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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 7 — Mar. 31, 2008
  • pp: 4452–4464
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Evolution from modal to spatially incoherent emission of a broad-area VCSEL

Shyam K. Mandre, Wolfgang Elsäßer, Ingo Fischer, Michael Peeters, and Guy Verschaffelt  »View Author Affiliations


Optics Express, Vol. 16, Issue 7, pp. 4452-4464 (2008)
http://dx.doi.org/10.1364/OE.16.004452


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Abstract

Broad-area vertical-cavity surface-emitting lasers (BA-VCSELs) can exhibit a state of spatially incoherent emission, as we recently reported in [M. Peeters et al., Opt. Express, 13, 9337 (2005)]. Here, we experimentally study the evolution of a BA-VCSEL under pulsed operation from well-defined modal emission with a multitude of transverse cavity modes to such spatially incoherent emission. The transition is studied using a high-speed intensified CCD camera and differential image analysis with which single-shot measurements of the imaged nearfield, farfield, spatial coherence, and spectral emission properties are acquired. This combination of experimental characterization tools allows for a detailed description of the BA-VCSEL’s emission behavior, which is necessary for an in-depth understanding of the processes involved. We find the interplay between the thermal chirp and the build-up of a spatially distributed thermal lens to be decisive for the break-up of the global cavity modes.

© 2008 Optical Society of America

1. Introduction

Over the last 20 years, some attractive emission properties of vertical-cavity surface-emitting lasers (VCSELs) compared to edge-emitting semiconductor lasers have led to VCSELs’ establishment as a prominent device within the semiconductor laser family. These properties include near-circular beam profiles, in contrast to astigmatic beams emitted by edge-emitting SLs. A further property of VCSELs, resulting from their short cavities, is single longitudinal mode emission. For a small enough aperture diameter (typically less than ~5 µm) the emission can even be in a single, fundamental transverse mode with an output power up to 7 mW [1

1. A. Haglund, J.S. Gustavsson, J. Vukusic, P. Modh, and A. Larsson, “Single Fundamental-Mode Output Power Exceeding 6 mW From VCSELs With a Shallow Surface Relief,” IEEE Photon. Technol. Lett. 16, 368–370 (2004). [CrossRef]

]. VCSELs find application in different fields, most noteworthy in data-communication and optical sensing. For other applications, such as printing or projection systems, higher output powers are required. This can be achieved by increasing the VCSELs’ aperture diameter up to 100 µm. With these larger devices, CW output powers up to 100 mW are attainable. However, the large Fresnel number of these broad-area VCSELs (BA-VCSELs) typically leads to the onset of a multitude of higher order transverse modes [2

2. C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, L. T. Florez, J. P. Harbison, and N. G. Stoffel, “Transverse mode characteristics of vertical cavity surface-emitting lasers,” Appl. Phys. Lett. 57, 218–220 (1990). [CrossRef]

, 3

3. C. Degen, I. Fischer, and W. Elsäßer, “Transverse modes in oxide confined VCSELs: influence of pump profile, spatial hole burning, and thermal effects,” Opt. Express 5, 38–47 (1999). [CrossRef] [PubMed]

]. This results in a structured and more divergent farfield (FF) beam profile, thus deteriorating the beam quality of the emitted light. The multi-transverse mode emission is accompanied by complex emission dynamics, which can be observed as a combination of spatiotemporal and polarization dynamics. The build-up of transverse modes, the spatiotemporal emission dynamics, and the interplay of the responsible mechanisms have been studied previously [4

4. A. Valle, J. Sarma, and K. A. Shore, “Dynamics of transverse mode competition in vertical cavity surface emitting laser diodes,” Opt. Commun. 115, 297–302 (1995). [CrossRef]

, 5

5. M. Giudici, J. R. Tredicce, G. Vaschenko, J. J. Rocca, and C. S. Menoni, “Spatio-temporal dynamics in vertical cavity surface emitting lasers excited by fast electrical pulses,” Opt. Commun. 158, 313–321 (1998). [CrossRef]

, 6

6. J. Mulet and S. Balle, “Transverse mode dynamics in vertical-cavity surface-emitting lasers: spatiotemporal versus modal expansion descriptions,” Phys. Rev. A 66, 053 802 (2002). [CrossRef]

, 7

7. A. Barchanski, T. Gensty, C. Degen, I. Fischer, and Elsäßer, “Picosecond emission dynamics of vertical-cavity surface-emitting lasers: spatial, spectral, and polarization-resolved characterization,” IEEE J. Quantum Electron. 39, 850–858 (2003). [CrossRef]

, 8

8. K. Becker, I. Fischer, and W. Elsäßer, “Spatio-temporal emission dynamics of VCSELs: modal competition in the turn-on behavior,” in Proceedings of SPIE, D. Lenstra, G. Morthier, T. Erneux, and M. Pessa, eds., 5452, 452 (2004).

, 9

9. P. Debernardi, G. P. Bava, C. Degen, I. Fischer, and Elsäßer, “Influence of anisotropies on transverse modes in oxide-confined VCSELs,” IEEE J. Quantum Electron. 38, 73–84 (2002). [CrossRef]

].

2. Characterization of the BA-VCSEL’s emission profile

2.1. Device characteristics

For our measurements we use an oxide confined BA-VCSEL with an aperture diameter of 50 µm. The BA-VCSEL emits at a wavelength of ~840 nm with a maximum cw output power of ~70 mW at an injected current of 80 mA. The VCSEL’s threshold current Ithr is approximately 14 mA. The experimental setup is modified depending on the measurement performed and will be discussed in the corresponding paragraphs.

2.2. Setup

The BA-VCSEL’s emission is recorded by a 10-bit, fast-gated intensified CCD-camera (iCCD, 4picos, Stanford Computer Optics, Inc.) with an exposure time of 300 ps. We can therefore observe the emission dynamics of the BA-VCSEL and the processes responsible for the observed emission behavior on these short ps timescales. The camera is used to trigger a pulse generator and, subsequently, the BA-VCSEL at 50 Hz. The pulse widths with which we drive the BA-VCSEL are chosen between 10 and 100 µs.With the resulting low duty cycle of less than 0.5%, we can assume that the VCSEL cools down to the sub-mount temperature (set at 28 °C) before the consecutive pulse is applied. Our setup allows us to acquire single-shot measurements at different temporal positions within the VCSEL-pulse by shifting the exposure gate-window with an adjustable delay line. Therefore, we can study the evolution of the BA-VCSEL’s emission during the entire pulse. To measure FF profiles of the emission, we place the camera at a distance of ~2.5 cm from the laser facet. Alternatively, we acquire magnified NF profiles of the emission by inserting an imaging lens into the setup.

Fig. 1. Upper row: a) Nearfield intensity distribution and b) corresponding farfield intensity distribution, both at τ~10 ns. Bottom row: c) Nearfield intensity distribution and d) corresponding farfield intensity distribution, both at τ ~5 µs. The pulse amplitude is 160 mA in all cases.

2.3. Nearfield - Farfield Comparison

Fig. 1 depicts single-shot measurements of the NF and corresponding FF intensity distributions at the onset of laser emission and after 5 µs. The measured profiles have been verified to be essentially the same for each pulse, therefore we can compare the NF and corresponding FF profiles even though they have not been measured simultaneously. We find that the BA-VCSEL’s emission undergoes a radical change during the first microseconds after turn-on. Figure 1(a) depicts the NF intensity distribution at τ~10 ns, where τ is the time after the onset of laser emission. At this early stage of emission, the NF emission is dominated by ring-shaped high order Gauss-Laguerre modes (daisy-modes) emitted at the periphery of the VCSEL-aperture. The transverse modes are more clearly resolved in the corresponding FF profile at τ~10 ns in Fig. 1(b). The FF exhibits structured and divergent emission and reflects the highly multimode emission of the corresponding NF profile. The FF and NF at τ~10 ns are in agreement with the results obtained so far for multimode VCSELs [6

6. J. Mulet and S. Balle, “Transverse mode dynamics in vertical-cavity surface-emitting lasers: spatiotemporal versus modal expansion descriptions,” Phys. Rev. A 66, 053 802 (2002). [CrossRef]

, 7

7. A. Barchanski, T. Gensty, C. Degen, I. Fischer, and Elsäßer, “Picosecond emission dynamics of vertical-cavity surface-emitting lasers: spatial, spectral, and polarization-resolved characterization,” IEEE J. Quantum Electron. 39, 850–858 (2003). [CrossRef]

, 8

8. K. Becker, I. Fischer, and W. Elsäßer, “Spatio-temporal emission dynamics of VCSELs: modal competition in the turn-on behavior,” in Proceedings of SPIE, D. Lenstra, G. Morthier, T. Erneux, and M. Pessa, eds., 5452, 452 (2004).

].

In contrast, the measurements at τ~5 µs shown in Fig. 1(c) and (d) exhibit a significantly different behavior. The NF intensity distribution at τ~5 µs (see Fig. 1(c)) is similar and only slightly blurred compared to the NF profile at τ~10 ns. The NF at τ~5 µs again suggests that high-order transverse modes dominate the BA-VCSEL’s emission. However, the corresponding FF profile at τ~5 µs does not confirm this assumption. Instead of a structured FF profile reflecting multi-mode emission, the FF in Fig. 1(d) exhibits a Gaussian intensity distribution. Moreover, while the structured FF profiles at the beginning of the emission [such as in Fig. 1(b)] can be obtained by Fourier transform of the corresponding NF profiles [such as in Fig. 1(a)], numerical analysis did not provide any phase profile with which the Gaussian FF profile such as in Fig. 1(d) could be constructed by Fourier transform of the corresponding NF such as in Fig. 1(c). The emission now corresponds to the spatially incoherent emission discussed in [10

10. M. Peeters, G. Verschaffelt, H. Thienpont, S. K. Mandre, I. Fischer, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express 13, 9337–9345 (2005). [CrossRef] [PubMed]

]. Because of the similarity between the NF profiles depicted in Figs. 1(a) at τ~10 ns and (c) at τ~5 µs, we can exclude significant changes in the carrier distribution from being responsible for the drastic change in the observed FF profile. In addition, the non-Gaussian NF profile in Fig. 1(c) proves that single, fundamental mode emission is not the origin of the Gaussian FF profile.

Fig. 2. Sequence of single-shot measurements of the BA-VCSEL’s farfield emission behavior. The BA-VCSEL was operated in quasi-cw mode with pulse widths of 30 µs and pulse amplitude of 160 mA. The times denoted below the images are the temporal positions of the single-shot measurements after turn-on.

2.4. Farfield evolution

The FF profile thus exhibits a drastic change during the emitted pulse, whereas the changes observed in the NF intensity profiles are rather minor. The changes happen on a time-scale much slower than the photon and carrier lifetimes, indicating that other effects play a dominant role. To understand the responsible processes we will investigate the spectral evolution during the pulse.

Fig. 3. Optical spectrum for a CW current of 60 mA.

3. Spectral Evolution

3.1. Spectrally dispersed NF

Before we look at the temporally resolved spectra during a pulse, we first plot in Fig. 3 the optical spectrum of the BA-VCSEL when it is driven by a CW current of 60 mA. This spectrum serves as a reference for the spectral measurements under pulsed conditions as presented below. In this optical spectrum, multiple groups of modes are clearly visible. The broad peaks in Fig. 3 (e.g. at 842.05 nm) indicate that some of the transverse modes are spectrally overlapping with each other. Not all of the modes can be resolved because of the limited resolution of the optical spectrum analyzer (0.01 nm or 4.25 GHz at a center wavelength of 842 nm).

Fig. 4. Sequence of single-shot measurements of the BA-VCSEL’s spectrally dispersed NF emission behavior. The BA-VCSEL was operated in quasi-cw mode with pulse widths of 30 µs and pulse amplitude of 160 mA. The times denoted in the upper right corner of the images are the positions of the single-shot measurements after turn-on. These temporal positions correspond approximately to the positions of the FF measurements in Fig. 2.
Fig. 5. Average wavelength shift of the BA-VCSEL’s emission. The BA-VCSEL was operated in quasi-cw mode with pulse widths of 30 µs and pulse amplitude of 160 mA. The spectral positions were obtained by determining the intensity-weighted averages of the single-shot measurements at different τ.
Fig. 6. Left: RF spectrum of the emitted intensity for CW, 70mA (light gray); and for 1 µs pulses with 1% duty cycle and 22 mA amplitude (dark gray) and 145 mA amplitude (black). The noise floor of the measurement setup is at -58 dBm. Right: Difference between the maximum and minimum amplitude of the RF spectrum as a function of the pulse amplitude for a fixed pulse width of 1 µs and duty cycle of 1% (circles). The line gives the trend and is intended to guide the eye.

3.2. RF spectrum

We measure the RF spectrum using a 12 GHz photodiode (1554-B, NewFocus) coupled via an amplifier (18 dB gain, 20 GHz bandwidth, 1422, NewFocus) to an electrical spectrum analyzer (30 GHz bandwidth, MS266C, Anritsu). As the duty cycle of the pulses we apply is only 1%, we get a very small signal (≈2dB above noise floor) if we measure the RF spectrum directly. In order to resolve the RF spectrum from the noise background, but also to measure the spectrum at different positions during a pulse, we use the gating capabilities of the electrical spectrum analyzer. That way, the spectrum will only be measured during the gate window (set to its minimum of 2 µs) after a trigger from the pulse driver has arrived. Both the resolution bandwidth and the video bandwidth of the electrical spectrum analyzer are set to 1 MHz as a compromise between high spectral resolution and high signal-to-noise ratio. For these settings, the time to measure one frequency component of the spectrum is 3 µs. Therefore, only one frequency component will be measured each time the gating window is open. Because of the minimum gate length of 2 µs, we are not able to measure the RF spectrum for pulses much shorter than 2 µs. Nevertheless, we can select different temporal positions within longer pulses by changing the delay of the trigger signal.

4. Temporally resolved spatial coherence

As stated before, we attribute the simultaneous appearance of the Gaussian FF, the parabola shaped spectrally dispersed NF and the flat RF spectrum to a loss of spatial coherence. To finally prove this, we directly characterize the transition to spatially incoherent emission by measuring the evolution of the NF spatial coherence. We do not measure the full spatial coherence function over the entire VCSEL aperture, but we use a pair of pinholes to select two positions in the imaged near field. The two pinholes are rectangular shaped with a size of 0.1 mm by 0.08 mm and have a center-to-center separation of 0.3 mm. The diameter of the imaged NF is 2.26 mm. Therefore, the magnification of the imaging setup is 45. The coherence radius that we extract from the measured FF divergence angle is 1.4 µm and the imaged coherence diameter is thus 0.126 mm. We have chosen the pinhole separation to be somewhat larger than the imaged coherence diameter (0.3 mm compared to 0.126 mm) such that the visibility of the interference fringes drops to almost zero in case of incoherent emission. The visibility of the resulting interference pattern is then used as a measure for the degree of spatial coherence.

Fig. 7. Evolution of the degree of spatial coherence during a pulse train for a pulse amplitude of 22 mA (gray circles) and of 145 mA (black squares). The lines are only intended to guide the eye. The BA-VCSEL was operated in quasi-cw mode with pulse widths of 40 ns.

5. Evolution back to continuous-wave operation

Fig. 8. Sequence of single-shot measurements of the BA-VCSEL’s spectrally dispersed NF emission behavior. The BA-VCSEL was operated in quasi-cw mode with pulse widths of approximately 105 µs. The times denoted in the upper right corner of the images are the positions of the single-shot measurements after turn-on.
Fig. 9. RF spectrum of the emitted intensity at different temporal position within a 100 µs pulse with amplitude of 145 mA amplitude: at 2 µs (black), at 50 µs (dark gray) and at 98 µs (light gray). The noise floor of the measurement setup is at -58 dBm

6. Discussion

We can now assume that the BA-VCSEL lases in a superposition of independent “coherence islands” instead of in cavity modes as was introduced in [10

10. M. Peeters, G. Verschaffelt, H. Thienpont, S. K. Mandre, I. Fischer, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express 13, 9337–9345 (2005). [CrossRef] [PubMed]

]. Thus, the BA-VCSEL can be considered a quasi-homogeneous Schell-model source characterized by spatially incoherent emission and a Gaussian FF distribution. The characteristic parabola-like structure of the spectrally dispersed NF profile shown in Fig. 4, e.g. at τ~1 µs, is a direct result of the inhomogeneous heating of the device and the emission in coherence islands. In [15

15. S. K. Mandre, W. Elsäßer, I. Fischer, M. Peeters, and G. Verschaffelt, “Determining the temporally and radially resolved temperature distribution inside a pulsed broad-area vertical-cavity surface-emitting laser cavity,” Appl. Phys. Lett. 89, 151 106 (2006). [CrossRef]

], we have shown how information on the radial temperature distribution within the VCSEL-cavity can be extracted from a sequence of such optical spectra.

7. Conclusions

We studied the evolution of a BA-VCSEL’s emission towards the spatially incoherent state of emission discussed in [10]. The measurements show the onset of high order transverse modes shortly after turn-on. Heating of the device then leads to the formation of a thermal lens which increases the time for modes to build-up in the cavity. Additionally, the device experiences a fast thermal chirp, which in combination with the thermal lens, prevents the build-up of cavity modes. Our measurements show that the transition to incoherent emission occurs within the first microsecond of emission. The transition back to emission in well-defined transverse modes takes place around 100 µs after turn-on.

The results presented here provide insight into the processes involved during the evolution of the emission. The emission properties may be harnessed for various applications where high output powers with low degree of spatial coherence are required. Possible applications for such a spatially incoherent light source may include illumination or utilization for projection beams, where the low spatial coherence of the emission might lead to a reduction in the speckle-contrast. Furthermore, our results might help to design laser structures suitable for spatially incoherent emission.

Acknowledgements

The authors are indebted to M. Grabherr of U.L.M.-Photonics for providing the devices. I. Fischer, M. Peeters and G. Verschaffelt acknowledge the IWT project 030501, the IAP P6/10 network “Photonics@be”, the VUB-IOF project “Micro-photonics” and EU FP6/NEMO network for financial support.

1.

A. Haglund, J.S. Gustavsson, J. Vukusic, P. Modh, and A. Larsson, “Single Fundamental-Mode Output Power Exceeding 6 mW From VCSELs With a Shallow Surface Relief,” IEEE Photon. Technol. Lett. 16, 368–370 (2004). [CrossRef]

2.

C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, L. T. Florez, J. P. Harbison, and N. G. Stoffel, “Transverse mode characteristics of vertical cavity surface-emitting lasers,” Appl. Phys. Lett. 57, 218–220 (1990). [CrossRef]

3.

C. Degen, I. Fischer, and W. Elsäßer, “Transverse modes in oxide confined VCSELs: influence of pump profile, spatial hole burning, and thermal effects,” Opt. Express 5, 38–47 (1999). [CrossRef] [PubMed]

4.

A. Valle, J. Sarma, and K. A. Shore, “Dynamics of transverse mode competition in vertical cavity surface emitting laser diodes,” Opt. Commun. 115, 297–302 (1995). [CrossRef]

5.

M. Giudici, J. R. Tredicce, G. Vaschenko, J. J. Rocca, and C. S. Menoni, “Spatio-temporal dynamics in vertical cavity surface emitting lasers excited by fast electrical pulses,” Opt. Commun. 158, 313–321 (1998). [CrossRef]

6.

J. Mulet and S. Balle, “Transverse mode dynamics in vertical-cavity surface-emitting lasers: spatiotemporal versus modal expansion descriptions,” Phys. Rev. A 66, 053 802 (2002). [CrossRef]

7.

A. Barchanski, T. Gensty, C. Degen, I. Fischer, and Elsäßer, “Picosecond emission dynamics of vertical-cavity surface-emitting lasers: spatial, spectral, and polarization-resolved characterization,” IEEE J. Quantum Electron. 39, 850–858 (2003). [CrossRef]

8.

K. Becker, I. Fischer, and W. Elsäßer, “Spatio-temporal emission dynamics of VCSELs: modal competition in the turn-on behavior,” in Proceedings of SPIE, D. Lenstra, G. Morthier, T. Erneux, and M. Pessa, eds., 5452, 452 (2004).

9.

P. Debernardi, G. P. Bava, C. Degen, I. Fischer, and Elsäßer, “Influence of anisotropies on transverse modes in oxide-confined VCSELs,” IEEE J. Quantum Electron. 38, 73–84 (2002). [CrossRef]

10.

M. Peeters, G. Verschaffelt, H. Thienpont, S. K. Mandre, I. Fischer, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express 13, 9337–9345 (2005). [CrossRef] [PubMed]

11.

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938). [CrossRef]

12.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967). [CrossRef]

13.

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams,” Opt. Lett. 2, 27–29 (1978). [CrossRef] [PubMed]

14.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Pr., 1995).

15.

S. K. Mandre, W. Elsäßer, I. Fischer, M. Peeters, and G. Verschaffelt, “Determining the temporally and radially resolved temperature distribution inside a pulsed broad-area vertical-cavity surface-emitting laser cavity,” Appl. Phys. Lett. 89, 151 106 (2006). [CrossRef]

16.

L. Wang and Q. Lin, “The evolutions of the spectrum and spatial coherence of laser radiation in resonators with hard apertures and phase modulation,” IEEE J. Quantum Electron. 39, 749–758 (2003). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.4070) Coherence and statistical optics : Modes
(190.3100) Nonlinear optics : Instabilities and chaos
(140.7260) Lasers and laser optics : Vertical cavity surface emitting lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: November 9, 2007
Revised Manuscript: January 8, 2008
Manuscript Accepted: January 8, 2008
Published: March 18, 2008

Citation
Shyam K. Mandre, Wolfgang Elsäßer, Ingo Fischer, Michael Peeters, and Guy Verschaffelt, "Evolution from modal to spatially incoherent emission of a broad-area VCSEL," Opt. Express 16, 4452-4464 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4452


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References

  1. A. Haglund, J. S. Gustavsson, J. Vukusic, P. Modh, and A. Larsson, "Single Fundamental-Mode Output Power Exceeding 6 mW From VCSELs With a Shallow Surface Relief," IEEE Photon. Technol. Lett. 16, 368-370 (2004). [CrossRef]
  2. C. J. Chang-Hasnain, M. Orenstein, A. Von Lehmen, L. T. Florez, J. P. Harbison, and N. G. Stoffel, "Transverse mode characteristics of vertical cavity surface-emitting lasers," Appl. Phys. Lett. 57, 218-220 (1990). [CrossRef]
  3. C. Degen, I. Fischer, and W. Elsaßer, "Transverse modes in oxide confined VCSELs: influence of pump profile, spatial hole burning, and thermal effects," Opt. Express 5, 38-47 (1999). [CrossRef] [PubMed]
  4. A. Valle, J. Sarma, and K. A. Shore, "Dynamics of transverse mode competition in vertical cavity surface emitting laser diodes," Opt. Commun. 115, 297-302 (1995). [CrossRef]
  5. M. Giudici, J. R. Tredicce, G. Vaschenko, J. J. Rocca, and C. S. Menoni, "Spatio-temporal dynamics in vertical cavity surface emitting lasers excited by fast electrical pulses," Opt. Commun. 158, 313-321 (1998). [CrossRef]
  6. J. Mulet and S. Balle, "Transverse mode dynamics in vertical-cavity surface-emitting lasers: spatiotemporal versus modal expansion descriptions," Phys. Rev. A 66,053 802 (2002). [CrossRef]
  7. A. Barchanski, T. Gensty, C. Degen, I. Fischer, and Elsäßer, "Picosecond emission dynamics of vertical-cavity surface-emitting lasers: spatial, spectral, and polarization-resolved characterization," IEEE J. Quantum Electron. 39, 850-858 (2003). [CrossRef]
  8. K. Becker, I. Fischer, and W. Elsäßer, "Spatio-temporal emission dynamics of VCSELs: modal competition in the turn-on behavior," in Proceedings of SPIE, D. Lenstra, G. Morthier, T. Erneux, and M. Pessa, eds., 5452, 452 (2004).
  9. P. Debernardi, G. P. Bava, C. Degen, I. Fischer, and Elsäßer, "Influence of anisotropies on transverse modes in oxide-confined VCSELs," IEEE J. Quantum Electron. 38, 73-84 (2002). [CrossRef]
  10. M. Peeters, G. Verschaffelt, H. Thienpont, S. K. Mandre, I. Fischer, and M. Grabherr, "Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers," Opt. Express 13, 9337-9345 (2005). [CrossRef] [PubMed]
  11. F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica 5, 785-795 (1938). [CrossRef]
  12. A. C. Schell, "A technique for the determination of the radiation pattern of a partially coherent aperture," IEEE Trans. Antennas Propag. AP-15, 187-188 (1967). [CrossRef]
  13. E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams," Opt. Lett. 2, 27-29 (1978). [CrossRef] [PubMed]
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Pr., 1995).
  15. S. K. Mandre, W. Elsaßer, I. Fischer, M. Peeters, and G. Verschaffelt, "Determining the temporally and radially resolved temperature distribution inside a pulsed broad-area vertical-cavity surface-emitting laser cavity," Appl. Phys. Lett.  89, 151 106 (2006). [CrossRef]
  16. L. Wang and Q. Lin, "The evolutions of the spectrum and spatial coherence of laser radiation in resonators with hard apertures and phase modulation," IEEE J. Quantum Electron. 39, 749-758 (2003). [CrossRef]

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