Polarization properties of vertical-cavitysurface-emitting lasers subject tofeedback with variably rotated
polarization angle
Shuiying Xiang,* Wei Pan, Lianshan Yan, Bin Luo, Ning Jiang, and Lei YangCenter for Information Photonics and Communications, School of Information Science
and Technology, Southwest Jiaotong University, Chengdu 610031, China
Semiconductor lasers subjected to optical feedbackhave attracted intensive research effort, owing totheir potential to generate high-dimensional chaosand their application in chaotic optical communica-tions [1–7]. Most existing studies on optical feedbackrelate to conventional edge-emitting lasers (EELs)and thus neglect the polarization nature. Vertical-cavity surface-emitting lasers (VCSELs) are promis-ing devices in many applications [2,3,8–11] becauseof their advantages of single longitudinal mode, sim-ple array fabrication, low threshold current, highmodulation rate, and low cost. Moreover, they prefer-entially emit linear polarized (LP) light along one ofthe two orthogonal directions (x and y) owing to weak
material and cavity anisotropies. Controllablepolarization switching (PS) may be useful to realizepolarization modulation for fiber-optic communica-tions [10] and reconfigurable optical interconnects[11]. In addition, polarization bistability VCSELswere also demonstrated to implement optical buffermemory [12,13].
Optical feedback strongly affects the dynamicsof VCSELs. Five feedback regimes of behavior forVCSELs subject to optical feedback were identified[14]. Since the first experimental report on PScharacteristics of optically injected VCSELs [15]and the proposal of a spin flip model (SFM) [16], thepolarization dynamics of single transverse modeVCSELs subject to optical feedback or injection havebeen investigated intensely [17–28]. Among thesestudies, a frequency-induced polarization bistabilityin VCSELs subject to orthogonal optical injectionwas observed [25]. Besides, the dynamics and
polarization characteristics of mutually coupledVCSELs were also investigated [26,27]. In addi-tion, polarization behaviors of multitransverse modeVCSELs with optical injections have also been inves-tigated both theoretically [29] and experimentally[30]. Recently, there was a report on the experimen-tal observation of dynamical collapses of an EELsubject to optical feedback with variably rotated po-larization, where the flexible polarization rotationangle was implemented by using a λ=4 wave platein the optical feedback path [31]. It is supposedthat, by introducing a polarization controller (PC)to vary the polarization angle of optical feedback,the VCSELs that hold the intrinsic polarizationproperties will exhibit much richer polarization prop-erties and provide more flexible polarization control,which deserves investigation in depth.In this paper, we concentrate on the effects of vari-
able polarization angle of feedback light on thepolarization properties of VCSELs subject to vari-able-angle polarization-rotated optical feedback (VA-PROF). It is observed that, as the polarization angleis varied from 0° to 90°, the pure polarization angleinduced dominant polarization mode switching is ob-served for the so-called case A feedback (i.e., only thexmode is selected to pass through the feedback loop),and the so-called critical angles are observed for theso-called case B feedback (i.e., the total outputs of aVCSEL pass through the feedback loop). The remain-der of this paper is organized as follows. In Section 2,a theoretical model is presented. The rate equationsthat include two polarization modes and the VA-PROF effect are given to describe our model. In Sec-tion 3, the validity of our model is demonstratedbriefly by investigating the PS of a VCSEL thathas been experimentally studied. Then the influ-ences of feedback strength and polarization angleon the polarization properties are discussed in detail.Conclusions are given in Section 4.
2. Theoretical Model
The schematic diagram for a VCSEL subjected to VA-PROF is shown in Fig. 1. Delayed variable-angle po-larization-rotated optical feedback is implementedby an external optical loop circuit, where the PCrotates the optical feedback to obtain the controlledpolarization angle (i.e., the piezoelectric nanoposi-
tioning stage utilized in Ref. [32] is a promising can-didate to control and measure the polarizationangles). For the purpose of comparison, we considertwo feedback cases called case A (x-polarization feed-back) and case B (xy-polarization feedback). In caseA, we utilize an x polarizer (XMP) to make sure thatonly the x-polarization (XP) mode is selected to passthrough the PC to reinject to the VCSEL. While inthe case B, we assume that the total outputs (i.e.,both XP and y-polarization (YP) modes) pass throughthe PC to be reinjected into the VCSEL, so the XMPshould be removed. The feedback strength can be ad-justed by the neutral density filter (NDF). The opti-cal isolator is used to guarantee the unidirectionaloptical circuit.
Here, we define the 0° polarization to be the “fun-damental” or first polarization mode, i.e., the XPmode, and the 90° polarization to be the second po-larization mode, i.e., the YP mode. Similar toRef. [32], the optical feedback is assumed to be ro-tated at a polarization angle θp varying from 0° to 90°with respect to the first VCSEL polarization mode.Hence, θp ¼ 0° represents the parallel optical feed-back [19–21], and θp ¼ 90° stands for the orthogonaloptical feedback [24,25].
Essentially, the purpose of introducing θp is to ob-tain controllable allocation of the fixed feedbackstrength to XP and YP modes. The decompositionof VAPROF light on the two orthogonal modes is illu-strated in Fig. 2. The variable Ex;y
��!is the slow varia-
tion of XP and YP of the electrical field, and τd is thefeedback delay. Note that we assume that every po-larization angle considered in this work was adjustedprior to investigating the polarization properties.The time to obtain the rotated polarization angle isnot taken into account. Thus the τd in our work de-pends only on the length of the feedback loop circuit.For case A feedback, the light feedback into the 0°
polarization is Exðt − τdÞ������!
cosðθpÞ and in the 90° polar-
ization is Exðt − τdÞ������!
sinðθpÞ. On the other hand, forcase B feedback, the light feedback into the 0° polari-
zation is Exðt − τdÞ������!
cosðθpÞ − Eyðt − τdÞ������!
sinðθpÞ and in
Fig. 1. Schematic diagram for the VCSEL subject to VPROF:VCSEL, vertical cavity surface emitting laser; ML, microscopiclens; BS, beam splitter; PC, polarization controller; M, mirror;NDF, neutral density filter; ISO, optical isolator; XMP, x polarizer.
Fig. 2. (Color online) Decomposition of the polarization-rotatedfeedback on the two orthogonal modes Ex and Ey.
cosðθpÞ.The Lang–Kobayashi equations have been shown
to be highly accurate in describing optical feedbackand injection effects in semiconductor lasers. Thishas been recently extended to enable treatment ofPS phenomena in VCSELs [18,19,22,27]. We havethus adopted this approach in our work. And thefeedback terms are included by assuming a singlereflection in the external cavity. We rewrite Ex;y
��! ¼Ex;yeiΦx;y , where Ex;y and Φx;y denote the amplitudeand phase of the XP and YP fields, respectively.Then, for case A feedback, we extend the Lang–Kobayashi equations to take into account two ortho-gonal linear polarizations [22,27,32,33] and theVAPROF as follows:
The variable N is the carrier density. The subscriptsx and y indicate the XP and YP modes, respectively.Note that gx and gy are the gain coefficients for XP
and YP modes, with gx ¼ gy þ g0ð1 − J=J0Þ [18],where J0 ¼ 1:7Jth is the switching current for thesolitary VCSEL; here we take a typical experi-mental value [20] without losing generality. And thefrequency detuning between XP and YP modes is de-fined as Δω ¼ ωx − ωy ¼ 2πΔf ¼ 2πðf x − f yÞ. Descrip-tions for other parameters and their values are listedin Table 1.
For case B feedback, the rate equations for the elec-trical amplitude and phase of VCSEL are similar toEqs. (1)–(4), except that the feedback terms from theYP mode of the VCSEL should be considered.
The rate equation of N for case B feedback is thesame as Eq. (5). So far, the rate equations of theVCSEL subjected to VAPROF for both case A andcase B feedback are obtained. We numerically solvethe rate equations by employing the 4th-orderRunge–Kutta method.
3. Results and Discussion
In this section, we first demonstrate briefly the valid-ity of our model via comparing the result with recentexperiments. Then we focus on the influence ofVAPROF on the polarization properties.We take θp ¼ 0° (i.e., parallel optical feedback) as
an example to discuss the polarization properties forcase B feedback when the injection current was in-creased continually for a relatively smaller kd soas to demonstrate briefly the validity of our model.The result shown in Fig. 3 agrees qualitatively wellwith the experimental finding of Hong et al. [19].In the following study, we fix the bias current to
guarantee that the VCSEL exhibits single XP modeemission with high YP mode suppression when las-ing as a solitary laser as suggested in Ref. [33].Here we discuss the effect of feedback strength on
the polarization properties of the VCSEL subject toVAPROF for different θp values. Figure 4 shows theaverage intensities of XP and YP modes versus thefeedback coefficient kd for case A feedback. Espe-cially, kd was varied in the range referred to inRef. [33]. Here, the average intensities are expressedas Ix ¼ hjExðtÞj2i and Iy ¼ hjEyðtÞj2i, where hi denotestime average. When θp ¼ 0°, the VCSEL emits singleXP mode all the time; there is no dominant modeswitching in the entire region of kd in Fig. 4(a). Thisis because Exðt − τdÞ cosðθpÞ ¼ Exðt − τdÞ, Exðt − τdÞsinðθpÞ ¼ 0; then the XP mode obtains more andmore feedback light with the increase of kd, whilethe YP mode obtains no feedback light in the entireregion of kd and is depressed. On the other hand, itcan be seen from Fig. 4(b) that, when θp ¼ 45°, theintensities of both modes increase, with the XP modealways being the dominant mode. For this θp, the op-tical feedback was allocated equally to the XP and YPmodes. For higher values of θp, i.e., 72° and 90°, withthe increase of feedback strength, the intensity of YPmode increases and the intensity of XP mode de-creases, and hence the dominant polarization modeswitching occurs. Moreover, it is illustrated fromFigs. 4(c) and 4(d) that larger θp leads to dominantpolarization mode switching occurring at a lowerkd, which indicates that the increase of θp favors
emitting of YP mode. For those θp values, the YPmode obtains more feedback light than XP modefor a given kd, and the feedback difference becomesmuch larger with an increase of kd. So the domi-nant mode switches to the YP mode. Furthermore,it is also observed that the total intensities inFigs. 4(a)–4(d) vary similarly, but are relativelysmaller for larger θp.
Correspondingly, Fig. 5 gives the average intensi-ties of XP and YP modes versus the feedback coeffi-cient kd for case B feedback. When θp ¼ 0°, it isshown from Fig. 5(a) that the variation of intensity
Fig. 3. (Color online) Polarization-resolved L–I curve of theVCSEL subjected to parallel optical feedback. The solid curveand the dashed curve correspond to XPmode and YPmode, respec-tively, with kd ¼ 2 × 109 s−1.
Fig. 4. (Color online) Polarization-resolved intensities versus op-tical feedback coefficient for case A (x-polarization feedback) forpolarization angle fixed at (a) θp ¼ 0°, (b) θp ¼ 45°, (c) θp ¼ 72°,and (d) θp ¼ 90°. The dotted (dashed) line with crosses (circles) cor-responds to the intensity of XP (YP) mode, and the solid line cor-responds to the total intensity.
exhibits a similar tendency to that of case A. This isbecause when θp ¼ 0°, the feedback light reinjectedinto the XP mode becomes Exðt − τdÞ and that intothe YP mode becomes Eyðt − τdÞ. Note that the biascurrent was fixed at a value to make sure the YPmode was highly depressed, leading to the feedbackobtained by YP mode being always Eyðt − τdÞ ¼ 0.However, when θp ≠ 0°, with an increase of kd, thecharacteristics are quite different. In Figs. 5(b)–5(d),the intensities of both XP and YP modes increasewith an increase of kd in the entire region. The domi-nant mode is always the XP mode. For these θpvalues, XP and YP modes obtain comparable feed-back for a given kd, which is presented in more detailbelow (see Fig. 10). And for a specific θp, as kd is in-creased, both the XP and YP modes obtain more op-tical feedback.Next we investigate the effects of variable polari-
zation angle on the polarization properties of aVCSEL. Here we set both the feedback strengthand bias current to be fixed values and vary the po-larization angle from 0° to 90°. Figure 6 shows thepolarization-resolved intensities as a function ofthe polarization angle θp for a VCSEL subject to caseA feedback. It can be observed from Figs. 6(a)–6(d)that, with an increase of θp, the intensity of XP modeand the total intensities decrease rapidly, while theintensity of YP mode increases slowly in Figs. 6(a)and 6(b) in the entire region but becomes saturatedin the range θp > 45° in Figs. 6(c) and 6(d). Moreover,for the larger kd, as the decreasing speed of the inten-sity of XP mode is much higher than the increasingspeed of YP mode, the dominant mode switches fromXP mode to YP mode. We call this polarization anglewhere the dominant mode is changed a switchingpoint. It can be found that the location of switchingpoint for different kd values is somewhat different. Itis located at about θp ¼ 63° in Fig. 6(c) and at aboutθp ¼ 54° in Fig. 6(d), which illustrates that a VCSELwith larger kd requires a smaller polarization angle
to implement the dominant polarization modeswitching.
To further explain the difference of variation of YPmode between Figs. 6(a)–6(d), as well as the domi-nant mode switching in Figs. 6(c) and 6(d), we definethe feedback strength obtained by XP and YP modesas FAx ¼ hjkdExðt − τdÞ cosðθpÞji, FAy ¼ hjkdExðt − τdÞsinðθpÞji. Figure 7 presents FAx and FAy as a functionof polarization angle. From Fig. 7(a), it can be ob-served that, with the increase of polarization angle,the FAx decreases and FAy increases. But as here kd isrelatively smaller, the optical feedback plays a lessimportant role than the gain coefficients. As a resultof gy < gx, the intensity of XPmode is still larger thanthat of YP mode. However, in Fig. 7(b), the variationof FAy is quite different from Fig. 7(a); it increaseswhen θp < 45° and becomes saturated after thatand finally decreases slowly. On the other hand,for the relatively larger kd, the optical feedbackand the gain coefficients compete for the dominanteffect. When the difference of feedback strength be-tween the two modes FAy − FAx is large enough, theoptical feedback plays the dominant role, leading tothe YP mode being the dominant mode. Obviously,the larger kd is, the larger the value of FAy − FAx.
We further present in Fig. 8 the outputs in the timedomain to illustrate the dominant polarization modeswitching process corresponding to Fig. 6(c). We cansee from Fig. 8 that, when θp ¼ 0°, the XPmode is thedominant mode. When θp ¼ 63° (i.e., the switchingpoint), the intensity of XP mode is almost equal tothat of YP mode, which denotes the switching point.When θp ¼ 90°, the intensity of YP mode is obviouslygreater than that of XPmode, leading to the YPmodebeing the dominant mode.Fig. 5. (Color online) As in Fig. 4 but for case B feedback.
Fig. 6. (Color online) Polarization-resolved intensities versus po-larization angel for case A (x-polarization feedback) for feedbackcoefficient fixed at (a) kd ¼ 2 × 1011, (b) kd ¼ 4 × 1011,(c) kd ¼ 8 × 1011, and (d) kd ¼ 10 × 1011. The other descriptionsare the same as in Fig. 4.
Correspondingly, Fig. 9 gives the polarization-resolved intensities as a function of polarization an-gle θp for a VCSEL subject to case B feedback. Itcan be found that the tendencies for smaller kd inFigs. 9(a) and 9(b) are similar to the counterpartin Fig. 6, except that there are some fluctuationsin Figs. 9(a) and 9(b). However, for larger kd, the po-larization properties are much different than that ofcase A feedback. In Fig. 9(c), as θp is increased, when0° ≤ θp ≤ 18°, the intensity of XP mode decreases andthe intensity of YP mode increases with the XP modebeing the dominant mode. When θp ¼ 27°, the inten-sity of YP mode becomes quite close to that of XPmode, and after that, both intensities vary near aconstant with slight fluctuations. We called this θp ¼27°, where both intensities begin to approach eachother as the critical point and the corresponding an-gle as critical angle θpc. As the difference between theintensities of the two modes is very small, it is mean-ingless to discuss the dominant polarization modehere. In Fig. 9(d), the trends are quite similar toFig. 9(c), except that the critical point occurs at asmaller angle θpc ¼ 18°.The feedback strengths obtained by XP mode and
YP mode for case B feedback are defined as FBx ¼hjkdExðt − τdÞ cosðθpÞ − kdEyðt − τdÞ sinðθpÞji, FBy ¼
hjkdExðt − τdÞ sinðθpÞþkdEyðt − τdÞ cosðθpÞji. Figure 10shows FBx and FBy as a function of polarization angle.It can be seen from Fig. 10(a) that, when 0° ≤ θp ≤
36°, FBx > FBy; after that, there is a sharp switchingfor the feedback strength leading to FBx < FBy, andboth keep around a constant value in the rest ofθp. Still, because of the gain coefficients, the XPmodeis the dominant mode in Fig. 9(a). For the other kd inFig. 10(b), the FBx decreases and FBy increases untilθp is increased to the critical angle. After that, thefeedback strengths obtained by XP and YP modesare comparable to each other, leading to the intensi-ties of XP mode and YP mode being close to eachother but with the intensity of XP mode beingslightly larger than that of YP mode owing to the ef-fects of gain coefficients gy < gx. Moreover, the com-parable feedback strength obtained by XP and YPmodes can also be used to explain the similar trendsfor different θp values in Figs. 5(b)–5(d).
Likewise, Fig. 11 presents the outputs in the timedomain for case B feedback. The results are in accor-dance with Fig. 9(c).
Fig. 7. (Color online) Feedback strength versus the polarizationangle for case A (x-polarization feedback: (a) kd ¼ 2 × 1011,(b) kd ¼ 8 × 1011.
Fig. 8. Outputs in the time domain for case A (x-polarizationfeedback), with kd ¼ 8 × 1011 s−1. For (a1)–(a3), θp ¼ 0°, 63°, and90°, respectively, and the same for (b1)–(b3).
Fig. 9. (Color online) As in Fig. 6 but for case B feedback.
Fig. 10. (Color online) Feedback strength versus the polarizationangle for case B (xy-polarization feedback): (a) kd ¼ 2 × 1011,(b) kd ¼ 10 × 1011.
To identify the emission states of a VCSEL andto measure the relative intensity of XP and YPmodes, we define a parameter called the normalizedintensity (NI) in the following way:
NIx;y ¼hjEx;yj2i
hjEx;yj2 þ jEy;xj2i: ð11Þ
When NIx ¼ 1 ðNIy ¼ 1Þ, the VCSEL emits in pureXP (YP) mode. NIx ¼ 0:5 ðNIy ¼ 0:5Þ is the switchingpoint. NIx > NIy ðNIy > NIxÞ means the XP (YP)mode is the dominant mode.Finally we present a two-dimensional view of the
normalized intensity, showing the dependence onboth feedback strength and polarization angle, soas to get a further understanding of the polarizationproperties. The normalized average intensities of theXP and YPmodes in the parameter space of θp and kdare shown in Figs. 12 and 13 for case A and B feed-back, respectively. It can be seen that, for case A,there are stable dominant mode switching points la-beled by the contour line at NIx ¼ NIy ¼ 0:5 inFig. 12. The contour line divides the picture into
two parts; in the left part, XP mode is the dominantmode, while in the right part, the YP mode becomesthe dominant one. However, for case B, we do not careabout the dominant mode switching but concentrateon the region where XP and YP have comparable in-tensities. From Fig. 13, we find that NIx > 0:5, NIy <0:5 for most parameters, so we assume that whenNIx ≤ 0:6, NIy ≥ 0:4, the XP and YP modes have closeintensities. We also labeled the region by the contourlines. It can be found that this region is much widerfor the relatively larger kd. Obviously, the polariza-tion properties are quite different for the two feed-back cases.
4. Conclusions
In summary, we have considered the influence ofvariably rotated polarization angle of optical feed-back on the polarization properties of a VCSEL fortwo feedback cases, case A and case B. By utilizinga polarization controller to obtain the variable polar-ization angle, the VCSEL can exhibit dominant po-larization mode switching even for fixed opticalfeedback strength and bias current for case A feed-back. Besides, it is also predicted that the largerθp favors the emitting of YP mode that forces thedominant mode switching. While for the case B feed-back, the dominant mode switching cannot be ob-served in the entire parameter space, but the so-called critical points can be found at larger kd. Theseresults show that the VCSEL subjected to VAPROFcan exhibit rich polarization properties apart fromits intrinsic properties, which provides a flexibleand reliable method to achieve polarization control.
S. Y. Xiang is sincerely grateful to Prof. L. Chros-towski and Dr. W. L. Zhang for their helpful guidanceand suggestions. This project is supported by the Na-tional Natural Science Foundation of China (No.60976039) and the Specialized Research Fund forthe Doctoral Program of Higher Education of China(20070613058).
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