An all optical prototype neuron based on optical Kerr material

Optik 126 (2015) 13–18

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Optik journal homepage: www.elsevier.de/ijleo

An all optical prototype neuron based on optical Kerr material Saumabha Bhattacharya a,∗ , Sankar Narayan Patra b , Sourangshu Mukhopadhyay a a b

Department of Physics, The University of Burdwan, Burdwan 713104, West Bengal, India Department of AEIE, The University Institute of Technology, Burdwan 713104, West Bengal, India

a r t i c l e

i n f o

Article history: Received 12 November 2013 Accepted 16 June 2014 Keywords: Nonlinear materials Optical Kerr effect Optical logic operations Neuron Parallelism in optics

a b s t r a c t Optical Kerr material is found popular for developing optical switching actions. In this paper a concept of developing all-optical neuron is first time reported with the exploitation of 3rd order nonlinearity of Kerr type of centro-symmetric nonlinear medium which is used as all-optical switches. This neuron can lead also to the all optical neural architecture. © 2014 Elsevier GmbH. All rights reserved.

1. Introduction Optics is found as a very strong and potential information carrying signal, which can be used with its inherent parallelism in superfast computation, data and signal processing, and also in image processing. Since in the middle of seventies there were many applications where optics was very successfully used for conducting arithmetic, algebraic and logic operations. All-optical parallel computation uses the parallelism of optics to overcome the limitations and restrictions of electronic data processing system. The spatial encoding technique is established as a very successful candidate for implementation of various arithmetic and logic operations. In the last three decades several proposals of all optical systems was reported for processing of digital operations [1–4]. In those works, implementation of all-optical logic family, optical combinational and sequential circuits, optical arithmetic units like adders, subtractor, multipliers, optical control units, optical memory units etc. were described [5–9]. Optical R–S flip-flop, optical J–K flip-flop etc. are also proposed for practical realization, where non-linear properties of some Kerr materials are exploited as far as practicable [10–15]. The present paper offers an all optical technique to develop a prototype neuron which conducts a specific logic operation depending on a preset input logic state. Generally a perceptron computes a linear combination of the inputs (possibly with an

∗ Corresponding author. Tel.: +91 8436396980. E-mail address: [email protected] (S. Bhattacharya). http://dx.doi.org/10.1016/j.ijleo.2014.06.174 0030-4026/© 2014 Elsevier GmbH. All rights reserved.

intercept or bias term) called net input. A basic neuron consists of a nonlinear activation function which is applied to the net input to produce a proper output. In an artificial neural network, a simple single layer of perceptron is designed multiplicatively as a set of highly interconnected processors. Each processing device is called a node. The connection among these nodes is sometimes identical with synaptic connection of biological neuron of brain. The processing ability of the neuron is fired in these synaptic connection or weights among the nodes. This kind of neural processing element is usually called threshold linear unit (TLU) or linear combiner unit (LCU). This unit is shown in Fig. 1. McCulloch and Pitts stimulated the understanding of the computational properties of biological nervous system in their fundamental paper “logical calculus of the idea immanent in nervous activity” [16]. It was readily understood the schemes to use in construction of computational system with elementary useful biological neurons and their interconnections (synapses) [17–20]. Our understanding of a simple model described by Hopfield and Tank has allowed us to plan a simple opto-neural model with the exploitation of third order non-linearity of Kerr type of centro-symmetric non-linear medium [21]. This simple optoneuro model is built with smooth and continuous optical signals introduced in a non linear medium. These weighted signals collectively produce the unit of activation. When this activation exceeds the threshold logic unit (TLU), it produces the output (Fig. 1) [16]. Each neuron is in either of two states at time t: Xj [t] = 1 or “firing” and Xj [t] = 0 or “not firing”; that is the signals are Boolean valued. A neuron Xi becomes active when the sum of those connections ωij coming from neurons j connected to it which are active,

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Z (AND) Y (EX-OR) M

NLM

M

CLS

X (NOR)

O BS2

A

Fig. 1. A schematic diagram of TLU or LCU of neuron.

BS1 B

Fig. 3. Optical non-linear medium (ONLM) as an optical switch.

The neuron with the hard limiter activation function is referred to as the McCulloch–Pitts model [16]. For our present purpose we have considered (x) as a hard limiter. Eq. (2) may be represented more compactly as (X) = Fig. 2. Graphical representation of the hard limiter type function [Xi = 1 when Xj ≥ 0 and Xi = 0 otherwise].

plus a bias bi , is larger than zero, which is followed by Eq. (1) [22,23].

⎡

xi (t) =  ⎣bi +



⎤

ωij (t − 1)⎦

(1)

xεC(i)

where (x) is the step function or hard limiter: Xi = 1 when Xj ≥ 0 and 0 otherwise i.e. the output Xi is then given by thresholding the activation function (hard limiter, shown in Fig. 2) [24].





Xi = 1 ifXj ≥ 0 or0 ifXj < 0

In this context, it should be remembered that conventional activation functions used in neural system are linear function, piecewise function, threshold function-hard limiter or step function, sigmoidal function, tangent hyperbolic function. The activation function is chosen depending on the type of problem. The threshold activation functions are two types; binary type threshold function and bipolar type threshold function. Binary type threshold is named as step function. A threshold (hard-limiter) activation function is either a binary type or a bipolar type respectively. The output of a binary threshold function can be written as:



y = f (u) =

0 if u < 0 1 if u ≥ 0

In the present work we have considered a prototypical biological model of all-optical neuron having inputs onto its dendrite furcation from other neuron and output to other neuron from synapses on its axon. The Kerr nonlinearity is used to implement prototype hard limiter based neuron. The intensity encoded light beams are introduced to the Kerr type NLM. The input channels are A and B (Fig. 3). Here absence of light represents the ‘0’logic state and presence of light represents the ‘1’ logic state. CLS is a constant light source which always gives a light of intensity I to the input channel of NLM. Let there are n inputs with input signals X1 , X2 ,. . .,Xn and weights W1 , W2 ,. . .,Wn (Fig. 1). The signals take only binary values either ‘1’ or ‘0’ only. This is given by (X) = W1 X1 + W2 X2 + · · · + Wn Xn (X) is actually activation function (discussed earlier).

(2)



Wi Xi

(3)

i

The output Y (Fig. 2) is then given by thresholding the activation function (hard limiter). The threshold function (x) is sometimes called a step function or hard-limiter for obvious reasons. If we are to use the analogy with a real neuron, the presence of an action-potential is denoted by ‘1’ and its absence by binary ‘0’ may be implemented by the hard limiter one. The whole operation is guided by a bias potential (b). The use of optical signal in neural network is a challenging idea for data processing [25–28]. Since the middle of seventies there started an effort to implement optical systems replacing the conventional electronic systems [29–31]. This is because of many inherent advantages (as photon is chargeless particle and therefore it does not create any cross-talk problem like electrons and it supports its inherent parallelism during information processing and computing) of optical signal over electronic one. 2. Optical non-linear material as optical switch Isotropic non-linear materials are those where the 1st order nonlinearity is absent, but the 2nd order is present. There are some isotropic non-linear materials (NLMs) (like pure fused silica glass, carbon-bi sulphite etc.) which show Kerr type of nonlinearity. This type of materials can show the character of self-focusing. For this type of material the 2nd order nonlinear term in the expression  has a significant value and cannot be neglected. of polarization (P) Considering up to the 2nd order term and neglecting the higher order terms the expression for polarization becomes [32] P = ε0 (1) E + ε0 (2) E × E + ε0 (3) E × E × E

(4)

For Isotropic materials   P(−E) = −P(E) Hence the refractive index of the isotropic non-linear material is given by

n=

n≈

ε = ε0





1 + (1) + ˛3

1 + (1)



1+



E02 ε0 ˛3

2ε0 1 + (1)



E02

as



˛3 E02

ε0 1 + (1)

 << 1 (5)

S. Bhattacharya et al. / Optik 126 (2015) 13–18

15

Hence, n = n0 + n2 I here n0 =



n2 = ˛3 /2ε0

(6)



1 + (1) is the constant linear refractive index term,



1 + (1)



Fig. 4. The fall of intensity of a beam in n no. of beam splitters.

is the non-linear correction term and

I = E02 is the intensity of the light signal passing through the NLM. Fused silica glass, carbon di sulphide (CS2 ) and many other dielectric materials can be used as a non-linear device. For fused silica n0 = 1.458, n2 = 2.7 × 1020 m2 /W and for CS2 , n0 = 1.63 and n2 = 514 × 1020 m2 /W. For an example the deviation of the light signal passing through CS2 for different intensity level of the light beam can be calculated very easily. For an ordinary CW laser of 100 mW power and 50 ␮m2 beam cross-section the intensity becomes 2 × 109 W/m2 . Now if the above continuous wave (CW) beam is changed to a pulsed beam of pulse duration 10−9 s the pulse power reaches a value of 2 × 1018 W/m2 . This pulsating beam can be obtained by the use of a suitable Q-switching or mode locking mechanism. Considering the average intensity of the pulse I = 2 × 1018 W/m2 the value of ‘n’ becomes 11.91 for CS2 . Now considering the angle of incidence at LM and NLM interface as 45◦ one can find the value of the angle of refraction in the NLM ( 2 ) as 4.529◦ . The above value of  2 can be obtained from Snell’s law. Now if the intensity of light passing through it is made doubled then ‘n’ reaches the value of 22.19 and the value of  2 goes to the value 2.429◦ . So the change of the angle of refraction ( 2 ) goes to the value of 2.1◦ . Here the light signal polarized perpendicular to the plane of paper (䊉) is considered as ‘logic 0’ state and the light beam polarized in the plane of paper () is treated as the state of ‘logic 1’. The different optical logic gates based on polarization encoding [8,9] have been proposed earlier. A simple half wave plate introduces a phase difference of  if the wavelength of the light signal remains constant. Thus when a light signal polarized perpendicular to the plane of paper (䊉) (logic 0), passes through the half-wave plate, it becomes polarized in the plane of paper () (logic 1) and vice-versa. 3. Optical switching by Kerr type of non-linear medium (NLM)

 I i

(8)

2n

The splitting of a light from n different 50:50 beam splitters is shown in Fig. 4. Here input has the intensity I and the output  beam n beam has the same as i I/(2 ). When both the input is at ‘0’, no light passes through the input channels A and B and one gets light in OX direction due to the constant light source. If the output is taken from OX channel we get NOR output, i.e. OX channel gives light only. In the same way for both the input ‘1’ i.e. an intensity of light ‘I’ is present in both the channel A and B one can get the output along OZ direction. If the output is taken from OZ channel one can get AND result of operation. This OZ channel gives light for A = I and B = I logic state and at that time it gives I intensity level. 4. Operation of all optical neurone

The equation of refractive index for some special type of centro symmetric non-linear material, having second order of nonlinearity can be written as n = n0 + n1 I

A and B are the two input channels through which light can be introduced to the NLM. Here absence of light represents the ‘0’logic state and presence of light having a present value of intensity ‘I’ represents the ‘1’ logic state. CLS is a constant light source which always gives a light of intensity I to the input channel of NLM. When a light beam of intensity I passes through a 50–50 beam splitter the intensity of the output light becomes I/2 in any one of the two out coming channels. Now if any one of the two above channels are passed through one more 50–50 beam splitter the intensity of the output light from any one channel of the 2nd beam splitter is 1/2 (I/2) = I/22 . For three 50–50 beam splitters the intensity of the output light in one out coming channel becomes I/23 . When light passes through n number of beam splitters the output intensity of a light beam becomes I/2n . Now if I number of beams are taken where each accesses the system of n number of beam splitters, then the added intensity of light from the last beam splitters becomes

(7)

where n0 is a constant term for refractive index at lower intensity, ni is a non-linear correction term. I is the intensity of light passing through it. From the above equation it can be realized that the refractive index of this special type of non-linear material changes with the intensity of light passing through it. Therefore the output beam path will also change if the intensity of light changes. This character of the non-linear materials can be applied in optical switching elements of all optical circuits. Fig. 3 utilizes the above equation of refractive index. Here CLS carries light of intensity “I” all the time. Now when no light is given in A and B, then the value of ‘n’ is lowest one and the angle of refraction is highest one, i.e. it exits from X channel. In the same way when light is present in CLS and in any of A and B channels, the ‘n’ takes higher value in comparison to the earlier one as the intensity at ‘O’ increases and the light exits through Y channel by reducing the angle of refraction. The third possibility is when CLS, A and B get light and the refractive index becomes highest and therefore according to Snell’s law it exits from Z channel. Realization of NOR, Ex-OR and AND operations using such type of non-linear materials are discussed now with help of an optical system described in Fig. 3.

The schematic diagram of a prototype and primary configuration of an all-optical forward feedback type neuron is given in Fig. 5. A and B are the two input channels through which light is introduced to the circuit. BS1, BS2 and BS3 are the 50–50 beam splitters. CLS is a constant light source of intensity I. Channel C is the preset channel. After passing through the beam splitters light enters to the Kerr type centro-symmetric non-linear material having the second order non-linearity. The photo refracted output light

m5

m4

m3

m2 m1

NLM Mirror 1

BS3

CLS(I) A

BS1 B

0

BS2 Mirror 2

I/4

C

I/2

Fig. 5. A prototype all-optical neurone giving output from m = 3.

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S. Bhattacharya et al. / Optik 126 (2015) 13–18

Table 1 Result of logical operations at different output channels (mi ) for the preset inputs. PRESET

Ch. no m1

Ch. no m2

Ch. no m3

Ch. no m4

Ch. no m5

0 I/4 I/2

NOR No light No light

Ex-OR NOR No light

AND Ex-OR NOR

No light AND Ex-OR

No light No light AND

is obtained from a channel marked by mi (i = 1 to 5). When no light is applied at the preset channel one gets NOR, Ex-OR and AND output at channel no. m1 , m2 and m3 respectively. Similarly while the intensity of light through the preset channel is I/4 one gets NOR, Ex-OR and AND output at channel no. m2 , m3 and m4 , respectively. When the intensity of light through the preset channel is I/2 NOR, Ex-OR and AND outputs are received at channel no. m3 , m4 and m5 , respectively. The result of logical operations from the NLM at different output channels (mi ) for the inputs at the preset channel (C) are given in Table 1. From Table 1 it can be concluded that due to changing the intensity level of the light at the preset channel (0, I/4 and I/2, respectively) the output of AND, Ex-OR and NOR, respectively, is obtained at a specific channel (m3 ). Thus, by changing the intensity level at the preset channel we can get a desired output in channel m3 (among AND, Ex-OR and NOR). 5. Theoretical model of the proposed scheme From the system described in Fig. 5 a generalized formula for expressing the output light intensity in different output channels for different signals of preset channel is proposed. Then derived generalized formula of output intensity (IOP ) IOP =

 I

(2n )

where n = number of beam splitters in the pathway of a light beam. I = intensity of a light beam applied at the input channel (A or B). For example if no light is applied at the preset channel m1 channel receives a light intensity I/23 for a light applied at either A or B (considering no light from CLS). Intensity of a light beam from CLS = I. Let the light intensity at the preset channel = 0, (1) Now for A = 0, B = 0; the light comes only from CLS and the number of beam splitters in the pathway of a light beam from CLS (n) = 2. Hence the output intensity in this case (IOP ) = I/22 = I/4.This output is obtained at m1. (2) For A = 0, B = I; number of beam splitters in the pathway of a light beam from CLS (n) = 2. Number of beam splitters in the pathway of a light beam from input B (n) = 3, Thus IOP = I/22 + I/23 = 3I/8.This output is obtained at m2 . (3) For A = I, B = 0; number of beam splitters in the pathway of a light beam from CLS (n) = 2 and number of beam splitters in the pathway of a light beam from input A (n) = 3, Therefore IOP = I/22 + I/23 = 3I/8.This output is obtained at m2 . (4) For A = I, B = I; number of beam splitters in the pathway of a light beam from CLS (n) = 2; number of beam splitters in the pathway of a light beam from input A and B (n) = 3, Therefore IOP = I/22 + 2I/23 = I/2.This output is obtained at m3 .

pathway of a light beam from preset channel (n) = 1, number of beam splitters in the pathway of a light beam from input B (n) = 3, Therefore IOP = I/22 + I/4 × (1/2) + I/23 = I/2.This output is obtained at m3 . (3) For A = I, B = 0; number of beam splitters in the pathway of a light beam from CLS (n) = 2, number of beam splitters in the pathway of a light beam from preset channel (n) = 1, number of beam splitters in the pathway of a light beam from input A (n) = 3, therefore IOP = I/22 + I/4. (1/2) + I/23 = I/2.This output is obtained at m3 . (4) For A = I, B = I; number of beam splitters in the pathway of a light beam from CLS (n) = 2 number of beam splitters in the pathway of a light beam from preset channel (n) = 1, number of beam splitters in the pathway of a light beam from input A and B (n) = 3, therefore IOP = I/22 + I/4. (1/2) + 2I/23 = 5I/8.This output is obtained at m4 . Let the preset channel intensity = I/2. (1) Now for A = 0, B = 0; number of beam splitters in the pathway of a light beam from CLS (n) = 2, number of beam splitters in the pathway of a light beam from preset channel (n) = 1, therefore, IOP = I/22 + I/2. (1/2) = I/4.This output is obtained at m3. (2) For A = 0, B = I; number of beam splitters in the pathway of a light beam from CLS (n) = 2, number of beam splitters in the pathway of a light beam from preset channel (n) = 1, number of beam splitters in the pathway of a light beam from input B (n) = 3, therefore IOP = = I/22 + I/2 × (1/2) + I/23 = 5I/8.This output is obtained at m4 . (3) For A = I, B = 0; number of beam splitters in the pathway of a light beam from CLS (n) = 2; Number of beam splitters in the pathway of a light beam from preset channel (n) = 1, number of beam splitters in the pathway of a light beam from input A, (n) = 3. Therefore IOP = = I/22 + I/2 × (1/2) + I/23 = 5I/8.This output is obtained at m4. (4) For A = I, B = I; number of beam splitters in the pathway of a light beam from CLS (n) = 2; number of beam splitters in the pathway of a light beam from preset channel (n) = 1, number of beam splitters in the pathway of a light beam from input A and B (n) = 3. Therefore IOP = = I/22 + I/2.(1/2) + 2I/23 = 3I/4.This output is obtained at m5 . Considering Eq. (8) for each beam (CLS, beams at A, B and C (preset) channel) the output intensities at mth channel (out of m1 , m2 , m3 , m4 and m5 ) can be expressed by a general expression IOP =

I A+B P + + 2 4 8

where P is the intensity level at the preset channel (C), A, B are the intensity levels of the two input channels respectively and IOP is the intensity level obtained at mth output channel. I is the light intensity at constant light source. At the same time it can also be derived the number of the output channel through which the optical output is received. The concerned no. of the output channel from which the output is obtained is given by the equation: m=8×

If the light intensity at preset channel = I/4, then (1) For A = 0, B = 0; number of beam splitters in the pathway of a light beam from CLS (n) = 2; number of beam splitters in the pathway of a light beam from preset channel (n) = 1,Therefore, IOP = I/22 + I/4. (1/2) = 3I/8.This output is obtained at m2 . (2) For A = 0, B = I; number of beam splitters in the pathway of a light beam from CLS (n) = 2, number of beam splitters in the

(9)

I  OP

I

−1

(10)

here I is the intensity level of CLS, and also that of the ‘logic 1’ state of the inputs (A, B) and preset channel (P). For the proposed optical circuit m must be an integer one. If the circuit is extended and if the value of ‘m’ is a fractional one then the nearest integer will give the value of ‘m’. In Table 3 it is seen that all the values of ‘m’ are integer, whereas in simulation result shown in Table 6, m values are very much close to integer values. R.H.S of Eq. (10) may take a fractional

S. Bhattacharya et al. / Optik 126 (2015) 13–18 Table 2 Intensity level at mi for different values of A, B and P (Here blank channels give no light). Preset channel

A

B

m1

0

0 0 I I

0 I 0 I

I/4

0 0 I I

0 I 0 I

I/4

I/2

0 0 I I

m2

m3

m4

m5

3I/8 3I/8

17

Table 4 Light intensities and the executing channel number for Ex-OR output. Preset channel

Output

Channel no. (m)

0 I/4 I/2

3I/8 I/2 5I/8

2 3 4

I/2 Table 5 Light intensities and the executing channel number for AND output.

3I/8 I/2 I/2 5I/8

0 I 0 I

I/2 5I/8 5I/8

Preset channel

Output

Channel no. (m)

0 I/4 I/2

I/2 5I/8 3I/4

3 4 5

3I/4

value but it is very close to the integer. Therefore, the proper choice of intensity levels at CLS, A, B and C can give non-fractional value of ‘m’. It can be found from Eqs. (9) and (10) that For P = 0 and i. for A = 0, B = 0, IOP = I/4 and the no. of output channel (m) = 1 i.e. m1 ii. for A = 0, B = I, IOP = 3I/8 and the no. of output channel (m) = 2 i.e. m2 iii. for A = I, B = 0, IOP = 3I/8 and the no. of output channel (m) = 2 i.e. m2 iv. for A = I, B = I, IOP = I/2 and the no. of output channel (m) = 3 i.e. m3

Table 6 The MATLAB simulated result of the prototype and primary optical neuron for the value of I optical unit. Preset channel

A

B

Output

m

0

0 0 I I

0 I 0 I

I/4 3I/8 3I/8 I/2

1 2 2 3

I/4

0 0 I I

0 I 0 I

3I/8 I/2 I/2 5I/8

2 3 3 4

I/2

0 0 I I

0 I 0 I

I/2 5I/8 5I/8 3I/4

3 4 4 5

For P = I/4 and i. for A = 0, B = 0, IOP = 3I/8 and the no. of output channel m = 2 i.e. m2 ii. for A = 0, B = I, IOP = I/2 and the no. of output channel m = 3 i.e. m3 iii. for A = I, B = 0, IOP = I/2 and the no. of output channel m = 3 i.e. m3 iv. for A = I, B = I, IOP = 5I/8 and the no. of output channel m = 4 i.e. m4 For P = I/2 and i. for A = 0, B = 0 IOP = I/2 and the no. of output channel m = 3 i.e. m3 ii. for A = 0, B = I IOP = 5I/8 and the no. of output channel m = 4 i.e. m4 iii. for A = I, B = 0 IOP = 5I/8 and the no. of output channel m = 4 i.e. m4 iv. for A = I, B = I IOP = 3I/4 and the no. of output channel m = 5 i.e. m5

in Fig. 6(a)–(c), which covers the intensity levels of the output only for the preset channel intensity value 0, i.e. no light is given into the preset channel. The simulated result confirm the graphical outcomes neither of NOR, EX-OR and AND operations as given in Fig. 6(a)–(c). We have used MATLAB R2012a to conduct our simulative experiment. We simulated our analysis considering two inputs (A, B) with initial synaptic weights and a fixed bias (i.e. given by preset channel). Final outcome of our simulation (done in MATLAB environment) represents binary patterns. That is after adjustment of the free parameters like synaptic weights and bias. This prototype model shows an output with particular intensity value at a

Considering all the above aspects the level of intensity obtained at different output channels for different input intensities and different preset values of intensities are given at Table 2. The nature of different logical outputs can be summarized as given in the following tables. Table 3 is meant for NOR output, Table 4 for Ex-OR output and Table 5 for AND output. The whole function or the prototype and primary optical neuron is simulated by MATLAB for a prefixed intensity value I optical unit. The simulated result is shown in Table 6. The 3-D graphical presentations of the outputs of the logical operations are shown Table 3 Light intensities and the executing channel number for NOR output. Preset channel

Output

Channel no. (m)

0 I/4 I/2

I/4 3I/8 I/2

1 2 3

Fig. 6. (a) Graphical representation of the output for m = 1 (NOR operation), (b) graphical representation of the output for m = 2 (EX-OR operation), (c) graphical representation of the output for m = 3 (AND operation).

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particular state (i.e. 1) or no output at other state (i.e. 0). The result is depicted in Table 6. 6. Justification of the theory The justification of the tables are described now. The justification of Table 1 is to show the operational results (between A and B) in the output channels (m1 to m5 ) for different values of intensities in the preset channel. The justification of Table 2 is to show the intensity values of the output light at different output channels for different preset values of intensity levels. This table confirms the operation shown in Table 1. Tables 3–5 justify Eq. (10) by giving the channel number for output intensity values of different logical operation. Table 6, justifies the function of the proposed all optical neuron as shown in Fig. 5 by MATLAB simulation results. As a whole the result supports all the tables from 1 to 5. Diagram shown in Fig. 6(a), (b), (c) also justifies the operation of the neuron circuit by supporting the 3-D graphs of the output intensities for NOR,EX-OR,AND operations obtained in channels 1, 2 and 3, respectively. These graphs are also originated by MATLAB simulation. In Table 2 the comparison among the intensity values obtained at different output channels (m1 to m5 ) for different preset values of intensity and the intensity values at A and B channels are given. 7. Conclusion The above schemes described an all-optical neuron with Kerr type non-linear material having strong third order non-linearity. The speed of operation of the whole neuron is tremendously high. It can be expressed as L × n/C, where L is the length of the Kerr material in presence of light intensity, and C is the free space velocity of light. This is less than 10−13 s. in reality. So the whole scheme functions with real time operation. The integrated all optical neurons can also be generated to develop a neural network. It can also be mentioned that by using semiconductor optical amplifier (SOA) and by using Erbium doped fiber amplifier the above scheme can be realized avoiding its practical limitations. The all optical neuron can also be extended for accommodating a large number of logic operations and optical memory cells to get a more generous operation with real time performance. Again it is interesting to note that the output channels are dedicated, i.e. each channel promises to give a desired logical output. The channel number cannot be a fraction. References [1] Raymond Arrathoon, Optical Computing: Digital and Symbolic, M. Dekker, New York, NY, 1989. [2] N. Pahari, D. Das, S. Mukhopadhyay, All-optical method for the addition of binary data by non-linear materials, Appl. Opt. 43 (2004) 6147–6150. [3] S.D. Smith, I. Janossy, H.A. Mackenzie, J.G.E. Reid, M.R. Taghizadeh, F.A.P. Tooley, A.C. Walker, Nonlinear optical circuits, logic gates for optical computers: the first digital optical circuits, Opt. Eng. 24 (1985) 569–573.

[4] G. Li, L. Liu, L. Shao, Y. Yin, J. Hua, Parallel optical negabinary arithmetic based on logic operation, Appl. Opt. 36 (1997) 1011–1016. [5] S. Mukhopadhyay, Optical Computation and Parallel Processing, Classique Books, Kolkata, 2000. [6] J.N. Roy, A.K. Maiti, D. Samanta, S. Mukhopadhyay, Tree-net architecture for integrated all-optical arithmetic operations and data comparison scheme with optical non-linear material, Opt. Switch. Net. 4 (2007) 231–237. [7] D. Samanta, S. Mukhopadhyay, All-optical method for maintaining a fixed intensity level of a light signal in optical computation, Opt. Commun. 281 (2008) 4851–4853. [8] Debajyoti Samanta, Sourangshu Mukhopadhyay, All optical method of developing parity generator and checker with polarization encoded light signal, J. Opt. 41 (2012) 167–172. [9] Debajyoti Samanta, S. Mukhopadhyay, A new scheme of implementing alloptical logic systems exploiting material nonlinearity and polarization based encoding technique, Optoelectron. Lett. (China) 4 (2008) 172–176. [10] Partha Ghosh, S. Mukhopadhyay, Implementation of tristate logic based all optical flip-flop with nonlinear material, Chin. Opt. Lett. 3 (2005) 478–479. [11] Sisir Kumar Garai, S. Mukhopadhyay, Method of implementing frequency encoded multiplexer and demultiplexer systems using nonlinear semiconductor optical amplifier, Opt. Laser Technol. 41 (2009) 972–976. [12] Sisir Kumar Garai, S. Mukhopadhyay, A novel method of developing all-optical frequency encoded memory unit exploiting nonlinear switching character of semiconductor optical amplifier, Opt. Laser Technol. 42 (2010) 1122–1127. [13] Ashish Pal, Sourangshu Mukhopadhyay, An alternative approach of developing a frequency encoded optical tri-state multiplexer with broad area semiconductor optical amplifier (BSOA), Opt. Laser Technol. 44 (2012) 281–284. [14] Nirmalya Pahari, S. Mukhopadhyay, New scheme for image edge detection using the switching mechanism of nonlinear optical material, Opt. Eng. (U.S.A.) 45 (2006) 037003-1–037003-4. [15] Nandita Mitra, S. Mukhopadhyay, A new scheme of an all-optical J–K flip-flop using nonlinear material, J. Opt. 37 (2008) 85–92. [16] Warren S. McCulloch, Walter H. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys. 5 (1943) 115–133. [17] A.O.D. Willows, D.A. Dorsett, G. Hoyle, The neuronal basis of behavior in Tritonia. I. Functional organization of the central nervous system, J. Neurobiol. 4 (1973) 207–237. [18] W.B. Kristan, in: H.M. Pinsker, W.D. Willis (Eds.), Information Processing in the Nervous System, Raven, New York, NY, 1980. [19] B.W. Knight, The Horseshoe Crab Eye: A Little Nervous System That is Solvable, American Mathematical Society, Providence, RI, 1973. [20] B.W. Knight, The relationship between the firing rate of a single neuron and the level of activity in a population of neurons. Experimental evidence for resonant enhancement in the population response, J. Gen. Physiol. 59 (1972) 767–778. [21] J.J. Hopfield, D. Tank, Computing with neural circuits: a model, Science 233 (1986) 625–633 (New series). [22] L. Mikel, Forcada, Neural Networks: Automata and Formal Models of Computation, University of Alacant, Alacant (Spain), 2002, Online link www.dlsi. ua.es/∼mlf/nnafmc/pbook.pdf . [23] Sebastian Seung, Lecture 1 Introduction to Neural Networks, The Massachusetts Institute of Technology, The Seung Lab, 2002, Link: http://hebb. mit.edu/courses/9.641 . [24] J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. U.S.A. 79 (1982) 2554–2558. [25] B.K. Jenkins, C.H. Wang, Model for an incoherent optical neuron that subtracts, Opt. Lett. 13 (1988) 892–894. [26] C.H. Wang, B.K. Jenkins, Subtracting incoherent optical neuron model: analysis, experiment and applications, Appl. Opt. 29 (1990) 2171–2186. [27] E.C. Mos, J.J.L. Hoppenbrouwers, M.T. Hill, Opticalk, Neur. Net. 11 (2000) 988–996. [28] A. Hurtado, I.D. Henning, M.J. Adams, Optical neuron using polarization switching in a 1550 nm-VCSEL, Opt. Express 18 (2010) 25170–25176. [29] S. Mukhopadhyay, A. Basuray, A.K. Dutta, New coding scheme for addition and subtraction using modified signed digit number representation in optical computation, Appl. Opt. (U.S.A.) 27 (1988) 1375–1376. [30] S. Mukhopadhyay, A. Basuray, A.K. Datta, A real-time optical parallel processor for binary addition with a carry, Opt. Commun. (The Netherlands) 66 (1988) 186–190. [31] S. Mukhopadhyay, An optical parallel processor of addition and subtraction of binary data, J. Opt. 18 (1989) 4622–4623. [32] A.N. Matveev, Optics, Mir Publishers, Moscow, 1988.