Abstract

A phase-locked diode-laser system based on master-slave coupling of two-dimensional vertical-cavity surface-emitting laser (VCSEL) arrays by injection locking is presented. Frequencies and phases are adjusted by laser-trimmed microresistors. Additional beam-transformation optics consisting of two diffractive optical elements (DOEs) and a Fourier lens concentrates most of the far-field power in a nearly diffraction-limited beam. Both the VCSEL array and the microlens array are monolithically integrated and mounted in a compact module. With an array of 21 slave lasers a system coherence of 95% (for several hours) and of nearly 90% (for several months) has been demonstrated without any active phase control. The scalability of the output power has been verified by locking of an array of 77 slave lasers with a system coherence of 78%. The optical system efficiency is 20–23%; with beam-transformation optics this efficiency could be improved to 44%.

© 2003 Optical Society of America

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References

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  1. H. J. Unold, S. W. Z. Mahmoud, R. Jäger, M. Kicherer, M. C. Riedl, K. J. Ebeling, “Improving single-mode VCSEL performance by introducing a long monolithic cavity,” IEEE Photon. Technol. Lett. 12, 939–941 (2000).
    [CrossRef]
  2. M. Mikulla, “Tapered high-power, high-brightness diode lasers: design and performance,” in High-Power Diode Lasers, R. Diehl, ed., Vol. 78 of Topics in Applied Physics (Springer-Verlag, 2000), pp. 265–288.
    [CrossRef]
  3. M. Kuznetsov, F. Hakimi, R. Sprague, A. Mooradian, “Design and characteristics of high-power (0.5-W cw) diode pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
    [CrossRef]
  4. R. H. Rediker, K. A. Rauschenbach, R. P. Schloss, “Operation of a coherent ensemble of five diode lasers in an external cavity,” IEEE J. Quantum Electron. 27, 1582–1593 (1991).
    [CrossRef]
  5. D. Botez, “High-power monolithic phase-locked arrays of antiguided semiconductor diode lasers,” IEE Proc. J 139, 14–23 (1992).
  6. J. Levy, K. Roh, “Coherent array of 900 semiconductor laser amplifiers,” in Laser Diodes and Applications, K. J. Linden, P. R. Akkapeddi, eds., Proc. SPIE2382, 58–69 (1995).
    [CrossRef]
  7. L. Bartelt-Berger, U. Brauch, A. Giesen, H. Huegel, H. Opower, “Power-scalable system of phase-locked single-mode diode lasers,” Appl. Opt. 38, 5752–5760 (1999).
    [CrossRef]
  8. B. Lücke, G. Hergenhan, U. Brauch, A. Giesen, “Phase tuning of injection locked VCSELs,” IEEE Photon. Technol. Lett. 13, 100–102 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. I. Anke, E.-B. Kley, H. Huebner, B. Schnabel, R. Poechlmann, “Replication of micro-optical profiles in ORMOCER and other polymers,” in Micro-Optical Technologies for Measurement, Sensors, and Microsystems. O. M. Parriaux, ed., Proc. SPIE2783, 325–332 (1996).
    [CrossRef]
  15. E.-B. Kley, “Continuous profile writing by electron and optical lithography,” Microelectron. Eng. 34, 261–298 (1997).
    [CrossRef]
  16. G. Hergenhan, B. Lücke, U. Brauch, “Experiments on the sealability of the coherent coupling of VCSEL arrays,” in Vertical-Cavity Surface-Emitting Lasers VI, C. Lei, S. P. Kilcoyne, eds., Proc. SPIE4649, 158–167 (2002).
    [CrossRef]

2001

B. Lücke, G. Hergenhan, U. Brauch, A. Giesen, “Phase tuning of injection locked VCSELs,” IEEE Photon. Technol. Lett. 13, 100–102 (2001).
[CrossRef]

2000

H. J. Unold, S. W. Z. Mahmoud, R. Jäger, M. Kicherer, M. C. Riedl, K. J. Ebeling, “Improving single-mode VCSEL performance by introducing a long monolithic cavity,” IEEE Photon. Technol. Lett. 12, 939–941 (2000).
[CrossRef]

1999

M. Kuznetsov, F. Hakimi, R. Sprague, A. Mooradian, “Design and characteristics of high-power (0.5-W cw) diode pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

L. Bartelt-Berger, U. Brauch, A. Giesen, H. Huegel, H. Opower, “Power-scalable system of phase-locked single-mode diode lasers,” Appl. Opt. 38, 5752–5760 (1999).
[CrossRef]

1997

E.-B. Kley, “Continuous profile writing by electron and optical lithography,” Microelectron. Eng. 34, 261–298 (1997).
[CrossRef]

1992

D. Botez, “High-power monolithic phase-locked arrays of antiguided semiconductor diode lasers,” IEE Proc. J 139, 14–23 (1992).

1991

R. H. Rediker, K. A. Rauschenbach, R. P. Schloss, “Operation of a coherent ensemble of five diode lasers in an external cavity,” IEEE J. Quantum Electron. 27, 1582–1593 (1991).
[CrossRef]

1988

Anke, I.

I. Anke, E.-B. Kley, H. Huebner, B. Schnabel, R. Poechlmann, “Replication of micro-optical profiles in ORMOCER and other polymers,” in Micro-Optical Technologies for Measurement, Sensors, and Microsystems. O. M. Parriaux, ed., Proc. SPIE2783, 325–332 (1996).
[CrossRef]

Bartelt-Berger, L.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1999).

Botez, D.

D. Botez, “High-power monolithic phase-locked arrays of antiguided semiconductor diode lasers,” IEE Proc. J 139, 14–23 (1992).

Brauch, U.

B. Lücke, G. Hergenhan, U. Brauch, A. Giesen, “Phase tuning of injection locked VCSELs,” IEEE Photon. Technol. Lett. 13, 100–102 (2001).
[CrossRef]

L. Bartelt-Berger, U. Brauch, A. Giesen, H. Huegel, H. Opower, “Power-scalable system of phase-locked single-mode diode lasers,” Appl. Opt. 38, 5752–5760 (1999).
[CrossRef]

G. Hergenhan, B. Lücke, U. Brauch, “Experiments on the sealability of the coherent coupling of VCSEL arrays,” in Vertical-Cavity Surface-Emitting Lasers VI, C. Lei, S. P. Kilcoyne, eds., Proc. SPIE4649, 158–167 (2002).
[CrossRef]

Bryngdahl, O.

Cumme, M.

E.-B. Kley, M. Cumme, L. Wittig, A. Tuennermann, “Fabrication and properties of refractive microoptical profiles for lenses, lens-arrays, and beam shaping elements,” in Advanced Optical Manufacturing and Testing Technology 2000. L. Yang, H. M. Pollicove, Q. Xin, J. C. Wyant, eds., Proc. SPIE4231, 144–152 (2000).
[CrossRef]

Ebeling, K. J.

H. J. Unold, S. W. Z. Mahmoud, R. Jäger, M. Kicherer, M. C. Riedl, K. J. Ebeling, “Improving single-mode VCSEL performance by introducing a long monolithic cavity,” IEEE Photon. Technol. Lett. 12, 939–941 (2000).
[CrossRef]

Giesen, A.

B. Lücke, G. Hergenhan, U. Brauch, A. Giesen, “Phase tuning of injection locked VCSELs,” IEEE Photon. Technol. Lett. 13, 100–102 (2001).
[CrossRef]

L. Bartelt-Berger, U. Brauch, A. Giesen, H. Huegel, H. Opower, “Power-scalable system of phase-locked single-mode diode lasers,” Appl. Opt. 38, 5752–5760 (1999).
[CrossRef]

Hakimi, F.

M. Kuznetsov, F. Hakimi, R. Sprague, A. Mooradian, “Design and characteristics of high-power (0.5-W cw) diode pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

Hergenhan, G.

B. Lücke, G. Hergenhan, U. Brauch, A. Giesen, “Phase tuning of injection locked VCSELs,” IEEE Photon. Technol. Lett. 13, 100–102 (2001).
[CrossRef]

G. Hergenhan, B. Lücke, U. Brauch, “Experiments on the sealability of the coherent coupling of VCSEL arrays,” in Vertical-Cavity Surface-Emitting Lasers VI, C. Lei, S. P. Kilcoyne, eds., Proc. SPIE4649, 158–167 (2002).
[CrossRef]

Huebner, H.

I. Anke, E.-B. Kley, H. Huebner, B. Schnabel, R. Poechlmann, “Replication of micro-optical profiles in ORMOCER and other polymers,” in Micro-Optical Technologies for Measurement, Sensors, and Microsystems. O. M. Parriaux, ed., Proc. SPIE2783, 325–332 (1996).
[CrossRef]

Huegel, H.

Jäger, R.

H. J. Unold, S. W. Z. Mahmoud, R. Jäger, M. Kicherer, M. C. Riedl, K. J. Ebeling, “Improving single-mode VCSEL performance by introducing a long monolithic cavity,” IEEE Photon. Technol. Lett. 12, 939–941 (2000).
[CrossRef]

Kicherer, M.

H. J. Unold, S. W. Z. Mahmoud, R. Jäger, M. Kicherer, M. C. Riedl, K. J. Ebeling, “Improving single-mode VCSEL performance by introducing a long monolithic cavity,” IEEE Photon. Technol. Lett. 12, 939–941 (2000).
[CrossRef]

Kley, E.-B.

E.-B. Kley, “Continuous profile writing by electron and optical lithography,” Microelectron. Eng. 34, 261–298 (1997).
[CrossRef]

I. Anke, E.-B. Kley, H. Huebner, B. Schnabel, R. Poechlmann, “Replication of micro-optical profiles in ORMOCER and other polymers,” in Micro-Optical Technologies for Measurement, Sensors, and Microsystems. O. M. Parriaux, ed., Proc. SPIE2783, 325–332 (1996).
[CrossRef]

E.-B. Kley, M. Cumme, L. Wittig, A. Tuennermann, “Fabrication and properties of refractive microoptical profiles for lenses, lens-arrays, and beam shaping elements,” in Advanced Optical Manufacturing and Testing Technology 2000. L. Yang, H. M. Pollicove, Q. Xin, J. C. Wyant, eds., Proc. SPIE4231, 144–152 (2000).
[CrossRef]

Kuznetsov, M.

M. Kuznetsov, F. Hakimi, R. Sprague, A. Mooradian, “Design and characteristics of high-power (0.5-W cw) diode pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

Levy, J.

J. Levy, K. Roh, “Coherent array of 900 semiconductor laser amplifiers,” in Laser Diodes and Applications, K. J. Linden, P. R. Akkapeddi, eds., Proc. SPIE2382, 58–69 (1995).
[CrossRef]

Lücke, B.

B. Lücke, G. Hergenhan, U. Brauch, A. Giesen, “Phase tuning of injection locked VCSELs,” IEEE Photon. Technol. Lett. 13, 100–102 (2001).
[CrossRef]

B. Lücke, “Kohärente Kopplung von Vertikalemitter-Arrays,” Ph.D. dissertation (Universität Stuttgart, Stuttgart, Germany, 2002).

G. Hergenhan, B. Lücke, U. Brauch, “Experiments on the sealability of the coherent coupling of VCSEL arrays,” in Vertical-Cavity Surface-Emitting Lasers VI, C. Lei, S. P. Kilcoyne, eds., Proc. SPIE4649, 158–167 (2002).
[CrossRef]

Mahmoud, S. W. Z.

H. J. Unold, S. W. Z. Mahmoud, R. Jäger, M. Kicherer, M. C. Riedl, K. J. Ebeling, “Improving single-mode VCSEL performance by introducing a long monolithic cavity,” IEEE Photon. Technol. Lett. 12, 939–941 (2000).
[CrossRef]

Mikulla, M.

M. Mikulla, “Tapered high-power, high-brightness diode lasers: design and performance,” in High-Power Diode Lasers, R. Diehl, ed., Vol. 78 of Topics in Applied Physics (Springer-Verlag, 2000), pp. 265–288.
[CrossRef]

Mooradian, A.

M. Kuznetsov, F. Hakimi, R. Sprague, A. Mooradian, “Design and characteristics of high-power (0.5-W cw) diode pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

Opower, H.

Poechlmann, R.

I. Anke, E.-B. Kley, H. Huebner, B. Schnabel, R. Poechlmann, “Replication of micro-optical profiles in ORMOCER and other polymers,” in Micro-Optical Technologies for Measurement, Sensors, and Microsystems. O. M. Parriaux, ed., Proc. SPIE2783, 325–332 (1996).
[CrossRef]

Rauschenbach, K. A.

R. H. Rediker, K. A. Rauschenbach, R. P. Schloss, “Operation of a coherent ensemble of five diode lasers in an external cavity,” IEEE J. Quantum Electron. 27, 1582–1593 (1991).
[CrossRef]

Rediker, R. H.

R. H. Rediker, K. A. Rauschenbach, R. P. Schloss, “Operation of a coherent ensemble of five diode lasers in an external cavity,” IEEE J. Quantum Electron. 27, 1582–1593 (1991).
[CrossRef]

Riedl, M. C.

H. J. Unold, S. W. Z. Mahmoud, R. Jäger, M. Kicherer, M. C. Riedl, K. J. Ebeling, “Improving single-mode VCSEL performance by introducing a long monolithic cavity,” IEEE Photon. Technol. Lett. 12, 939–941 (2000).
[CrossRef]

Roh, K.

J. Levy, K. Roh, “Coherent array of 900 semiconductor laser amplifiers,” in Laser Diodes and Applications, K. J. Linden, P. R. Akkapeddi, eds., Proc. SPIE2382, 58–69 (1995).
[CrossRef]

Schloss, R. P.

R. H. Rediker, K. A. Rauschenbach, R. P. Schloss, “Operation of a coherent ensemble of five diode lasers in an external cavity,” IEEE J. Quantum Electron. 27, 1582–1593 (1991).
[CrossRef]

Schnabel, B.

I. Anke, E.-B. Kley, H. Huebner, B. Schnabel, R. Poechlmann, “Replication of micro-optical profiles in ORMOCER and other polymers,” in Micro-Optical Technologies for Measurement, Sensors, and Microsystems. O. M. Parriaux, ed., Proc. SPIE2783, 325–332 (1996).
[CrossRef]

Siegman, A.

A. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

Sprague, R.

M. Kuznetsov, F. Hakimi, R. Sprague, A. Mooradian, “Design and characteristics of high-power (0.5-W cw) diode pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

Tuennermann, A.

E.-B. Kley, M. Cumme, L. Wittig, A. Tuennermann, “Fabrication and properties of refractive microoptical profiles for lenses, lens-arrays, and beam shaping elements,” in Advanced Optical Manufacturing and Testing Technology 2000. L. Yang, H. M. Pollicove, Q. Xin, J. C. Wyant, eds., Proc. SPIE4231, 144–152 (2000).
[CrossRef]

Unold, H. J.

H. J. Unold, S. W. Z. Mahmoud, R. Jäger, M. Kicherer, M. C. Riedl, K. J. Ebeling, “Improving single-mode VCSEL performance by introducing a long monolithic cavity,” IEEE Photon. Technol. Lett. 12, 939–941 (2000).
[CrossRef]

Wittig, L.

E.-B. Kley, M. Cumme, L. Wittig, A. Tuennermann, “Fabrication and properties of refractive microoptical profiles for lenses, lens-arrays, and beam shaping elements,” in Advanced Optical Manufacturing and Testing Technology 2000. L. Yang, H. M. Pollicove, Q. Xin, J. C. Wyant, eds., Proc. SPIE4231, 144–152 (2000).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1999).

Wyrowski, F.

Appl. Opt.

IEE Proc. J

D. Botez, “High-power monolithic phase-locked arrays of antiguided semiconductor diode lasers,” IEE Proc. J 139, 14–23 (1992).

IEEE J. Quantum Electron.

R. H. Rediker, K. A. Rauschenbach, R. P. Schloss, “Operation of a coherent ensemble of five diode lasers in an external cavity,” IEEE J. Quantum Electron. 27, 1582–1593 (1991).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

M. Kuznetsov, F. Hakimi, R. Sprague, A. Mooradian, “Design and characteristics of high-power (0.5-W cw) diode pumped vertical-external-cavity surface-emitting semiconductor lasers with circular TEM00 beams,” IEEE J. Sel. Top. Quantum Electron. 5, 561–573 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

H. J. Unold, S. W. Z. Mahmoud, R. Jäger, M. Kicherer, M. C. Riedl, K. J. Ebeling, “Improving single-mode VCSEL performance by introducing a long monolithic cavity,” IEEE Photon. Technol. Lett. 12, 939–941 (2000).
[CrossRef]

B. Lücke, G. Hergenhan, U. Brauch, A. Giesen, “Phase tuning of injection locked VCSELs,” IEEE Photon. Technol. Lett. 13, 100–102 (2001).
[CrossRef]

J. Opt. Soc. Am. A

Microelectron. Eng.

E.-B. Kley, “Continuous profile writing by electron and optical lithography,” Microelectron. Eng. 34, 261–298 (1997).
[CrossRef]

Other

G. Hergenhan, B. Lücke, U. Brauch, “Experiments on the sealability of the coherent coupling of VCSEL arrays,” in Vertical-Cavity Surface-Emitting Lasers VI, C. Lei, S. P. Kilcoyne, eds., Proc. SPIE4649, 158–167 (2002).
[CrossRef]

E.-B. Kley, M. Cumme, L. Wittig, A. Tuennermann, “Fabrication and properties of refractive microoptical profiles for lenses, lens-arrays, and beam shaping elements,” in Advanced Optical Manufacturing and Testing Technology 2000. L. Yang, H. M. Pollicove, Q. Xin, J. C. Wyant, eds., Proc. SPIE4231, 144–152 (2000).
[CrossRef]

I. Anke, E.-B. Kley, H. Huebner, B. Schnabel, R. Poechlmann, “Replication of micro-optical profiles in ORMOCER and other polymers,” in Micro-Optical Technologies for Measurement, Sensors, and Microsystems. O. M. Parriaux, ed., Proc. SPIE2783, 325–332 (1996).
[CrossRef]

J. Levy, K. Roh, “Coherent array of 900 semiconductor laser amplifiers,” in Laser Diodes and Applications, K. J. Linden, P. R. Akkapeddi, eds., Proc. SPIE2382, 58–69 (1995).
[CrossRef]

B. Lücke, “Kohärente Kopplung von Vertikalemitter-Arrays,” Ph.D. dissertation (Universität Stuttgart, Stuttgart, Germany, 2002).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1999).

A. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

M. Mikulla, “Tapered high-power, high-brightness diode lasers: design and performance,” in High-Power Diode Lasers, R. Diehl, ed., Vol. 78 of Topics in Applied Physics (Springer-Verlag, 2000), pp. 265–288.
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Two-dimensional array of VCSELs with a two-dimensional microlens array for collimation. Both laser and lens matrices are defined by lithography, facilitating production of well-collimated and parallel beams.

Fig. 2
Fig. 2

Limits of stable locking as a function of the injected master power relative to slave output power P M /P S . The slave current is I = 4 mA, i.e., 2.5 times threshold current I th. Solid curves, locking range calculated according to Eq. (1) with8 ΔνL=±105GHzPM/PS.

Fig. 3
Fig. 3

Measured and calculated maximal injection ratio for stable locking as a function of slave current I (normalized to threshold current I th = 1.5 mA).9

Fig. 4
Fig. 4

Electrical layout of the power supply for the VCSEL array when there is one voltage source for all lasers. Series resistors adjust individual current.

Fig. 5
Fig. 5

VCSEL chip (a) with circuit paths to connect each laser to its series resistor for current (frequency) tuning and (b) with plus/minus connectors only and on-chip series resistors for the current distribution.

Fig. 6
Fig. 6

Principal optical setup for superimposing the coherent beams of the VCSEL array in the Fourier (far-field) plane. Shaded areas, 1/e 2 diameters in the y direction along the z axis.

Fig. 7
Fig. 7

Beam-transformation optics. Top, optical setup for transforming an array of Gaussian beams into a single Gaussian beam by use of phase plates PP1 and PP2. Bottom, cross section of the power-density and phase distribution, respectively, in the near-field plane (a) just before and (b) just after phase modulation by plate PP1 and in the far-field plane (c) before and (d) after phase modulation by plate PP2.

Fig. 8
Fig. 8

Cross sections of top, the far-field power-density distribution and bottom, the far-field phase distribution Φ(X) of phase modulation PP2(X) by phase plate PP2 and of phase distribution Φout(X) [= (PP2(X) + Φ(X))] behind plate PP2.

Fig. 9
Fig. 9

Macroscopic optical setup of the phase-locked diode-laser system including optics for injection locking of the VCSEL array and for separation of master and slave beams.

Fig. 10
Fig. 10

Optomodule: (a) photo, (b) schematic drawing. PGA, pin-grid array.

Fig. 11
Fig. 11

Concept for a hybrid integrated module with all optical elements arranged on the surface of a glass-silica block.

Fig. 12
Fig. 12

Typical data plots of Avalon and Osram VCSELs. (a) Laser output power and voltage versus electrical current. Shaded areas, range usable for coupling from 1.5 times threshold current through multimode threshold current. (b) Electrical-to-optical efficiency versus electrical current. (c) Wavelength shift versus electrical current. ΔλAvalon and ΔλOsram mark the maximum wavelength shift within the usable working range. (d) Dissipated power as a function of electrical current. Highlighted are typical values under phase-locking conditions.

Fig. 13
Fig. 13

Microresistor array. The dark rectangles are the actual thin-film resistors; the central square is the common electrode. Inset, blow-up of one of the resistors including the laser-trim mark.

Fig. 14
Fig. 14

19 × 19 VCSEL chip (Osram) soldered onto a heat sink and bonded with gold wires to a PGA.

Fig. 15
Fig. 15

Far-field power-density distribution of the phase-locked 5 × 5 VCSEL array (Avalon) measured behind a lens with a 200-mm focal length. It is normalized to the maximum power density of the incoherent distribution.

Fig. 16
Fig. 16

Long-term stability of the coherent superposition determined by alternate measurement of coherent and incoherent on-axis power density versus time. The measured peak power density is given by the ratio coherent/incoherent of the corresponding data points. For comparison, also shown are the ideally coherent and the incoherent curves.

Fig. 17
Fig. 17

(a) Near-field and (b) far-field power-density distributions of the phase-locked 9 × 9 VCSEL array (Osram) measured in the focal plane of an f = 140 mm lens. On this scale the incoherent power-density distribution would have an amplitude of 1.

Fig. 18
Fig. 18

(a) Layout of the 9 × 9 beam splitting-combining optics. One tests the optics independently of the coherent coupling by injecting a Gaussian beam from right to left and examining the power-density distribution in (c) the near-field plane, (b) after superposition in the far-field plane, or (d) when the mirror has been inserted, after recombination.

Fig. 19
Fig. 19

Measured power-density distribution of the 9 × 9 VCSEL array output after beam transformation with the optics shown in Fig. 7.

Tables (3)

Tables Icon

Table 1 Data on the VCSEL Chips Used in the Experimentsa

Tables Icon

Table 2 Positioning Tolerancesa

Tables Icon

Table 3 Results of the Coherent Superpositiona

Equations (5)

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

ΔνL=±1-R2πτres1+α2PMPS,
ΔΦS=-arcsinΔνΔνL-arctan α.
IFX, Y=sinπafFLλ nxXsinπafFLλ XsinπafFLλ nyYsinπafFLλ Y,
Xi0=i fFLλnxa, i=±1, ±2, , ilnx, l  N,
Yj0=j fFLλnya, j=±1, ±2, , jlny, l  N.

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