Threshold Current Reduction Under External Light Injection

qVn qVBn

2.3 Threshold Current Reduction Under External Light Injection

As mentioned in the introduction, in 1994, Li [117] predicted that the gain is enhanced and hence threshold is decreased by including the spontaneous emission term. Later on, in 1998, Sivaprakasam and Singh [118] demonstrated the threshold current reduction experimentally by using two identical, single mode AlGaAs laser diodes (SDL 5412-H1,λ=850 nm ). Following the derivations given above, we will derive the expression about the threshold current reduction under injection locking condition.

From Eq. (2.43), in a free-running condition, the threshold current of diode lasers can be written as

th th

I qVn

= ητ

(2.43)

Herein, we introduce the concept of the gain compression [120]. The normalized gain of a free-running semiconductor laser can be written as

( ) ( )( )

wherenis the carrier density in the active region,∂ ∂g nis the gain coefficient,v is the group _g

velocity of light,τ_pis the photon lifetime, andn is the transparent carrier density. The gain _tr is also related to the photon numberS, in the laser cavity andk is the nonlinear gain _s coefficient.

Due to the influence of spontaneous emission, the normalized gain is always less than unity, regardless of whether the laser is biased below or above threshold [120]. This gain difference can be then expressed as

( ) ^, ¹

D

= G n S −

^(2.58)

When external light is injected into the laser, we assume that the carrier density isn and that _ui the photon number isS . Then, the normalized gain is _ui

(

^,

)

(

ui tr

)( ¹

s ui

)

G n S = G n − n − k S

^(2.59)

Under an injection-locking condition, the laser is biased at or above threshold. Therefore, the gain can be approximately given as [113, 120]

( ^, ) ( )

laser may be biased below threshold. Normally, Eq. (2.60) cannot be applied to this condition. Fortunately, semiconductor lasers applied to actual systems are not biased far below threshold (for example, higher than 95 percent of their threshold). Therefore, we can

use small-signal analysis in the following way:

to include the deviation from threshold, if the injected power is not very large so that 1 _g

G D

δ + . In this section, we shall show that this condition can be satisfied for mast actual cases.

If the cavity resonance frequency of a free-running laser corresponding to the threshold carrier density n_th is ω_th , according to Eqs. (2.57) and (2.58), the cavity frequency^ω

( )

ⁿ corresponding to the carrier densitynshould obey [120]

( ) n

D

²

ω = ω + α τ

(2.62)

if the laser is not biased far below threshold, whereα is the linewidth enhancement factor.

As discussed above, when Eq. (2.61) is satisfied, the cavity frequency ^ω

( )

ⁿ^inj considering external light injection is [108, 113, 120, 121]

( ) ⁿ

^ui ^th

( ^D

^{G n}

ⁿ

) ²

ω = ω + α + Δ τ

(2.63)

Equations (2.61) and (2.63) are two basic equations for our analysis.

Furthermore, by using Eqs. (2.57), (2.58) and (2.62), the electric fieldE of a free-running laser can be written as [120] is assumed to be a white Gaussian process with the spectral densityR , which can be _sp obtained from Eq. (2.16)

( )

For the same reason, considering (2.61) and (2.63), the slowly varying amplitude of the complex electric fieldE in a semiconductor laser with external light injection obeys [108, 113, 120]

the gain change due to external light injection, and

G G

T =D + ΔG ^(2.68)

the total gain difference of the laser from the threshold value 1τ_p , where

N n p

G =G τ V ,GS =G n S k

(

)

s τp ,Δ = Δ iN n V is the carrier number change due to optical injection,V is the volume of the active region,Δ is the angular frequency detuning between ω₀ the free-running laser and the external fieldE , and_i k is the coupling coefficient. _c

To include the possible excitation of the neighboring mode with increasing detuning, we present a rate equation for the photon numberS in that mode (unlocked mode) as [108] _p

(

)

Actually, the gain in this mode is different from that in the locked mode. However, to simplify the problem, we assume that they are the same, as done in [108, 113, 114, 121, 122].

The equation for the carrier numberN_ui =n V_ui in the active region is

( )

To compare the theory with experiments, we use the following notation for the carrier recombination effect. That is, ^{R N}

( )

⁼ ^{A N}^′ ⁺^{B N}^′ ²⁺^{C N}^′ ³^withA′ , B′ , and C′ being the nonradiative, radiative, and Auger recombination coefficients, respectively [114]. The spontaneous emission rate in Eqs. (2.64), (2.66), and (2.69) can therefore be represented as

2 2

sp sp sp sp sp

R = Γβ r V = Γβ Bn V = Γβ B N V′ .

As discussed above, when the external field is injected, the carrier number is changed from its free-running valueN =nV byΔN. Therefore, according to Eq. (2.70), the equation

The static properties of the laser can be obtained according to Eqs. (2.66), (2.69), and (2.71), assuming

( )

(2.66) can be written together as

whereS determines the relative strength of the injection. Equation (2.72) is an expression _iL forΔN. Formally, the solution of Eq. (2.72) is

According to Eq. (2.73), it is clear that ΔN has two possible solutions for each value of ω0

Δ . Also, this indicates that the gain change ΔG will have two roots. These two roots give rise to optical bistability, as predicted in [123, 124].

As discussed above, small-signal analysis is valid only if the normalized gain changeδ is much less than the normalized gain 1G +D_g [see Eq. (2.61)]. Therefore, we present some discussion for the maximum gain change δG_m due to external light injection.

Of course, if δG_m 1+D_g is satisfied, we can apply small-signal analysis.

The discriminant for Eq. (2.72) should not be less than zero becauseΔNmust be real. That is, the angular frequency detuningΔ must satisfy ω₀

(

) ( )

Obviously, the gain change ΔG reaches its maximum when the frequency detuningΔ = Δf₀ ω₀ 2π becomes [113]

This frequencyΔ is called the lower limit of the frequency detuning [113]. Also, the f

(2.73), is

where subscript m means the values corresponding to the maximum gain change.

When the gain differenceD from the threshold value1_G τ_pis taken into consideration by moving the term−D_Gto the left-hand side of Eq. (2.76) [see Eq. (2.77)], the total gain differenceT_G=D_G+ Δ from the threshold value can be increased by the external light G injection locking. It is noted that the gain differenceD is important only for the _G free-running laser biased below or far above threshold.

( )

Thus, the total gain differenceT will be increased under the condition external injection _G locking. Furthermore, we rewirte Eq. (2.77) by adding the loss rate 1τ_p to both sides of it:

Threshold gain for the Cavity loss rate Total gain difference under injection-locked laser for the free-running laser external injection locking

2 2

The first term on right-hand side of Eq. (2.78) is the cavity loss rate for the free-running laser.

The second term represents the influence of spontaneous emission. The last term represents the effect of external light injection on the threshold. We can see that its role is to reduce the threshold. According to the prediction by Sharfin and Dagenais [116], the optical injection can only decrease the gain from the zero field value but optical injection does not lower the

lasing threshold. In their model, they had calculated the threshold using the rate equation approach, neglecting the spontaneous emission part. However, by including the spontaneous emission part into the rate equation, we can find, from Eq. (2.78), that the threshold gain is reduced by external light injection from 1τ_p (free-running condition) to

( )

2 2 2

1τ_p R_sp S_Lm 4α k S_c _i 1 α S_Lm

⎡ − − + ⎤

⎢ ⎥

⎣ ⎦ (externally light injecting condition). Therefore,

under external light injection the threshold gain is indeed reduced.

Finally, according to Eq. (2.57), the normalized gain at threshold current can be expressed as

By combing Eqs. (2.78), (2.79) and (2.43), we can obtain the threshold current under external light injectionI_{th ui}_,

( )

Equation (2.80) accounts for the reduction of the threshold current under external light injection. We can also find that the magnitude of the reduced threshold current is related to the injection strength factorS_iLm.

In our operating condition, the FPLD is injection-locked at its threshold current.

Therefore, it is easy to obtain very large value ofS_iLm. By using the typical parameter values for a buried-heterostructure laser, we can obtain that if the power inside the FPLD is 2.25 μW, the reduction of 3 mA under injection power of 7 dBm can be achieved. The detailed parameters are listed in Table 2.1. These parameters show that the FPLD biased at threshold current and the very strong injection locking both cause the threshold current reduction.

Table 2.1

Typical parameter values for a 1.55 μm semiconductor laser in a injection-locking condition

Symbol Parameters Typical Values Units

I Injection current Amp

q Unit charge of electron 1.6e-19 Coulomb

d Thickness of active region 0.2 μm

w Width of active region 2 μm

L Length of active region 250 μm

V=dwL Volume of active region cm³

αm Facet loss 45 cm^-1

Γ=V/V_p Optical confinement factor 0.5 Vp=V/Γ The cavity volume occupied by

photons

cm³

βsp The spontaneous emission factor 1e-3 R Reflection coefficient of mirror 0.308

vg Group velocity 8.57e9 cm/sec

τp Photon lifetime 1.6e-12 sec

g_th Threshold gain 1/(Γvgτp)=145.9 cm^-1

τ Carrier lifetime at threshold 2e-9 sec

Rsp The spontaneous emission rate ₂

spBn Vth

Γ =1.254e13 sec^-1

Ith,free-running Threshold current 12.67 mA

α Linewidth enhancement factor 5

P0 Total output power 100 μW

SLm Photon number in the locked mode

1.011e13

S_iLm Relative injection strength 33.5 dB

ΔIinjection Threshold current reduction due

to injection locking

3 mA

Fig. 2.1 Light-current graph for free running, injected locked laser and the difference of the two.

[S. Sivaprakasam and Ranjit Singh, “Gain change and threshold reduction of diode laser by injection locking,” Optics Communications, 151, 253 (1998).]

Fig. 2.2 Electronic transitions between the conduction and valence bands. The first three (a-c) represent radiative transitions in which the energy to free or bind an electron is supplied by or given to a photon. The fourth illustrates two nonradiative processes.