Abstract

We investigate the fine structure of the optical spectrum of a broad-area laser diode with approximately 30-MHz resolution using spatially-resolved self-heterodyning technique. We show that this method is capable of measuring the relative powers and spacings of the individual lateral modes.

© 2014 Optical Society of America

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References

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  1. H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
    [CrossRef]
  2. D. F. Welch, “A brief history of high-power semiconductor lasers,” J. Sel. Top. Quantum Electron. 6(6), 1470–1477 (2000).
    [CrossRef]
  3. L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
    [CrossRef]
  4. H. Wenzel, “Basic aspects of high-power semiconductor laser simulation,” J. Sel. Top. Quantum Electron. 19(5), 1502913 (2013).
    [CrossRef]
  5. M. Achtenhagen, A. A. Hardy, C. S. Harder, “Coherent kinks in high-power ridge waveguide laser diodes,” J. Lightwave Technol. 24(5), 2225–2232 (2006).
    [CrossRef]
  6. L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
    [CrossRef]
  7. J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
    [CrossRef]
  8. N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
    [CrossRef]
  9. H. Partanen, J. Tervo, J. Turunen, “Spatial coherence of broad-area laser diodes,” Appl. Opt. 52(14), 3221–3228 (2013).
    [CrossRef] [PubMed]
  10. L. Büttner, J. Czarske, “Investigation of the influence of spatial coherence of a broad-area laser diode on the interference fringe system of a Mach-Zehnder interferometer for highly spatially resolved velocity measurements,” Appl. Opt. 44(9), 1582–1590 (2005).
    [CrossRef] [PubMed]
  11. N. Stelmakh and M. Vasilyev, “Spatially resolved spectroscopy of lateral modes of broad-area laser diodes by self-heterodyning,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), May 2008, San Jose, CA, paper CMN6.
    [CrossRef]
  12. N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19(18), 1392–1394 (2007).
    [CrossRef]
  13. N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
    [CrossRef]
  14. N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
    [CrossRef]
  15. J. K. Butler and H. Kressel, Semiconductor Lasers and Heterojunction LEDs (Academic Press, 1977), Ch. 6 and Ch. 7.

2013

H. Wenzel, “Basic aspects of high-power semiconductor laser simulation,” J. Sel. Top. Quantum Electron. 19(5), 1502913 (2013).
[CrossRef]

H. Partanen, J. Tervo, J. Turunen, “Spatial coherence of broad-area laser diodes,” Appl. Opt. 52(14), 3221–3228 (2013).
[CrossRef] [PubMed]

2011

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

2010

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
[CrossRef]

2009

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

2007

N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19(18), 1392–1394 (2007).
[CrossRef]

2006

N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
[CrossRef]

M. Achtenhagen, A. A. Hardy, C. S. Harder, “Coherent kinks in high-power ridge waveguide laser diodes,” J. Lightwave Technol. 24(5), 2225–2232 (2006).
[CrossRef]

2005

2000

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

D. F. Welch, “A brief history of high-power semiconductor lasers,” J. Sel. Top. Quantum Electron. 6(6), 1470–1477 (2000).
[CrossRef]

Achtenhagen, M.

Bai, C.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Bao, L.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Büttner, L.

Czarske, J.

DeVito, M.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Dong, W.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Elsaesser, T.

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Flowers, M.

N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
[CrossRef]

Gmachl, C.

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

Grimshaw, M.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Harder, C. S.

Hardy, A. A.

He, X.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Hempel, M.

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Kanskar, M.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Kovanis, V.

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

Leisher, P.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Lester, L. F.

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

Li, D.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Martinsen, R.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Mott, J.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

O’Reilly, E. P.

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

Ovtchinnikov, A.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Partanen, H.

Patterson, S.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Price, K.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Stelmakh, N.

N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
[CrossRef]

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19(18), 1392–1394 (2007).
[CrossRef]

N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
[CrossRef]

Tervo, J.

Tohmori, Y.

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

Tomm, J. W.

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Toor, F.

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

Treusch, H. G.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Turunen, J.

Vasilyev, M.

N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
[CrossRef]

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

Wang, J.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Welch, D. F.

D. F. Welch, “A brief history of high-power semiconductor lasers,” J. Sel. Top. Quantum Electron. 6(6), 1470–1477 (2000).
[CrossRef]

Wenzel, H.

H. Wenzel, “Basic aspects of high-power semiconductor laser simulation,” J. Sel. Top. Quantum Electron. 19(5), 1502913 (2013).
[CrossRef]

Wise, D.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Xu, D.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Yang, S.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Zhang, S.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Ziegler, M.

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

IEEE Photon. Technol. Lett.

N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
[CrossRef]

N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19(18), 1392–1394 (2007).
[CrossRef]

J. Lightwave Technol.

J. Sel. Top. Quantum Electron.

H. Wenzel, “Basic aspects of high-power semiconductor laser simulation,” J. Sel. Top. Quantum Electron. 19(5), 1502913 (2013).
[CrossRef]

D. F. Welch, “A brief history of high-power semiconductor lasers,” J. Sel. Top. Quantum Electron. 6(6), 1470–1477 (2000).
[CrossRef]

Laser & Photon. Rev.

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Opt. and Photon. News

N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
[CrossRef]

Proc. SPIE

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Other

N. Stelmakh and M. Vasilyev, “Spatially resolved spectroscopy of lateral modes of broad-area laser diodes by self-heterodyning,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), May 2008, San Jose, CA, paper CMN6.
[CrossRef]

J. K. Butler and H. Kressel, Semiconductor Lasers and Heterojunction LEDs (Academic Press, 1977), Ch. 6 and Ch. 7.

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Figures (6)

Fig. 1
Fig. 1

Simulated pictures of spatially-resolved RF spectra of a BALD (near field). Left: Spatial profiles of RF power corresponding to mode beating frequencies. The vertical axis of the graph has both RF frequency and intensity meanings. Number of shown lateral modes is P = 5. The corresponding number of the resulting beat frequencies is P(P–1)/2 = 10. Right: Same as picture on the left, but with intensity shown by color scale (i.e., vertical scale represents RF frequency only). The double-script notation of each plot indicates the corresponding indices of the lateral modes contributing to the given RF component.

Fig. 2
Fig. 2

Simulated picture of angularly-resolved RF spectra of a BALD (far field). Left: Spatial profiles of RF power corresponding to mode beating frequencies. The vertical axis of the graph has both RF frequency and intensity meanings. Number of shown lateral modes P is 5. The corresponding number of resulting strong beat frequencies is P–1 = 4 (shown in the bottom half of the picture). Right: Same as picture on the left, but with intensity shown by color scale (i.e., vertical scale represents RF frequency only). The double-script notation of each plot indicates the corresponding indices of the lateral modes contributing to the given RF component.

Fig. 3
Fig. 3

RF spectra measurement setup. “Near-field optical system” is a 1:5 telescope made of two achromatic doublet lenses with 40 mm and 200 mm focal distances. “Far-field optical system” is simply a free-space propagation distance of 480 mm.

Fig. 4
Fig. 4

(a) High-resolution detail of the optical spectrum of a 1-mm-long, 96-μm-wide BALD at 300 mA pump current. (b)–(d) Spatially-resolved near-field RF spectra of the same BALD. (b) Box model simulation. (c) Experimental results. (d) Box model modified by lowering the fundamental mode’s frequency by 370 MHz. Note that the horizontal scale in (c) is 5 × of that in (b) and (d), owing to the telescope magnification. The low-frequency (< 1 GHz) RF components in c) are most likely due to relaxation oscillations / mode partitioning noise.

Fig. 5
Fig. 5

Comparison of the relative mode frequencies predicted by the box model of Eq. (4) (solid black line) and observed in optical (blue circles) and RF (green diamonds) measurements. Purple line shows the box model modified by lowering the fundamental mode’s frequency by 370 MHz.

Fig. 6
Fig. 6

Experimental measurements (a), (c) and theoretical simulations (b), (d) of the angularly-resolved RF spectra (far field) for a 1-mm-long, 96-μm-wide BALD at: (a), (b) 300 mA and (c), (d) 450 mA pump currents. The simulations in (b), (d) represent the box model modified by lowering the fundamental mode’s frequency by 370 MHz. The low-frequency (< 1 GHz) RF components in a) and c) are most likely due to relaxation oscillations / mode partitioning noise.

Tables (2)

Tables Icon

Table 1 Results of RF Measurements

Tables Icon

Table 2 Comparison of the Results Obtained from RF and Direct Optical Measurements

Equations (47)

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

E mp (x,t)= A mp (t) e i ω mp t cos[ πpx W + π 2 (p1) ]= A mp (t) e i ω mp t Ψ p (x),
ω mp =m πc L n ph (λ) 1+ ( Lp Wm ) 2 ,
ω m0 =m πc L n ph ( λ m ) ,
ω mp ω m0 + πc λ m 4 n ph 2 ( λ m ) W 2 p 2 ,
E(x,t)= m= m 0 M/2 m 0 +M/2 p=1 P E mp (x,t).
S 2 RF (x,ω)= (2Rd/W) 2 | H(ω) | 2 p p | Ψ p (x) | 2 | Ψ p (x) | 2 m { S mp opt (ω) S m p opt (ω ω p p ) } ,
S RF (x,ω)=ξ | H(ω) | 2 p =2 P p=1 p | Ψ p (x) | 2 | Ψ p (x) | 2 g [(ω ω p p )/Δ ω ] g (0) S p opt S p opt ,
S RF,NF (x,ω)= ξ 4 | H(ω) | 2 p =2 P p=1 p S p opt S p opt g [(ω ω p p )/Δ ω ] g (0) × | cos[ ( p p) πx W +π p p 2 ]cos[ ( p +p) πx W +π p +p 2 ] | 2 .
S RF,NF (x,ω)= p =2 P p=1 p S p p RF | H(ω) | 2 | H( ω p p ) | 2 g [(ω ω p p )/Δ ω ] g (0) ,
S p p RF = max W 2 x W 2 [ S RF,NF (x, ω p p ) ]=ξ | H( ω p p ) | 2 μ p p S p opt S p opt
μ p p = max W 2 x W 2 { 1 4 | cos[ ( p p) πx W +π p p 2 ]cos[ ( p +p) πx W +π p +p 2 ] | 2 }.
S p opt = 1 ξ S p p RF S p p RF S p p RF μ p p μ p p μ p p |H( ω p p ) | 2 |H( ω p p ) | 2 |H( ω p p ) | 2 .
S RF,FF ( k x ,ω)= ξ | H(ω) | 2 p=1 P1 | 2 π 2 W 2 p(p+1)sin k x W ( π 2 p 2 k x 2 W 2 )[ π 2 (p+1) 2 k x 2 W 2 ] | 2 × g [ ω(2p+1)δω Δ ω ] g (0) S p opt S p+1 opt ,
ξ | H( ω 12 ) | 2 μ 12 S 1 opt S 2 opt = S 12 RF , ξ | H( ω 23 ) | 2 μ 23 S 2 opt S 3 opt = S 23 RF , ξ | H( ω 34 ) | 2 μ 34 S 3 opt S 4 opt = S 34 RF , ........ ξ | H( ω P1,P ) | 2 μ (P1),P S P1 opt S P opt = S (P1),P RF , p=1 P S p opt = S total opt ,
μ p,(p+1) = max 2π λ k x 2π λ { | 2 π 2 W 2 p(p+1)sin k x W ( π 2 p 2 k x 2 W 2 )[ π 2 (p+1) 2 k x 2 W 2 ] | 2 }.
ω mp = ω m1 + p =1 p1 ω p ,( p +1) ,
ε(x,y,t)=Φ(y) m,p A mp (t) e i ω mp t Ψ p (x)=Φ(y)E(x,t),
E(x,t)= m,p A mp (t) e i ω mp t Ψ p (x)= m,p E mp (x,t) ,
| Φ(y) | 2 dy = h y ,
A mp (t)=| A mp (t) | e i φ mp (t)
P(x,t)= 1 2 c ε 0 xd/2 x+d/2 d x h det /2 h det /2 dy | ε( x ,y,t) | 2 ,
P(x,t)=(1/2)c ε 0 h y d | E(x,t) | 2 =γ | E(x,t) | 2 ,
S opt (x,ω)=γ E * (x,t)E(x,t+τ) e iωτ dτ.
S opt (x,ω)= 2d W m,p | Ψ p (x) | 2 S mp opt (ω ω mp ) ,
S mp opt (ω)= γW 2d A mp * (t) A mp (t+τ) e iωτ dτ = γW 2d A mp * (0) A mp (τ) e iωτ dτ
i(x,t)=RP(x,t),
S 2 RF (x,ω)= i(x,t)i(x,t+τ) e iωτ dτ,
i(x,t)i(x,t+τ) = i(x,0)i(x,τ) .
i(x,t)=Rγ m, m ,p, p A mp (t) A m p * (t) e i( ω mp ω m p )t Ψ p (x) Ψ p * (x) .
i(x,0)i(x,τ) = R 2 γ 2 m,p, p m , p , p [ e i( ω m p ω m p )τ Ψ p (x) Ψ p * (x) Ψ p (x) Ψ p * (x) × A mp (0) A m p * (0) A m p (τ) A m p * (τ) ].
| A mp (0) A m p * (0) A m p (τ) A m p * (τ) | e i[ φ mp (0) φ m p (0)+ φ m p (τ) φ m p (τ) ] ,
i(x,0)i(x,τ) = R 2 γ 2 [ m, m ,p, p | A mp (0) | 2 | A m p (τ) | 2 | Ψ p (x) | 2 | Ψ p (x) | 2 + + m,p, p (p p ) A mp (0) A mp * (τ) A m p * (0) A m p (τ) | Ψ p (x) | 2 | Ψ p (x) | 2 e i( ω m p ω mp )τ ].
i(x,0)i(x,τ) RF = = R 2 γ 2 m,p p A mp (0) A mp * (τ) A m p * (0) A m p (τ) | Ψ p (x) | 2 | Ψ p (x) | 2 e i( ω m p ω mp )τ
A mp * (0) A mp (τ) = 2d γW S mp opt (ω) e iωτ dω 2π .
S 2 RF (x,ω)= = ( 2Rd W ) 2 m,p p | Ψ p (x) | 2 | Ψ p (x) | 2 S mp opt ( ω ) S m p opt ( ω ) e i( ω ω + ω mp ω m p +ω)τ d ω d ω 4 π 2 dτ = (2Rd/W) 2 m,p p | Ψ p (x) | 2 | Ψ p (x) | 2 S mp opt ( ω ) S m p opt ( ω + ω mp ω m p +ω) d ω 2π = (2Rd/W) 2 m,p p | Ψ p (x) | 2 | Ψ p (x) | 2 S mp opt ( ω ω) S m p opt [ ω ( ω m p ω mp )] d ω 2π = (2Rd/W) 2 m,p p | Ψ p (x) | 2 | Ψ p (x) | 2 { S mp opt (ω) S m p opt [ω( ω m p ω mp )] },
S 2 RF (x,ω)= ( 2Rd W ) 2 | H(ω) | 2 p p | Ψ p (x) | 2 | Ψ p (x) | 2 m { S mp opt (ω) S m p opt [ω( ω m p ω mp )] } .
S mp opt (ω)= S p opt g(ω/Δω) F m ,
g(ω/Δω) dω 2π =1,
m F m =1.
g (ω/Δ ω )=g(ω/Δω)g(ω/Δω)
S 2 RF (x,ω)= g (0) M ( 2Rd W ) 2 | H(ω) | 2 p p | Ψ p (x) | 2 | Ψ p (x) | 2 g [(ω ω p p )/Δ ω ] g (0) S p opt S p opt ,
m= m 0 M/2 m 0 +M/2 F m 2 = 1 M
S RF (x,ω)= S 2 RF (x,ω)+ S 2 RF (x,ω)=2 S 2 RF (x,ω)
S RF (x,ω)=ξ | H(ω) | 2 p =2 P p=1 p | Ψ p (x) | 2 | Ψ p (x) | 2 g [(ω ω p p )/Δ ω ] g (0) S p opt S p opt ,
S RF,NF (x,ω)= ξ 4 | H(ω) | 2 p =2 P p=1 p S p opt S p opt g [(ω ω p p )/Δ ω ] g (0) × | cos[ ( p p) πx W +π p p 2 ]cos[ ( p +p) πx W +π p +p 2 ] | 2 .
Ψ ˜ p ( k x )= 2πpW e i π 2 (p1) π 2 p 2 k x 2 W 2 sin( k x W+πp 2 ).
S RF,FF ( k x ,ω)= ξ | H(ω) | 2 p=1 P1 | Ψ ˜ p ( k x ) | 2 | Ψ ˜ p+1 ( k x ) | 2 g [(ω ω p,(p+1) )/Δ ω ] g (0) S p opt S p+1 opt = ξ | H(ω) | 2 p=1 P1 | 2 π 2 W 2 p(p+1)sin k x W ( π 2 p 2 k x 2 W 2 )[ π 2 (p+1) 2 k x 2 W 2 ] | 2 × g [ ω(2p+1)δω Δ ω ] g (0) S p opt S p+1 opt ,

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