Abstract

We present a new approach for extended-cavity diode-laser tuning to achieve wide mode-hop-free tuning ranges. By using a multiple piezoactuated grating mount, the cavity length and grating angle in the laser can be adjusted independently, allowing mode-hop-free tuning without the need for a mechanically optimized pivot-point mount. Furthermore, synchronized diode injection-current tuning allows diode lasers without antireflection coatings to be employed. In combination these two techniques make the construction of a cheap, efficient, and easily optimized extended-cavity diode laser possible. A theoretical analysis is presented for optimal control of piezoactuator displacements and injection current to achieve the widest possible mode-hop-free tuning ranges, and a comparison is made with measurements. The scheme is demonstrated for blue and violet GaN lasers operating at ∼450 nm and ∼410 nm, for which continuous tuning ranges exceeding 90 GHz have been achieved. Examples of applications of these lasers are also given.

© 2005 Optical Society of America

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  1. C. E. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
    [CrossRef]
  2. D. Sesko, C. G. Fan, C. E. Wieman, “Production of a cold atomic vapor using diode-laser cooling,” J. Opt. Soc. Am. B 5, 1225–1227 (1988).
    [CrossRef]
  3. P. W. Werle, P. Mazzinghi, F. D’Amato, M. De Rosa, K. Maurer, F. Slemr, “Signal processing and calibration procedures for in situ diode-laser absorption spectroscopy,” Spectrochim. Acta A 60, 1685–1705 (2004).
    [CrossRef]
  4. E. Schlosser, J. Wolfrum, L. Hildebrandt, H. Seifert, B. Oser, V. Ebert, “Diode laser based in-situ detection of alkali atoms: development of a new method for determination of residence-time distribution in combustion plants,” Appl. Phys. B 75, 237–247 (2002).
    [CrossRef]
  5. M. G. Allen, “Diode laser absorption sensors for gas-dynamic and combustion flows,” Meas. Sci. Technol. 9, 545–562 (1998).
    [CrossRef]
  6. S. Nakamura, “The roles of structural imperfections in InGaN-based blue light-emitting diodes and laser diodes,” Science 281, 956–961 (1998).
    [CrossRef]
  7. S. Nagahama, T. Yanamoto, M. Sano, T. Mukai, “Characteristics of laser diodes composed of GaN-based semiconductor,” Phys. Stat. Sol. A 190, 235–246 (2002).
    [CrossRef]
  8. K. B. MacAdam, A. Steinbach, C. E. Wieman, “A narrowband tunable diode laser system with grating feedback and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098–1111 (1992).
    [CrossRef]
  9. L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
    [CrossRef]
  10. G.-Y. Yan, A. L. Schawlow, “Measurement of diode laser characteristics affecting tunability with an external grating,” J. Opt. Soc. Am. B 9, 2122–2127 (1992).
    [CrossRef]
  11. F. Favre, D. Le Guen, J. C. Simon, B. Landousies, “External-cavity semiconductor laser with 15 nm continuous tuning range,” Electron. Lett. 21, 795–796 (1986).
    [CrossRef]
  12. M. de Labachelerie, G. Passedat, “Mode-hop suppression of littrow grating-tuned lasers,” Appl. Opt. 32, 269–274 (1993).
    [CrossRef] [PubMed]
  13. J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
    [CrossRef]
  14. T. Laurila, T. Joutsenoja, R. Hernberg, M. Kuittinen, “Tunable external-cavity diode laser at 650 nm based on a transmission diffraction grating,” Appl. Opt. 41, 5632–5637 (2002).
    [CrossRef] [PubMed]
  15. M. G. Boshier, D. Berkeland, E. A. Hinds, V. Sandoghdar, “External-cavity frequency stabilization of visible and infrared semiconductor lasers for high resolution spectroscopy,” Opt. Commun. 85, 355–359 (1991).
    [CrossRef]
  16. L. Hildebrandt, R. Knispel, S. Stry, J. R. Sacher, F. Schael, “Antireflection-coated blue GaN laser diodes in an external cavity and Doppler-free indium absorption spectroscopy,” Appl. Opt. 42, 2110–2118 (2003).
    [CrossRef] [PubMed]
  17. C. Petridis, I. D. Lindsay, D. J. M. Stothard, M. Ebrahimzadeh, “Mode-hop-free tuning over 80 GHz of an extended cavity diode laser without antireflection coating,” Rev. Sci. Instrum. 72, 3811–3815 (2001).
    [CrossRef]
  18. R. S. Conroy, J. J. Hewett, G. P. T. Lancaster, W. Sibbett, J. W. Allen, K. Dholakia, “Characterisation of an extended cavity violet diode laser,” Opt. Commun. 175, 185–188 (2000).
    [CrossRef]
  19. H. Leinen, D. Gläßner, H. Metcalf, R. Wynands, D. Haubrich, D. Meschede, “GaN blue diode lasers: a spectroscopist’s view,” Appl. Phys B 70, 567–571 (2000).
    [CrossRef]
  20. U. Gustafsson, J. Alnis, S. Svanberg, “Atomic spectroscopy with violet laser diodes,” Am. J. Phys. 68, 660–664 (2000).
    [CrossRef]
  21. W. R. Trutna, L. F. Stokes, “Continuously tuned external-cavity semiconductor-laser,” J. Lightwave Technol. 11, 1279–1286 (1993).
    [CrossRef]
  22. L. Hildebrandt, Sacher Lasertechnik Group, Hannah Arendt Strasse 3–7, D-35037 Marburg, Germany (personal communication, 2004).
  23. C. J. Hawthorn, K. P. Webber, R. E. Scholten, “Littrow configuration tunable external cavity diode laser with fixed direction output beam,” Rev. Sci. Instrum. 72, 4477–4479 (2001).
    [CrossRef]
  24. U. Tisch, B. Meyler, O. Katz, E. Finkman, J. Salzman, “Dependence of the refractive index of AlxGa1−xN on temperature and composition at elevated temperatures,” J. Appl. Phys. 89, 2676–2685 (2001).
    [CrossRef]
  25. H. X. Jiang, J. Y. Lin, “Mode spacing “anomaly” in InGaN blue lasers,” Appl. Phys. Letters 74, 1066–1068 (1999).
    [CrossRef]
  26. B. Boggs, C. Greiner, T. Wang, H. Lin, T. W. Mossberg, “Simple high-coherence rapidly tunable external-cavity diode laser,” Opt. Lett. 23, 1906–1908 (1998).
    [CrossRef]
  27. J. Cariou, P. Luc, Atlas du Spectre d’Absorption de la Molecule Tellure (Laboratoire Aimé-Cotton, CNRS II, Orsay, France, 1980).
  28. J. Hult, I. S. Burns, C. F. Kaminski, “Two-line atomic fluorescence thermometry using diode lasers,” Proc. Combust. Inst. 30, 1535–1543 (2005).
    [CrossRef]
  29. I. S. Burns, J. Hult, C. F. Kaminski, “Spectroscopic use of a novel blue diode laser in a wavelength region around 450 nm,” Appl. Phys. B. 79, 491–495 (2004).
    [CrossRef]
  30. J. Hult, I. S. Burns, C. F. Kaminski, “Measurements of the indium hyperfine structure in an atmospheric-pressure flame by use of diode-laser-induced fluorescence,” Opt. Lett. 29, 827–829 (2004).
    [CrossRef] [PubMed]
  31. J. E. Dec, J. O. Keller, “High speed thermometry using two-line atomic fluorescence,” Proc. Combust. Inst. 21, 1737–1745 (1986).
    [CrossRef]

2005 (1)

J. Hult, I. S. Burns, C. F. Kaminski, “Two-line atomic fluorescence thermometry using diode lasers,” Proc. Combust. Inst. 30, 1535–1543 (2005).
[CrossRef]

2004 (3)

I. S. Burns, J. Hult, C. F. Kaminski, “Spectroscopic use of a novel blue diode laser in a wavelength region around 450 nm,” Appl. Phys. B. 79, 491–495 (2004).
[CrossRef]

J. Hult, I. S. Burns, C. F. Kaminski, “Measurements of the indium hyperfine structure in an atmospheric-pressure flame by use of diode-laser-induced fluorescence,” Opt. Lett. 29, 827–829 (2004).
[CrossRef] [PubMed]

P. W. Werle, P. Mazzinghi, F. D’Amato, M. De Rosa, K. Maurer, F. Slemr, “Signal processing and calibration procedures for in situ diode-laser absorption spectroscopy,” Spectrochim. Acta A 60, 1685–1705 (2004).
[CrossRef]

2003 (1)

2002 (3)

T. Laurila, T. Joutsenoja, R. Hernberg, M. Kuittinen, “Tunable external-cavity diode laser at 650 nm based on a transmission diffraction grating,” Appl. Opt. 41, 5632–5637 (2002).
[CrossRef] [PubMed]

E. Schlosser, J. Wolfrum, L. Hildebrandt, H. Seifert, B. Oser, V. Ebert, “Diode laser based in-situ detection of alkali atoms: development of a new method for determination of residence-time distribution in combustion plants,” Appl. Phys. B 75, 237–247 (2002).
[CrossRef]

S. Nagahama, T. Yanamoto, M. Sano, T. Mukai, “Characteristics of laser diodes composed of GaN-based semiconductor,” Phys. Stat. Sol. A 190, 235–246 (2002).
[CrossRef]

2001 (3)

C. Petridis, I. D. Lindsay, D. J. M. Stothard, M. Ebrahimzadeh, “Mode-hop-free tuning over 80 GHz of an extended cavity diode laser without antireflection coating,” Rev. Sci. Instrum. 72, 3811–3815 (2001).
[CrossRef]

C. J. Hawthorn, K. P. Webber, R. E. Scholten, “Littrow configuration tunable external cavity diode laser with fixed direction output beam,” Rev. Sci. Instrum. 72, 4477–4479 (2001).
[CrossRef]

U. Tisch, B. Meyler, O. Katz, E. Finkman, J. Salzman, “Dependence of the refractive index of AlxGa1−xN on temperature and composition at elevated temperatures,” J. Appl. Phys. 89, 2676–2685 (2001).
[CrossRef]

2000 (3)

R. S. Conroy, J. J. Hewett, G. P. T. Lancaster, W. Sibbett, J. W. Allen, K. Dholakia, “Characterisation of an extended cavity violet diode laser,” Opt. Commun. 175, 185–188 (2000).
[CrossRef]

H. Leinen, D. Gläßner, H. Metcalf, R. Wynands, D. Haubrich, D. Meschede, “GaN blue diode lasers: a spectroscopist’s view,” Appl. Phys B 70, 567–571 (2000).
[CrossRef]

U. Gustafsson, J. Alnis, S. Svanberg, “Atomic spectroscopy with violet laser diodes,” Am. J. Phys. 68, 660–664 (2000).
[CrossRef]

1999 (1)

H. X. Jiang, J. Y. Lin, “Mode spacing “anomaly” in InGaN blue lasers,” Appl. Phys. Letters 74, 1066–1068 (1999).
[CrossRef]

1998 (3)

M. G. Allen, “Diode laser absorption sensors for gas-dynamic and combustion flows,” Meas. Sci. Technol. 9, 545–562 (1998).
[CrossRef]

S. Nakamura, “The roles of structural imperfections in InGaN-based blue light-emitting diodes and laser diodes,” Science 281, 956–961 (1998).
[CrossRef]

B. Boggs, C. Greiner, T. Wang, H. Lin, T. W. Mossberg, “Simple high-coherence rapidly tunable external-cavity diode laser,” Opt. Lett. 23, 1906–1908 (1998).
[CrossRef]

1995 (1)

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

1993 (2)

W. R. Trutna, L. F. Stokes, “Continuously tuned external-cavity semiconductor-laser,” J. Lightwave Technol. 11, 1279–1286 (1993).
[CrossRef]

M. de Labachelerie, G. Passedat, “Mode-hop suppression of littrow grating-tuned lasers,” Appl. Opt. 32, 269–274 (1993).
[CrossRef] [PubMed]

1992 (2)

G.-Y. Yan, A. L. Schawlow, “Measurement of diode laser characteristics affecting tunability with an external grating,” J. Opt. Soc. Am. B 9, 2122–2127 (1992).
[CrossRef]

K. B. MacAdam, A. Steinbach, C. E. Wieman, “A narrowband tunable diode laser system with grating feedback and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098–1111 (1992).
[CrossRef]

1991 (2)

C. E. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
[CrossRef]

M. G. Boshier, D. Berkeland, E. A. Hinds, V. Sandoghdar, “External-cavity frequency stabilization of visible and infrared semiconductor lasers for high resolution spectroscopy,” Opt. Commun. 85, 355–359 (1991).
[CrossRef]

1988 (2)

J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
[CrossRef]

D. Sesko, C. G. Fan, C. E. Wieman, “Production of a cold atomic vapor using diode-laser cooling,” J. Opt. Soc. Am. B 5, 1225–1227 (1988).
[CrossRef]

1986 (2)

J. E. Dec, J. O. Keller, “High speed thermometry using two-line atomic fluorescence,” Proc. Combust. Inst. 21, 1737–1745 (1986).
[CrossRef]

F. Favre, D. Le Guen, J. C. Simon, B. Landousies, “External-cavity semiconductor laser with 15 nm continuous tuning range,” Electron. Lett. 21, 795–796 (1986).
[CrossRef]

Al-Chalabi, S. A.

J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
[CrossRef]

Allen, J. W.

R. S. Conroy, J. J. Hewett, G. P. T. Lancaster, W. Sibbett, J. W. Allen, K. Dholakia, “Characterisation of an extended cavity violet diode laser,” Opt. Commun. 175, 185–188 (2000).
[CrossRef]

Allen, M. G.

M. G. Allen, “Diode laser absorption sensors for gas-dynamic and combustion flows,” Meas. Sci. Technol. 9, 545–562 (1998).
[CrossRef]

Alnis, J.

U. Gustafsson, J. Alnis, S. Svanberg, “Atomic spectroscopy with violet laser diodes,” Am. J. Phys. 68, 660–664 (2000).
[CrossRef]

Berkeland, D.

M. G. Boshier, D. Berkeland, E. A. Hinds, V. Sandoghdar, “External-cavity frequency stabilization of visible and infrared semiconductor lasers for high resolution spectroscopy,” Opt. Commun. 85, 355–359 (1991).
[CrossRef]

Boggs, B.

Boshier, M. G.

M. G. Boshier, D. Berkeland, E. A. Hinds, V. Sandoghdar, “External-cavity frequency stabilization of visible and infrared semiconductor lasers for high resolution spectroscopy,” Opt. Commun. 85, 355–359 (1991).
[CrossRef]

Brain, M. C.

J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
[CrossRef]

Burns, I. S.

J. Hult, I. S. Burns, C. F. Kaminski, “Two-line atomic fluorescence thermometry using diode lasers,” Proc. Combust. Inst. 30, 1535–1543 (2005).
[CrossRef]

J. Hult, I. S. Burns, C. F. Kaminski, “Measurements of the indium hyperfine structure in an atmospheric-pressure flame by use of diode-laser-induced fluorescence,” Opt. Lett. 29, 827–829 (2004).
[CrossRef] [PubMed]

I. S. Burns, J. Hult, C. F. Kaminski, “Spectroscopic use of a novel blue diode laser in a wavelength region around 450 nm,” Appl. Phys. B. 79, 491–495 (2004).
[CrossRef]

Cameron, K. H.

J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
[CrossRef]

Cariou, J.

J. Cariou, P. Luc, Atlas du Spectre d’Absorption de la Molecule Tellure (Laboratoire Aimé-Cotton, CNRS II, Orsay, France, 1980).

Conroy, R. S.

R. S. Conroy, J. J. Hewett, G. P. T. Lancaster, W. Sibbett, J. W. Allen, K. Dholakia, “Characterisation of an extended cavity violet diode laser,” Opt. Commun. 175, 185–188 (2000).
[CrossRef]

D’Amato, F.

P. W. Werle, P. Mazzinghi, F. D’Amato, M. De Rosa, K. Maurer, F. Slemr, “Signal processing and calibration procedures for in situ diode-laser absorption spectroscopy,” Spectrochim. Acta A 60, 1685–1705 (2004).
[CrossRef]

de Labachelerie, M.

De Rosa, M.

P. W. Werle, P. Mazzinghi, F. D’Amato, M. De Rosa, K. Maurer, F. Slemr, “Signal processing and calibration procedures for in situ diode-laser absorption spectroscopy,” Spectrochim. Acta A 60, 1685–1705 (2004).
[CrossRef]

Dec, J. E.

J. E. Dec, J. O. Keller, “High speed thermometry using two-line atomic fluorescence,” Proc. Combust. Inst. 21, 1737–1745 (1986).
[CrossRef]

Devlin, W. J.

J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
[CrossRef]

Dholakia, K.

R. S. Conroy, J. J. Hewett, G. P. T. Lancaster, W. Sibbett, J. W. Allen, K. Dholakia, “Characterisation of an extended cavity violet diode laser,” Opt. Commun. 175, 185–188 (2000).
[CrossRef]

Ebert, V.

E. Schlosser, J. Wolfrum, L. Hildebrandt, H. Seifert, B. Oser, V. Ebert, “Diode laser based in-situ detection of alkali atoms: development of a new method for determination of residence-time distribution in combustion plants,” Appl. Phys. B 75, 237–247 (2002).
[CrossRef]

Ebrahimzadeh, M.

C. Petridis, I. D. Lindsay, D. J. M. Stothard, M. Ebrahimzadeh, “Mode-hop-free tuning over 80 GHz of an extended cavity diode laser without antireflection coating,” Rev. Sci. Instrum. 72, 3811–3815 (2001).
[CrossRef]

Esslinger, T.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

Fan, C. G.

Favre, F.

F. Favre, D. Le Guen, J. C. Simon, B. Landousies, “External-cavity semiconductor laser with 15 nm continuous tuning range,” Electron. Lett. 21, 795–796 (1986).
[CrossRef]

Finkman, E.

U. Tisch, B. Meyler, O. Katz, E. Finkman, J. Salzman, “Dependence of the refractive index of AlxGa1−xN on temperature and composition at elevated temperatures,” J. Appl. Phys. 89, 2676–2685 (2001).
[CrossRef]

Gläßner, D.

H. Leinen, D. Gläßner, H. Metcalf, R. Wynands, D. Haubrich, D. Meschede, “GaN blue diode lasers: a spectroscopist’s view,” Appl. Phys B 70, 567–571 (2000).
[CrossRef]

Greiner, C.

Gustafsson, U.

U. Gustafsson, J. Alnis, S. Svanberg, “Atomic spectroscopy with violet laser diodes,” Am. J. Phys. 68, 660–664 (2000).
[CrossRef]

Hänsch, T. W.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

Haubrich, D.

H. Leinen, D. Gläßner, H. Metcalf, R. Wynands, D. Haubrich, D. Meschede, “GaN blue diode lasers: a spectroscopist’s view,” Appl. Phys B 70, 567–571 (2000).
[CrossRef]

Hawthorn, C. J.

C. J. Hawthorn, K. P. Webber, R. E. Scholten, “Littrow configuration tunable external cavity diode laser with fixed direction output beam,” Rev. Sci. Instrum. 72, 4477–4479 (2001).
[CrossRef]

Hemmerich, A.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

Hernberg, R.

Hewett, J. J.

R. S. Conroy, J. J. Hewett, G. P. T. Lancaster, W. Sibbett, J. W. Allen, K. Dholakia, “Characterisation of an extended cavity violet diode laser,” Opt. Commun. 175, 185–188 (2000).
[CrossRef]

Hildebrandt, L.

L. Hildebrandt, R. Knispel, S. Stry, J. R. Sacher, F. Schael, “Antireflection-coated blue GaN laser diodes in an external cavity and Doppler-free indium absorption spectroscopy,” Appl. Opt. 42, 2110–2118 (2003).
[CrossRef] [PubMed]

E. Schlosser, J. Wolfrum, L. Hildebrandt, H. Seifert, B. Oser, V. Ebert, “Diode laser based in-situ detection of alkali atoms: development of a new method for determination of residence-time distribution in combustion plants,” Appl. Phys. B 75, 237–247 (2002).
[CrossRef]

L. Hildebrandt, Sacher Lasertechnik Group, Hannah Arendt Strasse 3–7, D-35037 Marburg, Germany (personal communication, 2004).

Hinds, E. A.

M. G. Boshier, D. Berkeland, E. A. Hinds, V. Sandoghdar, “External-cavity frequency stabilization of visible and infrared semiconductor lasers for high resolution spectroscopy,” Opt. Commun. 85, 355–359 (1991).
[CrossRef]

Hollberg, L.

C. E. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
[CrossRef]

Hult, J.

J. Hult, I. S. Burns, C. F. Kaminski, “Two-line atomic fluorescence thermometry using diode lasers,” Proc. Combust. Inst. 30, 1535–1543 (2005).
[CrossRef]

J. Hult, I. S. Burns, C. F. Kaminski, “Measurements of the indium hyperfine structure in an atmospheric-pressure flame by use of diode-laser-induced fluorescence,” Opt. Lett. 29, 827–829 (2004).
[CrossRef] [PubMed]

I. S. Burns, J. Hult, C. F. Kaminski, “Spectroscopic use of a novel blue diode laser in a wavelength region around 450 nm,” Appl. Phys. B. 79, 491–495 (2004).
[CrossRef]

Jiang, H. X.

H. X. Jiang, J. Y. Lin, “Mode spacing “anomaly” in InGaN blue lasers,” Appl. Phys. Letters 74, 1066–1068 (1999).
[CrossRef]

Joutsenoja, T.

Kaminski, C. F.

J. Hult, I. S. Burns, C. F. Kaminski, “Two-line atomic fluorescence thermometry using diode lasers,” Proc. Combust. Inst. 30, 1535–1543 (2005).
[CrossRef]

J. Hult, I. S. Burns, C. F. Kaminski, “Measurements of the indium hyperfine structure in an atmospheric-pressure flame by use of diode-laser-induced fluorescence,” Opt. Lett. 29, 827–829 (2004).
[CrossRef] [PubMed]

I. S. Burns, J. Hult, C. F. Kaminski, “Spectroscopic use of a novel blue diode laser in a wavelength region around 450 nm,” Appl. Phys. B. 79, 491–495 (2004).
[CrossRef]

Katz, O.

U. Tisch, B. Meyler, O. Katz, E. Finkman, J. Salzman, “Dependence of the refractive index of AlxGa1−xN on temperature and composition at elevated temperatures,” J. Appl. Phys. 89, 2676–2685 (2001).
[CrossRef]

Keller, J. O.

J. E. Dec, J. O. Keller, “High speed thermometry using two-line atomic fluorescence,” Proc. Combust. Inst. 21, 1737–1745 (1986).
[CrossRef]

Knispel, R.

König, W.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

Kuittinen, M.

Lancaster, G. P. T.

R. S. Conroy, J. J. Hewett, G. P. T. Lancaster, W. Sibbett, J. W. Allen, K. Dholakia, “Characterisation of an extended cavity violet diode laser,” Opt. Commun. 175, 185–188 (2000).
[CrossRef]

Landousies, B.

F. Favre, D. Le Guen, J. C. Simon, B. Landousies, “External-cavity semiconductor laser with 15 nm continuous tuning range,” Electron. Lett. 21, 795–796 (1986).
[CrossRef]

Laurila, T.

Le Guen, D.

F. Favre, D. Le Guen, J. C. Simon, B. Landousies, “External-cavity semiconductor laser with 15 nm continuous tuning range,” Electron. Lett. 21, 795–796 (1986).
[CrossRef]

Leinen, H.

H. Leinen, D. Gläßner, H. Metcalf, R. Wynands, D. Haubrich, D. Meschede, “GaN blue diode lasers: a spectroscopist’s view,” Appl. Phys B 70, 567–571 (2000).
[CrossRef]

Lin, H.

Lin, J. Y.

H. X. Jiang, J. Y. Lin, “Mode spacing “anomaly” in InGaN blue lasers,” Appl. Phys. Letters 74, 1066–1068 (1999).
[CrossRef]

Lindsay, I. D.

C. Petridis, I. D. Lindsay, D. J. M. Stothard, M. Ebrahimzadeh, “Mode-hop-free tuning over 80 GHz of an extended cavity diode laser without antireflection coating,” Rev. Sci. Instrum. 72, 3811–3815 (2001).
[CrossRef]

Luc, P.

J. Cariou, P. Luc, Atlas du Spectre d’Absorption de la Molecule Tellure (Laboratoire Aimé-Cotton, CNRS II, Orsay, France, 1980).

MacAdam, K. B.

K. B. MacAdam, A. Steinbach, C. E. Wieman, “A narrowband tunable diode laser system with grating feedback and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098–1111 (1992).
[CrossRef]

Maurer, K.

P. W. Werle, P. Mazzinghi, F. D’Amato, M. De Rosa, K. Maurer, F. Slemr, “Signal processing and calibration procedures for in situ diode-laser absorption spectroscopy,” Spectrochim. Acta A 60, 1685–1705 (2004).
[CrossRef]

Mazzinghi, P.

P. W. Werle, P. Mazzinghi, F. D’Amato, M. De Rosa, K. Maurer, F. Slemr, “Signal processing and calibration procedures for in situ diode-laser absorption spectroscopy,” Spectrochim. Acta A 60, 1685–1705 (2004).
[CrossRef]

Mellis, J.

J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
[CrossRef]

Meschede, D.

H. Leinen, D. Gläßner, H. Metcalf, R. Wynands, D. Haubrich, D. Meschede, “GaN blue diode lasers: a spectroscopist’s view,” Appl. Phys B 70, 567–571 (2000).
[CrossRef]

Metcalf, H.

H. Leinen, D. Gläßner, H. Metcalf, R. Wynands, D. Haubrich, D. Meschede, “GaN blue diode lasers: a spectroscopist’s view,” Appl. Phys B 70, 567–571 (2000).
[CrossRef]

Meyler, B.

U. Tisch, B. Meyler, O. Katz, E. Finkman, J. Salzman, “Dependence of the refractive index of AlxGa1−xN on temperature and composition at elevated temperatures,” J. Appl. Phys. 89, 2676–2685 (2001).
[CrossRef]

Mossberg, T. W.

Mukai, T.

S. Nagahama, T. Yanamoto, M. Sano, T. Mukai, “Characteristics of laser diodes composed of GaN-based semiconductor,” Phys. Stat. Sol. A 190, 235–246 (2002).
[CrossRef]

Nagahama, S.

S. Nagahama, T. Yanamoto, M. Sano, T. Mukai, “Characteristics of laser diodes composed of GaN-based semiconductor,” Phys. Stat. Sol. A 190, 235–246 (2002).
[CrossRef]

Nakamura, S.

S. Nakamura, “The roles of structural imperfections in InGaN-based blue light-emitting diodes and laser diodes,” Science 281, 956–961 (1998).
[CrossRef]

Oser, B.

E. Schlosser, J. Wolfrum, L. Hildebrandt, H. Seifert, B. Oser, V. Ebert, “Diode laser based in-situ detection of alkali atoms: development of a new method for determination of residence-time distribution in combustion plants,” Appl. Phys. B 75, 237–247 (2002).
[CrossRef]

Passedat, G.

Petridis, C.

C. Petridis, I. D. Lindsay, D. J. M. Stothard, M. Ebrahimzadeh, “Mode-hop-free tuning over 80 GHz of an extended cavity diode laser without antireflection coating,” Rev. Sci. Instrum. 72, 3811–3815 (2001).
[CrossRef]

Regnault, J. C.

J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
[CrossRef]

Ricci, L.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

Sacher, J. R.

Salzman, J.

U. Tisch, B. Meyler, O. Katz, E. Finkman, J. Salzman, “Dependence of the refractive index of AlxGa1−xN on temperature and composition at elevated temperatures,” J. Appl. Phys. 89, 2676–2685 (2001).
[CrossRef]

Sandoghdar, V.

M. G. Boshier, D. Berkeland, E. A. Hinds, V. Sandoghdar, “External-cavity frequency stabilization of visible and infrared semiconductor lasers for high resolution spectroscopy,” Opt. Commun. 85, 355–359 (1991).
[CrossRef]

Sano, M.

S. Nagahama, T. Yanamoto, M. Sano, T. Mukai, “Characteristics of laser diodes composed of GaN-based semiconductor,” Phys. Stat. Sol. A 190, 235–246 (2002).
[CrossRef]

Schael, F.

Schawlow, A. L.

Schlosser, E.

E. Schlosser, J. Wolfrum, L. Hildebrandt, H. Seifert, B. Oser, V. Ebert, “Diode laser based in-situ detection of alkali atoms: development of a new method for determination of residence-time distribution in combustion plants,” Appl. Phys. B 75, 237–247 (2002).
[CrossRef]

Scholten, R. E.

C. J. Hawthorn, K. P. Webber, R. E. Scholten, “Littrow configuration tunable external cavity diode laser with fixed direction output beam,” Rev. Sci. Instrum. 72, 4477–4479 (2001).
[CrossRef]

Seifert, H.

E. Schlosser, J. Wolfrum, L. Hildebrandt, H. Seifert, B. Oser, V. Ebert, “Diode laser based in-situ detection of alkali atoms: development of a new method for determination of residence-time distribution in combustion plants,” Appl. Phys. B 75, 237–247 (2002).
[CrossRef]

Sesko, D.

Sibbett, W.

R. S. Conroy, J. J. Hewett, G. P. T. Lancaster, W. Sibbett, J. W. Allen, K. Dholakia, “Characterisation of an extended cavity violet diode laser,” Opt. Commun. 175, 185–188 (2000).
[CrossRef]

Simon, J. C.

F. Favre, D. Le Guen, J. C. Simon, B. Landousies, “External-cavity semiconductor laser with 15 nm continuous tuning range,” Electron. Lett. 21, 795–796 (1986).
[CrossRef]

Slemr, F.

P. W. Werle, P. Mazzinghi, F. D’Amato, M. De Rosa, K. Maurer, F. Slemr, “Signal processing and calibration procedures for in situ diode-laser absorption spectroscopy,” Spectrochim. Acta A 60, 1685–1705 (2004).
[CrossRef]

Steinbach, A.

K. B. MacAdam, A. Steinbach, C. E. Wieman, “A narrowband tunable diode laser system with grating feedback and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098–1111 (1992).
[CrossRef]

Stokes, L. F.

W. R. Trutna, L. F. Stokes, “Continuously tuned external-cavity semiconductor-laser,” J. Lightwave Technol. 11, 1279–1286 (1993).
[CrossRef]

Stothard, D. J. M.

C. Petridis, I. D. Lindsay, D. J. M. Stothard, M. Ebrahimzadeh, “Mode-hop-free tuning over 80 GHz of an extended cavity diode laser without antireflection coating,” Rev. Sci. Instrum. 72, 3811–3815 (2001).
[CrossRef]

Stry, S.

Svanberg, S.

U. Gustafsson, J. Alnis, S. Svanberg, “Atomic spectroscopy with violet laser diodes,” Am. J. Phys. 68, 660–664 (2000).
[CrossRef]

Tisch, U.

U. Tisch, B. Meyler, O. Katz, E. Finkman, J. Salzman, “Dependence of the refractive index of AlxGa1−xN on temperature and composition at elevated temperatures,” J. Appl. Phys. 89, 2676–2685 (2001).
[CrossRef]

Trutna, W. R.

W. R. Trutna, L. F. Stokes, “Continuously tuned external-cavity semiconductor-laser,” J. Lightwave Technol. 11, 1279–1286 (1993).
[CrossRef]

Vuletic, V.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

Wang, T.

Webber, K. P.

C. J. Hawthorn, K. P. Webber, R. E. Scholten, “Littrow configuration tunable external cavity diode laser with fixed direction output beam,” Rev. Sci. Instrum. 72, 4477–4479 (2001).
[CrossRef]

Weidemüller, M.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

Werle, P. W.

P. W. Werle, P. Mazzinghi, F. D’Amato, M. De Rosa, K. Maurer, F. Slemr, “Signal processing and calibration procedures for in situ diode-laser absorption spectroscopy,” Spectrochim. Acta A 60, 1685–1705 (2004).
[CrossRef]

Wieman, C. E.

K. B. MacAdam, A. Steinbach, C. E. Wieman, “A narrowband tunable diode laser system with grating feedback and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098–1111 (1992).
[CrossRef]

C. E. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
[CrossRef]

D. Sesko, C. G. Fan, C. E. Wieman, “Production of a cold atomic vapor using diode-laser cooling,” J. Opt. Soc. Am. B 5, 1225–1227 (1988).
[CrossRef]

Wolfrum, J.

E. Schlosser, J. Wolfrum, L. Hildebrandt, H. Seifert, B. Oser, V. Ebert, “Diode laser based in-situ detection of alkali atoms: development of a new method for determination of residence-time distribution in combustion plants,” Appl. Phys. B 75, 237–247 (2002).
[CrossRef]

Wyatt, R.

J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
[CrossRef]

Wynands, R.

H. Leinen, D. Gläßner, H. Metcalf, R. Wynands, D. Haubrich, D. Meschede, “GaN blue diode lasers: a spectroscopist’s view,” Appl. Phys B 70, 567–571 (2000).
[CrossRef]

Yan, G.-Y.

Yanamoto, T.

S. Nagahama, T. Yanamoto, M. Sano, T. Mukai, “Characteristics of laser diodes composed of GaN-based semiconductor,” Phys. Stat. Sol. A 190, 235–246 (2002).
[CrossRef]

Zimmermann, C.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

Am. J. Phys. (2)

K. B. MacAdam, A. Steinbach, C. E. Wieman, “A narrowband tunable diode laser system with grating feedback and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098–1111 (1992).
[CrossRef]

U. Gustafsson, J. Alnis, S. Svanberg, “Atomic spectroscopy with violet laser diodes,” Am. J. Phys. 68, 660–664 (2000).
[CrossRef]

Appl. Opt. (3)

Appl. Phys B (1)

H. Leinen, D. Gläßner, H. Metcalf, R. Wynands, D. Haubrich, D. Meschede, “GaN blue diode lasers: a spectroscopist’s view,” Appl. Phys B 70, 567–571 (2000).
[CrossRef]

Appl. Phys. B (1)

E. Schlosser, J. Wolfrum, L. Hildebrandt, H. Seifert, B. Oser, V. Ebert, “Diode laser based in-situ detection of alkali atoms: development of a new method for determination of residence-time distribution in combustion plants,” Appl. Phys. B 75, 237–247 (2002).
[CrossRef]

Appl. Phys. B. (1)

I. S. Burns, J. Hult, C. F. Kaminski, “Spectroscopic use of a novel blue diode laser in a wavelength region around 450 nm,” Appl. Phys. B. 79, 491–495 (2004).
[CrossRef]

Appl. Phys. Letters (1)

H. X. Jiang, J. Y. Lin, “Mode spacing “anomaly” in InGaN blue lasers,” Appl. Phys. Letters 74, 1066–1068 (1999).
[CrossRef]

Electron. Lett. (2)

F. Favre, D. Le Guen, J. C. Simon, B. Landousies, “External-cavity semiconductor laser with 15 nm continuous tuning range,” Electron. Lett. 21, 795–796 (1986).
[CrossRef]

J. Mellis, S. A. Al-Chalabi, K. H. Cameron, R. Wyatt, J. C. Regnault, W. J. Devlin, M. C. Brain, “Miniature packaged external-cavity semiconductor-laser with 50-GHz continuous electrical tuning range,” Electron. Lett. 24, 988–989 (1988).
[CrossRef]

J. Appl. Phys. (1)

U. Tisch, B. Meyler, O. Katz, E. Finkman, J. Salzman, “Dependence of the refractive index of AlxGa1−xN on temperature and composition at elevated temperatures,” J. Appl. Phys. 89, 2676–2685 (2001).
[CrossRef]

J. Lightwave Technol. (1)

W. R. Trutna, L. F. Stokes, “Continuously tuned external-cavity semiconductor-laser,” J. Lightwave Technol. 11, 1279–1286 (1993).
[CrossRef]

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

Meas. Sci. Technol. (1)

M. G. Allen, “Diode laser absorption sensors for gas-dynamic and combustion flows,” Meas. Sci. Technol. 9, 545–562 (1998).
[CrossRef]

Opt. Commun. (3)

M. G. Boshier, D. Berkeland, E. A. Hinds, V. Sandoghdar, “External-cavity frequency stabilization of visible and infrared semiconductor lasers for high resolution spectroscopy,” Opt. Commun. 85, 355–359 (1991).
[CrossRef]

R. S. Conroy, J. J. Hewett, G. P. T. Lancaster, W. Sibbett, J. W. Allen, K. Dholakia, “Characterisation of an extended cavity violet diode laser,” Opt. Commun. 175, 185–188 (2000).
[CrossRef]

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. 117, 541–549 (1995).
[CrossRef]

Opt. Lett. (2)

Phys. Stat. Sol. A (1)

S. Nagahama, T. Yanamoto, M. Sano, T. Mukai, “Characteristics of laser diodes composed of GaN-based semiconductor,” Phys. Stat. Sol. A 190, 235–246 (2002).
[CrossRef]

Proc. Combust. Inst. (2)

J. E. Dec, J. O. Keller, “High speed thermometry using two-line atomic fluorescence,” Proc. Combust. Inst. 21, 1737–1745 (1986).
[CrossRef]

J. Hult, I. S. Burns, C. F. Kaminski, “Two-line atomic fluorescence thermometry using diode lasers,” Proc. Combust. Inst. 30, 1535–1543 (2005).
[CrossRef]

Rev. Sci. Instrum. (3)

C. Petridis, I. D. Lindsay, D. J. M. Stothard, M. Ebrahimzadeh, “Mode-hop-free tuning over 80 GHz of an extended cavity diode laser without antireflection coating,” Rev. Sci. Instrum. 72, 3811–3815 (2001).
[CrossRef]

C. E. Wieman, L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
[CrossRef]

C. J. Hawthorn, K. P. Webber, R. E. Scholten, “Littrow configuration tunable external cavity diode laser with fixed direction output beam,” Rev. Sci. Instrum. 72, 4477–4479 (2001).
[CrossRef]

Science (1)

S. Nakamura, “The roles of structural imperfections in InGaN-based blue light-emitting diodes and laser diodes,” Science 281, 956–961 (1998).
[CrossRef]

Spectrochim. Acta A (1)

P. W. Werle, P. Mazzinghi, F. D’Amato, M. De Rosa, K. Maurer, F. Slemr, “Signal processing and calibration procedures for in situ diode-laser absorption spectroscopy,” Spectrochim. Acta A 60, 1685–1705 (2004).
[CrossRef]

Other (2)

L. Hildebrandt, Sacher Lasertechnik Group, Hannah Arendt Strasse 3–7, D-35037 Marburg, Germany (personal communication, 2004).

J. Cariou, P. Luc, Atlas du Spectre d’Absorption de la Molecule Tellure (Laboratoire Aimé-Cotton, CNRS II, Orsay, France, 1980).

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Figures (9)

Fig. 1
Fig. 1

Schematic illustration of the multiple-actuator extended-cavity diode-laser system.

Fig. 2
Fig. 2

Detailed view of the grating translation with respect to the incident laser beam, as the angle of the grating is changed.

Fig. 3
Fig. 3

Schematic illustration of the extended-cavity diode-laser system.

Fig. 4
Fig. 4

Spectral output of the Fabry–Perot diode lasers as a function of (a) temperature for the 450 nm laser, (b) temperature for the 410 nm laser, (c) current for the 450 nm laser, and (d) current for the 410 nm laser. Spectral output below lasing threshold for (e) the 450 nm diode and (f) the 410 nm diode.

Fig. 5
Fig. 5

(a) Transmitted fringe pattern of a 450 nm ECDL wavelength scan through two different Fabry–Perot interferometers (FSR = 3.1 GHz and 7.5 GHz). 37 fringes from the lower FSR etalon are observed, corresponding to 110 GHz mode-hop-free tuning. The corresponding laser output power is also shown. (b) Mirror spacing of scanning Fabry–Perot interferometer (FSR = 7.5 GHz) used to validate single-mode emission during ECDL scanning. (c)–(f) Scanning Fabry–Perot traces recorded at four different times during a single laser wavelength scan; the corresponding times are indicated in (b).

Fig. 6
Fig. 6

Schematic diagram of the diode- and extended-cavity mode structure of an ECDL; the grating-feedback profile is also indicated.

Fig. 7
Fig. 7

(a) FP etalon transmission as a function of laser wavelength, (b) time series of FP etalon transmission, with the laser tuned to the location indicated by the gray box in (a), (c) histogram of instantaneous laser frequencies corresponding to the transmission trace in (b). From the width of the frequency distribution in (c) the 450 nm ECDL line width was determined to be below 8 MHz over a 150 ms interval.

Fig. 8
Fig. 8

Transmitted fringe pattern of a 410 nm extended-cavity diode-laser wavelength scan through a low-finesse Fabry–Perot interferometer (FSR = 3.1 GHz). 31 fringes can be distinguished, corresponding to a 93 GHz wide mode-hop-free scan. The corresponding laser output power is also shown.

Fig. 9
Fig. 9

Simultaneously recorded Te2 absorption and indium LIF spectrum at ∼451 nm. The Te2 absorption spectrum was acquired in a low-pressure cell at a temperature of 600 °C. The arrows indicate peaks that could be identified in the tellurium reference atlas.27 The LIF spectrum corresponds to the 52P3/2 → 62S1/2 transition in indium and was recorded in a flame seeded with indium. A composite fit consisting of six Voigt profiles is shown superimposed on the spectrum. Below the spectrum the positions and relative strengths of the six individual hyperfine transitions are indicated.

Equations (14)

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

2 d sin θ = λ ,
l 2 = N 2 λ / 2 n ,
tan ( Δ θ ) = Δ a Δ b l A ,
Δ θ = Δ λ 2 d cos ( θ ) .
Δ a Δ b = l A tan ( Δ λ 2 d cos ( θ ) ) l A 2 d cos ( θ ) Δ λ ,
Δ l 2 = Δ l 2 , eff = l C ( 1 cos ( Δ θ ) ) l B sin ( Δ θ ) Δ b l B Δ θ Δ b = l B 2 d cos ( θ ) Δ λ Δ b ,
Δ l 2 = N 2 2 Δ λ = l 2 λ Δ λ ,
Δ b = l B 2 d cos ( θ ) Δ λ l 2 λ Δ λ ,
Δ a = l A 2 d cos ( θ ) Δ λ + l B 2 d cos ( θ ) Δ λ l 2 λ Δ λ .
Δ a Δ b = l B l A 2 l 2 d cos ( θ ) / λ l B 2 l 2 d cos ( θ ) / λ = 1 + l A 2 l 2 d cos ( θ ) / λ l B = 1 + l A 2 l 2 d 1 sin 2 ( θ ) / λ l B = 1 + l A l 2 4 d 2 λ 2 ( 2 d sin ( θ ) λ ) 2 l B = 1 + l A l 2 4 d 2 λ 2 1 l B .
l 1 = N 1 λ / 2 n ,
Δ λ = β Δ I = 2 n N 1 Δ l 1 ,
Δ I Δ V b = 1 β Δ λ 1 α Δ b = α β ( l B 2 d cos θ l 2 λ ) = α λ β ( l B 4 d 2 λ 2 1 l 2 ) .
Δ I Δ V b = α Δ λ β Δ b = α λ Δ l 2 β l 2 Δ b = α λ β l 2 ( l B l A ( 1 Δ V a Δ V b ) 1 ) ,

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