Polarization bistability of vertical-cavity surface-emitting lasers (VCSELs) is the situation in which two orthogonal linear polarization states can selectively and stably exist for a single operation condition (e.g., injected current and temperature) and provide switching between the two orthogonal linear polarizations and complex polarization dynamics. This is attractive for photonic computing in the sense that the complexity of a nonlinear node can be enhanced. However, polarization bistability is considered inherent to the material properties and cavity structure of VCSELs, which makes it difficult to optimize the complex dynamics to achieve the best computational performance. We show that the polarization bistability of VCSELs can be controlled by manipulating electron spins in the active region. We achieve such manipulation by using the transverse external magnetic field, i.e., Larmor precession of electron spins. We reveal that the control of the Larmor-precession frequency induces a polarization switching and its hysteresis behavior of VCSELs without changing current, heat, and external light, demonstrating that the manipulation of electron spins can control polarization bistability of VCSELs. This finding is a novel phenomenon in spin-controlled VCSELs, which will contribute to the development of interdisciplinary research fields in computing between photonics and spintronics.

Optical bistability refers to the situation in which two stable optical output states are associated with an input state1 and has attracted the interest of many researchers.1–3 Vertical-cavity surface-emitting lasers (VCSELs) have been known to exhibit polarization bistability, which is a type of optical bistability.1 The isotropic cavity structure of a VCSEL that is different from those of conventional edge-emitting lasers can be a cause of polarization bistability,4 in which one of the two output linear polarization states is selected by an input state such as current, heat, or external light.5–10 For example, polarization switching (PS) between the two orthogonal linear polarizations and its hysteresis behavior were observed when the injection current was swept upward and then downward,5–7 which confirms the existence of polarization bistability. Polarization bistability is useful to maintain information in the input state for a long time, and VCSELs are capable of high-density integration. Hence, various applications of VCSELs that take advantage of polarization bistability have been proposed, such as optical buffer memory, random number generation, and photonic neurons.11–13 The spin–flip model,14 which takes into account the interaction between polarizations of light and electron spins, is a theoretical model to simulate the polarization bistability of VCSELs. As shown in Fig. 1, the polarization property of a VCSEL is closely related to the spin polarization of electrons in a quantum-well active region because of the optical selection rule.15 Angular momentum conservation between photons and electrons allows specific transitions with right- (E+) and left-handed (E) circular polarizations depending on the spin polarization of electrons. Polarization bistability is considered to originate from the saturable dispersion and linear birefringence of VCSELs and is determined using the material and structural parameters of a VCSEL on the basis of the spin–flip model.14–16 Thus, polarization bistability is believed to be inherent to each VCSEL, but its control has received little attention. However, it should be possible to control the condition of polarization bistability and induce a PS by intentionally manipulating electron spins because the polarization properties originate from the interaction between the polarizations of light and electron spin.

FIG. 1.

Schematic of polarization dynamics in VCSELs based on the spin–flip model. Ωx: Larmor-precession frequency, γs: spin-decay rate, γp: linear birefringence, Jz: Z-direction angular momentum, E+: electric field with right-handed circular polarization, and E: electric field with left-handed circular polarization. Heavy holes are assumed to lose their spin polarization much faster than electrons in the quantum well active region of VCSELs.17 

FIG. 1.

Schematic of polarization dynamics in VCSELs based on the spin–flip model. Ωx: Larmor-precession frequency, γs: spin-decay rate, γp: linear birefringence, Jz: Z-direction angular momentum, E+: electric field with right-handed circular polarization, and E: electric field with left-handed circular polarization. Heavy holes are assumed to lose their spin polarization much faster than electrons in the quantum well active region of VCSELs.17 

Close modal

Spin-controlled VCSELs in which spin polarization of electrons is intentionally manipulated have been widely investigated.18–29 The injection of spin polarized electrons into VCSELs has led to unique features such as lasing with circular polarization, threshold-current reduction, and high-speed polarization modulation.30–37 The ability of high-speed polarization modulation stimulated various applications such as optical communications and photonic computing.38–42 Another type of spin manipulation is the Larmor precession of electron spins caused by an external magnetic field, as illustrated in Fig. 1, which has not yet been extensively studied in VCSELs. Hallstein et al. reported that the Larmor precession induces the polarization oscillation of VCSELs depending on its precession frequency, which is proportional to the strength of the transverse external magnetic field.43 In a theoretical study, the spin–flip model predicted a self-sustained Larmor oscillation (SSLO), which is a resonant phenomenon related to linear birefringence (γp) and Larmor-precession frequency (Ωx).15 Ωx is given by Ωx = BB/, where g is the Landé g factor of electrons, μB is the Bohr magneton, B is the external magnetic field, and is the reduced Planck constant. However, the effects of electron spin manipulation, such as the Larmor precession on the polarization bistability of VCSELs, have not yet been clarified.

We present a novel route of controlling the polarization bistability of VCSELs by electron spin manipulation. The Larmor precession under external magnetic fields is exploited as a manipulation tool of electron spins that exist as noise even in the absence of intentional spin injections. We theoretically and experimentally investigated the polarization bistability of VCSELs under the Larmor precession of electron spins. We confirmed a modified current condition of polarization bistability depending on Larmor-precession frequency, which provided a PS induced by swept Larmor-precession frequency without tuning current, heat, and external light. These results indicate that spin manipulation can control polarization bistability beyond limitations imposed by the material and structural parameters of VCSELs and pave the way to interdisciplinary research of photonics and spintronics.

The polarization bistability of VCSELs can be confirmed by observing PS and its hysteresis behavior during the sweeping of current (pump rate). However, we investigated PS and its hysteresis behavior by observing a sweeping current with a constant magnetic field or by observing a sweeping magnetic field with a constant current. We reveal the effects of the Larmor precession of electron spins on the polarization bistability of VCSELs by visualizing PS conditions in a graph.

We conducted simulations using the following equations of the spin–flip model, which takes into account the x-direction external magnetic field with Ωx:15,
(1)
(2)
(3)
(4)
where E+ and E are the electric fields with right- and left-handed circular polarizations, N is the normalized carrier density, mz and my are the spin polarizations of z- and y-components, and η is the pump rate normalized to a threshold. Other parameters are summarized in Table I. We basically used the parameter values reported for a 1.55-μm InAlGaAs quantum-well VCSEL manufactured by RayCan Co., Ltd.44 The γp, γa, and α were adjusted to reasonably reproduce the measured results presented in Sec. III. The ξ± was implemented as a complex value with a distinct random seed to consider both amplitude and phase noise for each electric field. A β for conventional VCSELs15 is not critical to reproduce our measured results, but a non-zero value must be used to induce PS. Although an explicit spin-noise component is not included in Eq. (3), the last term on the right side of Eq. (1) induces the spin noise. A γs is specified in each simulation.
TABLE I.

Parameters for simulations.

SymbolMeaningValue
γ Carrier decay rate 0.83 ns−1 
γs Spin decay rate Variable 
γp Linear birefringence π × 24 ns−1 
γa Linear dichroism 0.25 ns−1 
κ Cavity decay rate 26 ns−1 
α Linewidth enhancement factor 2.0 
β Spontaneous emission factor 1 × 10−5 
ξ± White Gaussian noise 0 dBW 
SymbolMeaningValue
γ Carrier decay rate 0.83 ns−1 
γs Spin decay rate Variable 
γp Linear birefringence π × 24 ns−1 
γa Linear dichroism 0.25 ns−1 
κ Cavity decay rate 26 ns−1 
α Linewidth enhancement factor 2.0 
β Spontaneous emission factor 1 × 10−5 
ξ± White Gaussian noise 0 dBW 

Our simulations included a condition under which the SSLO15x/2γp| = 1 is satisfied because we use various Ωx. This condition corresponds to the resonant coupling of the two orthogonal polarization modes in a birefringent VCSEL due to the Larmor precession of electron spins.15 The SSLO highly modulates polarizations of light and electron spin in the VCSEL, which may play an important role in controlling polarization bistability. Therefore, we briefly discuss the occurrence of the SSLO with the parameters in Table I; a γs of 5 ns−1 was selected since the SSLO is observable only when γs is much lower than |Ωx|. The value can be achieved at room temperature in VCSELs with an active region comprising (110)-oriented quantum wells or quantum dots.31,32

The calculated time evolution of the degree of circular polarization (Pc), defined as Pc = (E+E)/(E+ + E) when η = 3.0 and Ωx/2γp = 1, is shown in Fig. 2(a). The initial conditions for E+, E, N, mz, and my were set to 1, 1, 1, 0, and 0, respectively; |Pc| was amplified from 0 to 1 over time with an oscillation frequency of γp/π (oscillation period of 42 ps), and the time evolution after ∼20 ns indicated steady oscillation. This oscillatory amplification of Pc accompanied that of mz at the same frequency of γp/π, as shown in Fig. 2(b); thus, polarizations of light and electron spin amplified each other from their noise components. These results indicate the occurrence of the SSLO, which is observable when |Ωx| > γs.15 In the following simulations, we used such low γs of 5 ns−1 to include the effects of the SSLO unless otherwise stated.

FIG. 2.

Time evolutions of (a) Pc and (b) mz at η of 3.0 simulated when Ωx/2γp = 1 and γs = 5 ns−1. Insets of (a) and (b) indicate enlarged waveforms from 20.0 to 20.5 ns.

FIG. 2.

Time evolutions of (a) Pc and (b) mz at η of 3.0 simulated when Ωx/2γp = 1 and γs = 5 ns−1. Insets of (a) and (b) indicate enlarged waveforms from 20.0 to 20.5 ns.

Close modal

We investigated polarization-resolved light–current (L–I) characteristics by sweeping η with a constant Ωx to clarify the polarization bistability of VCSELs under the Larmor precession of electron spins. Figure 3 shows the polarization-resolved L–I characteristics simulated with different Ωx/2γp. Electric fields with X and Y polarizations were calculated from Ex=(E++E)/2 and Ey=i(E+E)/2, and |Ex|2 and |Ey|2 were averaged over time in a steady state between 95 and 100 ns. The η was swept from 0.5 to 5.5 and then swept again from 5.5 to 0.5 to confirm hysteresis behavior. When Ωx/2γp = 0, the VCSEL was lasing with Y polarization just above the threshold current (η = 1), then PS occurred at η of 3.0. PS occurred again by decreasing η to 2.6, and a typical hysteresis loop of PS appeared.

FIG. 3.

Polarization-resolved L–I characteristics with γs of 5 ns−1 and different Ωx/2γp. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. Arrows indicate the direction of PS.

FIG. 3.

Polarization-resolved L–I characteristics with γs of 5 ns−1 and different Ωx/2γp. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. Arrows indicate the direction of PS.

Close modal

The orthogonal linear polarization modes can be classified as gain-favored or gain-disfavored modes due to the linear dichroism of a VCSEL.45 The gain-favored mode, which exhibits higher gain, results in lasing when the pump rate is near the threshold. Switching to the gain-disfavored mode may occur at higher pump rates due to an interplay of phase-amplitude coupling and linear birefringence,45 which can be considered a result of the cross-gain saturation effect under certain approximations in the spin–flip model.46 In our simulations, Y and X polarization modes correspond to gain-favored and gain-disfavored modes, respectively. Thus, |Ex|2 is slightly lower than |Ey|2 just after PS due to the higher threshold current for the X polarization mode.45 The frequency of the Y polarization mode is higher than that of the X polarization mode. Consequently, PS from Y to X (from X to Y) polarization modes correspond to type I (type II).47 Conditions of η for obtaining PSs were shifted by increasing Ωx/2γp. The L–I curve showed a special behavior when Ωx/2γp = 1.0. Under this resonant condition, |Ex|2 and |Ey|2 converged to the same value when η > 2.3. This means that the SSLO occurs when η > 2.3, and this requirement of high η is attributed to the fact that an achievable maximum frequency of the SSLO depends on η.15 Neither the polarization oscillation nor PS was observed when Ωx/2γp = 1.2. Therefore, if Ωx/2γp is swept from 0 to 1.2 with a high η, there can be PS even without sweeping η.

Next, we simulated the polarization-resolved light–magnetic field (L–B) characteristics of the VCSEL by sweeping Ωx/2γp under fixed η, which enables us to directly investigate PS induced by the Larmor precession of electron spins. Polarization-resolved L–B characteristics with different η are shown in Fig. 4. The |Ex|2 and |Ey|2 were averaged over time in a steady state between 95 and 100 ns, and Ωx/2γp was swept as 0 → 1.5 → 0 → −1.5 → 0. There was no PS when η = 1.25, and the VCSEL was lasing with Y polarization. When η = 1.50, we observed PSs and two different types of hysteresis loops (Nos. 1 and 2), which are symmetric about the sign of Ωx, as shown in the enlarged view of the figure. Hysteresis loop No. 1 was triggered by Ωx/2γp of 0.91, which corresponds to type I PS from gain-favored (Y polarization) to gain-disfavored (X polarization) modes. Hysteresis loop No. 2 was triggered by Ωx/2γp of 1.0, which corresponds to type II PS from gain-disfavored to gain-favored modes. Hysteresis loop No. 2 disappeared when η = 2.00, and PS changed from step-like to continuous. When η ≥ 3.00, the remaining hysteresis loop (No. 1) disappeared. These results indicate that hysteresis loop No. 2 turns into continuous PS without a hysteresis loop due to the occurrence of the SSLO, which is consistent with the observations shown in Fig. 3.

FIG. 4.

Polarization-resolved L–B characteristics with γs of 5 ns−1 and different η. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. There are two types of hysteresis loops (Nos. 1 and 2) when η = 1.50, as shown in the enlarged view. Arrows indicate the direction of PS.

FIG. 4.

Polarization-resolved L–B characteristics with γs of 5 ns−1 and different η. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. There are two types of hysteresis loops (Nos. 1 and 2) when η = 1.50, as shown in the enlarged view. Arrows indicate the direction of PS.

Close modal

We investigated the PS conditions of the simulated results to gain insight into the polarization bistability of the VCSEL under the Larmor precession of electron spins. As PS conditions, the starting points of PSs given by η and Ωx/2γp were obtained from the L–I and L–B characteristics, respectively. Figure 5 shows the PS conditions in relation to η and Ωx/2γp. The PS conditions obtained from the L–B characteristics exhibiting the SSLO were designated as “L–B w/SSLO.” The hysteresis loops of PSs formed a bistable region due to the effect of η and Ωx/2γp. No difference in PS conditions was observed with respect to the sign of Ωx since the direction of spin precession is not distinguished in a VCSEL. The overlap of PS conditions obtained from the L–I and L–B characteristics indicates that these PS conditions originate from the same polarization dynamics of electron spin and light in a VCSEL at specific η and Ωx/2γp.

FIG. 5.

Simulated PS conditions of VCSEL. The black stars and green circles indicate the PS conditions obtained from the L–I and L–B characteristics. Unfilled green circles were obtained from the L–B characteristics with SSLO. The red and blue arrows indicate the stable regions of X and Y polarizations.

FIG. 5.

Simulated PS conditions of VCSEL. The black stars and green circles indicate the PS conditions obtained from the L–I and L–B characteristics. Unfilled green circles were obtained from the L–B characteristics with SSLO. The red and blue arrows indicate the stable regions of X and Y polarizations.

Close modal

Next, we discuss the case with a higher γs of 50 ns−1, which cannot achieve the SSLO. Figure 6(a) shows the PS conditions simulated with γs of 50 ns−1. The suppression of the SSLO led to the emergence of PS with a hysteresis loop in the L–B characteristics at η ≥ 3.5. As shown in Fig. 6(b), the polarization-resolved L–B characteristic with η of 3 indicated a single PS without a hysteresis loop at Ωx/2γp of 1.1 when the initial lasing polarization was set to X polarization and Ωx/2γp was swept as 0 → 1.5 → 0 → −1.5 → 0. Similarly, PS was observed at Ωx/2γp of −1.1 when Ωx/2γp was swept as 0 → −1.5 → 0 → 1.5 → 0 (not shown). These PSs were type II and from gain-disfavored (X polarization) to gain-favored (Y polarization) modes, and the parameter range of η was within the bistable region. The PSs accompanied a hysteresis loop for both signs of Ωx when η = 4, as shown in Fig. 6(c).

FIG. 6.

(a) Simulated PS conditions of VCSEL. Polarization-resolved L–B characteristics when (b) η = 3 and (c) η = 4 under swept Ωx/2γp of 0 → 1.5 → 0 → −1.5 → 0. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. Arrows indicate the transition of each polarization component. Time evolutions of (d) |Ex|2 and |Ey|2 and (e) mz with η of 4 and Ωx/2γp of 1.12. The inset of (e) indicates an enlarged waveform from 125.0 to 125.5 ns.

FIG. 6.

(a) Simulated PS conditions of VCSEL. Polarization-resolved L–B characteristics when (b) η = 3 and (c) η = 4 under swept Ωx/2γp of 0 → 1.5 → 0 → −1.5 → 0. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. Arrows indicate the transition of each polarization component. Time evolutions of (d) |Ex|2 and |Ey|2 and (e) mz with η of 4 and Ωx/2γp of 1.12. The inset of (e) indicates an enlarged waveform from 125.0 to 125.5 ns.

Close modal

The time evolutions of |Ex|2 and |Ey|2 simulated when η = 4 and Ωx/2γp = 1.12 are shown in Fig. 6(d). PS occurred ∼120 ns after the initial condition in which E+, E, N, mz, and my were set to 1.22, 1.22, 1, 0, and 0, respectively. The switching delay was related to the resonant amplification of spin noise in the VCSEL, as shown in Fig. 6(e). The transient increase in mz during PS accompanied oscillation at 24 GHz, corresponding to γp/π. Therefore, the type II PS from gain-disfavored to gain-favored modes triggered by Ωx is attributed to enhanced spin noise due to the resonant coupling of two orthogonal polarization modes by the Larmor precession. This assumption is reasonable by considering the resonant enhancement of spin polarization modulation sensitivity at around the modulation frequencies of γp/π.36,37 The spin polarization modulation is a differential modulation scheme of up- and down-spin electron densities in a VCSEL, while in this study, the Larmor precession of electron spin noise resulted in the internal spin polarization modulation, the modulation frequency of which depends on precession frequency. The cause of PS under Larmor precession of electron spins can be classified as the nonlinear effect, whereby nonlinear dynamics are induced by saturable dispersion and coupling of inversion populations with opposite spins.45 Manipulation of electron spins has an advantage in high-speed control of PS compared to that of temperature and strain. In addition, variations in temperature and total electron density in a VCSEL can be suppressed in contrast to the current tuning.

Finally, we compared PS conditions simulated with different γs, as shown in Fig. 7. Only positive Ωx was considered because of the symmetric characteristics. Compared with γs of 5 ns−1, higher η was required to obtain the SSLO, which causes the loss of polarization bistability with γs of 10 ns−1. The SSLO was not observed within the maximum η of 7 when γs ≥ 25 ns−1, and wider bistable regions were observed. Note that the width of the hysteresis loop obtained from the L–I characteristic with Ωx/2γp of 0 increased with increasing γs, which can be predicted from the stability analysis of the spin–flip model.16,46 Although the qualitative explanation of polarization bistability depending on Ωx/2γp requires further study, the spin precession probably affects the effective dichroism of a VCSEL depending on γs. Our numerical observations indicate that the manipulation of electron spins can aid in determining the polarization bistability of VCSELs and be useful for switching the polarization state.

FIG. 7.

PS conditions simulated with γs of 5, 10, 25, 50, and 100 ns−1.

FIG. 7.

PS conditions simulated with γs of 5, 10, 25, 50, and 100 ns−1.

Close modal

We conducted experiments to confirm the polarization bistability of VCSELs under the Larmor precession of electron spins predicted in Sec. II. Three VCSELs were measured to confirm the reproducibility of the measured results. The VCSELs were purchased from RayCan Co., Ltd. and have an all-epitaxial structure with a strain-compensated InAlGaAs quantum-well active region and distributed Bragg reflectors.48 One of the reflectors is coated with Au to enhance reflectivity and conduct flip-chip bonding. This flip-chip-bonding process induces a certain amount of strain-induced linear birefringence of VCSELs even when intentional stress is not applied.

The experimental setup for measuring the polarization properties of the VCSELs is shown in Fig. 8(a). The VCSEL with the TO-CAN package was held using a copper holder and placed between electromagnets. We used a dipole electromagnet (Model 3480, GMW Associates) with a pole diameter of 5 mm. The field uniformity was better than ±0.5% over a diameter of 3 mm. The magnetic field at the sample position was calibrated prior to the experiment using a gauss meter. The magnetic field was applied in the in-plane direction (x direction) of this VCSEL. The output light of the VCSEL was collimated using a lens and then input into a polarization-insensitive isolator to suppress any instabilities due to unintentionally reflected light from optical elements. The X and Y polarization components were separated using a polarization beam splitter (PBS), and each component was detected using two optical power meters. An optical spectrum analyzer was used to confirm the optical spectra of the VCSEL. All experiments were conducted at room temperature, and no active temperature controls were conducted. The injection current of the VCSEL and external magnetic field were swept with steps of 0.05 mA and 0.01 T, respectively, to measure the L–I and L–B characteristics. We next present the results of one of the three VCSELs, “No. 1,” but the data on all three VCSELs can be found in the supplementary material.

FIG. 8.

(a) Experimental setup for measuring polarization properties of VCSELs. (b) Optical spectra at 0 T with stepwise increment (left graph) and decrement (right graph) of injection current. (c) Optical spectra at 6.5 mA with stepwise increment (left graph) and decrement (right graph) of the external magnetic field.

FIG. 8.

(a) Experimental setup for measuring polarization properties of VCSELs. (b) Optical spectra at 0 T with stepwise increment (left graph) and decrement (right graph) of injection current. (c) Optical spectra at 6.5 mA with stepwise increment (left graph) and decrement (right graph) of the external magnetic field.

Close modal

Figure 8(b) shows the optical spectra of VCSEL No. 1 measured at 0 T with stepwise increment and decrement of current. The increment of current induced lasing above 1.5 mA and wavelength shift due to ohmic heating. The estimated temperature variation with current was ∼3 °C/mA. An abrupt wavelength shift at around 6.2 mA during stepwise increments of current was due to type I PS from Y to X polarizations. The frequency separation of the two linear polarizations was measured as 23.9 GHz at the current twice the threshold. The type II PS was also confirmed during a stepwise decrement of current at around 5.2 mA. Figure 8(c) shows the optical spectra at 6.5 mA with stepwise increment and decrement of the external magnetic field. Type II and type I PSs were observed at around 1.5 and 1.0 T during the increment and decrement of the magnetic field, respectively. Contrary to the current sweep, the magnetic-field sweep did not affect the lasing wavelength, which indicates that applying a magnetic field can exclude the thermal effect.

The polarization-resolved L–I characteristics of VCSEL No. 1 under different magnetic fields are shown in Fig. 9. The current was swept from 0 to 9 mA and then swept from 9 to 0 mA to confirm hysteresis behaviors. Under zero magnetic field, the VCSEL was lasing with Y polarization above the threshold current until the current reached 6.2 mA, at which point type I PS occurred. Type II PS occurred by decreasing the current to 5.2 mA, and a typical hysteresis loop of PS was obtained. The hysteresis loop was not observed at 1.3 T within the upper limit of 9 mA. The results were almost the same even when the magnetic field was inverted from positive to negative, as shown in the supplementary material. Note that our measurement setup shown in Fig. 8(a) was not accurate enough to evaluate the optical power difference between X and Y polarizations due to polarization-dependent losses in the setup, although the gain-favored and gain-disfavored modes may exhibit a slight optical power difference between them.45 

FIG. 9.

Polarization-resolved L–I characteristics under different magnetic fields. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. Arrows indicate the direction of PS.

FIG. 9.

Polarization-resolved L–I characteristics under different magnetic fields. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. Arrows indicate the direction of PS.

Close modal

The polarization-resolved L–B characteristics of VCSEL No. 1 under different currents are shown in Fig. 10. The magnetic field was swept as 0 → 1.5 → 0 → −1.5 → 0 T. There was no PS when the VCSEL was lasing with Y polarization at 6.00 mA. The polarization switched from Y to X polarizations (type I) when the VCSEL was biased above 6.21 mA. Under this condition, we observed PSs and hysteresis loops with regard to the magnetic field, which are symmetric about the sign of the magnetic field. No PS was confirmed at 6.80 mA and above within the maximum magnetic field of ±1.5 T.

FIG. 10.

Polarization-resolved L–B characteristics under different currents. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. Arrows indicate the direction of PS.

FIG. 10.

Polarization-resolved L–B characteristics under different currents. The red dashed and blue solid lines indicate the X and Y polarization components, respectively. Arrows indicate the direction of PS.

Close modal

Although there was no PS in the L–B characteristics at 6.00 mA, as shown in Fig. 10, we observed type II PS when initial lasing polarization was set to X polarization by sweeping current as 0 → 9 → 6 mA. As shown in Fig. 11(a), PS occurred only at 1.37 T, and a hysteresis loop was not observed when the magnetic field was swept as 0 →1.5 → 0 → −1.5 → 0 T. Similarly, PS occurred only at −1.37 T when the magnetic field was swept as 0 → −1.5 → 0 → 1.5 → 0 T, as shown in Fig. 11(b). These results correspond to the simulated results of Fig. 6(b) in which the unidirectional type II PS occurred from gain-disfavored to gain-favored modes.

FIG. 11.

Polarization-resolved L–B characteristics at 6.00 mA for (a) 0 → 1.5 → 0 → −1.5 → 0 T and (b) 0 → −1.5 → 0 →1.5 → 0 T. Lasing polarization was set as X polarization before sweeping the magnetic field. Arrows indicate the transition of each polarization component.

FIG. 11.

Polarization-resolved L–B characteristics at 6.00 mA for (a) 0 → 1.5 → 0 → −1.5 → 0 T and (b) 0 → −1.5 → 0 →1.5 → 0 T. Lasing polarization was set as X polarization before sweeping the magnetic field. Arrows indicate the transition of each polarization component.

Close modal

Figure 12 summarizes the PS conditions measured for the three VCSELs. The threshold current (Ith), lasing wavelength, and linear birefringence of these VCSELs are listed in Table II. Additional data not included in Figs. 911 are shown in the supplementary material. Figure 12(a) shows the PS conditions measured for VCSEL No. 1 in relation to the current normalized by the Ith and magnetic field. The bottom horizontal axis indicates Ωx/2γp, and a specific g of 1.3 was assumed for calculating Ωx. As shown in Fig. 12(a), hysteresis loops of PSs formed a bistable region due to the current and magnetic field. The overlap of PS conditions obtained from different measurements of L–I and L–B characteristics was confirmed. No significant difference in PS conditions was observed with respect to the direction of the magnetic field. The SSLO was not confirmed within the measurement conditions. As shown in Figs. 12(b) and 12(c), similar results were also confirmed for VCSEL Nos. 2 and 3. When g of 1.3 was assumed for each VCSEL, the simulated results with γs of 25 or 50 ns−1 in Fig. 7 indicate reasonable agreement with the measured ones. These findings support the hypothesis that polarization bistability is controlled by the Larmor precession of electron spins, as evidenced by the observation of PSs that are independent of current, heat, and external light.

FIG. 12.

PS conditions measured for VCSEL Nos. (a) 1, (b) 2, and (c) 3. The vertical axis indicates the current normalized by threshold current. The top and bottom horizontal axes, respectively, indicate the magnetic field and Ωx/2γp calculated under the assumption of g = 1.3. The black stars and green circles, respectively, indicate the PS conditions obtained from the L–I and L–B characteristics. The red and blue arrows, respectively, indicate the regions of X and Y polarizations.

FIG. 12.

PS conditions measured for VCSEL Nos. (a) 1, (b) 2, and (c) 3. The vertical axis indicates the current normalized by threshold current. The top and bottom horizontal axes, respectively, indicate the magnetic field and Ωx/2γp calculated under the assumption of g = 1.3. The black stars and green circles, respectively, indicate the PS conditions obtained from the L–I and L–B characteristics. The red and blue arrows, respectively, indicate the regions of X and Y polarizations.

Close modal
TABLE II.

Comparison of three VCSELs. Values of lasing wavelength and linear birefringence were obtained from optical spectra at a current twice the threshold.

VCSELThreshold current (mA)Lasing wavelength (nm)Linear birefringence (π ns−1)
No. 1 1.5 1545 23.9 
No. 2 1.4 1546 22.6 
No. 3 0.9 1549 21.3 
VCSELThreshold current (mA)Lasing wavelength (nm)Linear birefringence (π ns−1)
No. 1 1.5 1545 23.9 
No. 2 1.4 1546 22.6 
No. 3 0.9 1549 21.3 

There can be PSs based on a thermal effect in which wide temperature variation leads to a wavelength shift of gain and cavity mode with different rates.8 An increase or decrease in injection current to a VCSEL induces temperature variation due to ohmic heating and can be a cause of PS. The thermal lensing effect may affect the PS condition because some parameters of a VCSEL depend on the modal volume.49 Frequency-dependent losses and strain effects in a quantum-well active region can also be a cause of PS.50 There was non-negligible temperature variation in the measurements of L–I characteristics, which may have induced variation of parameters, such as α, γa, and γs, and thus can be a cause of discrepancies between measured and simulated PS conditions. However, we did not observe temperature variation of the VCSELs when measuring the L–B characteristics. The measured feature, in which the Larmor precession of electron spins modifies the current required for the polarization bistability to occur, shown in Fig. 12, exhibited reasonable agreement with the simulated one in Fig. 6(a). Therefore, it is reasonable to consider that the measured variation of the bistable region due to the external magnetic field is dominated by not the thermal effect but the Larmor precession of electron spins. Saturation of output power was observed from the measured L–I characteristics of the VCSELs under high bias conditions, while the spin–flip model used in this study did not take into account such an effect. This factor is also a possible cause of discrepancies between measured and simulated results. The SSLO may be observable even at room temperature when a VCSEL has a (110)-oriented quantum-well or quantum-dot active region with a lower γs.31,32 In addition, γs may depend on the external magnetic field in the quantum well active region.51 In this case, the condition |Ωx| > γs for occurrence of the SSLO is modified.

VCSEL No. 3 indicated slightly different PS conditions and parameters from those of Nos. 1 and 2. This may be due to the difference in the process batch for VCSEL No. 3 from the others. Note that all the VCSELs showed type I PS during the upward sweep of the current when measuring the L–I characteristics, and the photoluminescence peak of the active region of the VCSELs was designed to be 1540 nm.48 

Although the distributed Bragg reflector of the VCSELs prohibited measuring g in the quantum-well active region, the assumed |g| of 1.3 shown in Fig. 12 is not unrealistic. In extensively studied GaAs, electrons have a g of about −0.3 at room temperature.52 For InGaAs/InAlAs quantum wells with a bandgap wavelength of ∼1.55 μm, a g of about −3 has been reported at 200 K.53 The VCSELs for our study had seven pairs of strain-compensated InAlGaAs/InAlGaAs quantum wells with a bandgap wavelength of ∼1.55 μm but with an undisclosed composition ratio. A relatively high absolute g is expected in our VCSELs compared with GaAs VCSELs due to the narrow gap materials.

We theoretically and experimentally investigated the polarization bistability of VCSELs under the Larmor precession of electron spins. We theoretically revealed that a PS was induced when the Larmor-precession frequency was swept. This phenomenon was attributed to the fact that the Larmor precession of electron spins modifies the current required for polarization bistability to occur. When the Larmor-precession frequency was swept, the SSLO induced with a low γs compared with γp could be a cause of continuous PS from gain-disfavored to gain-favored modes without a hysteresis loop, and its suppression with a higher γs resulted in a transition from continuous PS to step-like PS with a hysteresis loop. We experimentally demonstrated the case with a high γs, namely, the PS with a hysteresis loop controlled by the Larmor-precession frequency of VCSELs. Our observations suggest that the polarization bistability of VCSELs can be controlled by electron spin manipulation beyond the limitation of the inherent properties of material and structure. Technological progress in electron-spin injection and transport in semiconductors54–57 has given rise to expectations for practical spin manipulations. Thus, further investigation of polarization bistability and its new functions of VCSELs based on the interplay with photonics and spintronics is anticipated.

The supplementary material includes the measured L–I characteristics, L–B characteristics, and optical spectra of each VCSEL.

This work was supported by The Murata Science Foundation, Shimadzu Science Foundation, and JSPS KAKENHI Grant Nos. 22H01536 and 24H00426. N.Y. acknowledges the technical support from the Fundamental Technology Center, Research Institute of Electrical Communication, Tohoku University. N.Y. and H.Y. acknowledge the stimulated discussion in the meeting of the Cooperative Research Project of the Research Institute of Electrical Communication, Tohoku University.

The authors have no conflicts to disclose.

Nobuhide Yokota: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Kazuhiro Ikeda: Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Satoshi Iba: Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Takeo Katayama: Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Hiroshi Yasaka: Conceptualization (supporting); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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