. INTRODUCTIONertical-cavity surface-emitting lasers (VCSELs) areromising devices in many applications [1–4] because ofheir advantages of single longitudinal mode, simple ar-ay fabrication, low threshold current, high modulationate, low cost, and so on. It is known that VCSELs pref-rentially emit linearly polarized (LP) light along one ofhe two orthogonal directions (x and y) that coincide withhe crystal axes owing to weak material and cavitynisotropies and can be easily switched between the tworthogonal LP states by modifying the bias current or theevice temperature or by applying additional degrees ofreedom. The realization of controllable polarizationwitching is useful to implement reconfigurable optical in-erconnects [1] and optical buffer memory [2]. On thether hand, the polarization dynamics of VCSELs ashese of fiber ring lasers have potential applications inhe communication systems [3,4]. A. Scire et al. theoreti-ally predicted polarization message encoding throughectorial chaos synchronization in self-pulsation VCSELsnd proposed the phase modulation of the vectorial field,hich allowed for transmission of secure data at high bit
ates up to 100 Gbits/s [4].Since the first experimental report on the polarization
witching characteristics of optical injected VCSELs [5]nd the proposal of a spin-flip model (SFM) [6], the polar-zation properties of single transverse mode VCSELs sub-ected to optical feedback or injection have been investi-ated intensely [7–23]. Moreover, the polarizationroperties of multi-transverse mode VCSELs with optical
njection have also been theoretically analyzed [24] andxperimentally investigated [25]. On the other hand, forptical feedback or injection the parallel optical feedback/njection and the orthogonal optical feedback/injection aresually adopted [13–21]. Interestingly, an intracavity ro-ating polarizer (RP) was introduced in the external cav-ty to obtain optical feedback with variably rotated polar-zation and was experimentally used to force the laser tomit in the TE, TM, and elliptical polarization states. Theuthors furthermore theoretically explained the experi-ental results based on the Jones matrix and steady-
tate rate equations [26]. Recently there has been a reportn the experimental observation of dynamical collapses of
semiconductor laser, which addressed selectively theM mode subject to optical feedback with variably rotatedolarization [27]. However, the study on the polarizationroperties of VCSELs subjected to optical feedback withontrollable polarization remains scarce. It is supposedhat by introducing an RP into the external cavity to ad-ust the polarization state of feedback, rich properties onhe polarization states of VCSELs would result and morensights into the polarization dynamics would bechieved, which deserves a detailed investigation.Unlike most reports where the time-averaged intensi-
ies of both LP modes are considered, we concentrate onhe polarization degree of VCSELs subjected to opticaleedback with controllable polarization. For the purposef comparison, we consider the polarization properties ofwo VCSELs corresponding to differently spaced fre-uency splitting between the two LP modes (i.e., VCSEL1
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ith closely spaced frequency splitting and VCSEL2 withidely spaced frequency splitting). The remainder of thisaper is organized as follows. In Section 2, a theoreticalodel that includes the variable polarization state of op-
ical feedback is derived from the SFM. In Section 3, theolarization switching induced by varying the polarizerngle for both VCSEL1 and VCSEL2 are discussed andompared in detail. To explain the different polarizationroperties between VCSEL1 and VCSEL2, the degrees ofolarization are calculated, and the effects of the initialonditions, spin-flip rate, and feedback strength, areaken into account. To show better the polarizationwitching process, the outputs in the time domain areompared, and the representations of the time evolutionsf the polarization state during polarization switching onhe Poincaré sphere are further presented. Finally, con-lusions are drawn in Section 4.
. THEORY AND MODELhe schematic diagram for a VCSEL subjected to optical
eedback with controllable polarization is shown in Fig. 1.he output of the VCSEL passes through the RP and theneinjects into the VCSEL. Especially, we adopt the ring-oop model to avoid a multiple-reflection scheme withinhe optical feedback loop. The feedback strength can bedjusted by the neutral density filter (NDF). Note that,imilar to [26], the optical feedback with controllable po-arization is obtained by means of introducing an RP inhe extended cavity. We define the 0° polarization to behe x-polarization (XP) mode of the VCSEL and the 90°olarization to be the y-polarization (YP) mode. Supposehat the transmission axis of the polarizer (i.e., RP) is ori-nted at a polarizer angle �p with respect to the XP modef the VCSEL, which can be rotated between 0° and 90°.hen only the components Ex cos��p� of the XP mode andy sin��p� of the YP mode are allowed to pass through theP, where Ex and Ey denote the electric fields of the XPnd YP modes, respectively. In particular, when �p=0°the E-field of the XP mode parallel to the RP), all thelectric field of the XP mode but none of the YP mode willass, which corresponds to the XP mode optical feedback.n the other hand, when �p=90° (the E-field of the YPode parallel to RP), all the electric field of the YP mode
ut none of XP mode will pass, which is the YP mode op-ical feedback. Thus, the linearly polarized emergingeam from the RP becomes Ef=Ex cos��p�+Ey sin��p�, hav-ng field oscillations along the transmission axis.
ig. 1. (Color online) Schematic illustration of VCSELs with ro-ating polarizer in the external cavity. VCSEL, vertical cavityurface emitting laser; ML,: microscopic lens; BS, beam splitter;P, rotating polarizer; M, mirror; NDF, neutral density filter;
SO, optical isolator.
When the feedback light is incident into the VCSEL atngle �p with respect to the XP mode field, the feedbackights reinjected into the XP and YP modes can be ex-ressed as Ex��t−��=Ef�t−��cos��p� and Ey��t−��=Ef�t��sin��p� [23], where � is the feedback delay. Especiallyhen �p=0°, the reinjected feedback lights are Ex��t−��Ef�t−��cos��p�= �Ex�t−��cos��p�+Ey�t−��sin��p��cos��p�Ex�t−�� along the XP mode and Ey��t−��=Ef�t−��sin��p�0 along the YP mode. These feedback terms are theame as those of �p=0° in the case A feedback in [23].hen �p=90°, the reinjected feedback lights are Ex��t−��Ef�t−��cos��p�=0 along the XP mode and Ey��t−��=Ef�t��sin��p�= �Ex�t−��cos��p�+Ey�t−��sin��p��sin��p�=Ey�t�� along the YP mode. Note that we assume that eacholarizer angle considered in this work was adjusted prioro the investigation on the polarization properties. And,he calculations are redone with the same initial condi-ions for each polarizer angle if there is no special instruc-ion. Then the time to obtain the polarizer angle is notaken into account. Thus, the delay � in our work only de-ends on the length of the feedback loop circuit. It isorth mentioning that these expressions are also in ac-
ordance with the expressions using the Jones matrixheory in [26] [See Eq. (9)].
Our theoretical model is based on the SFM of a solitaryCSEL operating in the fundamental transverse-modeegime [6]. These equations can be rewritten in terms ofwo orthogonal LP components Ex and Ey and two carrieropulations, i.e., the total carrier inversions between con-uction and valence bands N and the difference betweenarrier inversions with opposite spins n. Then we extendhe SFM to take into account the optical feedback withontrollable polarization as follows [6,7,23]:
Ex = k�1 + i����N − 1�Ex + inEy�
− ��a + i�p�Ex + �Ex�t − ��cos2��p�e−i��
+ �Ey�t − ��cos��p�sin��p�e−i��, �1�
Ey = k�1 + i����N − 1�Ey − inEx�
+ ��a + i�p�Ey + �Ex�t − ��sin��p�cos��p�e−i��
+ �Ey�t − ��sin2��p�e−i��, �2�
N = �N�� − N�1 + �Ex�2 + �Ey�2� + in�ExEy* − EyEx
*��, �3�
n = − �sn − �N�n��Ex�2 + �Ey�2� + iN�EyEx* − ExEy
*��, �4�
here k is the field decay rate, �N is the decay rate of N,s is the spin-flip rate, � is the linewidth enhancementactor, � is the normalized injection current (�=1 is at thehreshold), � is the center frequency of VCSEL, � is theeedback strength, �p is the linear birefringence, and �a ishe linear dichroism. It is clear that �p leads to a fre-uency difference of 2�p between XP and YP modes (withhe XP mode having the lower frequency when �p is posi-ive) and that �a leads to different thresholds for the twoodes, with the YP mode having the lower thresholdhen �a is positive [7]. In the following discussion, theangevin noise terms are not included in the calculation
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478 J. Opt. Soc. Am. B/Vol. 27, No. 3 /March 2010 Xiang et al.
or the sake of simplicity. We numerically solve the equa-ions by the fourth order Runge–Kutta method by usingn integration step of 0.5 ps with typical VCSEL param-ters [7]: �=300 ns−1, �=3, �N=1 ns−1, �s=50 ns−1, �a=0.1 ns−1, �=1.5, and �=850 nm,�=3 ns. The parametersand �p are chosen freely to study their effects on the po-
arization properties.
. RESULTS AND DISCUSSIONSn this section, for the purpose of comparison VCSEL1ith closely spaced frequency splitting (e.g., �p=6 ns−1)nd VCSEL2 with widely spaced frequency splittinge.g.,�p=40 ns−1) are analyzed. First, the polarization de-rees of VCSELs are discussed in detail for both VCSELs,nd the effects of initial conditions, spin-flip rate, as wells the feedback strength � are considered. Second, theutputs in the time domain and the polarization evolu-ions by the representation on the Poincaré sphere areurther presented to explain the polarization switchingrocess.To quantify the degree of polarization for a given pair of
eld amplitude components Ex�t� and Ey�t�, we give theractional polarization (FP) defined by the four Stokes pa-ameters in Cartesian coordinate system as [7]
FP =�S1�t��2 + �S2�t��2 + �S3�t��2
�S0�t��2 , �5�
here � � denotes time average, S0�t�= �Ex�t��2+ �Ey�t��2,1�t�= �Ex�t��2− �Ey�t��2, S2�t�=2 Re�Ex�t�Ey
*�t��, S3�t�2 Im�Ex�t�Ey
*�t��, and the time used for averaging is t�20 ns,400 ns�, which can let transients die away. Thealue of the FP ranges from 0 (natural unpolarized light)o 1 (polarized light).
Figure 2 shows the polarization-resolved intensitiesnd the defined FP of VCSEL1 with �p=6 ns−1 and of VC-EL2 with �p=40 ns−1, respectively, where the initial con-itions are taken in both the x- and y-polarized states ofhe solitary laser for each polarizer angle. The feedbacktrength is set to be �=30 ns−1 to make sure that the VC-ELs exhibit chaotic output. The intensities are ex-ressed as Ix,y= ��Ex,y�t��2�. Note that the feedbacktrength and injection current are set to be fixed here toocus on the effect of polarizer angle. For VCSEL1, it cane seen from Fig. 2 (a1) that with the increase of polarizerngle, the intensity of the XP mode decreases almost lin-arly and the intensity of the YP mode increases almostinearly. Especially when �p=0°, only the XP mode existsith the YP mode being suppressed, and when �p=90°
nly the YP mode exists, which denotes the polarizationtate switches from the XP mode to the YP mode as theolarizer angle is varied from �p=0° to �p=90°. For thosentermediate values of �p, both the XP and the YP modesoexist and have comparable intensities at about �p=40°,howing that the output of VCSEL1 is neither purely theP mode nor the YP mode but a mixed state of polariza-
ion. In addition, it can be seen from Fig. 2 (a2) that theefined FP varies around a relatively larger value that islose to 1. However, for VCSEL2 with �p=40 ns−1, the po-arization properties are quite different. Within the range
40°, the intensity of the XP mode is almost a constant
p
hat is much greater than that of the YP mode; therefore,he YP mode is almost negligible. But there is relativelyharp polarization switching that begins at approxi-ately �p=40°. The intensities of the XP and YP modes
re almost equal at �p=50°. In the range �p60°, the in-ensity of the YP mode is much greater than that of theP mode such that the XP mode is almost negligible.oreover, the defined FP decreases drastically in the
ange of 40° �p60° and is close to 0 at about �p=50°.e define this specific polarizer angle where the FP
eaches its minimum value FPmin as the critical anglepm. It can be found that when the intensities of the XPnd YP modes are comparable, the value of the FP is rela-ively small and reaches its minimum value when the in-ensities are equal. Moreover, a smaller absolute value ofhe intensity difference gives rise to a smaller value of FP.ut for the �p=6 ns−1, the FP varies slowly, which iseaningless for discussing the �pm. Interestingly, the re-
ults shown in Fig. 2 (a1) and Fig. 2 (b1) are also quiteimilar to the experimental findings that dealt with thedge-emitting lasers [26]. Namely, the result for �p6 ns−1 is similar to that of the laser diode with closelypaced parallel and perpendicular spontaneous emissionurves by 5.6 nm, and the result for �p=40 ns−1 is similaro that of the laser diode with widely spaced componentsf the spectrum by 12.6 nm.
Note that in [23], we adopted the Lang–Kobayashiquations to study the polarization properties, which mo-ivates us to analyze the difference between the SFModel in this paper and the simpler model considered in
23] and to study the effect of the spin-flip rate briefly. Weonsider the case of the limit of a very large spin-flip rates value (very fast mixing of populations with differentpins), which is equivalent to setting n equal to zero inqs. (1)–(4), and then the equations become
Ex = k�1 + i���N − 1�Ex − ��a + i�p�Ex
+ �Ex�t − ��cos2��p�e−i��
+ �Ey�t − ��cos��p�sin��p�e−i��, �6�
Ey = k�1 + i���N − 1�Ey + ��a + i�p�Ey
+ �Ex�t − ��sin��p�cos��p�e−i��
+ �Ey�t − ��sin2��p�e−i��, �7�
N = �N�� − N�1 + �Ex�2 + �Ey�2��. �8�
t can be seen that Eqs. (6)–(8) are the two-mode Lang–obayashi equations. We integrate Eqs. (6)–(8) with the
ame parameters and initial conditions as in Fig. 2, andhe corresponding results for the case of the limit of a veryarge spin-flip rate ��s→�� are shown in Fig. 3. It can beeen that the polarization properties are quite similar tohe results shown in Fig. 2. Nevertheless, the criticalngle for the case of �p=40 ns−1 moves to a lower valueround �pm=45°.Instead of taking the initial condition in both the XPode and the YP mode for each polarizer angle as it was
one in Fig. 2, we change the polarizer angle in steps toheck the dependence of FP on the initial conditions. That
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Xiang et al. Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. B 479
s to say, the initial condition for the next value of �p is thenal condition of the previous value of �p. In particular,he polarizer angle is increased in steps, first from �p0° to �p=90° and then decreased in steps from �p=90° top=0°, so as to discuss the hysteresis properties as well.ere, each new value of �p is held in a time duration of0 ns, and the FP is calculated by averaging over the last0 ns for each new polarizer angle. We just concentrate onhe polarization properties of VCSEL2 with �p=40 ns−1,nd the corresponding results are shown in Fig. 4. It cane observed that for all the cases of initial conditions, theritical angle �pm and the corresponding FPmin are stillound. When the initial condition is chosen in the XPode, the FP reaches its minimal value at about �pm51° during the increase process but at about �pm=48°uring the decrease process. That is to say, the hysteresisccurs and a greater value of FPmin is obtained in the de-rease process. When the initial condition is chosen inoth the XP and YP modes, the results shown in Fig. 4(b)re similar to those in Fig. 4(a). However, when the initialondition is chosen in the YP mode, the �pm obtained inhe increase and decrease processes are almost identical.oreover, the value of FPmin for increasing �p is relatively
arger than that of decreasing �p.Next, we further study how the polarization properties
re affected by the feedback strength. Figure 5 and Fig. 6
ig. 2. (Color online) Polarization-resolved intensities (a1, b1)nd FP (a2, b2) as a function of the polarizer angles, with �p6 ns−1, �=30 ns−1 for the left column and �p=40 ns−1, �30 ns−1 for the right column.
Fig. 3. (Color online) Same as Fig. 2, with � →�.
s
how the defined FP as a function of the polarizer angleor different feedback strengths for both cases of �p6 ns−1 and �p=40 ns−1, respectively. For both cases, four
eedback strengths at �=10 ns−1, 20 ns−1, 40 ns−1, and0 ns−1 are introduced. For �p=6 ns−1, it can be observedrom Fig. 5 that there is no sharp decrease of FP for allhe feedback strengths. When �=10 ns−1, the values of FPre relative small (i.e., smaller than 0.6) in the range0° �p80°, while for the rest of �, the values of FPary near a relatively larger value that is close to 1 at thehole range of polarizer angles. A higher � gives rise to a
arger FP. On the other hand, for �p=40 ns−1, the sharpecrease of FP occurs under all the feedback strengths, soe just concentrate on the critical angle �pm and its cor-
esponding minimum value of FP (i.e. FPmin) for each �. Itan be observed from Fig. 6 that with the increase of � theritical angle �pm moves to a lower polarizer angle, andhe resulting FPmin increases.
ig. 4. (Color online) FP as a function of the polarizer angles forifferent initial conditions. (a) The initial condition is chosen inhe XP mode; (b) the initial condition is chosen in both the XPnd YP modes; (c) the initial condition is chosen in the YP mode.
ig. 5. (Color online) FP as a function of the polarizer angles forifferent �, with � =6 ns−1.
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For the analyses above, we studied the polarization de-ree of the two VCSELs without including the time evo-utions for the polarization properties. Next, the outputsn the time domain and the polarization evolutions by theepresentation on the Poincaré sphere would be furtherresented to provide more physical insight into the polar-zation properties.
Figure 7 and Fig. 8 show the outputs in the time do-ain for specific polarizer angle for both �p=6 ns−1 and
p=40 ns−1 corresponding to Fig. 2, respectively. It can beeen from Fig. 7 that when �p=0°, the XP mode is theominant polarization mode and exhibits chaotic dynam-cs with the YP mode being completely suppressed, whichefers to purely the XP mode. Interestingly, these resultsre quite similar to the results of �p=0° for the case Aeedback in [23]. For �p=40°, both the XP and YP modesxhibit chaotic dynamics, and their intensities are close toach other. When �p=90°, only the YP mode exhibits cha-tic dynamics and the XP mode is completely suppressed,hich refers to purely the YP mode. The time series for
ig. 6. (Color online) FP as a function of the polarizer angles forifferent �, with �p=40 ns−1.
ig. 7. Output in the time domain for both XP (left column) andP (right column) modes, (a1–a2) �p=0°, (b1–b2) �p=40°, (c1–c2)=90°, with � =6 ns−1, �=30 ns−1.
p p
p=0° and �p=90° shown in Fig. 8 are quite similar tohose shown in Fig. 7. However, it can be found that forp=50° there are pulse packages for both the XP and YPodes, and anti-correlation characteristics seem to exist
etween the x-polarized packages and the y-polarizedackages [28]. We further calculate the time-dependentormalized cross-correlation function C��t� between theP and YP modes to clarify the differences between Figs.(b1)–(b2) and Figs. 8 (b1)–(b2). The cross-correlation co-
here �t denotes the lag time. The results are shown inig. 9, where Fig. 9(a) corresponds to Figs. 7 (b1)–(b2) andig. 9(b) corresponds to Figs. 8 (b1)–(b2). From Fig. 9(a)e observe a clear maximum of the cross-correlation coef-cient (positive C) around zero time lag and some positiveeaks of C at the delay time (and multiples thereof),
ig. 8. Output in the time domain for both XP (left column) andP (right column) modes, (a1–a2) �p=0°, (b1–b2) �p=50°, (c1–c2)p=90°, with �p=40 ns−1, �=30 ns−1.
Xiang et al. Vol. 27, No. 3 /March 2010/J. Opt. Soc. Am. B 481
hich means the XP mode and the YP mode exhibit in aorrelated manner. However, in Fig. 9(b), a clear mini-um of the cross-correlation coefficient (negative C)
round zero time lag and some negative peaks of C at theelay time (and multiples thereof) are found, whicheans the XP mode and the YP mode exhibit in an anti-
orrelated manner.To characterize the evolution of the polarization state of
ight and to further explain the polarization switchingrocess, we use the Poincaré sphere plot, where for aiven pair of field amplitude components Ex�t� and Ey�t�e assign a point on the Poincaré sphere whose coordi-ates are [9,29] x�t�=S1�t� /S0�t�, y�t�=S2�t� /S0�t�, and�t�=S3�t� /S0�t�. The points in the Poincaré sphere of unitadius are in one-to-one correspondence with the differentolarization states of the laser beam. The northern andouthern hemispheres correspond to right- and left-lliptical polarizations. The south pole, the north pole,nd the equatorial plane represent the left-circularly po-arized, the right-circularly polarized, and the linearly po-arized lights. In particular, the positive x axis and theegative x axis represent two orthogonal polarizationtates, x polarization and y polarization.
Figure 10 shows the time evolutions of a polarizationtate on the normalized Poincaré sphere. For �p=6 ns−1,e consider the polarization evolutions under the condi-
ions �p=0° ,10° ,40° ,80° ,90°, which correspond tohese of Fig. 2 (a1). When �p=0°, only one point is locatedt the positive x-axis on the Poincaré sphere surface,hich denotes the purely XP mode. When �p=90°, a
ingle point located at the negative x-axis on the Poincaréphere surface denotes the purely YP mode. As the pic-
ig. 10. (Color online) Evolutions of polarization state plottedn the normalized Poincaré sphere for VCSEL1 with �p=6 ns−1
left column) and VCSEL2 with �p=40 ns−1 (right column), with=30 ns−1.
ures for these two �p are quite simple, we do not presenthem in Fig. 10. When �p=10°, it can be seen that the po-arization output from a VCSEL changes rapidly withime, but the points are mainly localized around a certainegion near the equator with a slight deviation from theositive x-axis on the Poincaré sphere, which denotes ainearly polarized state with a slight different orientationn respect to the XP mode. Therefore all the points on theoincaré sphere imply that there exists a competition be-
ween this linearly polarized state and a few ellipticallyolarized states. When �p=40°, the competition between ainearly polarized state and elliptically polarized statesecomes more frequent. When �p=80°, note that the-axis is reversed with respect to that of �p=10° for theake of convenient visualization. The polarization evolu-ions are similar to the result induced by �p=10° excepthat the points mainly lie in the region near the equatorith a slight deviation from the negative x-axis, which in-icates a linearly polarized state with a slight differentrientation in respect to the YP mode.
Again, for �p=40 ns−1 we consider the polarization evo-utions under the conditions �p=0° ,10° ,50° ,80° ,90°,hich are consistent with those used in Fig. 2 (b1). The
epresentations on the normalized Poincaré sphere forp=0° and �p=90° are the same as those in VCSEL1 withp=6 ns−1. When �p=10°, it can be seen that the pointsainly locate on the region near the positive x-axis, and
here seems to be a competition between the XP mode andhe elliptically polarized states. When �p=50°, the pointsistribute on almost the whole sphere, indicating quite ir-egular polarization fluctuations, which could be causedy the anti-correlation characteristics between the-polarized packages and the y-polarized packages shownn Fig. 7(b). When �p=80°, the x-axis is also reversed, theoints mainly locate on the region near the negative-axis, and there seems to be a competition between theP mode and the elliptically polarized states.By comparing the time evolutions of the polarization
tates in VCSEL1 and VCSEL2 represented on the nor-alized Poincaré sphere, it can be observed that although
he polarization states switch from the XP mode to the YPode as the polarizer angle is varied from �p=0° to �p90° for both VCSELs, the polarization evolutions during
he polarization switching process are quite different,hich cannot be obtained by using time-averaged inten-
ities analyses.
. CONCLUSIONn summary, we have investigated the polarization de-rees of VCSELs subjected to optical feedback with con-rollable polarization. The results show that as the polar-zer angle is varied from 0° to 90°, polarization switchingrocesses are observed in both VCSELs with different fre-uency splitting, VCSEL1 with �p=6 ns−1 and VCSEL2ith �p=40 ns−1. However, the switching processes areuite different. For VCSEL1, the polarization state varieslowly from the XP mode along the linearly polarizedtate oriented at the direction between the XP and YPodes and finally switches to the YP mode. But for VC-EL2, the polarization switching from the XP mode to theP mode that occurred at around the critical angle � is
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482 J. Opt. Soc. Am. B/Vol. 27, No. 3 /March 2010 Xiang et al.
uch sharper. When the polarizer angle is less than �pm,he linearly polarized state keeps locating around the XPode, while for the polarizer angles greater than �pm the
inearly polarized state turns to locate around the YPode. It is also predicted that in VCSEL1 a higher � gives
ise to a larger FP, while in VCSEL2 the critical angle �pmoves to a lower value and the FPmin increases with the
ncrease of �. The outputs in the time domain furtherhow that when �p=�pm the anti-correlation property oc-urs between the XP and the YP polarized packages inCSEL2. From the representation on the normalizedoincaré sphere, it can be observed that the two time evo-
utions of polarization states during the polarizationwitching process for VCSEL1 and VCSEL2 are also dif-erent. Furthermore, for both VCSELs the competition be-ween linearly polarized states and elliptically polarizedtates are observed. In particular, the polarization stateompetition is more frequent when the polarizer angle isear the critical angle, leading to an almost random dis-ribution on the Poincaré sphere surface. These resultsresent more insight into the polarization properties ofCSELs and may provide more flexible polarization con-
rol method.
CKNOWLEDGMENTS. Y. Xiang is sincerely grateful to Dr. W. L. Zhang for hiselp. The authors thank all reviewers for their helpfuluggestions. This project is supported by the Nationatural Science Foundation of China (NNSFC) project0976039 and the Specialized Research Fund for the Doc-oral Program of Higher Education of China project0070613058.
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