Abstract

We study coherence properties of a χ(2) optical parametric oscillator (OPO), which produces 2/3-octave-wide spectrum centered at the subharmonic (3120 nm) of the femtosecond pump laser. Our method consists of interfering the outputs of two identical, but independent OPOs pumped by the same laser. We demonstrate that the two OPOs show stable spatial and temporal interference and are mutually locked in frequency and in phase. By observing a collective heterodyne beat signal between the two OPOs we show that one can deterministically choose, by cavity length adjustment, between the two frequency states corresponding to the two sets of modes shifted with respect to each other by half of the laser pulse repetition rate. Moreover, we observe that the existence of two opposite phase states, a known common feature of a parametrically driven n = 2 subharmonic oscillator, reveals itself in our experiment as a common phase, 0 or π, being established through the whole set of some 300 thousand longitudinal modes.

© 2012 OSA

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  3. C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  5. S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
    [CrossRef] [PubMed]
  6. B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
  9. C. Erny, K. Moutzouris, J. Biegert, D. Kühlke, F. Adler, A. Leitenstorfer, and U. Keller, “Mid-infrared difference-frequency generation of ultrashort pulses tunable between 3.2 and 4.8 microm from a compact fiber source,” Opt. Lett. 32(9), 1138–1140 (2007).
    [CrossRef] [PubMed]
  10. A. Gambetta, R. Ramponi, and M. Marangoni, “Mid-infrared optical combs from a compact amplified Er-doped fiber oscillator,” Opt. Lett. 33(22), 2671–2673 (2008).
    [CrossRef] [PubMed]
  11. P. Malara, P. Maddaloni, G. Gagliardi, and P. De Natale, “Absolute frequency measurement of molecular transitions by a direct link to a comb generated around 3-microm,” Opt. Express 16(11), 8242–8249 (2008).
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  12. J. H. Sun, B. J. S. Gale, and D. T. Reid, “Composite frequency comb spanning 0.4-2.4 microm from a phase-controlled femtosecond Ti:sapphire laser and synchronously pumped optical parametric oscillator,” Opt. Lett. 32(11), 1414–1416 (2007).
    [CrossRef] [PubMed]
  13. F. Adler, K. C. Cossel, M. J. Thorpe, I. Hartl, M. E. Fermann, and J. Ye, “Phase-stabilized, 1.5 W frequency comb at 2.8-4.8 microm,” Opt. Lett. 34(9), 1330–1332 (2009).
    [CrossRef] [PubMed]
  14. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2011 (3)

2010 (3)

S. T. Wong, K. L. Vodopyanov, and R. L. Byer, “Self-phase-locked divide-by-2 optical parametric oscillator as a broadband frequency comb source,” J. Opt. Soc. Am. B 27(5), 876–882 (2010).
[CrossRef]

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

2009 (1)

2008 (5)

2007 (4)

2004 (1)

2001 (1)

J.-J. Zondy, A. Douillet, A. Tallet, E. Ressayre, and M. Le Berre, “Theory of self-phase-locked optical parametric oscillators,” Phys. Rev. A 63(2), 023814 (2001).
[CrossRef]

2000 (1)

1990 (1)

1962 (2)

A. E. Siegman, “Nonlinear optical effects: an optical power limiter,” Appl. Opt. 1(6), 739–744 (1962).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

1831 (1)

M. Faraday, “On the forms and states of fluids on vibrating elastic surfaces,” Philos. Trans. R. Soc. Lond. 121, 319–340 (1831).

Adler, F.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Arpin, P.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Balslev-Clausen, D.

Becker, T.

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

Bernhardt, B.

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

Biegert, J.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Byer, R. L.

Chen, M.-C.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Colby, E.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Corkum, P. B.

P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007).
[CrossRef]

Cossel, K. C.

Day, T.

De Natale, P.

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[CrossRef] [PubMed]

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[CrossRef] [PubMed]

Digonnet, M.

Douillet, A.

J.-J. Zondy, A. Douillet, A. Tallet, E. Ressayre, and M. Le Berre, “Theory of self-phase-locked optical parametric oscillators,” Phys. Rev. A 63(2), 023814 (2001).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

England, R. J.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Erny, C.

Faraday, M.

M. Faraday, “On the forms and states of fluids on vibrating elastic surfaces,” Philos. Trans. R. Soc. Lond. 121, 319–340 (1831).

Fermann, M. E.

Gagliardi, G.

Gale, B. J. S.

Gambetta, A.

Gohle, C.

Hamm, P.

Hänsch, T. W.

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

Hartl, I.

Hollberg, L.

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[CrossRef] [PubMed]

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[CrossRef] [PubMed]

F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. 29(13), 1542–1544 (2004).
[CrossRef] [PubMed]

Ischebeck, R.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Jacquet, P.

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

Kaindl, R. A.

Kapteyn, H. C.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Keilmann, F.

Keller, U.

Kippenberg, T. J.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[CrossRef] [PubMed]

Kirchner, M. S.

Krausz, F.

P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007).
[CrossRef]

Kühlke, D.

Le Berre, M.

J.-J. Zondy, A. Douillet, A. Tallet, E. Ressayre, and M. Le Berre, “Theory of self-phase-locked optical parametric oscillators,” Phys. Rev. A 63(2), 023814 (2001).
[CrossRef]

Leindecker, N.

Leitenstorfer, A.

Maddaloni, P.

Malara, P.

Marandi, A.

Marangoni, M.

Mbele, V.

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[CrossRef] [PubMed]

McGuinness, C.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Moutzouris, K.

Murnane, M. M.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Nabors, C. D.

Nelson, J.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Noble, R.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[CrossRef]

Picqué, N.

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

Plettner, T.

S. T. Wong, T. Plettner, K. L. Vodopyanov, K. Urbanek, M. Digonnet, and R. L. Byer, “Self-phase-locked degenerate femtosecond optical parametric oscillator,” Opt. Lett. 33(16), 1896–1898 (2008).
[CrossRef] [PubMed]

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Popmintchev, T.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Ramponi, R.

Reid, D. T.

Reimann, K.

Ressayre, E.

J.-J. Zondy, A. Douillet, A. Tallet, E. Ressayre, and M. Le Berre, “Theory of self-phase-locked optical parametric oscillators,” Phys. Rev. A 63(2), 023814 (2001).
[CrossRef]

Schunemann, P. G.

Sears, C. M. S.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Siegman, A. E.

Siemann, R. H.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Sorokin, E.

K. L. Vodopyanov, E. Sorokin, I. T. Sorokina, and P. G. Schunemann, “Mid-IR frequency comb source spanning 4.4-5.4 μm based on subharmonic GaAs optical parametric oscillator,” Opt. Lett. 36(12), 2275–2277 (2011).
[CrossRef] [PubMed]

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

Sorokina, I. T.

K. L. Vodopyanov, E. Sorokin, I. T. Sorokina, and P. G. Schunemann, “Mid-IR frequency comb source spanning 4.4-5.4 μm based on subharmonic GaAs optical parametric oscillator,” Opt. Lett. 36(12), 2275–2277 (2011).
[CrossRef] [PubMed]

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

Spencer, J.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Sun, J. H.

Tallet, A.

J.-J. Zondy, A. Douillet, A. Tallet, E. Ressayre, and M. Le Berre, “Theory of self-phase-locked optical parametric oscillators,” Phys. Rev. A 63(2), 023814 (2001).
[CrossRef]

Thon, R.

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

Thorpe, M. J.

Urbanek, K.

Vodopyanov, K. L.

Walz, D.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. Special Topics – Accelerators and Beams 11(10), 101301 (2008).
[CrossRef]

Weiner, A. M.

Woerner, M.

Wong, S. T.

Wurm, M.

Yang, S. T.

Ye, J.

Zondy, J.-J.

J.-J. Zondy, A. Douillet, A. Tallet, E. Ressayre, and M. Le Berre, “Theory of self-phase-locked optical parametric oscillators,” Phys. Rev. A 63(2), 023814 (2001).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

B. Bernhardt, E. Sorokin, P. Jacquet, R. Thon, T. Becker, I. T. Sorokina, N. Picqué, and T. W. Hänsch, “Mid-infrared dual-comb spectroscopy with 2.4 μm Cr2+:ZnSe femtosecond lasers,” Appl. Phys. B 100(1), 3–8 (2010).
[CrossRef]

J. Opt. Soc. Am. B (3)

Nat. Photonics (1)

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[CrossRef]

Nat. Phys. (1)

P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007).
[CrossRef]

Nature (1)

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445(7128), 627–630 (2007).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (7)

S. T. Wong, T. Plettner, K. L. Vodopyanov, K. Urbanek, M. Digonnet, and R. L. Byer, “Self-phase-locked degenerate femtosecond optical parametric oscillator,” Opt. Lett. 33(16), 1896–1898 (2008).
[CrossRef] [PubMed]

A. Gambetta, R. Ramponi, and M. Marangoni, “Mid-infrared optical combs from a compact amplified Er-doped fiber oscillator,” Opt. Lett. 33(22), 2671–2673 (2008).
[CrossRef] [PubMed]

F. Adler, K. C. Cossel, M. J. Thorpe, I. Hartl, M. E. Fermann, and J. Ye, “Phase-stabilized, 1.5 W frequency comb at 2.8-4.8 microm,” Opt. Lett. 34(9), 1330–1332 (2009).
[CrossRef] [PubMed]

F. Keilmann, C. Gohle, and R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. 29(13), 1542–1544 (2004).
[CrossRef] [PubMed]

C. Erny, K. Moutzouris, J. Biegert, D. Kühlke, F. Adler, A. Leitenstorfer, and U. Keller, “Mid-infrared difference-frequency generation of ultrashort pulses tunable between 3.2 and 4.8 microm from a compact fiber source,” Opt. Lett. 32(9), 1138–1140 (2007).
[CrossRef] [PubMed]

J. H. Sun, B. J. S. Gale, and D. T. Reid, “Composite frequency comb spanning 0.4-2.4 microm from a phase-controlled femtosecond Ti:sapphire laser and synchronously pumped optical parametric oscillator,” Opt. Lett. 32(11), 1414–1416 (2007).
[CrossRef] [PubMed]

K. L. Vodopyanov, E. Sorokin, I. T. Sorokina, and P. G. Schunemann, “Mid-IR frequency comb source spanning 4.4-5.4 μm based on subharmonic GaAs optical parametric oscillator,” Opt. Lett. 36(12), 2275–2277 (2011).
[CrossRef] [PubMed]

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Phys. Rev. (1)

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[CrossRef]

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Modes of a synchronously pumped degenerate OPO. Only two sets of longitudinal modes (A and B) are allowed by photon energy conservation.

Fig. 2
Fig. 2

Ring-cavity degenerate OPO, synchronously pumped by a femtosecond erbium fiber laser. PZT – piezoelectric transducer, BS- pellicle beamsplitter, PD- InAs photodiode, and F – longpass filter.

Fig. 3
Fig. 3

OPO spectrum measured with a Fourier-transform spectrometer. The inset shows the OPO beam profile.

Fig. 4
Fig. 4

(a) Schematic diagram of the experimental setup. λ/2 – halfwave plate, PBS – polarizing beamsplitter, BS- pellicle beamsplitter, PD- fast InAs photodiode. (b,c) Complementary interference patterns obtained with an infrared camera. Dashed lines are guides for an eye to show that maximum intensity in (c) corresponds to minimum in (b).

Fig. 5
Fig. 5

Time and frequency domain interference between the two identical OPOs. (a) RF spectrogram of the detector signal when both OPOs were in the same (AA or BB) frequency state and (c) in different (AB) frequency states; (b) and (d) are the corresponding time-domain detector signals. 50-MHz (frep/2) beats were observed in the case (c) corresponding to the sequence of pulses separated by 20 ns, which is twice the repetition period of the pump (d).

Equations (6)

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φ p = φ s + φ i + π / 2 + integer × 2 π ,
φ p = 2 φ s , i + π / 2 + integer × 2 π ,
ν n = f C E O + n f r e p ,
ν m , A = f C E O 2 + m f r e p ,
ν m , B = f C E O 2 + ( m + 1 2 ) f r e p ,
E B ( t ) = E A ( t ) e i 2 π f r e p 2 t = E A ( t ) e i π f r e p t .

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