coefficient set S2, BW4 with QWPT and 1 ° -misaligned QWPC for thicknesses of 1 and 2 mm .

Fig. 10
Fig. 10

Normalized total SHG responses generated from coefficient set S1 with QWPT, BW1 and 3 ° misaligned QWPC, BW4 for thicknesses of 1 and 2 mm .

Tables (3)

Tables Icon

Table 1 Spectral BWs ( Δ λ ) and Equivalent Durations ( Δ t ) Studied

Tables Icon

Table 2 Basic QWP Configurations a

Tables Icon

Table 3 Coefficient Sets from Eq. (5) Studied

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

R ( θ ) = [ cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ] .
W = [ e i ( 2 π λ ) n o ( λ ) d 0 0 e i ( 2 π λ ) n e ( λ ) d ] .
M = R ( θ ) R ( α ) W 2 R ( α ) W 1 R ( θ ) .
N = 2 L Δ λ λ 0 2 = 40,000 .
E i ( Ω ) = ω 1 [ f i E p ( ω 1 ) E p ( Ω ω 1 ) + g i E s ( ω 1 ) E s ( Ω ω 1 ) + h i E p ( ω 1 ) E s ( Ω ω 1 ) ] .
Δ λ = β λ 0 2 c Δ t ,
ϵ = 1 ( b a ) 2 .

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