Modified small signal response taking PPR into consideration

The analysis of modified small signal response is mainly based on references [13] and [16]. Both practical and theoretical examples [2-8, 10-12] have shown that the most important prerequisite for the existence of PPR is to have multiple resonance cavities. Such cavity can be couple cavity injection grating (CCIG) type, 2 section DBR type [10-15], or some other inner reflection cavity [17].

3.2.1 Conventional S21 parameter of laser diode

S21 parameter reflects the potential bandwidth of the lasers. To obtain S21 parameter, dynamic analysis on laser system is necessary. The dynamic behavior of laser system can be derived from the rate equation. The density rate equation of laser diode can be written as:

!"

!# = ^%₍₎^&'− +𝑅_-.+ 𝑅_%01 − 𝑣₃𝑔𝑁_. (3.1)

!"₆

!# = 7Γ𝑣₃𝑔 −_:⁹

6; 𝑁_.+ Γ𝑅_-.^< (3.2) The meanings of each parameters are shown in Table 3.1. To evaluate the dynamical behavior of laser diode in response to some perturbation to the laser diode system, for instance as a

exact analytical solution of equation group of (3.1) and (3.2) is impossible in fact. Therefore, some approximations must be done in order to obtain the solution. Considering that the dynamic changes in carriers and photons are quite small, i.e., the perturbation to the system is quite weak in amplitude. Under such situation, the evaluated response is call small signal response.

The small signal response can be derived by taking differential of both rate equations of (3.1) and (3.2). Considering the I, N, NP, and g as variables. The differential of (3.1) and (3.2) become:

𝑑 >^!"_!#? =₍₎^@& 𝑑𝐼 −_:⁹

BC𝑑𝑁 − 𝑣₃𝑔𝑑𝑁_.− 𝑁_.𝑣₃𝑑𝑔 (3.3)

𝑑 >^!"_!#^D? = 7Γ𝑣₃𝑔 −_:⁹

6; 𝑑𝑁_E+ 𝑁_EΓ𝑣₃𝑑𝑔 +_:^F

BGH 𝑑𝑁 (3.4) where

9

:BC= ^!I_!"^J6+^!I_!"^KL (3.5)

9

:_BG^H = ^!I_!"^J6H (3.6) The Eqs. (3.5) and (3.6) can be replaced by a simple matrix equation as follows:

!

!#7𝑑𝑁

𝑑𝑁_.; = >−𝛾_"" −𝛾_"E 𝛾_E" −𝛾_EE? 7𝑑𝑁

𝑑𝑁_.; + N

%_&' ()

(𝑁_.𝑣₃𝑔 + 𝑅_-.^< )𝑑ΓQ (3.7)

where

𝛾_"" = 1

𝜏_TU+ 𝑣₃𝑎𝑁_. 𝛾_"E = 1

Γ𝜏_.−𝑅_-.^<

𝑁_. − 𝑣₃𝑎𝑁_. 𝛾_E"= Γ

𝜏_T"^< + Γ𝑣₃𝑎𝑁_. 𝛾_EE =Γ𝑅_-.^<

𝑁_. + Γ𝑣₃𝑎𝑁_.

In the matrix form above, the modulation current is considered to be driving term or forcing

instance noise, loss, gain or other related laser parameters. As it is small signal response, we can assume the solutions of the form as:

𝑑𝐼(𝑡) = 𝐼₉𝑒^YZ#

𝑑𝑁(𝑡) = 𝑁₉𝑒^YZ#

𝑑𝑁_.(𝑡) = 𝑁_.9𝑒^YZ#

(3.8)

Here, N1 and Np1 are the original carrier density and photon density, respectively. By setting 𝑑[ → 𝑗𝜔, we can obtain 𝑡

_7𝛾_""+ 𝑗𝜔 𝛾_"E

−𝛾_E" 𝛾_EE + 𝑗𝜔; 7 𝑁₉

𝑁_.9; =^@₍₎^&'`>10?b (3.9)

The determination of the matrix above is given by Δ ≡ e𝛾_""+ 𝑗𝜔 𝛾_"E

−𝛾_E" 𝛾_EE + 𝑗𝜔e = −𝜔^f + 𝑗𝜔(𝛾_"" + 𝛾_EE) + 𝛾_"E𝛾_E"+ 𝛾_""𝛾_EE (3.10) The photon and carrier density are obtained by Cramer’s rule as

𝑁₉ = ^@₍₎^&'`∗^hDD_Z^iYZ

jk 𝐻(𝜔)

𝑁_E9=^@₍₎^&'`∗^h_Z^DG

jk 𝐻(𝜔) (3.11) Here, the modulation response is conventionally described as modulation transfer function

𝐻(𝜔) = 𝜔_I^f/Δ ≡_Z ^Zjk

jknZ^kiYZh (3.12) The 𝜔_I here is the so-called relaxation resonance frequency and 𝛾 is the damping factor.

𝜔_I^f ≡ 𝛾_"E𝛾_E"+ 𝛾_""𝛾_EE

𝛾 ≡ 𝛾_""𝛾_EE (3.13)

The density of carrier and photon both follow the frequency response of the modulation transfer function. Therefor the 𝐻(𝜔) parameter largely affects the frequency response. As can be seen from Eq. 3.12, the bandwidth is largely limited by the relaxation resonance frequency. Fig. 3.1 illustrate a typical modulation transfer function for increasing the values of relaxation resonance frequency.

3.2.2 S21 parameter of laser diode when PPR is taken into consideration

In Eq. 3.7, the Γ as the optical confinement is regarded as constant, therefor 𝑑Γ is zero.

In the case of multiple cavity lasers, however, the Γ term is not a constant anymore. This is quite reasonable: the optical field distribution is different in different cavity are quite different and will change as perturbation changes. Thus, derivation from (3.7) to (3.9) must be rearranged. The cross section of the laser is quite stable, however, the optical confinement changes along the propagation direction, i.e., the ‘z’ direction. Hence, the longitudinal confinement factor Γ is considered as Γ = Γ_pqΓ(z, t). The equation (3.9) is re-arranged as:

𝛾_""+ 𝑗𝜔 𝛾_"E 𝑁₉ ^@&'`𝑑𝐼

𝑙𝑜𝑔|𝐻(𝜔)|^f

~(𝜔_I/𝛾)^f 𝜔I

-3 dB bandwidth

𝑙𝑜𝑔𝜔

Fig. 3.1 Sketch of the modulation transfer function for increasing values of relaxation resonance frequency and damping factor. 3 dB bandwidth is enhanced while increasing relaxation resonance frequency.

For this case, the small signal photon density is obtained by Cramer’s rule as:

𝑁_E9= ^@&'`

() ∗^hDG

∆ +^(hGGiYZ)+("₆z_{3iI_J6^H 1

∆ ∗ ^|F

}^~•€ (3.15) The modulation transfer function is then derived as:

𝐻(𝜔) =^@₍₎^&'`∗^hDG_∆ +_'⁹

`∫ ‚^!Fƒ„_!#^(…,#)∗^(hGGiYZ)+("₆z_{3iI_J6^H 1F_ƒ„

∆}^~•€ † 𝑑𝑡

‡

ˆ (3.16)

= 𝜂_Š𝐼₉ 𝑞𝑉 ∗𝛾_E"

∆ +(𝛾_"" + 𝑗𝜔)+(𝑁_.𝑣₃𝑔 + 𝑅_-.^< 1

∆𝐼₉

× 𝑗∆𝜔𝑇𝑒^{Y(∆ZnZ)‡/f}sin 𝑐((∆𝜔 − 𝜔)𝑇/2) ” 𝑐(𝑧)𝑑𝑧^–

ˆ

Where ∆𝜔 is the frequency difference between the longitudinal modes inside the laser diode.

𝜔 is the modulation frequency. Here, the c(z) is complex amplitude integration of all longitudinal modes.

𝑐(𝑧) = ∫ 𝑃(𝜆)𝑑𝜆_n™^™ (3.17) 𝑃(𝜆) represents the power density as a function of wavelength. Generally speaking, there are multiple longitudinal modes existing inside the laser system, i.e., different wavelength emission. This quite understandable for regular FP lasers, because multiple emission wavelength exists. In the case of single wavelength emission lasers, although single wavelength is emitting, the other wavelength 𝜆_Š that has roundtrip phase of multiple 2π are also the

“matched wavenumbers”. Such 𝜆_Š can be taken into consideration in Eq. (3.17).

Consider a very simple case with two longitudinal modes, the 𝐻(𝜔) can be analyzed then. As can be seen in Eq. (3.16), the transfer function gets maximum value when the modulation frequency 𝜔 = ∆𝜔. Such max value is induced by mode coupling between the two longitudinal modes, i.e., the photon photon resonance. Furthermore, the resonance intensity is largely affected by the value of ∫ 𝑐(𝑧)𝑑𝑧_ˆ^– . The PPR peak gets stronger when value of the overlap

integral ∫ 𝑐(𝑧)𝑑𝑧_ˆ^– increases. Assume that the field amplitudes of two longitudinal modes are a1 and a2. The overlap integral is

∫ 𝑐(𝑧)𝑑𝑧_ˆ^– = ∫ 𝑎_ˆ^– 91𝑒^{Yš`…</sup>𝑎<sub>f</sub><sup>∗</sup>𝑒<sup>nYš`…}𝑑𝑧 =^›_YTš^`›k∗(𝑒^YTš– − 1) (3.18) Where the Δ𝑘 = 𝑘₉− 𝑘_f, and 𝑘₉, 𝑘_f are the wavenumber of the longitudinal modes. In general,

Δ𝑘 = 𝑘₉− 𝑘_f = 2π ‚^%_ž^`

` −^%_ž^k

k† =^Ÿ_–(𝑚₉− 𝑚_f) (3.19) Where 𝑛₉ and 𝑛_f are the effective refractive index and 𝑚₉ and 𝑚_f are mode orders. Therefor,

∫ 𝑐(𝑧)𝑑𝑧_ˆ^– = ¢

Yf›`›_k^∗–

Ÿ(£_`n£_k), 𝑤ℎ𝑒𝑛 𝑚₉ − 𝑚_f = ±1, ±3, ±5, ⋯

0, 𝑤ℎ𝑒𝑛 𝑚₉− 𝑚_f = ±2, ±4, ±6, ⋯ (3.20) As the side mode suppression ratio (SMSR) is calculated as:

SMSR(dB) = 20 log_9ˆ(^›_´f^`) (3.21)

The final transfer function then become:

𝐻(𝜔) =^@₍₎^&'`∗^hDG_∆ +^(hGGiYZ)+("₆z_{3iI_J6^H 1

∆'`

× 𝑗∆𝜔𝑇𝑒^{Y(∆ZnZ)‡/f}sin 𝑐((∆𝜔 − 𝜔)𝑇/2)_Ÿ^Y_9i9ˆ^{f∗9ˆµ¶µj/k·}_µ¶µj/`·

(3.21)

Based on Eq. (3.21), the modified small signal response can be simulated. Figure 3.2 shows such result. The black line represents the measured S21 response of active-MMI LD [18].

Yellow line represents the simulated small signal response taking PPR phenomenon into consideration. Brown line shows the simulated small signal response without PPR. The frequency difference is replaced by wavelength difference, which is intrinsically the same parameter. Wavelength difference used for simulation is 0.12 nm, which is evaluated from the

measured spectrum data shown in Fig. 3.3. As can be seen from Fig. 3.3, the main lasing mode and its nearest sub-longitudinal mode has a wavelength of 0.12 nm and SMSR of 30 dB.

According to the results from Fig. 3.2 and Fig. 3.3, the experimental data and simulation data

Frequency [GHz]

0 5 10 15

-5 -10 -15

0 5 10 15 20

17 GHz

Small signal response [dB]

Fig. 3.2. The simulated small signal response when PPR is taken into consideration. Yellow line is simulated result with PPR. Brown line is result without PPR. Black line shows measured small signal response.

-10 -20 -30 -40 -50 -60 -70

1555 1556 1557 1558 1559

1555.8 1556 1556.2 1556.4 -10

-30 -50 -70

Wavelength [nm]

Output power [dBm]

Wavelength [nm]

~0.12 nm

Output power [dBm]

Fig. 3.3. Measured spectrum of active-MMI LD. Single wavelength emission with 30 dB SMSR is confirmed. Wavelength difference of 0.12 nm has been used for simulation in Fig. 3.2.

matches with each other quite well. Thanks to PPR, the modulation bandwidth is enhanced to 17 GHz.