**2.3 Photonic neuron**

The neuron consists of an input stage that is a linear combination (weighted addition) of the outputs of the neurons feeding it. The combined signals from the input stage are integrated to produce a nonlinear response as dictated by the activation function see **Figure 2**.

This neuron must perform three mathematical operations:


All inputs must be of the same nature as outputs.

**Figure 2.** *Neuron signal flow model [6].*

*T s*ðÞ¼ *<sup>y</sup>*

half of the s-plane in **Figure 5**.

*Neuromorphic Photonics*

*DOI: http://dx.doi.org/10.5772/intechopen.94297*

**Figure 5.** *S-plane plot.*

**7**

**Figure 4.**

*Negative feedback loop system [8].*

Criterion [10] by investigating pole location.

*<sup>x</sup>* <sup>¼</sup> *A s*ð Þ

This system can be linearized by making the gain product *A s*ð Þ ∗ *B s*ð ) ≫ 1. With this condition, the transfer function becomes dependent solely on the feedback gain coefficient and response of the feedback loop, which becomes linear:

*T s*ðÞ¼ <sup>1</sup>

A feedback loop can oscillate if its open-loop gain exceeds unity and simultaneously its open-loop phase shift exceeds π. There are poles that are present in the closed loop configuration with at least one pole of an unstable loop lies in the right

It is based on Nyquist plot where the open-loop transfer function is analyzed with a plot of real and imaginary parts. Where stability of the closed-loop system is

If poles are present in the Right Hand Plane (RHP), the closed-loop system becomes unstable. In brief, the Nyquist criterion is a method for the determination of the stability of feedback loop systems as a function of an adjustable gain and delay parameters in the feedback section. It simply determines if the system is

For the self-pulsation mode, it is well known that an active circuit with feedback

Barkhausen [10] is fulfilled. This criterion is defined as when the denominator of

Analysis of stability of this system can be done according to the Nyquist

determined If poles are present in the Left Half (LHP) of the s-Plane.

stable for any specified value of the feedback transfer function *B(s)*.

can produce self-sustained oscillations only if the criterion formulated by

<sup>1</sup> <sup>þ</sup> *A s*ð Þ <sup>∗</sup> *B s*ð Þ (1)

*B s*ð Þ (2)

**Figure 3.**

*Spiking neuron information encoding scheme of events rather than bits [6].*

#### **2.4 Information coding**

In order to have an efficient signal modulation scheme, a hybrid combination of analog and digital signal representation need to be implemented that mimics the way the brain encodes information as events in time. The type of a well-known modulation scheme used in optical communication systems called Pulse Position Modulation (PPM) [6, 7].

This modulation format exploits the efficiency of analog signals and at the same time reduces the noise accumulation that distorts analog signals (see **Figure 3**).

## **3. Semiconductor lasers with optoelectronic feedback as photonic neuron**

Semiconductor lasers are widely used in many applications for both digital and analog signal processing. For the past two decades, specifications of these lasers have addressed many specific applications by tailoring the laser design parameters to meet specific performance target. While the aim of this work is to use low cost lasers with generic specifications, modify and enhance their essential performance parameters to behave as a photonic neuron in addition to the applications addressed in [8] by using electronic feedback for triggering self-pulsating behavior necessary for spiking neuron model. The main driver behind this work is to facilitate photonic integration, improving laser modulation bandwidth and increasing laser relaxation oscillation frequency. The performance for all these applications is analyzed in both time and frequency domains. Mainly, by adjusting the feedback loop settings so it can operate outside its stable regime but just ahead of the chaos mode, so the laser can run in self-pulsation mode. This provides the use of the laser drive current as a single point of control for the pulsation rep-rate and.

#### **3.1 Basic control theory**

The theory behind this work is based on the classical control theory of negative feedback. Recent work by [9], has presented a rigorous, yet simple and intuitive, non-linear analysis method for predicting injection locking in LC oscillators.

A system with a negative feedback control loop is shown in **Figure 4**.

It consists of a forward-gain element with transfer function *A(s)*, with s is the Laplace operator and can be replaced by *(jω)* feedback element with transfer function *B(s)*.

Where *A(s)* represents laser transfer function, *B(s)* represents the feedback loop transfer function, *x* is the injection Current and *y* is the Optical output power.

The closed loop transfer function of such system is:

*Neuromorphic Photonics DOI: http://dx.doi.org/10.5772/intechopen.94297*

$$T(s) = \frac{\mathcal{Y}}{\mathcal{X}} = \frac{A(s)}{1 + A(s) \* B(s)}\tag{1}$$

This system can be linearized by making the gain product *A s*ð Þ ∗ *B s*ð ) ≫ 1.

With this condition, the transfer function becomes dependent solely on the feedback gain coefficient and response of the feedback loop, which becomes linear:

$$T(s) = \frac{1}{B(s)}\tag{2}$$

A feedback loop can oscillate if its open-loop gain exceeds unity and simultaneously its open-loop phase shift exceeds π. There are poles that are present in the closed loop configuration with at least one pole of an unstable loop lies in the right half of the s-plane in **Figure 5**.

Analysis of stability of this system can be done according to the Nyquist Criterion [10] by investigating pole location.

It is based on Nyquist plot where the open-loop transfer function is analyzed with a plot of real and imaginary parts. Where stability of the closed-loop system is determined If poles are present in the Left Half (LHP) of the s-Plane.

If poles are present in the Right Hand Plane (RHP), the closed-loop system becomes unstable. In brief, the Nyquist criterion is a method for the determination of the stability of feedback loop systems as a function of an adjustable gain and delay parameters in the feedback section. It simply determines if the system is stable for any specified value of the feedback transfer function *B(s)*.

For the self-pulsation mode, it is well known that an active circuit with feedback can produce self-sustained oscillations only if the criterion formulated by Barkhausen [10] is fulfilled. This criterion is defined as when the denominator of

**Figure 4.** *Negative feedback loop system [8].*

**Figure 5.** *S-plane plot.*

**2.4 Information coding**

*Spiking neuron information encoding scheme of events rather than bits [6].*

**Figure 3.**

*Optoelectronics*

Modulation (PPM) [6, 7].

**3.1 Basic control theory**

function *B(s)*.

**6**

**neuron**

In order to have an efficient signal modulation scheme, a hybrid combination of analog and digital signal representation need to be implemented that mimics the way the brain encodes information as events in time. The type of a well-known modulation scheme used in optical communication systems called Pulse Position

This modulation format exploits the efficiency of analog signals and at the same time reduces the noise accumulation that distorts analog signals (see **Figure 3**).

Semiconductor lasers are widely used in many applications for both digital and analog signal processing. For the past two decades, specifications of these lasers have addressed many specific applications by tailoring the laser design parameters to meet specific performance target. While the aim of this work is to use low cost lasers with generic specifications, modify and enhance their essential performance parameters to behave as a photonic neuron in addition to the applications addressed in [8] by using electronic feedback for triggering self-pulsating behavior necessary for spiking neuron model. The main driver behind this work is to facilitate photonic integration, improving laser modulation bandwidth and increasing laser relaxation oscillation frequency. The performance for all these applications is analyzed in both time and frequency domains. Mainly, by adjusting the feedback loop settings so it can operate outside its stable regime but just ahead of the chaos mode, so the laser can run in self-pulsation mode. This provides the use of the laser drive current as a

The theory behind this work is based on the classical control theory of negative feedback. Recent work by [9], has presented a rigorous, yet simple and intuitive, non-linear analysis method for predicting injection locking in LC oscillators. A system with a negative feedback control loop is shown in **Figure 4**. It consists of a forward-gain element with transfer function *A(s)*, with s is the Laplace operator and can be replaced by *(jω)* feedback element with transfer

Where *A(s)* represents laser transfer function, *B(s)* represents the feedback loop

transfer function, *x* is the injection Current and *y* is the Optical output power.

**3. Semiconductor lasers with optoelectronic feedback as photonic**

single point of control for the pulsation rep-rate and.

The closed loop transfer function of such system is:

the closed loop transfer function is zero. The poles in this self-pulsation mode need to be located outside RHP (Chaos mode) and LHP (Stable mode), only present up and down on the imaginary axis of the s-plane plot with a zero value on the real axis and where the phase of this transfer function:

$$
\Delta T(\text{ jao}) = \mathbf{0} \Rightarrow \mathbf{o} = a\mathbf{o}\_0 \tag{3}
$$

$$|T(j a\_0)| = \mathbf{1} \tag{4}$$

The optical output power from the laser is represented in Eq. (8)

**Symbol Value Dimension Description**

α 3.1 — Line-width enhancement factor P(t) — W Optical power from laser Q 1*:*602*x*10�<sup>19</sup> C Electronic charge η 0.1 — Total quantum efficiency h 6*:*624*x*10�<sup>34</sup> J.s Plank's constant

*ω<sup>n</sup>* 75*:*4*x*109 Rad/s 3 dB Bandwidth of amplifier Circuit

*σ<sup>g</sup>* 2*x*10�<sup>20</sup> *m*<sup>2</sup> Gain cross section

*<sup>f</sup> <sup>r</sup>* <sup>¼</sup> **<sup>1</sup> 2***π*

*H j*ð Þ¼ *<sup>ω</sup> <sup>K</sup>*

The laser transfer function *H* is of the form:

r

shown in **Figure 6**.

*Paramters used in this work [14].*

*Neuromorphic Photonics*

*DOI: http://dx.doi.org/10.5772/intechopen.94297*

**Table 1.**

in the following forms:

**Figure 6.**

**9**

*Laser system with feedback.*

**P t**ðÞ¼ **S t**ð Þ <sup>∗</sup> **Va** <sup>∗</sup> <sup>η</sup>**<sup>0</sup>** <sup>∗</sup> **<sup>h</sup>** <sup>∗</sup> <sup>ν</sup> **2** ∗ Γ ∗ **τ<sup>p</sup>**

The system being analyzed which includes the laser and the feedback loop is

This system consists of a wideband monitor diode located at the back facet of the laser cavity and electrical amplifier. This implementation using the wideband back-facet monitor [15], provides the means to control and manage the short propagation delay in the feedback loop, this is necessary layout in order to achieve the desired performance characteristics. It also provides a mechanically stable system. The key parameters calculated from the model equations are the relaxation oscillation frequency (ROF), and the damping factor. The system is configured to account for the delay, gain and bandwidth of the feedback loop and are expressed

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>K</sup>* � **<sup>1</sup>**

**<sup>2</sup>** *<sup>γ</sup><sup>d</sup>* ð Þ**<sup>2</sup>**

ð Þ *<sup>j</sup><sup>ω</sup>* <sup>∗</sup> *<sup>j</sup><sup>ω</sup>* <sup>þ</sup> *<sup>γ</sup><sup>d</sup>* <sup>½</sup> ð Þ� þ *<sup>K</sup>* (10)

(8)

(9)

Eqs. (3) and (4) are the phase and gain conditions, respectively. Based on Barkhausen criterion, the oscillation frequency is determined by the phase condition (3).

#### **3.2 Self-pulsating system**

There have been numerous publications on the effects of lasers with electronic feedback [11–13], covering mainly the various states from operating this system. The in this work is on increasing the feedback loop delay to achieve self-pulsation and chaos modes.

Starting with the DFB laser characteristics that are modeled using the well-known rate equations [8, 14] that have been modified to include the electronic feedback parameters.

$$\frac{dN(t)}{dt} = \frac{I(t)}{q \ast V\_a} - \mathbf{g}\_0 \frac{[N(t) - N\_0] \ast S(t)}{1 + \varepsilon \ast S(t)} - \frac{N(t)}{\tau\_n} + \left[\frac{o\iota\_n}{2\pi} \ast (\rho \ast S[t-\tau])\right] \tag{5}$$

$$\frac{d\mathbf{S}(t)}{dt} = \Gamma \ast \mathbf{g}\_0 \frac{[\mathbf{N}(t) - \mathbf{N}\_0] \ast \mathbf{S}(t)}{\mathbf{1} + \boldsymbol{\varepsilon} \ast \mathbf{S}(t)} - \frac{\mathbf{S}(t)}{\tau\_p} + \frac{\Gamma \ast \beta}{\tau\_n} \ast \mathbf{N}(t) \tag{6}$$

$$\frac{d\phi(t)}{dt} = \frac{1}{2}a\left[\Gamma \ast \mathbf{g}\_0[N(t) - N\_0] - \frac{1}{\tau\_p}\right] \tag{7}$$

Where Eq. (5), represents the carrier density equation with the feedback terms, **ρ** represents the feedback loop gain, **τ** represents the feedback loop propagation delay and *ω<sup>n</sup>* represents the 3 dB bandwidth of the amplifier circuit. Eq. (6), represents the photon density, and Eq. (7) the optical phase.

The laser parameters included in these equations are listed in **Table 1**.



#### **Table 1.**

the closed loop transfer function is zero. The poles in this self-pulsation mode need to be located outside RHP (Chaos mode) and LHP (Stable mode), only present up and down on the imaginary axis of the s-plane plot with a zero value on the real axis

Eqs. (3) and (4) are the phase and gain conditions, respectively. Based on Barkhausen criterion, the oscillation frequency is determined by the phase

There have been numerous publications on the effects of lasers with electronic feedback [11–13], covering mainly the various states from operating this system. The in this work is on increasing the feedback loop delay to achieve self-pulsation

Starting with the DFB laser characteristics that are modeled using the well-known rate equations [8, 14] that have been modified to include the electronic

<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>* <sup>∗</sup> *S t*ð Þ � *N t*ð Þ

<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>* <sup>∗</sup> *S t*ð Þ � *S t*ð Þ

Where Eq. (5), represents the carrier density equation with the feedback terms, **ρ** represents the feedback loop gain, **τ** represents the feedback loop propagation delay and *ω<sup>n</sup>* represents the 3 dB bandwidth of the amplifier circuit. Eq. (6),

*α* Γ ∗ *g*0½ �� *N t*ðÞ� *N*<sup>0</sup>

½ � *N t*ðÞ� *N*<sup>0</sup> ∗ *S t*ð Þ

The laser parameters included in these equations are listed in **Table 1**.

**Symbol Value Dimension Description** I(t) — A Laser current S(t) — m�<sup>3</sup> Photon density Γ 0.44 — Optical confinement factor

N(t) — m�<sup>3</sup> Carrier density N0 1*:*2*x*1018 cm�<sup>3</sup> Carrier density at transparency ε 3*:*4*x*10�<sup>17</sup> cm<sup>3</sup> Gain saturation parameter *τ<sup>p</sup>* 1*:*0*x*10�<sup>12</sup> s Photon lifetime β 4*:*0*x*10�<sup>4</sup> — Spontaneous emission factor *τ<sup>n</sup>* 3*:*0*x*10�<sup>9</sup> s Carrier lifetime Va 9*:*0*x*10�<sup>11</sup> cm<sup>3</sup> Volume of the active region *Φ* — — Phase of the electric field from the laser

*τn*

� �

<sup>þ</sup> *<sup>ω</sup><sup>n</sup>*

1 *τp*

/s Gain slope

*τp* þ Γ ∗ *β τn*

<sup>2</sup>*<sup>π</sup>* <sup>∗</sup> ð Þ *<sup>ρ</sup>* <sup>∗</sup> *S t*½ � � *<sup>τ</sup>* h i

(5)

(7)

∗ *N t*ð Þ (6)

½ � *N t*ðÞ� *N*<sup>0</sup> ∗ *S t*ð Þ

Δ*T j* ð Þ¼ ω 0 ) ω ¼ *ω*<sup>0</sup> (3)

j*T j*ð *ω*0j ¼ 1 (4)

and where the phase of this transfer function:

condition (3).

*Optoelectronics*

and chaos modes.

*dN t*ð Þ

**8**

feedback parameters.

*dt* <sup>¼</sup> *I t*ð Þ *q* ∗*Va*

� *g*<sup>0</sup>

*dt* <sup>¼</sup> <sup>Γ</sup> <sup>∗</sup> *<sup>g</sup>*<sup>0</sup>

g0 3*x*10�<sup>6</sup> cm�<sup>3</sup>

*d*ϕð Þ*t dt* <sup>¼</sup> <sup>1</sup> 2

represents the photon density, and Eq. (7) the optical phase.

*dS t*ð Þ

**3.2 Self-pulsating system**

*Paramters used in this work [14].*

The optical output power from the laser is represented in Eq. (8)

$$\mathbf{P(t)} = \frac{\mathbf{S(t)} \* \mathbf{V\_a} \* \eta\_0 \* \mathbf{h} \* \nu}{\mathbf{2} \* \Gamma \* \mathbf{r\_p}} \tag{8}$$

The system being analyzed which includes the laser and the feedback loop is shown in **Figure 6**.

This system consists of a wideband monitor diode located at the back facet of the laser cavity and electrical amplifier. This implementation using the wideband back-facet monitor [15], provides the means to control and manage the short propagation delay in the feedback loop, this is necessary layout in order to achieve the desired performance characteristics. It also provides a mechanically stable system. The key parameters calculated from the model equations are the relaxation oscillation frequency (ROF), and the damping factor. The system is configured to account for the delay, gain and bandwidth of the feedback loop and are expressed in the following forms:

$$f\_r = \frac{1}{2\pi} \sqrt{\mathbf{K} - \frac{1}{2} (\mathbf{y}\_d)^2} \tag{9}$$

The laser transfer function *H* is of the form:

$$H(j\omega) = \frac{K}{[(j\omega)\*(j\omega + \chi\_d)]+K} \tag{10}$$

**Figure 6.** *Laser system with feedback.*

Where K is:

$$K = \left[\frac{\Gamma \* \mathbf{g}\_0}{q \* V\_a} (I\_{Bias} - I\_{th})\right] \left[1 - \frac{\Gamma}{q \* V\_a} \* \varepsilon \* \tau\_p (I\_{Bias} - I\_{th})\right] \tag{11}$$

and γd, the damping factor, is of the form:

$$\gamma\_d = \frac{\mathbf{1}}{\tau\_n} + \left[ \frac{\Gamma \ast \mathbf{g}\_0}{q \ast V\_a} \left( \tau\_p + \frac{\mathbf{e}}{\mathbf{g}\_0} \right) ((I\_{Bias} - I\_{th})) \left[ \mathbf{1} - \frac{\Gamma}{q \ast V\_a} \ast \mathbf{e} \ast \tau\_p (I\_{Bias} - I\_{th}) \right] \right] \tag{12}$$

In the feedback loop, the amplifier transfer function A is of the form

$$\mathbf{A} = \frac{-\rho}{\mathbf{1} + \left(\frac{j\omega}{\omega\_{\text{u}}}\right)}\tag{13}$$

with 50 mA bias current was 80 ps with pulse width of 30 ps. These limitations on the pulse width are governed mainly by the laser carrier lifetime in the laser structure. This is a crucial feature for photonic neurons, as the pulse interval can be adjusted by modulating the laser current, where asymmetric spacing is needed

*Time domain plot for self-pulsation case (delay = 50 ps, gain = 0.05) for 50 mA bias current where the pulse*

*Magnitude and phase plots of the laser transfer function with feedback loop in self-pulsation regime for various*

*Pulse interval adjustment as a function of bias current for 50 ps delay and gain of 0.05.*

based on specific events that lead to neuron firing.

*DOI: http://dx.doi.org/10.5772/intechopen.94297*

*Neuromorphic Photonics*

*current values (FB loop gain = 0.05, FB loop delay = 50 ps).*

**Figure 8.**

**Figure 7.**

**Figure 9.**

**11**

*interval is 147 ps.*

Where *ρ* is the feedback gain, and *ω<sup>n</sup>* is the 3 dB bandwidth of the amplifier circuit.

For the delay transfer function *B* is of the form

$$B = \mathfrak{e}^{-j\alpha\pi} \tag{14}$$

Where *τ* is the propagation time delay of the feedback loop system.

Based on the well-known control theory of systems with negative feedback [16], the complete transfer function on this complete laser system *Y* is of the form

$$Y(j\omega) = \frac{H}{1 + (H \ast A \ast B)} \tag{15}$$

Using the parameters listed in **Table 1**, the calculated laser threshold current is 9.4 mA. The slope efficiency is calculated at 0.04 mW/mA.

#### **4. Simulation results and discussion**

Setting up the system to operate in self-pulsating state with fixed FB loop gain of �0.05 and loop delay of 50 ps, the physical phenomenon of self-pulsation process is described as the sharpening and extraction of the first spike of the relaxation oscillation frequency (ROF) of the laser cavity. The feedback sharpens the falling edge of the first spike and suppresses the subsequent spikes.

Hence, lasers with stronger ROF generate shorter pulses. We show the system transfer function (Y(jω)) magnitude and phase plots in **Figure 7**. What we see is in the case where the feedback loop is applied an enhanced second peak in the magnitude transfer function plot of **Figure 7** which indicates the generation of sharp pulsation. The inverse of the frequency peak corresponds to the pulse interval in the time domain.

A capture of the time-domain picture of the self-pulsation mode, is shown in **Figure 8** where the set points are at 50 mA bias current with feedback delay of 50 ps and feedback gain of �0.05. This plot shows the output power of the system where the pulse interval is 147 ps and the pulse width is 50 ps.

**Figure 9** shows the change of the pulse interval (Free Spectral Range) as a function of the bias current. The pulse interval can be fine-tuned over a range > 50 ps. The shortest pulse interval was achieved for these particular laser parameters from **Table 1**. When setting the delay at 30 ps and the gain at �0.05 with 50 mA bias current was 80 ps with pulse width of 30 ps. These limitations on the pulse width are governed mainly by the laser carrier lifetime in the laser structure. This is a crucial feature for photonic neurons, as the pulse interval can be adjusted by modulating the laser current, where asymmetric spacing is needed based on specific events that lead to neuron firing.

**Figure 7.**

Where K is:

*Optoelectronics*

*<sup>γ</sup><sup>d</sup>* <sup>¼</sup> **<sup>1</sup>** *τn* þ

circuit.

time domain.

**10**

*<sup>K</sup>* <sup>¼</sup> <sup>Γ</sup> <sup>∗</sup> *<sup>g</sup>*<sup>0</sup> *q* ∗*Va*

Γ ∗ *g***<sup>0</sup>** *q* ∗ *Va*

and γd, the damping factor, is of the form:

*τ<sup>p</sup>* þ *ε g***0** 

For the delay transfer function *B* is of the form

ð Þ *IBias* � *Ith* 

<sup>1</sup> � <sup>Γ</sup> *q* ∗*Va*

ð � ð Þ *IBias* � *Ith* **<sup>1</sup>** � <sup>Γ</sup>

In the feedback loop, the amplifier transfer function A is of the form

*<sup>A</sup>* <sup>¼</sup> �*<sup>ρ</sup>* **<sup>1</sup>** <sup>þ</sup> *<sup>j</sup><sup>ω</sup> ω<sup>n</sup>*

Where *ρ* is the feedback gain, and *ω<sup>n</sup>* is the 3 dB bandwidth of the amplifier

*B* ¼ *e*

Based on the well-known control theory of systems with negative feedback [16],

Using the parameters listed in **Table 1**, the calculated laser threshold current is

Setting up the system to operate in self-pulsating state with fixed FB loop gain of �0.05 and loop delay of 50 ps, the physical phenomenon of self-pulsation process is described as the sharpening and extraction of the first spike of the relaxation oscillation frequency (ROF) of the laser cavity. The feedback sharpens the falling

Hence, lasers with stronger ROF generate shorter pulses. We show the system transfer function (Y(jω)) magnitude and phase plots in **Figure 7**. What we see is in the case where the feedback loop is applied an enhanced second peak in the magnitude transfer function plot of **Figure 7** which indicates the generation of sharp pulsation. The inverse of the frequency peak corresponds to the pulse interval in the

A capture of the time-domain picture of the self-pulsation mode, is shown in **Figure 8** where the set points are at 50 mA bias current with feedback delay of 50 ps and feedback gain of �0.05. This plot shows the output power of the system where

**Figure 9** shows the change of the pulse interval (Free Spectral Range) as a

range > 50 ps. The shortest pulse interval was achieved for these particular laser parameters from **Table 1**. When setting the delay at 30 ps and the gain at �0.05

function of the bias current. The pulse interval can be fine-tuned over a

*H*

Where *τ* is the propagation time delay of the feedback loop system.

*Y j*ð Þ¼ *ω*

9.4 mA. The slope efficiency is calculated at 0.04 mW/mA.

edge of the first spike and suppresses the subsequent spikes.

the pulse interval is 147 ps and the pulse width is 50 ps.

**4. Simulation results and discussion**

the complete transfer function on this complete laser system *Y* is of the form

∗ *ε* ∗ *τp*ð Þ *IBias* � *Ith*

∗ *ε* ∗ *τp*ð Þ *IBias* � *Ith*

(13)

�*jωτ* (14)

**<sup>1</sup>** <sup>þ</sup> ð Þ *<sup>H</sup>* <sup>∗</sup> *<sup>A</sup>* <sup>∗</sup> *<sup>B</sup>* (15)

(11)

(12)

*q* ∗ *Va*

*Magnitude and phase plots of the laser transfer function with feedback loop in self-pulsation regime for various current values (FB loop gain = 0.05, FB loop delay = 50 ps).*

#### **Figure 8.**

*Time domain plot for self-pulsation case (delay = 50 ps, gain = 0.05) for 50 mA bias current where the pulse interval is 147 ps.*

**Figure 9.** *Pulse interval adjustment as a function of bias current for 50 ps delay and gain of 0.05.*

### **4.1 Pulsed-source noise analysis**

In section (2) of this chapter, we analyzed the rate Eqs. (5)–(7) without the inclusion of the Langevin noise terms *FN*ð Þ*t* , *FS*ð Þ*t and Fφ*ð Þ*t* are the noise terms added respectively to the rate equations. These noise terms are Gaussian random processes with zero mean value under the Markovian assumption (memory-less system) [17]. The Markovian approximation of this correlation function is of the form:

$$
\langle \mathbf{F\_i(t)F\_j(t')} \rangle = \mathbf{2D}\_{\text{ij}} \mathbf{\delta(t-t')} \tag{16}
$$

Where *i,j = S,N, or ϕ.*

*Dij* is the diffusion coefficient with full derivation presented in [17].

The other type of noise effect analyzed is the system phase noise, which has dramatic effects on the performance of pulsed laser sources especially when it comes to timing jitter. The system phase noise *L(f)* is produced from the effect of the laser linewidth *δν* and the power spectral density *Sφ*ð Þ*f* of that linewidth.

$$\mathcal{S}\_{\boldsymbol{\uprho}}(f) = \frac{\mathbf{1}}{\mathbf{1} + \frac{2 \ast f}{\delta \boldsymbol{\uprho}}} \tag{17}$$

For a pulsed source with a pulse interval of 80 ps, the maximum tolerated rms jitter for sampling application is 120th the pulse interval according to [19]. The listed requirements of maximum tolerated rms jitter is 667 fs while our calculated

We also analyzed how certain laser physical design parameters presented can enhance further the performance of this self-pulsating laser structure with feedback for neuromorphic application. Our analysis determined that increasing the laser cavity length can produce a narrower linewidth by increasing the photon lifetime which will enhance further the timing jitter performance, another approach is to use quantum dot based laser structures which can deliver close to zero or negative

Based on the modified rate equations for analyzing DFB laser system with electronic feedback, this work addresses the need for self-pulsating laser behaving as a photonic neuron, this work provides detailed requirements for feedback loop delay, bandwidth, and gain ranges required to operate the laser self-pulsation modes. These effects were simulated numerically and guidelines were generated for the list of recommended parameters necessary to realize such system. The time domain pulse interval which is crucial for neumorphic photonics was also analyzed using only the laser drive current for tuning the pulse interval of 2 ps/mA for the realization of variable spaced pulses necessary for this application with pulse spikes as narrow as 30 ps. We also provided analysis of phase noise and rms jitter. These results also show that a pulse train can be generated and controlled with only the laser bias current without the use of external clocking or signaling sources, while PPM signals can ride on top of the laser current modulation to code signals into the laser output, which now provides to degrees of adjustments, one for the pulse grouping (interval) and one for the information to be transmitted using PPM.

jitter shown in **Figure 11** is around 15 fs.

*rms timing jitter over the entire frequency range.*

linewidth enhancement factor (α parameter).

**5. Conclusion**

**13**

**Figure 11.**

*Neuromorphic Photonics*

*DOI: http://dx.doi.org/10.5772/intechopen.94297*

The system phase noise *L(f)* shown in **Figure 10** is related to the linewidth power spectral density as follow [18]:

$$L(f) = \frac{\mathcal{S}\_{\boldsymbol{\theta}}(f)}{2} \tag{18}$$

The integrated rms timing jitter *σ <sup>j</sup>* which represents the upper bound of the timing jitter of the oscillator shown in **Figure 11** is calculated as follow [19].

$$\sigma\_j = \frac{\text{Pulse Interval}}{2\pi} \sqrt{2 \ast \int\_{f\text{min}}^{f\text{max}} L(f) df} \tag{19}$$

Where *f min* and *f max* are the boundary of the frequency range.

**Figure 10.** *Laser phase noise plot derived from the spectral density of the line-width.*

#### **Figure 11.**

**4.1 Pulsed-source noise analysis**

Where *i,j = S,N, or ϕ.*

power spectral density as follow [18]:

form:

*Optoelectronics*

**Figure 10.**

**12**

In section (2) of this chapter, we analyzed the rate Eqs. (5)–(7) without the inclusion of the Langevin noise terms *FN*ð Þ*t* , *FS*ð Þ*t and Fφ*ð Þ*t* are the noise terms added respectively to the rate equations. These noise terms are Gaussian random processes with zero mean value under the Markovian assumption (memory-less system) [17]. The Markovian approximation of this correlation function is of the

<sup>0</sup> ð Þ � � <sup>¼</sup> **2Dij<sup>δ</sup> <sup>t</sup>** � **<sup>t</sup>**

The other type of noise effect analyzed is the system phase noise, which has dramatic effects on the performance of pulsed laser sources especially when it comes to timing jitter. The system phase noise *L(f)* is produced from the effect of the laser linewidth *δν* and the power spectral density *Sφ*ð Þ*f* of that linewidth.

*<sup>S</sup>φ*ð Þ¼ *<sup>f</sup>* <sup>1</sup>

The system phase noise *L(f)* shown in **Figure 10** is related to the linewidth

*L f*ð Þ¼ *<sup>S</sup>φ*ð Þ*<sup>f</sup>*

The integrated rms timing jitter *σ <sup>j</sup>* which represents the upper bound of the timing jitter of the oscillator shown in **Figure 11** is calculated as follow [19].

> *<sup>σ</sup> <sup>j</sup>* <sup>¼</sup> *Pulse Interval* 2*π*

*Laser phase noise plot derived from the spectral density of the line-width.*

Where *f min* and *f max* are the boundary of the frequency range.

<sup>1</sup> <sup>þ</sup> <sup>2</sup> <sup>∗</sup> *<sup>f</sup> δ*ν

2 ∗

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*L f* ð Þ*df*

ð *fmax fmin*

*Dij* is the diffusion coefficient with full derivation presented in [17].

<sup>0</sup> ð Þ (16)

**<sup>2</sup>** (18)

(17)

(19)

**Fi**ð Þ**t Fj t**

*rms timing jitter over the entire frequency range.*

For a pulsed source with a pulse interval of 80 ps, the maximum tolerated rms jitter for sampling application is 120th the pulse interval according to [19]. The listed requirements of maximum tolerated rms jitter is 667 fs while our calculated jitter shown in **Figure 11** is around 15 fs.

We also analyzed how certain laser physical design parameters presented can enhance further the performance of this self-pulsating laser structure with feedback for neuromorphic application. Our analysis determined that increasing the laser cavity length can produce a narrower linewidth by increasing the photon lifetime which will enhance further the timing jitter performance, another approach is to use quantum dot based laser structures which can deliver close to zero or negative linewidth enhancement factor (α parameter).

### **5. Conclusion**

Based on the modified rate equations for analyzing DFB laser system with electronic feedback, this work addresses the need for self-pulsating laser behaving as a photonic neuron, this work provides detailed requirements for feedback loop delay, bandwidth, and gain ranges required to operate the laser self-pulsation modes. These effects were simulated numerically and guidelines were generated for the list of recommended parameters necessary to realize such system. The time domain pulse interval which is crucial for neumorphic photonics was also analyzed using only the laser drive current for tuning the pulse interval of 2 ps/mA for the realization of variable spaced pulses necessary for this application with pulse spikes as narrow as 30 ps. We also provided analysis of phase noise and rms jitter. These results also show that a pulse train can be generated and controlled with only the laser bias current without the use of external clocking or signaling sources, while PPM signals can ride on top of the laser current modulation to code signals into the laser output, which now provides to degrees of adjustments, one for the pulse grouping (interval) and one for the information to be transmitted using PPM.

*Optoelectronics*
