3.1. Characteristics of MM-VCSELs in Four Designs
The power-to-current (P-I) curve and dP/dI slope of the MM-VCSEL with the design of L-OA-7 are shown in
Figure 3a. The MM-VCSEL with the design of L-OA-7 exhibits a threshold current of 0.6 mA and maximal optical output power of 4.1 mW before the Auger effect significantly reduces the radiative emission quantum efficiency. The voltage-to-current (V-I) curve and the differential resistance for MM-VCSEL with the design of L-OA-7 are also depicted in
Figure 3b. The differential resistance of 80.3 Ω is obtained by the first-order derivative of the voltage with respect to the current. In addition, the threshold current of 0.6 mA and the maximal optical output power of 3.5 mW are obtained for MM-VCSEL with the design of S-OA-7, as shown in
Figure 3c. The differential resistance of 76.4 Ω for MM-VCSEL with the design of S-OA-7 is calculated from the V-I curve in
Figure 3d. From the abovementioned description, the MM-VCSELs with the designs of L-OA-7 and S-OA-7 have similar slope efficiency. The summary for the basic characteristic parameters of the MM-VCSELs with four designs is listed in
Table 1. Due to the difference in the oxide aperture of the MM-VCSELs, the MM-VCSEL with an oxide aperture of 7 µm (OA-7) has a higher photon density than that with an oxide aperture of 7.5 µm (OA-7.5) under the same operating current. Therefore, the MM-VCSEL with the design of OA-7 exhibits a lower threshold current because the smaller oxide aperture limits the current direction to enhance the larger current density. Furthermore, the MM-VCSEL with the design of OA-7 has a thicker oxide aperture to increase the internal serial resistance, which further enlarges the larger differential resistance. Also, the optical output power and slope efficiency of the MM-VCSELs with the larger oxide aperture are higher than those with the smaller oxide aperture because of the different photon mode volumes for the MM-VCSEL with the various designs.
The differential quantum efficiency (ηed) can be defined as ηed = (q/hυ) (dPout/dIbias) where q, h, ν, Pout, and Ibias, respectively denote the electron charge, the Planck constant, the frequency, the output power, and the biased current. The differential quantum efficiency of the MM-VCSELs with four designs of L-OA-7/L-OA-7.5/S-OA-7/S-OA-7.5 can be calculated as 0.18/0.32/0.17/0.25 from the dP/dI slope. The resistance of the MM-VCSEL with the L-type design is larger than that with the S-type design because of the larger serial resistance induced by the longer transmission line in the MM-VCSEL with the L-type design. Generally, the characteristic impedance (ZRF) in the microwave circuits is set as 50 Ω.
When the resistance at the load end is not 50 Ω to generate the impedance mismatch, this phenomenon will cause the signal reflection to degrade the transmission performance. The reflection coefficient (Γ) of the MM-VCSEL can be represented as Γ = (ZVCSEL − ZRF)/(ZVCSEL + ZRF) with the ZVCSEL denoting the impedance of the MM-VCSEL. The reflection coefficients of the MM-VCSELs with different designs of L-OA-7/L-OA-7.5/S-OA-7/S-OA-7.5 are respectively obtained as 0.23/0.18/0.20/0.15 from their differential resistances by using the abovementioned formula. With defining the voltage standing wave ratio (VSWR) as VSWR = (1 + Г)/(1 − Г) and the return loss (RL) as RL = −20log|Г|, the corresponding VSWR and return loss for the MM-VCSEL with four designs of L-OA-7/L-OA-7.5/S-OA-7/S-OA-7.5 can be respectively evaluated as 1.59/1.44/1.5/1.35 and 12.7/14.9/14.0/16.5 dB. The greater return loss leads to little signal reflection, which easily maintains the transmission performance.
The optical spectra of the MM-VCSEL with the design of S-OA-7 under different biased currents from I
th to 20I
th are presented in
Figure 4a. In our work, the wavelength resolution of the OSA (Ando, AQ6317B) is set as the limited resolution of 0.01 nm. From
Figure 4a, the numbers of the red-shifted lasing modes for the MM-VCSEL under the biased current below 5I
th are few because of the less thermal effect induced by the current injection. Because the effective refractive index in the oxide region is lower than the core region, the high-order mode of MM-VCSEL will appear at the short-wavelength region in
Figure 4a. The mode spacing can be represented as [
35]:
where
neff denotes the average effective refractive index in the central part before oxidation; Δ
neff the difference in effective index between the core and the oxidation region; Δ
λ mode spacing;
λ the resonance wavelength. Increasing the bias current enlarges the high-order transverse modes at the short-wavelength region to induce the wider root-mean-square (RMS) spectral. When the bias current increases beyond 5I
th, the thermal effect becomes larger to lead the more energy conversion to phonons in the quantum well, which further contributes to the redshift of the overall wavelength. The fundamental mode wavelength and the RMS spectral width of the MM-VCSEL with the design of S-OA-7 under the different bias-current operations are depicted in
Figure 4b. The fundamental mode wavelength increases from 855.1 (I
th) to 859.1 nm (20I
th) because of the higher thermal effect induced by the bias current. An overall wavelength displacement of the MM-VCSEL with the design of S-OA-7 can be achieved to 4 nm by increasing the bias current from I
th to 20I
th. The only mode number of the VCSEL in the long-reach fiber transmission is used to hardly evaluate the precise modal dispersion. The wavelength and power distribution are concurrently considered to estimate the modal dispersion. Therefore, the RMS spectral width is adopted to determine the modal dispersion in MM-VCSEL. The RMS spectral width (Δ
λRMS) of the MM-VCSEL can be expressed as [
15]:
where
Pi and
λi, respectively, indicate the peak power and central wavelength of the i
th modes. From Equation (2), the RMS spectral width of the MM-VCSEL with the design of S-OA-7 expands from 0.22 (I
th) to 0.91 nm (6I
th) and maintains a stable width under the biased current operation beyond 6I
th. The optical spectra of the MM-VCSEL with four designs of L-OA-7/S-OA-7/L-OA-7.5/S-OA-7.5 under the operation at 15I
th are, respectively, shown in
Figure 4c–f.
The maximal peak wavelength, RMS spectral width, mode spacing, and effective mode number of the MM-VCSEL with four designs of L-OA-7/S-OA-7/L-OA-7.5/S-OA-7.5 are summarized in
Table 2. Besides, the mode distribution in the optical spectrum can be described by the mode spacing and the RMS spectral width. The mode spacing of the MM-VCSEL is defined as [
36]:
where
a denotes the aperture diameter. The mode spacings of the MM-VCSELs with four designs of L-OA-7/L-OA-7.5/S-OA-7/S-OA-7.5 can be respectively obtained as 0.56/0.44/0.52/0.48 nm. In this work, the larger diameter of the oxide aperture can contribute to the narrower mode spacing, as confirmed by the previous works [
2,
36]. Considering the maximal peak value range to −20 dB less than the maximal peak value, the calculated RMS spectral widths reveal similar values of 0.8970 and 0.8975 nm for the MM-VCSEL with the designs of S-OA-7 and S-OA-7.5. Therefore, the effective mode numbers (Mode#) of the MM-VCSEL with the designs of S-OA-7 and S-OA-7.5 are, respectively, obtained as 1.71 and 1.85. From the abovementioned results, the oxide aperture as a spatial filter effectively detunes its diameter to precisely control the transverse mode in MM-VCSEL.
3.2. Modulation Responses of the MM-VCSELs
To analyze the small-signal modulation response, the equivalent circuits of the MM-VCSELs with different designs are constructed to include the intrinsic and extrinsic responses. The equivalent circuit of the MM-VCSEL is described in
Figure 5a. L indicates the inductance of the microstrip transmission line between contacting pad and VCSEL electrode. R
m is the resistance across the DBR layer of the MM-VCSEL. C
p is the frequency-dependent pad capacitance between the metal contact and the bottom mirror stack layer. R
p is the resistance produced by the frequency-dependent dielectric losses in the benzocyclobutene layer. R
a represents that the serial resistance in the active region includes the current path limited by the oxide aperture. C
a is the total capacitance including the p-n junction, the oxide aperture, and the intrinsic layer capacitance [
32,
37]. By using the commercial software (Keysight, Advanced Design System, Santa Rosa, CA, USA) to simulate, the simulated frequency responses based on the equivalent circuit of MM-VCSEL by varying the effective inductance are demonstrated in
Figure 5b. The widest 3 dB bandwidth will be simulated by tuning the effective inductance, as shown in
Figure 5b. The impedance between the extrinsic inductance of the transmission microstrip and the capacitance of the intrinsic active region can be matched to reduce the imaginary impedance. This phenomenon significantly mitigates the power loss to achieve the maximal bandwidth. From the inset of
Figure 5b, the inductance value of this MM-VCSEL locates at the right side of the maximal 3-dB bandwidth. In addition, the simulated frequency responses can be varied by changing the C
a, as shown in
Figure 5c. In
Figure 5c, the 3-dB bandwidth of the MM-VCSEL shrinks with increasing capacitance. The smaller oxide aperture design has less capacitance to achieve a more broadened modulation bandwidth. Furthermore, the smaller oxide aperture size enhances the carrier density to provide a larger relaxation oscillation frequency. These phenomena can effectively improve the modulation capacity of the MM-VCSEL.
The frequency responses of the MM-VCSELs with the designs of L-OA-7 and S-OA-7 are displayed in
Figure 6a,b, respectively. The sinusoidal wave for modulating the MM-VCSEL was generated from a radio-frequency (RF) synthesizer (Agilent, E4433B, Santa Clara, CA, USA) with a scanning range from 250 kHz to 22 GHz. After directly modulating the MM-VCSEL, the device output was received by a PD. Then, the converted signal was analyzed by an electrical spectrum analyzer (ESA, Agilent, 8565E). In addition, the LabVIEW program was used to control the RF synthesizer and ESA. Increasing the bias current upshifts the resonance frequency to the high frequency. This also enhances the damping effect to induce a flatten and broadened modulation response. The 3-dB bandwidths of the MM-VCSELs with designs of L-OA-7 and S-OA-7 under the 15I
th operation are, respectively, measured as 18.0 and 19.2 GHz. Further increasing the bias current to 20I
th suppresses the 3-dB modulation bandwidths to 17.34 and 15.7 GHz for the MM-VCSEL with designs of L-OA-7 and S-OA-7, respectively, because of the overdamped effect in VCSEL under the high-current operation. From the abovementioned results, the critically damped point is located around 15I
th to provide a reference for the bias optimization in the high-speed transmission test.
Table 3 summarizes the 3-dB modulation bandwidths of the MM-VCSELs with four designs under the 15I
th operation. The MM-VCSELs with four designs of L-OA-7/L-OA-7.5/S-OA-7/S-OA-7.5 reveal their respective 3-dB bandwidths of 18.75/18.51/19.21/18.98 GHz under the 15I
th operation. Regardless of the MM-VCSELs with L- or S-type designs, the difference of 3-dB bandwidth for the MM-VCSELs with different oxide apertures is about 0.2 GHz. Furthermore, different transmission lines of the MM-VCSEL induce the additional equivalent inductance and serial resistance to generate the 0.5-GHz change of the modulation bandwidth. Therefore, the MM-VCSEL with the design of S-OA-7 is the best design among all the devices.
For the abovementioned results, the transmission microstrip connected between the VCSEL and metal padding can be used to sufficiently optimize impedance matching and improve the 3-dB modulation bandwidth because the transmission microstrip can be regarded as the series resistance and inductance to detune the impedance via adjusting its length. Furthermore, the 3-dB modulation bandwidth of the MM-VCSEL can be enhanced by changing the oxide aperture to vary the capacitance in the active region matching the series inductance.
3.3. Bias Current Optimization for High-Speed Transmission
The D factor indicates the modulation efficiency of an intrinsic laser to represent the capability to reach the high relaxation oscillation frequency at a low bias current. The D factor can be expressed as [
38]:
where
νg denotes the group velocity of light; ∂
g/∂
N the differential gain;
Vp the effective mode volume;
ηi the current injection efficiency as the fraction of the terminal current generating the carrier in the active layer. A drastic phase change can be induced when the largest resonance amplitude occurs. The damping rapidly attenuates the oscillation in the laser to result in a flattened modulation response. The bias current controls the carrier and photon densities and the relaxation oscillation frequency. The relaxation oscillation frequency can be described as [
38]:
where
fR denoting the relaxation oscillation frequency is proportional to the root of bias current when bias current is much greater than the threshold current. Furthermore, the damping factor and relaxation oscillation frequency will simultaneously limit the modulation bandwidth of the VCSEL. In addition to high relaxation oscillation frequency, the low damping factor is essential for high-speed modulation. The
K factor can be expressed as the following formula [
38],
where
τp denotes the photon lifetime;
ε the gain compression factor. In addition, the damping factor can also be represented as [
38]:
where
γ denotes the damping factor approximately proportional to the square of the relaxation oscillation frequency;
γ the damping factor offset; Γ the optical confinement factor;
Rsp the rate per unit volume of spontaneous emission into the lasing mode;
τΔN the differential carrier lifetime;
Np the photon density. The damping factor offset usually appears only near the threshold current. When VCSEL operates under the high-current region, the excessive damping induced by the gain compression effect will slow down the response of the VCSEL.
Figure 7 exhibits the eye diagrams and Q-factor of the MM-VCSEL with designs of S-OA-7 and S-OA-7.5 under different bias currents and the same input amplitude for 10-Gbit/s and 20-Gbit/s NRZ-OOK transmission. The overshoot effect is correlated to the operating bias current and the modulated input amplitude. The overshoot effect is related to the bias condition and the modulated input amplitude. To optimize the operating condition, the bias current of MM-VCSEL is optimized by the Q factor from the eye diagram to compare the damping level [
39]. The overshoot induced by underdamping dominates the Q factor at low bias. Under the high-bias condition, the decreased slope efficiency by the self-heating effect and the compressed 3-dB bandwidth by the overdamping effect will attenuate the amplitude of the signal to limit the Q factor. The overshoot in the eye diagram is avoided at the critically damping point by detuning the bias current.
In
Figure 7, the damping effect has almost the same influence on the MM-VCSELs with two designs from the optimization curve because of the tiny design difference between these two devices. Therefore, the optimized bias currents are located at 16.66I
th and 16.92I
th for the MM-VCSELs with two designs of S-OA-7 and S-OA-7.5, respectively. For 10 Gbit/s transmission, the lower damping decided by the lower D factor for the MM-VCSEL with the design of S-OA-7.5 causes the more significant overshoot in the eye diagram. In addition, the low damping also induces a long oscillation to degrade the transmission performance. With increasing bias current below 8 mA, the relaxation oscillation frequency, damping factor, and modulation bandwidth increase to suppress the duration of oscillation and overshoot. Because the pulsewidth for each bit is shortened by half for the 20 Gbit/s transmissions, the overshoot duration is severely deformed in the eye diagram and interferes with the next bit. Therefore, it is necessary to overcome the overshoot in high-speed transmission.
Figure 8a shows the horizontal BER bathtubs of the 40-Gbit/s NRZ-OOK data carried by the MM-VCSEL with the design of S-OA-7 under different bias currents. From
Figure 8a, the lowest root-mean-square jitter value can be obtained as 1.70 ps under a BER of 10
−12 when the MM-VCSEL is operated at 10 mA. The underdamping effect still exists at 8 mA to degrade the root-mean-square jitter (RMSJ) to 1.93 ps by the excessive data history-dependent timing jitter. By further enlarging the bias current to 12 mA, the RMSJ also decays to 1.84 ps because of the rise time limited by the reduced 3-dB modulation bandwidth. The vertical BER bathtub of the 40-Gbit/s NRZ-OOK data carried by the MM-VCSEL with the design of S-OA-7 under different bias currents are displayed in
Figure 8b.
The limitation for overshoot is more critical in high-speed transmission to make the BER value under the 8-mA operation higher than those under other current operations. Increasing the bias current to 10 mA suppresses the overshoot effect to improve the BER. The BER becomes higher when the MM-VCSEL is operated at 12 mA because of the decreased slope efficiency and 3-dB bandwidth. In the inset of
Figure 8b, the eye diagrams under different bias operations have a similar phenomenon to confirm the abovementioned results. Furthermore, the modulation frequency and Q factor of the MM-VCSELs with four designs are further optimized via the bias current optimization.
Figure 9 shows the frequency responses of the MM-VCSELs with four designs operated at their optimized Q factor conditions. The optimized 3-dB bandwidths of the MM-VCSELs with four designs of L-OA-7/L-OA-7.5/S-OA-7/S-OA-7.5 are respectively obtained as 19.2/18.2/21.3/21.1 GHz under their corresponding optimized bias currents of 8/9/10/11 mA, as summarized as
Table 4. The reflection coefficient of the MM-VCSEL with the L-type design is larger than that of the MM-VCSEL with the S-type design. That is because the extra transmission microstrip length induces impedance mismatching. After the tradeoff between the modulation slope efficiency and the resonance frequency, the high signal reflection for the MM-VCSEL with the L-type design needs the appropriate bias to overcome the insufficient output signal amplitude. To avoid reducing slope efficiency by self-heating, the optimized bias current for the MM-VCSEL with the L-type design is lower than that for the MM-VCSEL with the S-type design. Besides, the optimized bias current for the MM-VCSEL with the 7.5-µm-oxide-aperture designs is higher than that for the MM-VCSEL with the 7-µm-oxide-aperture designs. This is because the devices with larger oxide aperture have lower carrier density to require a higher bias current for achieving the critically damping operation.
3.4. RIN and MPN of the MM-VCSELs with Different Designs
Figure 10a–d exhibit the relative intensity noise (RIN) spectra ranging from 0 to 20 GHz for the MM-VCSELs with different designs of L-OA-7/S-OA-7/L-OA-7.5/S-OA-7.5. Because the primary source of RIN is originated from the spontaneous emission of the MM-VCSEL, the operation current approaches the threshold current to increase the RIN value [
40]. In addition, the resonance frequency of the MM-VCSEL upshifts toward the high frequency by increasing the bias current. The noise peak around the resonance frequency gradually immerses into shot noise because of the damping enhancement. From
Figure 10a–d, the MM-VCSEL with the smaller oxide aperture has a higher photon density to induce the faster frequency shift. Because of the larger optical volume and lower photon density, the higher RIN values are observed for the MM-VCSELs with the larger oxide aperture from the experimental results. At the low bias current, the lower damping factor and lower resonance frequency induced by the lower photon density for the MM-VCSEL with the designs of L-OA-7.5/S-OA-7.5 increase the RIN value [
41,
42], as confirmed by
Figure 7.
The resonance frequency for the MM-VCSELs with four designs under different bias currents is shown in
Figure 11. The D factor can be obtained from the slope to indicate the modulation efficiency of an intrinsic laser. The MM-VCSELs with the design of L-OA-7/L-OA-7.5/S-OA-7/S-OA-7.5, respectively, obtain their D factors of 3.95/3.53/4.21/3.53 GHz/mA
0.5. Because of the higher photon density in the MM-VCSEL with the smaller oxide aperture design, the D factor for the MM-VCSEL with the OA-7 design is larger than for the MM-VCSEL with the OA-7.5 design. The mode partition noise (MPN) is induced by an interaction between the carrier distribution and mode intensity. Transverse modes will compete in inhomogeneous carrier distribution to affect the dynamic response in VCSELs.
At a high mode intensity point, the stimulated recombination depletes the local carriers and produces a hole in the carrier distribution. This phenomenon is called the spatial hole burning effect (SHB) [
43,
44,
45,
46,
47]. The lacking carriers in the hole compress the current mode and then enhance the other modes. This effect is also called mode switching. The VCSEL usually operates at the high-bias operation for sufficient frequency response under the high-speed transmission. The resonance noise peak immerses into the shot noise to only slightly affect the transmission performance. The remaining MPN mainly limits the high-speed transmission performance to induce a worse root-mean-square noise (RMSN) in two levels of the OOK modulation. To analyze MPN appearing from the RIN spectrum in the low-frequency region, the zoomed-in RIN spectra ranging from 10 to 100 MHz for the MM-VCSELs with different designs of L-OA-7/S-OA-7/L-OA-7.5/S-OA-7.5 are shown in
Figure 12a–d. The MPN occurs in the low-frequency region with mode-selective coupling. The high RIN value also appears in the low-frequency region from 10 to 100 MHz with poor alignment when the relaxation oscillation frequency already upshifts to the high-frequency region. The less mode competition happens in the smaller oxide aperture because of the less spatial overlap by transverse modes [
41,
48,
49,
50]. The overlap of the transverse mode in the MM-VCSEL with the large oxide aperture is more prominent with less current confinement, which indicates the more severe mode competition. Therefore, the MM-VCSELs with the larger oxide aperture have higher RIN in the low-frequency region because of the stronger mode competition.
Table 5 summarizes the average RIN values ranging from 10 to 100 MHz at the bias currents of 1/5/10I
th for the MM-VCSELs with four designs. By increasing the bias current, the greater MPN is caused by locally higher stimulated recombination [
38]. The discrepancy of RIN for the MM-VCSELs with four designs is considered to analyze the MPN. The RINs of the MM-VCSELs with designs of L-OA-7/L-OA-7.5/S-OA-7/S-OA-7.5 are, respectively, increased to 6.2/11.4/4.1/9.1 dBc/Hz as the bias current enlarges to 10I
th. The abovementioned results verify that the mode competition will become more severe with the expansion of the oxide aperture size.
3.5. Transmission Performances in the MM-VCSELs with Four Designs
The eye diagrams of the 40 Gbit/s NRZ-OOK data carried by the MM-VCSELs with four designs of L-OA-7/S-OA-7/L-OA-7.5/S-OA-7.5 under their optimized bias current and the same signal amplitude are shown in
Figure 13a–d. After receiving the photodetector with a conversion gain of 65 V/W, the signal amplitudes of the modulated outputs are respectively obtained as 204/444/516/602 mV for the MM-VCSELs with four designs of L-OA-7/L-OA-7.5/S-OA-7/S-OA-7.5 with an electrical input signal amplitude of 800 mV. Because the MM-VCSEL with the design of L-OA-7.5 exhibits nearly twice larger slope efficiency and lower reflection coefficient than that with the design of L-OA-7, the eye diagram amplitude of the MM-VCSEL with the design of L-OA-7.5 is larger than that of the MM-VCSEL with the design of L-OA-7, as shown in
Figure 13a,c. In
Figure 13b,d, the same trend is observed for the MM-VCSEL with the S-type design.
The performances of the NRZ-OOK and PAM-4 transmissions carried by the MM-VCSEL with the design of S-OA-7 are shown in
Figure 14a,b. By using the pre-emphasis technology to pre-distort the waveform, the data rate of the NRZ-OOK transmission can be achieved to 50 Gbit/s to pass the telecom criterion. The amplitude and RMSN of the received data can be measured as 61 and 4.9 mV, respectively. Furthermore, the 42 GBaud PAM-4 data transmission is demonstrated with a corresponding data rate of 84 Gbit/s, an amplitude of 80 mV, and an RMSN of 6.3 mV to pass the KP4 criterion when the MM-VCSEL with the design of S-OA-7 is used as the optical carrier.
Figure 14c,d exhibit the MM-VCSEL with the design of S-OA-7.5 carrying the NRZ-OOK and PAM-4 data to compare the high-speed capability with the MM-VCSEL with the design of S-OA-7. The MM-VCSEL with the design of S-OA-7 can reach the 48-Gbit/s NRZ-OOK transmission with an amplitude of 74 mV and an RMSN of 5.8 mV. In addition, the 41 GBaud PAM-4 data transmission with a corresponding data rate of 82 Gbit/s can be demonstrated with an amplitude of 95 mV and an RMSN of 8.5 mV. The maximal data rates of the NRZ-OOK and PAM-4 data carried by the MM-VCSELs with four designs are summarized in
Table 6. The MM-VCSEL with the design of L-OA-7 possesses the worst transmission performance among the four devices because of its most significant reflection coefficient, insufficient bandwidth, and slope efficiency. The MM-VCSEL with the design of L-OA-7.5 exhibits its best slope efficiency to achieve better performance than that with the design of L-OA-7. The 3-dB bandwidth of the MM-VCSEL with the S-type design is broader than that with the L-type design to demonstrate better transmission performance. The MM-VCSEL with the design of S-OA-7.5 possesses a large oxide aperture to enhance the MPN. This enhanced MPN can degrade the SNR to suppress the transmission rate, even though this device has a larger modulated output amplitude because of its smallest reflection coefficient and better slope efficiency. Therefore, the MM-VCSEL with the design of S-OA-7 is the most suitable device to achieve the highest speed transmission capability in this work after considering the reflection coefficient, the slope efficiency, RMS spectral width, 3-dB bandwidth, D factor, and RIN noise.