Abstract

We present experimental and numerical study of temporal characteristics of injection-locked polarization switching of a conventional-type 1.55-μm wavelength single-mode vertical cavity surface-emitting laser (VCSEL). Delayed recovery response of the main-mode of the VCSEL was observed when short and strong optical injection pulses of an orthogonal polarization to the VCSEL’s main polarization-mode were applied. Numerical analysis based on a spin-flip model describes that the relatively long upper level lifetimes compared to a short injection-pulse width and long cavity photon lifetimes cause delayed recovery response of the main-mode of the VCSEL. An optimum bias current of the VCSEL was also observed for the shortest recovery time of its free-running polarization mode after the orthogonal polarization beam pulse injection.

© 2011 OSA

1. Introduction

Polarization switching (PS) of VCSELs has been of significant interest of many researchers recently. Many experimental and theoretical investigations [18] have been performed for multimode VCSELs mostly with continuous wave (CW) injection locking technique, but detailed temporal behavior of the PS in single-mode (SM) VCSEL under short-pulse injection locking is not yet uncovered fully.

For digital and analogue communication application, knowledge of temporal characteristics of the PS is very important. Besides many advantages of VCSELs over edge emitting lasers such as low threshold, circular output beam, easy fabrication in 2D structures and low cost, the polarization property of the VCSEL output is being pursued for potential all-optical switching applications [9,10]. Recently we have observed some temporal delay on the recovered signals after the PS in VCSELs was induced by very short pulse injection [11]. In addition, our recent measurement has shown that the temporal delay on the recovered signals after the injection-induced PS becomes more significant when the VCSEL chip is more isotropic in optical gain and cavity geometry and the injection beam is more strong to induce the polarization switching. It is reported that even cylindrical-shaped VCSELs with a circular symmetric output aperture have a polarization bistability during change of injection current when the wavelength separation between the main polarization-mode and side-mode is very close [12]. The VCSEL’s bistability disappears and the VCSEL lases at a particularly well-defined polarization direction when the wavelengths between the two polarization modes are well separated due to its asymmetric shape and/or internal birefringence. Even the circularly cylindrical-shaped VCSELs with a well-defined polarization output show a significant temporal delay on the recovered original polarization mode after the injection-induced PS when the injection beam power becomes sufficiently strong compared to the VCSEL’s operating condition at a given bias current. Symmetricity and gain uniformity of the circularly cylindrical-shaped VCSELs vary from chip to chip although the VCSEL chips are taken from the same wafer. Therefore, in this paper we have investigated experimentally and numerically the temporal delay characteristics of the recovered original polarization mode after the optical pulse injection-induced PS of a SM VCSEL of well-defined polarization output with a well-separated side-mode wavelength.

Since the SM VCSEL has one prominent mode depending on the small gain anisotropy due to birefringence and spatial hole burning in the fundamental transverse mode of its cavity, we let the prominent main mode polarization in X-direction and the other suppressed mode, which has an orthogonal polarization to that of the main mode, in Y-direction. While the VCSEL is emitting a polarized beam in the X-direction, we inject an external pulse injection into the Y-direction. This pulse injection causes the VCSEL output switched from the X-polarization to Y-polarization direction during the short pulse period. When the injection pulse duration becomes very short (in order of shorter than or equal to 50 ps) and its peak power is sufficiently strong, it is observed that a temporal delay occurs in recovery of the original polarization mode after switching. This delay in recovery of the polarization mode varies with various parameters, such as bias current, injection optical power, and detuning between the master and slave laser wavelengths, which is explained on the basis of a spin-flip model (SFM).

2. Experimental setup

The experimental setup used in our measurement is shown in Fig. 1 . A distributed feedback laser diode (DFB LD) was used as a tunable master laser whose wavelength was varied by changing its operating temperature. A conventional-type 1.55 μm-wavelength SM VCSEL was a slave laser which was driven at a fixed bias current of 2.7 mA and maintained at a constant temperature of 27 °C with a thermoelectric cooler packaged TO-CAN package. The output spectrum of the free-running VCSEL measured with an optical spectrum analyzer (OSA) is shown in Fig. 2 , which has the two dominant peaks, each corresponding to the fundamental mode in the X-polarization direction and to the side mode in the Y-polarization direction. The X- and Y-polarization peaks were observed at 1553.73 nm and 1554.15 nm, respectively, with a spectral separation of 0.42 nm. The resolution of the OSA used was 0.07 nm. The SMSR (side-mode suppression ratio) was maintained at 30.22 dB for separated detection of the two polarization-modes with a fiber-type polarization-beam-splitter (PBS) and a digital communication analyzer (DCA, HP 83485A) under polarization control with the polarization controller PC2. The threshold of this VCSEL was 2.25 mA at the operating temperature of 27 °C.

 

Fig. 1 A schematic diagram of experimental setup. PC: polarization controller, PBS: polarization beam splitter, and DCA: digital communication analyzer.

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Fig. 2 Measured optical spectrum of the SM VCSEL in a free running mode, showing the X- and Y-polarization modes.

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The L-I characteristics of a stand-alone VCSEL used in this measurent showed that no polarization switching took place under change of its driving current, thus illustrating no inherent bi-stability in our VCSEL. The CW VCSEL output has a X-polarized mode beam of 44.90 μW power and a Y-polarized mode at 2.27 μW measured with two optical channels of the DCA at a fixed bias current of 2.7 mA with a polarization controller PC2 adjusted to the X-polarization maximum condition when there is no external beam injection. This small power ratio between the two polarization outputs may be attributed mainly to the poor polarization isolation (~18 dB) of the PBS used in our measurement.

For the PS experiment with short-pulse injection, the DFB laser was gain-switched with a 500 MHz pulse generator to deliver laser pulses of temporal pulse width 51.70 ps. The peak power of the injection pulses was 189.32 μW. The wavelength of the injection DFB laser was tuned over 1553.99 nm ~1554.41 nm with its operating temperature control from 11.5°C through 15.8°C. The polarization of the DFB Laser was matched with the side-mode of the VCSEL in the Y-polarization direction using PC1. The power meter connected to the 20% output port of a 80:20 beam splitter monitored the average power of the DFB LD’s injection pulses. The injection-locked VCSEL output was passed through an optical circulator and then into the PBS.

The separated X- and Y-polarized outputs from the PBS were analyzed using the optical input channels of the DCA. Figure 3 shows the actual pulse traces of the two polarization-mode outputs measured on the DCA. The traces show the delayed X-polarization output recovery time of longer than 500 ps after removal of the injection pulse in the orthogonal polarization direction.

 

Fig. 3 Measured oscilloscope traces of the X- and Y-polarization outputs of the VCSEL beam from the PBS with the DCA at the zero-detuned injection wavelength. The injection pulse had a pulse width of 51.7 ps. (Time scale = 500 ps/DIV, Vertical scale for X-polarized output = 20 µW/DIV and that for Y-polarized output is 50 µW/DIV.)

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As shown in Figs. 4 , 5 and 6 , depending on the bias currents of the VCSEL and the optical injection powers the different delay time of the main-mode recovery after the polarization switching as well as some relaxation oscillation were observed. The amplitude of the relaxation oscillation of the X-polarized output observed during the recovery process after the end of the injection pulse was relatively very small. So, we have considered the rise time of the recovery signal of the X-polarized output as the recovery time of the X-polarized output. The rise time of the recovery signal of the X-polarized output is defined as the time taken by the X-polarized output to rise from the 10% to 90% of the steady state value of the X-polarized output. To find the origin of the delay in the X-polarization output recovery time, effects of variation of the bias current of the VCSEL and peak power of the injection pulse on the delay in the recovery time of the X-polarization pulse were studied. Figures 4 and 5 show the measured traces of X- and Y-polarized intensity outputs for various bias currents of the VCSEL and for various peak powers of the Y-polarized injection pulse, respectively. For a constant injection peak power of 128.55 μW at the injection wavelength corresponding to the Y-polarization peak of VCSEL (1554.15 nm), the polarization recovery time changes from 270.1 ps to 266.3 ps, 819 ps, and 922.3 ps as the bias current increases from 3.0 mA to 4.5 mA in a step of 0.5 mA. On the other hand, for a constant bias current of 4.5 mA the recovery time changes from 1012 ps to 922.3 ps, 1341 ps, and 1013 ps as the injection power reduces from 182 μW to 127 μW, 82 μW and 55 μW, respectively. While for a constant bias current of 3.5 mA, the recovery time changes from 266.3 ps to 226.8 ps, 282.4 ps, and 235.7 ps as the injection power reduces from 129 μW to 92 μW, 57 μW and 39 μW, respectively, which is shown in Fig. 6.

 

Fig. 4 Variation of recovery time of the X- and Y-polarization-mode output intensities (yellow and violet color traces, respectively) with the bias current of the VCSEL at a constant injection power of 127 μW at the injection wavelength corresponding to the Y-polarization peak of 1554.15 nm. (Time scale = 500 ps/DIV, Vertical scale = 20 µW/DIV, the X-polarized output offset was adjusted to (a) −50 μW, (b) −25 μW, (c) 0 μW and (d) + 25 μW for the VCSEL bias currents of (a) 3.0 mA, (b) 3.5 mA, (c) 4.0 mA and (d) 4.5 mA, respectively). The marks “X off’ indicate the zero-levels of the X-polarization output intensities which correspond the upper traces (yellow color).

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Fig. 5 Temporal variation of the X- and Y-polarization-mode output intensities (yellow and violet color traces, respectively) of the SM VCSEL under a Y-polarization pulse beam injection of various peak powers ((a)~(d)) at a constant bias current of 4.5 mA. (Time scale = 500 ps/DIV, Vertical scale = 20 µW/DIV.) P inj: Injected power in µW. The offset for X-polarized output power was adjusted to 25 μW. The marks “X off’ indicate the zero-levels of the X-polarization output intensities which correspond the upper traces (yellow color).

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Fig. 6 Temporal variation of the X- and Y-polarization-mode output intensities (yellow and violet color traces, respectively) of the SM VCSEL under a Y-polarization pulse beam injection of various peak powers ((a)–(d)) at a constant bias current of 3.5 mA. (Time scale = 500 ps/DIV, Vertical scale = 20 µW/DIV.), P inj: Injection pulse power in µW. The marks “X off’ indicate the zero-levels of the X-polarization output intensities which correspond the upper traces (yellow color).

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The details of the observed phenomena are explained in section 5.

3. Theoretical background

For numerical analysis of the temporal characteristics of the PS action in VCSELs, the theoretical model given by Miguel, Feng and Molony [1] has been used. This model is extended for study of the time-varying optical injection effect into the orthogonal Y-polarized mode (i.e. side mode) of a VCSEL output at detuned injection wavelength [2].

The output beam of a VCSEL without an external optical beam injection is constant in time under a constant continuous wave (CW) driving current, and the normalized electrical field values of the optical beam intensity are taken as Ex = 1 for the X-polarization mode and Ey = 0 for the Y-polarization mode. The injection pulse is modeled as a Gaussian-shaped pulse of 50 ps FWHM pulse width which is injected at a time of 0 ns.

The rate equations for the electrical fields of the X- and Y-polarizations, the excited population and the population difference between spin-up and spin down radiation channels are as follows [2]:

dExdt=k(1+jα)(NExEx+jnEy)(γa+jγp+jΔω)Ex+βspξx
dEydt=k(1+jα)(NEyEyjnEx)+(γa+jγpjΔω)Ey+βspξy+kinjEinj
dNdt=γeN(1+P)+γeμjγen(EyExExEy)
dndt=γsnγenPjγen(EyEx*ExEy*)
Here, E x and E y are the complex amplitudes of the VCSEL output at X- and Y-polarization directions, respectively. N and n are the total population inversion and the difference between the population inversions for spin-up and spin-down radiation channels, respectively. k is the decay rate of the electric field in the VCSEL cavity. α is the linewidth enhancement factor. β sp is the strength of the spontaneous emission. μ is the normalized injection current (μ = 1 for threshold). P = |E x|2 + |E y|2 is the normalized output power. ξ x and ξ y are independent Gaussian white noise sources with a zero mean and a unit variance in the X- and Y-polarization directions, respectively. E inj is the electric field amplitude of the injection beam, which is modeled as a Gaussian-shaped time-dependent pulse. Δω is the frequency detuning between the master laser (DFB-LD) and the suppressed Y-polarization mode of the slave VCSEL. kinj is the coupling coefficient of the injection beam into the VCSEL. γ a is the linear dichroism, γ p is the linear birefringence, γ e is the decay rate of the total population inversion N, and γ s is the spin-flip rate.

In order to find the proper value of kinj, the X-polarized intensity is plotted for various values of kinj at k = 25 ns−1 and γa = −1.25 ns−1, and the result is shown in Fig. 7 . It is found that when kinj = k, calculated values for the X-polarized intensity and other parameters, such as the turn-off pulse width of the X-polarized output and the turn-on pulse width of the Y-polarized intensity, are close to the experimentally measured values. Thus, the value of kinj is taken to be equal to k as in Refs [2] and [3].

 

Fig. 7 Variation of the X-polarized intensity versus time for various values of kinj in ns−1 at k = 25 ns−1.

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The value of birefringence (γp) is a half of the frequency difference value between the two spectral modes of the VCSEL [3]. So the two spectral mode separation was measured to be 0.42 nm from the observed VCSEL’s output spectrum, and thus γp was calculated as 26.09 ns−1. Since our VCSEL showed no polarization switching for change of its driving current only, the normalized bias current for the VCSEL with respect to the threshold current was kept constant at 1.2, i.e. at 2.7 mA. Other constants used are explained in Table 1 below. The effect of variation of γ a and γ e on the VCSEL’s polarization output intensity is studied, and the results are shown in Fig. 8 .

Tables Icon

Table 1. Parameter Values used in Theoretical Analysis

 

Fig. 8 Temporal intensity variation of the X- and Y-polarization-mode outputs (black and red colors, respectively) of the SM VCSEL under a Y-polarization pulse beam injection for various values of the linear dichroism (γa) and carrier decay rate (γe).

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γ a is related to the power difference between the X- and Y-polarization mode outputs. When this difference is very small, it is very easy to switch the polarization output. Since γ e is the decay rate of the total excited population, its effect can be seen as the output intensity measured on the DCA.

For small γ a value, as γ e increases (i.e. the carrier decay time decreases), the polarization switching takes place more easily [Figs. 8(a), 8(b) and 8(c)] and the polarization recovery time for the X-polarized output increases. On the other hand, for a large γ a value, we need a short decay time (i.e. a large decay rate γ e) to ensure the polarization switching and shorter X-polarized output recovery time. Under the pulse injection of an orthogonal polarization mode, the VCSEL’s output polarization is switched temporarily, and as soon as injection pulse is over, it switches back to the initial polarization conditions.

In our case, we use a single-mode VCSEL. Therefore, the value of γ a is taken as γa=1 .25ns1, and from consideration of the X-polarization mode as a dominant fundamental mode, the value of γ a is taken negative [3].

The optimum value of inverse of the photon life-time is calculated, by using various values of the field decay rate (also called as inverse of the photon lifetime, which is given by 1/2k) [3]. From Fig. 9 , it is clear that, when the photon life-time is close to the spin flip time, the relaxation oscillation is the minimum and the numerically calculated parameters match with those experimentally measured. Hence, the value of the field decay rate, for our VCSEL, is chosen as 25 ns−1 for the analysis of X- and Y-polarized output variation with wavelength detuning from the peak value of its Y-polarized output.

 

Fig. 9 Variation of the X-polarization mode intensity of the VCSEL output under a pulsed Y-polarization beam injection for various field decay rates at a fixed spin-flip rate of 50 ns−1. The photon life time is calculated as (1/2k), where k is field decay rate in ns−1.

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4. Comparison of numerically calculated and experimentally measured results

4.1 Effect of wavelength detuning

When the injection beam wavelength is detuned from the side-mode wavelength of the SM VCSEL, the VCSEL’s X- and Y-polarization mode outputs are analyzed numerically using Eqs. (1) through (4) and the parameters shown in Table 1.

Figure 10 shows comparison of numerically calculated and experimentally observed results for the X- and Y-polarized beam intensity dynamics of the SM VCSEL under an external pulse injection in the Y-polarization direction at various detuned wavelengths. The measured and calculated results indicate that strong polarization switching occurs at the detuned wavelengths of the injection laser beam from 0 to 0.08 nm. However, when the injection beam is detuned to shorter wavelengths toward Δλ = −0.08 nm, the polarization switching mechanism becomes suppressed. The calculated results match well with the measured ones for γ a = −1.25 ns−1.

 

Fig. 10 Comparison of experimentally observed oscilloscope traces and numerically calculated plots of the X- and Y-polarization beam dynamics of a SM VCSEL under short pulse injection of a Y-polarization mode beam at various detuning wavelengths. The injection pulse width is 51.70 ps and its peak power is 129 µW. For numerically calculated plots, the injection pulse width is considered as 50 ps. γa is taken as −1 ns−1 for Figs. (g) through (l) and as −1.25 ns−1 for Figs. (m) through (r).

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4.2 Pulse width variation

Figure 11(a) shows comparative plots of the measured and calculated pulse widths (Full Width at Half Maximum; FWHM) of the X-polarization output turn-off pulse as functions of the injection wavelength detuning from the Y-polarization peak wavelength, while Fig. 11(b) shows those for the Y-polarization turn-on output pulses. The two different calculation results are plotted for the linear dichroisms of −1.0 ns−1 and −1.25 ns−1. The calculated and measured turn-off pulse widths of the X-polarization output pulse show a similar trend. The calculated turn-on pulse width plot of the Y-polarization pulse output as a function of the wavelength detuning shows a similar trend to the measured results, but it is shifted toward a long wavelength side compared to the measured one.

 

Fig. 11 Comparison of the measured and calculated (a) turn-off pulse widths of the X-polarization output and (b) turn-on pulse widths of Y-polarized output as functions of the injection wavelength detuning from the Y-polarization peak wavelength of the VCSEL. Black lines with open squares represent experimental results, red lines with open circles represent numerically calculated results for γa = −1 ns−1, and blue lines with open triangles represent numerically calculated results for γa = −1.25 ns−1.

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4.3 Intensity variation

Figures 12(a) and 12(b) show the measured and calculated intensities of the turn-off pulse minimum of the X-polarization output and those of the turn-on pulse maximum of the Y-polarization output pulse, respectively, as functions of wavelength detuning from the Y-polarization output’s spectral peak. The intensity values are taken in a normalized linear unit which is obtained by dividing the each intensity value by the maximum output intensity obtained during the wavelength detuning from −0.16 nm to 0.26 nm.

 

Fig. 12 Comparison of the measured and calculated (normalized) intensities of (a) the turn-off pulse minimum of the X-polarization output and (b) the turn-on pulse maximum of the Y- polarization output of the VCSEL as functions of wavelength detuning from the Y-polarization output’s spectral peak. Black lines with open squares represent experimental results, red lines with open circles represent numerically calculated results for γa = −1 ns−1, and blue lines with open triangles represent numerically calculated results for γa = −1.25 ns−1.

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The numerically calculated results shown in Figs. 12(a) and 12(b) indicate similar trends to the experimentally measured results though their shapes do not match exactly to each other. Both of the numerical and experimental results show that detuning of the injection beam wavelength to slightly longer wavelength side than the Y-polarization spectral peak of the VCSEL allows strong polarization switching. This phenomena may be explained with the fact that the excited electrons are likely to be more populated in lower levels of the conduction band and the holes are likely to be more populated in upper levels of the valence band.

5. Discussion

5.1 Spin flip model

The spin-flip model (Fig. 13 ) considers magnetic sublevels of the conduction electron band and the valance hole band [5]. The electron transition from the conduction band to the heavy hole state is 3 times more probable than that from the conduction band to the light hole state [4]. Therefore, the transition to the light hole state is neglected in our analysis. We also assume that the spin relaxation of holes are instantaneous, and hence the time taken for the holes’ spin relaxation can be neglected.

 

Fig. 13 Band structure of a quantum well and selection rules.

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When there is a transition from the upper + 1/2 (electron’s spin up) state to the lower + 3/2 (heavy hole) state corresponding ΔJ = + 1, a left circularly polarized light is emitted, while, for the transition from the upper −1/2 (electron’s spin down) state to the lower −3/2 (heavy hole) state corresponding ΔJ = −1, a right circularly polarized light is emitted [3].

The output light is combination of the left- and right-circularly polarized lights, giving a plane polarized light. The complex amplitudes of the plane polarized light in terms of the circularly polarized lights are given by,

Ex=E++E2
Ey=iE+E2
Since we are observing the VCSEL’s output light intensity in either the X- or Y-polarization mode, we consider only the plane polarized light for our analysis.

5.2 Effect of Y-polarized pulse injection on the X- and Y-polarized outputs of the VCSEL

As an initial condition, we consider that the X-polarized light is emitted as the fundamental mode while the Y-polarized light as a side mode. The side-mode suppression ratio is high enough (30.22 dB in our case) so that we can neglect the intensity of the Y-polarized light. It can be understood as the complex amplitudes of the right- and left-circularly polarized light are equal (E+ = E-) i.e. the spin-up and spin-down populations at the conduction band are the same. The above equation indicates that the X-polarized beam intensity is twice of the left- or right-circularly polarized intensity, and is taken as 1 (normalized), while the Y-polarized intensity is considered as zero, for numerical calculations. When we inject an external Y-polarized light, it increases the difference between right and left circularly polarized light amplitudes. The injected light disturbs the distribution of the spin-down and spin-up populations. Let us consider a situation where the spin-up population is swiped out by the injection, and the spin-down is untouched. (Even the opposite situation can take place.) As the total population in the conduction band is decreased, it gives reduction in the X-polarized beam intensity, while increase in the Y-polarization beam intensity.

When the injection pulse is over or after the spin-flip time, the spin-down population flips back into the spin-up state. This tries to equalize the population distributions. However, if the injection pulse is longer than the spin-relaxation time, then the flipped spin-up electrons are swiped out by the injection pulse. This again increase the Y-polarization beam intensity and the output Y-polarized beam pulse is temporally expanded as compared with the injected pulse. Figure 14 illustrates the temporal variations of the X- and Y-polarization beam intensities, the total population inversion variation, and the difference variation between the population inversions for spin-up and spin-down radiation channels under injection of a 50-ps pulse in the Y-polarization direction at the time zero. Flipping of the spin polarization from one state to the other can be expected from the temporal oscillation of the population inversion difference between the spin-up and spin-down radiation channels shown in Fig. 14(d). Thus, the relaxation oscillations in the X- and Y-polarization output intensities are expected from the spin-polarization flipping dynamics.

 

Fig. 14 Intensity variations of the (a) X-polarization and (b) Y-polarization modes, (c) the total population inversion variation, and (d) the difference variation between the population inversions for spin-up and spin-down radiation channels, all as functions of time, under a 50-ps pulse injection in the Y-polarization direction at time zero. The scales on the Y-axes are in normalized linear units.

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Once the injection pulse is over, the population inversion starts to build up. The decay rate of total population is of the order of 1 ns−1 [1]. The X-polarization beam intensity regains its steady-state value after some time which is comparable to the cavity build-up time. The cavity buildup time is related to the recovery time of the excited state population inversion which is substantially longer than the photon lifetime t c [13]. The photon lifetime can be calculated as t c = t RT/(1 – R 1 R 2) assuming no other loss except the mirror reflectivities, where t RT is the round trip time of photons in the laser cavity and R 1 and R 2 are the reflectivities of two cavity mirrors, respectively [13]. Since the cavity round trip time t RT is expressed as t RT = 2nd/c with refractive index of n of the cavity medium, cavity length of d, and the speed of light in a free space c. The laser cavity parameters of the cavity length and mirror reflectivities of our VCSEL are listed in Refs [14] and [15]. The cavity length d is about 10 μm which corresponds approximately to the separation between the middle of the top and bottom distributed Bragg reflectors (DBRs), and R 1 = 0.999 and R 2 = 0.9983 for the DBRs. The photon lifetime is calculated to be about an order of 90 ps which is longer than the injection pulse width. Thus, the build-up time of the X-polarization recovery is longer than the photon lifetime, and the delay recovery is observed as shown in Fig. 3. The photon lifetime calculated from the cavity length and mirror reflectivities is approximately four times greater than the obtained photon lifetime from Fig. 9 from comparison of the experimentally observed results and numerically calculated plots. We can expect two major causes for this difference. One is potentially different mirror reflectivities of the Bragg reflectors because they vary slightly from batch to batch even through a same wafer fabrication recipe is used. The other one is potential variation of the mirror reflectivities as well as the refractive index of the cavity materials during the external laser beam injection because the injection beam changes the carrier distributions in the semiconductor materials. The injection beam may lower the mirror reflectivities or/and reduce the effective cavity length, which results in a shortened photon lifetime.

For a Y-polarized injection beam close the zero wavelength detuning (i.e. for the injection beam at the side-mode wavelength of the VCSEL), almost all the carriers from the conduction band are removed. This situation is similar to when the VCSEL is not driven with any current before the current supply is turned on. Thus, once the injection beam is removed (i.e. the situation similar to VCSEL current supply is turned on), the carrier build-up starts to take places through a relaxation oscillation. This phenomena can be seen in Fig. 3.

5.3 X-polarized output recovery time versus VCSEL bias current and Y-polarized injection pulse power

The recovery time of the X-polarized output of the VCSEL shown in Figs. 4 and 5 is plotted against its bias current and the peak power of the Y-polarized injection pulse at the Y-polarized output peak wavelength (i.e. at the zero-detuning wavelength) and are shown in Figs. 15(a) and 15(b), respectively.

 

Fig. 15 (a) Recovery time of the X-polarization output as a function of the VCSEL bias current at various peak powers of the Y-polarized injection pulse and (b) recovery time for the X-polarization output as a function of peak power of the Y-polarized injection pulse at various VCSEL bias currents.

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It is known that as the VCSEL bias current increases, number of carriers in the upper level (population inversion) increases. When the injection power and the injection pulse width are kept constant, number of the photons injected is constant. These photons (or some fraction of them, due to the loss of photons in the injection process) cause the carriers to flip their spin. When the bias current is small, it takes some time for the VCSEL to have a sufficient carrier population in the upper conduction level to induce the X-polarization laser output after exhausting its population due to the injection pulse. When the number of the injection photons is comparable with that of the carriers in the upper level, the entire number of the carriers flip their spin and make transitions into the valance band by giving out the Y-polarized output pulse. Especially the bias current of 3.5 mA is sufficient enough to provide the upper level carrier population in the original X-polarization state within a short period of time even after the Y-polarization injection pulse. Thus, the recovery time of the polarized output becomes short as shown in Fig. 15(a). As the bias current increases above 3.5 mA, the current becomes more than the maximum carrier population which can be excited into the upper conduction level within the carrier lifetime and keep pumping the carriers again right after an induced Y-polarized beam emission of the initially excited carrier population due to the injection pulse. Thus, this Y-polarization emission keeps inducing the stimulated emission to the rest of the upper level carriers during the cavity lifetimes of the cascadedly emitted photons. It also indicates that the redistribution of the excited electrons in the conduction band takes place faster than cavity build-up time to provide the cascaded process of the stimulated emission. This process causes the delay of the recovery time of the laser output to the X-polarization mode as shown in Figs. 4 and 15(a) for the large bias current cases of 4.0 mA and above. Thus, the optimum VCSEL bias current, for which the recovery time of the VCSEL’s output to the X-polarized beam after the polarization switching is the minimum, is found to be 3.5 mA.

Figures 5 and 15(b) show variation of the recovery time of the VCSEL output to the original X-polarization mode after the Y-polarized beam emission as a function of the peak power of the Y-polarization injection pulse for various bias currents. The results show no significant injection pulse power dependence of the recovery time. This phenomenon can be described with a similar explanation done for Figs. 4 and 15(a). The bias current of 3.5 mA is enough to keep most of the carriers available in the active region within the upper conduction band, and the injection peak power above 30 μW is enough to induce the stimulated emission of all of the excited carriers in the energy band corresponding to the injection beam wavelength into the Y-polarization mode output during the injection pulse width. Thus, once the injection pulse is ceased, the excited carrier population is recovered immediately into the X-polarization mode because of the continuous bias current, and provides the X-polarized beam output. As the peak power of the Y-polarized injection pulse increases, more number of the excited electrons flip their spin and recombine with the holes in the valance band. This decreases number of the excited electrons recombining with holes in the valance band to deliver the X-polarized output. Thus, the X-polarized output intensity decreases with the increasing peak injection pulse power as shown in Fig. 5. However, since the cavity build-up time is longer than the injection pulse width of 51.70 ps, the stronger injection pulse powers do not improve the recovery time of the VCSEL output into the X-polarization mode after the injection pulse.

Once the bias current is below 3.5 mA, the build-up time for the upper level population becomes long, and the recovered X-polarization output appears slow. On the other hand, when the bias current is above 3.5 mA, the electron collision keeps fast recycling the upper level population even right after the injection-pulse induced Y-polarization emission and providing extended emission of the Y-polarization beam through stimulated emission processes of the cascadedly excited electrons. That is why we observe the delayed recovery times of the VCSEL output into the original X-polarization mode compared to that for the bias current 3.5 mA as shown in Fig. 15(a).

6. Conclusion

We have studied temporal dynamics of polarization switching of a single-mode VCSEL under injection of short laser pulses of an orthogonal polarization direction to its free-running output polarization mode. The short pulse injection beam was obtained from a gain-switched DFB-LD and its wavelength is tuned by changing its operation temperature. The peak intensity and pulse width of the polarization-switched pulses as well as the recovery time of the returning signals of the original free-running polarization-mode of the VCSEL were measured experimentally and calculated numerically for various detuned injection wavelengths from the side-mode beam wavelength of the VCSEL in the orthogonal polarization direction. Both calculated and measured results show similar trends except slight discrepancy in their detailed dependencies on the injection beam’s wavelengths. The difference between the numerical and experimental results may be due to several reasons like dependence of the birefringence and dichroism of the VCSEL on injection intensity, spectral/spatial hole burning, and difficulty in matching the polarization of the injection pulse with the side-mode polarization of the VCSEL in the actual experiment. The loss of the injection beam’s power at the DBR layers of the VCSEL itself and the detector response have not been included in our numerical simulation, which may be included in further elaborate analyses. The recovery time of the SM VCSEL to its original free-running polarization mode after polarization switching due to injection pulses of an orthogonal polarization mode has also been studied for various bias currents and injection pulse powers. The results indicate that there is an optimum bias current for the given VCSEL to deliver fastest polarization recovery time.

Acknowledgments

This work was supported by the Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Korean Ministry of Education, Science and Technology under Grants 2009-0073617 and 2009-0084514. The authors gratefully thank Drs. Byeung-Soo Yoo and Jay Roh of Raycan Co., Ltd. for providing us the VCSELs.

References and links

1. M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995). [CrossRef]  

2. W. L. Zhang, W. Pan, B. Luo, M. Y. Wang, and X. H. Zou, “Polarization switching and hysteresis of VCSELs with time-varying optical injection,” IEEE J. Sel. Top. Quantum Electron. 14(3), 889–894 (2008). [CrossRef]  

3. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997). [CrossRef]  

4. A. Dyson and M. J. Adams, “Spin-polarized properties of optically pumped vertical-cavity surface-emitting lasers,” J. Opt. B Quantum Semiclassical Opt. 5(3), 222–226 (2003). [CrossRef]  

5. A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped VCSELs,” IEEE J. Quantum Electron. 35(3), 342–351 (1999). [CrossRef]  

6. Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, “Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback,” IEEE J. Quantum Electron. 41(5), 619–624 (2005). [CrossRef]  

7. M. Sciamanna and K. Panajotov, “Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers,” Phys. Rev. A 73(2), 023811 (2006). [CrossRef]  

8. J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002). [CrossRef]  

9. I. Gatare, K. Panajotov, and M. Sciamanna, “Frequency induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. A 75(2), 023804 (2007). [CrossRef]  

10. S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997). [CrossRef]  

11. K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008). [CrossRef]  

12. S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010). [CrossRef]  

13. A. E. Siegman, Lasers (Univ. Sci. Books, 1986), Chaps. 25 and 13.

14. J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002). [CrossRef]  

15. M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006). [CrossRef]  

References

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  1. M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).
    [CrossRef]
  2. W. L. Zhang, W. Pan, B. Luo, M. Y. Wang, and X. H. Zou, “Polarization switching and hysteresis of VCSELs with time-varying optical injection,” IEEE J. Sel. Top. Quantum Electron. 14(3), 889–894 (2008).
    [CrossRef]
  3. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
    [CrossRef]
  4. A. Dyson and M. J. Adams, “Spin-polarized properties of optically pumped vertical-cavity surface-emitting lasers,” J. Opt. B Quantum Semiclassical Opt. 5(3), 222–226 (2003).
    [CrossRef]
  5. A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped VCSELs,” IEEE J. Quantum Electron. 35(3), 342–351 (1999).
    [CrossRef]
  6. Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, “Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback,” IEEE J. Quantum Electron. 41(5), 619–624 (2005).
    [CrossRef]
  7. M. Sciamanna and K. Panajotov, “Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers,” Phys. Rev. A 73(2), 023811 (2006).
    [CrossRef]
  8. J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002).
    [CrossRef]
  9. I. Gatare, K. Panajotov, and M. Sciamanna, “Frequency induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. A 75(2), 023804 (2007).
    [CrossRef]
  10. S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
    [CrossRef]
  11. K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008).
    [CrossRef]
  12. S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
    [CrossRef]
  13. A. E. Siegman, Lasers (Univ. Sci. Books, 1986), Chaps. 25 and 13.
  14. J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002).
    [CrossRef]
  15. M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006).
    [CrossRef]

2010

S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
[CrossRef]

2008

K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008).
[CrossRef]

W. L. Zhang, W. Pan, B. Luo, M. Y. Wang, and X. H. Zou, “Polarization switching and hysteresis of VCSELs with time-varying optical injection,” IEEE J. Sel. Top. Quantum Electron. 14(3), 889–894 (2008).
[CrossRef]

2007

I. Gatare, K. Panajotov, and M. Sciamanna, “Frequency induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. A 75(2), 023804 (2007).
[CrossRef]

2006

M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006).
[CrossRef]

M. Sciamanna and K. Panajotov, “Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers,” Phys. Rev. A 73(2), 023811 (2006).
[CrossRef]

2005

Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, “Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback,” IEEE J. Quantum Electron. 41(5), 619–624 (2005).
[CrossRef]

2003

A. Dyson and M. J. Adams, “Spin-polarized properties of optically pumped vertical-cavity surface-emitting lasers,” J. Opt. B Quantum Semiclassical Opt. 5(3), 222–226 (2003).
[CrossRef]

2002

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002).
[CrossRef]

J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002).
[CrossRef]

1999

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped VCSELs,” IEEE J. Quantum Electron. 35(3), 342–351 (1999).
[CrossRef]

1997

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[CrossRef]

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

1995

M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).
[CrossRef]

Abraham, N. B.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[CrossRef]

Adams, M. J.

A. Dyson and M. J. Adams, “Spin-polarized properties of optically pumped vertical-cavity surface-emitting lasers,” J. Opt. B Quantum Semiclassical Opt. 5(3), 222–226 (2003).
[CrossRef]

Albert, J.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002).
[CrossRef]

Balle, S.

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped VCSELs,” IEEE J. Quantum Electron. 35(3), 342–351 (1999).
[CrossRef]

Berger, J. D.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Danckaert, J.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002).
[CrossRef]

Dyson, A.

A. Dyson and M. J. Adams, “Spin-polarized properties of optically pumped vertical-cavity surface-emitting lasers,” J. Opt. B Quantum Semiclassical Opt. 5(3), 222–226 (2003).
[CrossRef]

Erneux, T.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002).
[CrossRef]

Feng, Q.

M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).
[CrossRef]

Gahl, A.

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped VCSELs,” IEEE J. Quantum Electron. 35(3), 342–351 (1999).
[CrossRef]

Gatare, I.

I. Gatare, K. Panajotov, and M. Sciamanna, “Frequency induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. A 75(2), 023804 (2007).
[CrossRef]

Gibbs, H. M.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Hallstein, S.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Han, W.-S.

M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006).
[CrossRef]

J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002).
[CrossRef]

Hilpert, M.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Hong, Y.

Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, “Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback,” IEEE J. Quantum Electron. 41(5), 619–624 (2005).
[CrossRef]

Jahnke, F.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Jeong, K. H.

K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008).
[CrossRef]

Ju, R.

Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, “Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback,” IEEE J. Quantum Electron. 41(5), 619–624 (2005).
[CrossRef]

Ju, Y.-G.

J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002).
[CrossRef]

Jung, H. W.

S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
[CrossRef]

Khitrova, G.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Kim, K. H.

S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
[CrossRef]

K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008).
[CrossRef]

Koch, S. W.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Kwon, O.-K.

M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006).
[CrossRef]

J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002).
[CrossRef]

Lee, J.-H.

J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002).
[CrossRef]

Lee, K.-H.

M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006).
[CrossRef]

Lee, M. H.

S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
[CrossRef]

K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008).
[CrossRef]

Lee, S. H.

S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
[CrossRef]

K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008).
[CrossRef]

Luo, B.

W. L. Zhang, W. Pan, B. Luo, M. Y. Wang, and X. H. Zou, “Polarization switching and hysteresis of VCSELs with time-varying optical injection,” IEEE J. Sel. Top. Quantum Electron. 14(3), 889–894 (2008).
[CrossRef]

Martin-Regalado, J.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[CrossRef]

Miguel, M. S.

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped VCSELs,” IEEE J. Quantum Electron. 35(3), 342–351 (1999).
[CrossRef]

Moloney, J.

M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).
[CrossRef]

Nagler, B.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002).
[CrossRef]

Oestreich, M.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Pan, W.

W. L. Zhang, W. Pan, B. Luo, M. Y. Wang, and X. H. Zou, “Polarization switching and hysteresis of VCSELs with time-varying optical injection,” IEEE J. Sel. Top. Quantum Electron. 14(3), 889–894 (2008).
[CrossRef]

Panajotov, K.

I. Gatare, K. Panajotov, and M. Sciamanna, “Frequency induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. A 75(2), 023804 (2007).
[CrossRef]

M. Sciamanna and K. Panajotov, “Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers,” Phys. Rev. A 73(2), 023811 (2006).
[CrossRef]

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002).
[CrossRef]

Park, M.-R.

M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006).
[CrossRef]

Park, S.-J.

M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006).
[CrossRef]

Prati, F.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[CrossRef]

Roh, J.

S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
[CrossRef]

Ruhle, W. W.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

San Miguel, M.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[CrossRef]

M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).
[CrossRef]

Schneider, H. C.

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Sciamanna, M.

I. Gatare, K. Panajotov, and M. Sciamanna, “Frequency induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. A 75(2), 023804 (2007).
[CrossRef]

M. Sciamanna and K. Panajotov, “Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers,” Phys. Rev. A 73(2), 023811 (2006).
[CrossRef]

Shin, J.-H.

J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002).
[CrossRef]

Shore, K. A.

S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
[CrossRef]

K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008).
[CrossRef]

Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, “Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback,” IEEE J. Quantum Electron. 41(5), 619–624 (2005).
[CrossRef]

Spencer, P. S.

Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, “Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback,” IEEE J. Quantum Electron. 41(5), 619–624 (2005).
[CrossRef]

Veretennicoff, I.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002).
[CrossRef]

Wang, M. Y.

W. L. Zhang, W. Pan, B. Luo, M. Y. Wang, and X. H. Zou, “Polarization switching and hysteresis of VCSELs with time-varying optical injection,” IEEE J. Sel. Top. Quantum Electron. 14(3), 889–894 (2008).
[CrossRef]

Yoo, B. S.

K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008).
[CrossRef]

Yoo, B.-S.

S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
[CrossRef]

M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006).
[CrossRef]

J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002).
[CrossRef]

Zhang, W. L.

W. L. Zhang, W. Pan, B. Luo, M. Y. Wang, and X. H. Zou, “Polarization switching and hysteresis of VCSELs with time-varying optical injection,” IEEE J. Sel. Top. Quantum Electron. 14(3), 889–894 (2008).
[CrossRef]

Zou, X. H.

W. L. Zhang, W. Pan, B. Luo, M. Y. Wang, and X. H. Zou, “Polarization switching and hysteresis of VCSELs with time-varying optical injection,” IEEE J. Sel. Top. Quantum Electron. 14(3), 889–894 (2008).
[CrossRef]

IEEE J. Quantum Electron.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[CrossRef]

A. Gahl, S. Balle, and M. S. Miguel, “Polarization dynamics of optically pumped VCSELs,” IEEE J. Quantum Electron. 35(3), 342–351 (1999).
[CrossRef]

Y. Hong, R. Ju, P. S. Spencer, and K. A. Shore, “Investigation of polarization bistability in vertical-cavity surface-emitting lasers subjected to optical feedback,” IEEE J. Quantum Electron. 41(5), 619–624 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

W. L. Zhang, W. Pan, B. Luo, M. Y. Wang, and X. H. Zou, “Polarization switching and hysteresis of VCSELs with time-varying optical injection,” IEEE J. Sel. Top. Quantum Electron. 14(3), 889–894 (2008).
[CrossRef]

IEEE Photon. Technol. Lett.

K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5-μm wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20(10), 779–781 (2008).
[CrossRef]

S. H. Lee, H. W. Jung, K. H. Kim, M. H. Lee, B.-S. Yoo, J. Roh, and K. A. Shore, “1-GHz all-optical flip-flop operation of conventional cylindrical-shaped single-mode VCSELs under low power optical injection,” IEEE Photon. Technol. Lett. 22(23), 1759–1761 (2010).
[CrossRef]

J.-H. Shin, B.-S. Yoo, W.-S. Han, O.-K. Kwon, Y.-G. Ju, and J.-H. Lee, “CW operation and threshold characteristics of all-monolithic InAlGaAs 1.55-μm VCSELs grown by MOCVD,” IEEE Photon. Technol. Lett. 14(8), 1031–1033 (2002).
[CrossRef]

J. Opt. B Quantum Semiclassical Opt.

A. Dyson and M. J. Adams, “Spin-polarized properties of optically pumped vertical-cavity surface-emitting lasers,” J. Opt. B Quantum Semiclassical Opt. 5(3), 222–226 (2003).
[CrossRef]

Jpn. J. Appl. Phys.

M.-R. Park, O.-K. Kwon, W.-S. Han, K.-H. Lee, S.-J. Park, and B.-S. Yoo, “All-monolithic 1.55 µm InAlGaAs/InP vertical cavity surface emitting lasers grown by metal organic chemical vapor deposition,” Jpn. J. Appl. Phys. 45(1), L8–L10 (2006).
[CrossRef]

Opt. Commun.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, “Minimal rate equations describing polarization switching in vertical-cavity surface-emitting lasers,” Opt. Commun. 201(1-3), 129–137 (2002).
[CrossRef]

Phys. Rev. A

I. Gatare, K. Panajotov, and M. Sciamanna, “Frequency induced polarization bistability in vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. A 75(2), 023804 (2007).
[CrossRef]

M. Sciamanna and K. Panajotov, “Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers,” Phys. Rev. A 73(2), 023811 (2006).
[CrossRef]

M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).
[CrossRef]

Phys. Rev. B

S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rűhle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, “Manifestation of coherent spin precession in stimulated semiconductor emission dynamics,” Phys. Rev. B 56(12), R7076–R7079 (1997).
[CrossRef]

Other

A. E. Siegman, Lasers (Univ. Sci. Books, 1986), Chaps. 25 and 13.

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Figures (15)

Fig. 1
Fig. 1

A schematic diagram of experimental setup. PC: polarization controller, PBS: polarization beam splitter, and DCA: digital communication analyzer.

Fig. 2
Fig. 2

Measured optical spectrum of the SM VCSEL in a free running mode, showing the X- and Y-polarization modes.

Fig. 3
Fig. 3

Measured oscilloscope traces of the X- and Y-polarization outputs of the VCSEL beam from the PBS with the DCA at the zero-detuned injection wavelength. The injection pulse had a pulse width of 51.7 ps. (Time scale = 500 ps/DIV, Vertical scale for X-polarized output = 20 µW/DIV and that for Y-polarized output is 50 µW/DIV.)

Fig. 4
Fig. 4

Variation of recovery time of the X- and Y-polarization-mode output intensities (yellow and violet color traces, respectively) with the bias current of the VCSEL at a constant injection power of 127 μW at the injection wavelength corresponding to the Y-polarization peak of 1554.15 nm. (Time scale = 500 ps/DIV, Vertical scale = 20 µW/DIV, the X-polarized output offset was adjusted to (a) −50 μW, (b) −25 μW, (c) 0 μW and (d) + 25 μW for the VCSEL bias currents of (a) 3.0 mA, (b) 3.5 mA, (c) 4.0 mA and (d) 4.5 mA, respectively). The marks “X off’ indicate the zero-levels of the X-polarization output intensities which correspond the upper traces (yellow color).

Fig. 5
Fig. 5

Temporal variation of the X- and Y-polarization-mode output intensities (yellow and violet color traces, respectively) of the SM VCSEL under a Y-polarization pulse beam injection of various peak powers ((a)~(d)) at a constant bias current of 4.5 mA. (Time scale = 500 ps/DIV, Vertical scale = 20 µW/DIV.) P inj: Injected power in µW. The offset for X-polarized output power was adjusted to 25 μW. The marks “X off’ indicate the zero-levels of the X-polarization output intensities which correspond the upper traces (yellow color).

Fig. 6
Fig. 6

Temporal variation of the X- and Y-polarization-mode output intensities (yellow and violet color traces, respectively) of the SM VCSEL under a Y-polarization pulse beam injection of various peak powers ((a)–(d)) at a constant bias current of 3.5 mA. (Time scale = 500 ps/DIV, Vertical scale = 20 µW/DIV.), P inj: Injection pulse power in µW. The marks “X off’ indicate the zero-levels of the X-polarization output intensities which correspond the upper traces (yellow color).

Fig. 7
Fig. 7

Variation of the X-polarized intensity versus time for various values of kinj in ns−1 at k = 25 ns−1.

Fig. 8
Fig. 8

Temporal intensity variation of the X- and Y-polarization-mode outputs (black and red colors, respectively) of the SM VCSEL under a Y-polarization pulse beam injection for various values of the linear dichroism (γa) and carrier decay rate (γe).

Fig. 9
Fig. 9

Variation of the X-polarization mode intensity of the VCSEL output under a pulsed Y-polarization beam injection for various field decay rates at a fixed spin-flip rate of 50 ns−1. The photon life time is calculated as (1/2k), where k is field decay rate in ns−1.

Fig. 10
Fig. 10

Comparison of experimentally observed oscilloscope traces and numerically calculated plots of the X- and Y-polarization beam dynamics of a SM VCSEL under short pulse injection of a Y-polarization mode beam at various detuning wavelengths. The injection pulse width is 51.70 ps and its peak power is 129 µW. For numerically calculated plots, the injection pulse width is considered as 50 ps. γa is taken as −1 ns−1 for Figs. (g) through (l) and as −1.25 ns−1 for Figs. (m) through (r).

Fig. 11
Fig. 11

Comparison of the measured and calculated (a) turn-off pulse widths of the X-polarization output and (b) turn-on pulse widths of Y-polarized output as functions of the injection wavelength detuning from the Y-polarization peak wavelength of the VCSEL. Black lines with open squares represent experimental results, red lines with open circles represent numerically calculated results for γa = −1 ns−1, and blue lines with open triangles represent numerically calculated results for γa = −1.25 ns−1.

Fig. 12
Fig. 12

Comparison of the measured and calculated (normalized) intensities of (a) the turn-off pulse minimum of the X-polarization output and (b) the turn-on pulse maximum of the Y- polarization output of the VCSEL as functions of wavelength detuning from the Y-polarization output’s spectral peak. Black lines with open squares represent experimental results, red lines with open circles represent numerically calculated results for γa = −1 ns−1, and blue lines with open triangles represent numerically calculated results for γa = −1.25 ns−1.

Fig. 13
Fig. 13

Band structure of a quantum well and selection rules.

Fig. 14
Fig. 14

Intensity variations of the (a) X-polarization and (b) Y-polarization modes, (c) the total population inversion variation, and (d) the difference variation between the population inversions for spin-up and spin-down radiation channels, all as functions of time, under a 50-ps pulse injection in the Y-polarization direction at time zero. The scales on the Y-axes are in normalized linear units.

Fig. 15
Fig. 15

(a) Recovery time of the X-polarization output as a function of the VCSEL bias current at various peak powers of the Y-polarized injection pulse and (b) recovery time for the X-polarization output as a function of peak power of the Y-polarized injection pulse at various VCSEL bias currents.

Tables (1)

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Table 1 Parameter Values used in Theoretical Analysis

Equations (6)

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d E x d t = k ( 1 + j α ) ( N E x E x + j n E y ) ( γ a + j γ p + j Δ ω ) E x + β s p ξ x
d E y d t = k ( 1 + j α ) ( N E y E y j n E x ) + ( γ a + j γ p j Δ ω ) E y + β s p ξ y + k i n j E i n j
d N d t = γ e N ( 1 + P ) + γ e μ j γ e n ( E y E x E x E y )
d n d t = γ s n γ e n P j γ e n ( E y E x * E x E y * )
E x = E + + E 2
E y = i E + E 2

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