**2.2 Fundamentals of SOA and RSOA**

Gain in a semiconductor material results from current injection into the PIN structure. The relationship between the current I and the carrier density (n) is given by the rate-equation. The rate-equation should include the stimulated emission as well as the spontaneous and absorption rate. R�n� is the rate of carrier recombination including the spontaneous emission and excluding the stimulated emission. Electrons can recombine radiatively and nonradiatively therefore R�n� can be written as:

$$R(n) = R\_{rad}(n) + R\_{non-rad}(n) \tag{1}$$

When an electron from the Conduction Band (CB) recombines with a Valence Band (VB) hole and this process leads to the emission of a photon, it is called the radiative recombination. The rate of radiative recombination is:

$$\mathbf{R}\_{\rm rad}(\mathbf{n}) = \mathbf{B}.\mathbf{n}^2\tag{2}$$

Next Generation of Optical Access Network Based on Reflective-SOA 5

but with no wavelength dependence. Amplified spontaneous emission power noise is assumed to be a white noise, with an equivalent optical noise bandwidth. When the current is modulated in a RSOA, the carrier density and the photon density are varying with time and position. This is caused by the optical wave propagation and the carrier/photon interaction. The carrier density variation is introduced in the model by dividing the total device length into smaller sections. For each section the carrier density is assumed to remain constant along the longitudinal direction. The equations are linking the driving current, the carrier density and the photon density. Figure 1 represents the model elementary section. It includes ports representing the input and output photon density (forward and backward), input and output amplified spontaneous emission (forward and backward), the input electrical current injection and carrier density. We do not consider the phase shift of the

Fig. 1. RSOA elementary representation for numerical modelling

*out*

*ASE out*

*ASE in*

*in*

material gain equation becomes:

Where � is the gain saturation parameter.

*I/e.V n*

*in S+*

*out S‐*

*ASE in S+*

*ASE out S‐*

*LS*

Where R1 and R2 are power reflection coefficients.

The forward and backward propagating optical fields (excluding spontaneous emission) are

*d*

The material gain (gm) is usually approximated by a linear function of the carrier density. In general the material gain also depends on the photon density S. For high photon density, the gain saturates and this phenomenon is described by the gain compression factor. Then, the

�� � ��

P��0� � ������P�� � ��P��0�

The boundary conditions for the device input and output facets, are given by:

P��� � ��� � P����������� (8)

*w*

*z*

*LS*

*L*

����� (9)

<sup>P</sup>��L� � ��P��L� (10)

described by the relation between the input optical power and output optical power.

signal.

*S+*

*S‐*

*S+*

*S‐*

This term corresponds to the spontaneous emission recombination. Non-radiative processes deplete the carrier density population in the CB then fewer carriers remain available for the stimulated emission and the generated amount of light is limited. The three main nonradiative recombination mechanisms in semiconductor are:


$$\mathbf{R}\_{\rm SRH} = \mathbf{A}. \mathbf{n} \tag{3}$$


$$\mathbf{R\_{Aug}} = \mathbf{C.n}^3\tag{4}$$


$$\mathbf{R}\_{\text{leak}} = \mathbf{D}\_{\text{leak}} \cdot \mathbf{n}^{3.5} \\ \text{for diffusion and } \mathbf{R}\_{\text{leak}} = \mathbf{D}\_{\text{leak}} \cdot \mathbf{n}^{5.5} \text{ for drift} \tag{5}$$

The dominant leakage current is usually due to carrier drift. Therefore the total recombination rate is given by:

$$\mathbf{R(n)} = \mathbf{R\_{rad}(n)} + \mathbf{R\_{non-rad}(n)} = \mathbf{A.n} + \mathbf{B.n^2} + \mathbf{C.n^3} + \mathbf{D\_{leak}.n^{5.5}} \tag{6}$$

The carrier leakage is usually neglected. So the rate-equation states that the resulting change of the carrier density in the active zone is equal to the difference between the carrier supplied by electrical injection and the carrier's recombination. Amplification results from stimulated recombination of the electrons and holes due to the presence of photons. The interaction between photons and electrons inside the active region depends on the position and the time. Therefore the carrier density at z and t is governed by the final rate-equation. We neglect carrier diffusion in order to simplify the carrier density rate-equation. This assumption is valid as long as the amplifier length L is much longer than the diffusion length, which is typically on the order of microns. We also assume that the carrier density is independent of the lateral dimensions.

$$\frac{d\ln(\mathbf{z},t)}{dt} = \frac{\mathbf{l}(t)}{\mathbf{e}\mathbf{V}} - \left(\mathbf{A}.\ln(\mathbf{z},t) + \mathbf{B}.\mathbf{n}(\mathbf{z},t)^2 + \mathbf{C}.\mathbf{n}(\mathbf{z},t)^3\right) - \mathbf{v}\_\mathbf{g}\mathbf{g}\_{\text{net}}.\mathbf{S}(\mathbf{z},t) \tag{7}$$

Where n(z,t) is the carrier density, I(t) is the applied bias current, S(z,t) is the photon density, gnet is the net gain and vg is the light velocity group.

A time domain model for reflective semiconductor optical amplifiers (RSOAs) was developed based on the carrier rate and wave propagation equations. The non linear gain saturation and the amplified spontaneous emission have been considered and implemented together in a current injected RSOA model (Liu et al., 2011). This approach follows the same analytical formalism as Connelly's static model (Connelly, 2007).

To make the model suitable for static analysis some assumptions have been made and simplifications have been introduced. Since, as a modulator, the RSOA is mainly illuminated by a CW optical source, the material gain is assumed to vary linearly with the carrier density

This term corresponds to the spontaneous emission recombination. Non-radiative processes deplete the carrier density population in the CB then fewer carriers remain available for the stimulated emission and the generated amount of light is limited. The three main non-




The dominant leakage current is usually due to carrier drift. Therefore the total

The carrier leakage is usually neglected. So the rate-equation states that the resulting change of the carrier density in the active zone is equal to the difference between the carrier supplied by electrical injection and the carrier's recombination. Amplification results from stimulated recombination of the electrons and holes due to the presence of photons. The interaction between photons and electrons inside the active region depends on the position and the time. Therefore the carrier density at z and t is governed by the final rate-equation. We neglect carrier diffusion in order to simplify the carrier density rate-equation. This assumption is valid as long as the amplifier length L is much longer than the diffusion length, which is typically on the order of microns. We also assume that the carrier density is

Where n(z,t) is the carrier density, I(t) is the applied bias current, S(z,t) is the photon

A time domain model for reflective semiconductor optical amplifiers (RSOAs) was developed based on the carrier rate and wave propagation equations. The non linear gain saturation and the amplified spontaneous emission have been considered and implemented together in a current injected RSOA model (Liu et al., 2011). This approach follows the same

To make the model suitable for static analysis some assumptions have been made and simplifications have been introduced. Since, as a modulator, the RSOA is mainly illuminated by a CW optical source, the material gain is assumed to vary linearly with the carrier density

R���� � �����. n�.�for diffusion and R���� � �����. n�.� for drift (5)

R�n� � R����n� � R��������n� � A. n � �. n� � �. n� � �����. n�.� (6)

�.� � �A. n�z, t� � �. n�z, t�� � �. n�z, t��� � ��g���. S�z, t� (7)

R��� � A. n (3)

R��� � �. n� (4)

radiative recombination mechanisms in semiconductor are:

crystal lattice. The rate of the Auger transitions is:

recombination rate is given by:

independent of the lateral dimensions.

density, gnet is the net gain and vg is the light velocity group.

analytical formalism as Connelly's static model (Connelly, 2007).

����,�� �� � ����

(SHR) recombination. The rate of SRH recombination is:

of the carriers therefore is given by (Olshansky et al., 1984):

but with no wavelength dependence. Amplified spontaneous emission power noise is assumed to be a white noise, with an equivalent optical noise bandwidth. When the current is modulated in a RSOA, the carrier density and the photon density are varying with time and position. This is caused by the optical wave propagation and the carrier/photon interaction. The carrier density variation is introduced in the model by dividing the total device length into smaller sections. For each section the carrier density is assumed to remain constant along the longitudinal direction. The equations are linking the driving current, the carrier density and the photon density. Figure 1 represents the model elementary section. It includes ports representing the input and output photon density (forward and backward), input and output amplified spontaneous emission (forward and backward), the input electrical current injection and carrier density. We do not consider the phase shift of the signal.

Fig. 1. RSOA elementary representation for numerical modelling

The forward and backward propagating optical fields (excluding spontaneous emission) are described by the relation between the input optical power and output optical power.

$$\mathbf{P}^{\pm}(\mathbf{z} \pm \Delta \mathbf{z}) = \mathbf{P}^{\pm}(\mathbf{z}) \mathbf{e}^{\text{g}\_{\text{net}} \Delta \mathbf{z}} \tag{8}$$

The material gain (gm) is usually approximated by a linear function of the carrier density. In general the material gain also depends on the photon density S. For high photon density, the gain saturates and this phenomenon is described by the gain compression factor. Then, the material gain equation becomes:

$$g\_S = \frac{g\_m}{1 + \varepsilon S} \tag{9}$$

Where � is the gain saturation parameter.

The boundary conditions for the device input and output facets, are given by:

$$\begin{aligned} \mathbf{P}^+ \text{(0)} &= (\mathbf{1} - \mathbf{R\_1}) \mathbf{P\_{ln}} + \mathbf{R\_1} \mathbf{P^-} \text{(0)}\\ \mathbf{P^-} \text{(L)} &= \mathbf{R\_2} \mathbf{P^+} \text{(L)} \end{aligned} \tag{10}$$

Where R1 and R2 are power reflection coefficients.

Next Generation of Optical Access Network Based on Reflective-SOA 7

along the RSOA. At the mirror and input facets, the signal photon density becomes larger with the injected current as it has been more amplified during the forward and backward

From this preliminary analysis, a general conclusion can be deduced. RSOAs should saturate faster than classic SOAs. The overall photon density inside a RSOA is larger than in a classic SOA, reducing the material gain available for signal amplification. However the forward and backward signal amplifications could compensate for this effect. Large photon density should also affect the E/O bandwidth and could be useful to obtain high speed devices. All these effects are stronger at high input electrical injection and high input optical

Fig. 3. Carrier and photon density spatial distribution in RSOA device. (a) and (b) represents

Optical gain measurements depending on the input current and optical power were realized. Figure 4 shows the experimental setup which is used to perform static measurements. The required wavelength controlled by an external cavity laser is launched into the RSOA through an optical circulator (OC). A combined power meter and attenuator is used to control the input power to the RSOA. An optical spectrum analyser and a power

the simulation from Pin = -40 dBm ; (c) and (d) for 0 dBm

**3. RSOA devices static characteristics** 

propagations.

power.

The amplified spontaneous emission is the main noise source in an RSOA and determines the RSOA static and dynamic performances under low input optical power. For high stimulated emission output power the spontaneous emission drops significantly and its impact on the device performances is less significant. For a section of length Δz the ASE power spectral density generated within that section is given by the following equation :

$$\mathbf{P\_{ASE}} = \eta\_{Sp}(\mathbf{G\_S(z)} - 1)\mathbf{h}\mathbf{v}\mathbf{B\_0} \tag{11}$$

Where Gs is the single pass gain of one section and *η*sp is the spontaneous emission factor. The spontaneous emission factor can be approximated by (D'Alessandro et al., 2011):

$$
\eta\_{Sp} = \frac{n}{n - \mathbf{n}\_0} \tag{12}
$$

In our model we have assumed a constant noise power spectral density over an optical bandwidth *Bo*. The bandwidth *Bo* is estimated at 5x1012 Hz. The implementation of the ASE noise travelling wave follows a procedure similar to the optical signal travelling wave. The spontaneous emission output power for the forward and backward noise signals has two contributions: the amplified input noise and the generated spontaneous emission component within the section. The gain variations with the bias current (Experimental and modelled) are compared in Figure 2.

Fig. 2. Fibre-to-fibre gain for RSOA versus bias current

The carrier density profile is represented in figure 3 as well as the total photon density. At low input injection (Pin = -40 dBm), the carrier density profile is in this case not symmetrical due to the high reflection of the second facet. Strong depletion occurs from the ASE and the signal double propagation (reflective behaviour of the device). Also at Pin = -40 dBm, the ASE power dominates the signal power which explains that the RSOA device saturates more at high input electrical current.

At high input optical power, the carrier density in an RSOA is flattened due to the forward and backward propagations of the signal inside the device. The saturation effect occurs all

The amplified spontaneous emission is the main noise source in an RSOA and determines the RSOA static and dynamic performances under low input optical power. For high stimulated emission output power the spontaneous emission drops significantly and its impact on the device performances is less significant. For a section of length Δz the ASE power spectral density generated within that section is given by the following equation :

Where Gs is the single pass gain of one section and *η*sp is the spontaneous emission factor. The spontaneous emission factor can be approximated by (D'Alessandro et al., 2011):

> ��� � � ����

modelled) are compared in Figure 2.

Fibre-to-fibre Gain (dB)

Fig. 2. Fibre-to-fibre gain for RSOA versus bias current





0

10

20

30

40

more at high input electrical current.

In our model we have assumed a constant noise power spectral density over an optical bandwidth *Bo*. The bandwidth *Bo* is estimated at 5x1012 Hz. The implementation of the ASE noise travelling wave follows a procedure similar to the optical signal travelling wave. The spontaneous emission output power for the forward and backward noise signals has two contributions: the amplified input noise and the generated spontaneous emission component within the section. The gain variations with the bias current (Experimental and

The carrier density profile is represented in figure 3 as well as the total photon density. At low input injection (Pin = -40 dBm), the carrier density profile is in this case not symmetrical due to the high reflection of the second facet. Strong depletion occurs from the ASE and the signal double propagation (reflective behaviour of the device). Also at Pin = -40 dBm, the ASE power dominates the signal power which explains that the RSOA device saturates

20 40 60 80 100

Current (mA)

At high input optical power, the carrier density in an RSOA is flattened due to the forward and backward propagations of the signal inside the device. The saturation effect occurs all

P��� � ��������� � ��hνB� (11)

(12)

 Measured Modelled

along the RSOA. At the mirror and input facets, the signal photon density becomes larger with the injected current as it has been more amplified during the forward and backward propagations.

From this preliminary analysis, a general conclusion can be deduced. RSOAs should saturate faster than classic SOAs. The overall photon density inside a RSOA is larger than in a classic SOA, reducing the material gain available for signal amplification. However the forward and backward signal amplifications could compensate for this effect. Large photon density should also affect the E/O bandwidth and could be useful to obtain high speed devices. All these effects are stronger at high input electrical injection and high input optical power.

Fig. 3. Carrier and photon density spatial distribution in RSOA device. (a) and (b) represents the simulation from Pin = -40 dBm ; (c) and (d) for 0 dBm
