Quantifying photodiode nonlinear characteristic induced by optical power and voltage

A new mathematical model, optical-power-dependent model R(P), is proposed by investigating the nonlinearity of modified uni-traveling carrier photodiodes (MUTC-PD). Benefiting from this model, the harmonic power and output 3rd order intercept point (OIP3) of a photodiode can be calculated. Considering a bias voltage shift of the photodiode caused by a large voltage swing effect, the R(P) mathematical model is expanded into a voltage-related one, which is the optical-power-and-voltage-dependent model, R(P,V). The calculate results of both mathematical models are completely in accord with simulation results.


Introduction
As one of the core devices in an optical communication system, photodiode is crucial to the effectiveness of data reception. The nonlinearity is an important property of the photodiode, which illustrates the time-domain distortion of the output signal and the high order harmonics in a frequency domain ).
On one hand, in order to obtain a larger dynamic range in the optical link, we need to reduce the nonlinearity of photodiodes ).On the other hand, when using the photodiode for the purpose of up/down conversion e.g. optoelectronic mixer

Device structure
The device under test was a back-illuminated mesa MUTC-PD as reported in Xin et al. (2007). The specific structure is shown in Table 1 and Fig. 1.
It contains a 850 nm thick InGaAs absorption layer and a 605 nm thick n-doped InP collection layer. The absorption layer contains a 200 nm InGaAs depletion absorption region and a 650 nm thick graded doped InGaAs absorption region ranging from 2.5 × 10 17 to 2 × 10 18 cm −3 to generate a self-built electric field which assists electron transition. The 605 nm InP electron drift layer and 200 nm InGaAs non-depleted absorption layer are lightly n-type doped for space charge compensation. The 30 nm InGaAsP space layer is placed at the interface between the absorption layer and the transition layer, elevating the electric field at the interface as well as smoothing the band gap and the 20 nm InP electronblocking layer is place next to the absorption layer.

Nonliearity model
Responsivity is a physical quantity representing the photoelectric conversion capability of a photodiode, as shown in Eq. (1). where I out ,P in , , h and f represent the output current, incident optical power, internal quantum efficiency, Planck constant and frequency, respectively. When receiving optical signal, the fluctuation of the input optical power influence the responsibility of the photodiode (Fu et al. 2011). Therefore, the output of the photodiode appears nonlinear characteristics, showing high-order harmonics.

R(P) model
The response of MUTC-PD under different input optical power and bias voltage are shown in Fig. 2. It reveals that there is a small increase in the responsivity of the MUTC-PD along with the input optical power increase. Since a self-induced electric field (Shimizu et al. 1998) in the non-depleted absorption region is created via incident optical power. It will generate band bending which accelerates the photo-excited electron and allows more photogenerated electrons to pass through the absorption region. This optical power-dependent responsibility is used to establish the mathematically model R(P) to calculate photodiode distortion. It is realized by taking a Taylor series expansion of the optical power-dependent responsivity around average incident optical power, as shown in (2).
where P and P 0 in (2) are the incident optical power and average incident optical power respectively.
When considering the nonlinearity of the photodiode, the third-order intermodulation distortions (IMD3) are particularly important, since their frequencies are very close to the fundamental modulation frequencies. Therefore, we use the two-tone modulation to calculate the nonlinearity of the photodetector. The incident optical power can be expressed as (3). where m, f 1 and f 2 are the modulation depth and the frequency of the two modulated optical signals. According to the (1) and (2), the output photocurrent can be expressed by (4). (2)

Fig. 2 Photodiode responsivity versus optical input power under different voltage
Since the 3rd order intermodulation product is the highest intermodulation component required to measure the nonlinearity of photodiode, it is reasonable to take a second order Taylor expansion of R(P) as shown in (5) and shorten the R(P 0 ) to R to get (6).
The amplitude of the fundamental tone, second and third harmonics can be obtained through (6) as shown in (7,8,9). And the OIP3 is calculated to measure the nonlinearity of the MUTC-PD through the (10).
It can be seen that the greater the response changes with the input optical power, the more obvious the nonlinearity of the MUTC-PD.

R(P,V) model
There will be a serious voltage swing effect when the output signal amplitude is large (Li et al. 2016). Since the output RF signal causes a voltage drop at the load resistor R L and series resistor R S as show in Fig. 3 which in turn influence the bias voltage of the photodiode.
V b and V L in Fig. 3 is the bias voltage and voltage drop respectively, I is the photogenerated current, C is the device capacitance, R p , R s and R L is the parallel resistance, series resistance and external load which usually 50 Ω. The voltage drop V L is created by the output photocurrent and can be expressed as (11).
where I avg means average output photocurrent. Thus, we can calculate the photodiode's voltage through (12).
The responsivity of MUTC-PD versus voltage is shown in Fig. 4 which indicate that the responsivity increases with the voltage. Since the bias voltage can change the velocity of the carriers and the trapping of the carriers between the absorbing layer and the collecting layer.
Therefore, we expand the R(P) mathematiclal model into a R(P,V) one for photodiode nonlinearity calculation in the case of large voltage swing effect. Two-dimensional Taylor series expansion of the responsivity is taken to the calculate the nonlinearity of MUTC-PD in the R(P,V) mathematical model as shown in the (13). where the V 0 is the average bias voltage. We can get the amplitude factor of the fundamental tone, second and third harmonics as shown in (14, 15 and 16) by shorten the R(P 0 ) to R.

Results and discussion
In summary, we establish a new mathematical model, optical-intensity dependent responsivity (R(P)), to explain the generation of the nonlinearity of the photodiode and calculated the nonlinearity of the MUTC-PD. In the case of large voltage swing effects, the R(P) model is expanded into a R(P,V) one. The calculation results of both mathematical models agree well with the TCAD simulation results. The material parameters of In 0.47 GaAs, InP and InGaAsP are listed in Table 2.  Thus, the responsivity of the photodiode will not only be used to describe the ability of the photoelectric conversion. Its variation of responsivity also contains information about the nonlinearity of the photodiode as shown in (13)(14). Therefore, by observing the change in the responsivity with the optical intensity and bias voltage, the high-order harmonics of the photodiode can be calculated quickly.

Result of R(P) mathematical model
We use the R(P) mathematical model to quantify the nonlinearity of the MUTC-PD at different average input light rates, when setting the modulation depth of the input signal in (3) to 0.4. and the bias voltage to 4 V. Figure 5 shows the calculated results of R(P) model and its comparison with the SILVACO TCAD simulation when the bias voltage is 4 V and the modulation depth is 0.4. According to the results, the magnitudes of the fundamental tone, IMD2 and IMD3 obtained from this mathematical model are basically consistent with the simulation result of TCAD. The difference between the OIP3 obtained using the R(P) model and the simulated result is within 0.4 dBm and the mean error of OIP3 is 0.23dbm. Therefore, the R(P) model have the ability to quantify the nonlinearity of photodiode accurately when the voltage swing effect is small.

Result of R(P,V) mathematical model
The R(P,V) model is used under the condition of large voltage swing effect. We calculated results of R(P,V) model and its comparison with the SILVACO TCAD simulation when the bias voltage is 4 V and the modulation depth is 0.8, as shown in Fig. 6.
According to Fig. 6a and b, the magnitudes of the fundamental tone and IMD2 consist well with the simulation result of TCAD. Figure 6c and d indicates that the calculated results of IMD3 and OIP3 of the R(P,V) model have the same trend as those obtained from the TCAD. The IMD3 increase with incident optical power and reach a minimal value at incident optical power of 1.1 W and increase again afterward, while the OIP3 presents the opposite trend as IMD3. The mean error of OIP3 between R(P,V) model and TCAD simulation is3.1dbm, which means the R(P,V) model have ability to quantify the nonlinearity of MUTC-PD under large voltage swing effect.
One of the reasons for the calculation error of OIP3 between R(P,V) model and TCAD simulation can be the inaccuracy introduced by the number of polynomial fits. When performing the derivation of the responsivity in (16), it may produce poor-fitting if the number of fits is too small or the overfitting if the number of fits is too large. In this paper, the ordinary least squares method is used to fit the responsivity as a function of the variation of incident optical power and voltage. The results of R (P,V) model can be more accurate if a better fitting algorithm can be adopted.
Another reason for the calculation error is that the model ignores the carrier transport time. If the carrier transport time is considered, there will be a time delay Δt in the voltage drop at RL. Then the (12) should be expressed as (17). And this time delay introduce additional nonlinearity since it varies with the input optical power and bias voltage.

Conclusion
In summary, we establish a new mathematical model, optical-power-dependent responsivity (R(P)), to explain the generation of the nonlinearity of the photodiode and calculated the nonlinearity of the MUTC-PD. The R(P) model is expanded it into R(P,V) model in the case of large voltage swing effect. The calculation results of both mathematical model agree well with the TCAD simulation results.
Thus, the responsivity of the photodiode will not only be used to describe the ability of the photoelectric conversion. Its variation of responsivity also contains information about the nonlinearity of the photodiode as shown in (13,14). Therefore, the high-order harmonics of the photodiode can be quickly calculated by the change of the responsivity with the optical power and bias voltage.
Authors' contributions JY and YQH designed research, analyzed data, and wrote the paper. ZCW and JWD helped perform the analysis with constructive discussions. CJ contributed analysis and manuscript preparation. Data availability All data generated or analysed during this study are included in this published article.