METHOD OF DEVELOPING ALL OPTICAL HALF-ADDER BASED ON NONLINEAR DIRECTIONAL COUPLER

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1 Optics and Photonics Letters Vol. 6, No. (203) (0 pages) c World Scientific Publishing Company DOI: 0.42/S METHOD OF DEVELOPING ALL OPTICAL HALF-ADDER BASED ON NONLINEAR DIRECTIONAL COUPLER PRASANTA MANDAL Department of Physics & Technophysics Vidyasagar University Midnapur-7202, West Bengal, India prasanta_mandal2004@yahoo.co.in Received 27 July 202 Revised 3 October 202 Accepted 2 January 203 Published 8 February 203 The transfer of light energy from one wave guide to another due to direct coupling mechanism can be used as a directional coupler switch. Several all optical switching operations have already been proposed by scientists and technologists. In this paper we report the advantageous aspects of nonlinear coupling mechanisms for developing all optical half-adder systems. The basic principle behind this optical switch is the reduction of coupling length due to the nonlinear interaction of the wave guide with the intensity. Keywords: Nonlinear medium; all optical switch; directional coupler.. Introduction Today most computing devices largely depend on the electronic switch. In case of optoelectronic devices optical signals are converted to electronic ones. They are then amplified and regenerated. After a switching operation, those signals are again converted to optical signals. This type of switching is called \optical-to-electronic-to-optical" (OEO) conversion. There are some major limitations to this switching technique. Repeated conversion of optical to electronic or vice versa makes the switching action slow compared to purely optical data processing systems. The main benefit of all-optical switching technology is that one can use it in optoelectronic circuits by replacing the existing electronic switches. Therefore, OEO conversion is not required at all. Switching costs will be cheaper compared to its equivalent electronic counterparts. Moreover, one can expect tremendous operational speed (far above GHz) compared to electronic equivalent switching. 5 Parallel operation can also be realized in the case of all optical switching. 6 8 There have been numerous proposals on the development of optical switching systems and switching-based devices, but the applications are still at the primary stage. Many scientists

2 P. Mandal have already developed different logic operations by optical nonlinear material-based switches and also all-optical methods for conduction of arithmetic operation of Binary data. 9 7 In this paper the authors report a method of optical directional coupler half-adder based on optical switching techniques. The main feature behind this type of switching is the periodic exchange of energy from one optical fiber to another due to coupling of radiation by emergent waves. The significant advantage of the system is that one can conduct an addition between two bits by using only a coupler. No conventional optical or opto-electrical switch is used to do the job and hence no time delay will affect the system. Hence one can realize a real time addition between the two bits. 2. Optical Directional Coupler Utilizing the Nonlinearity of an Optical Fiber In the case of a nonlinear optical switch, the optical nonlinear material plays an important role for intensity dependent switching action. 2,5 The refractive index for some specific types of well-known isotropic nonlinear materials is written as n NL ¼ n o þ n 2 I where n o is the constant refractive index at low level intensity, n 2 is the third order nonlinear correction factors of the isotropic nonlinear medium and I ðtþ is the electric field intensity of the light pulse passing through the material. Again in the case of a nonlinear material-based optical fiber the intensity I p t, where, p t is the power carried by a mode in an optical fiber and is the effective area of the fiber core. Hence, n ¼ n 0 þ n 2 p t þ n 3 So, the final propagation constant for the wave in the media may approximately written as p 2 t A 2 eff : ðþ ¼ 0 þ k 0 n 2 p t þ k 0 n 3 p 2 t A 2 eff ; ð2þ where 0 is the propagation constant for linear case, k 0 is the free space wave number and n 2 and n 3 are the third and fifth order nonlinear correction terms respectively. Thus it can be seen that the propagation constant ðþ depends on the intensity of the input pulse. So in the case of a directional coupler made of two wave guides of the same type the propagation constants are not equal ð 6¼ 2 Þ if the intensities of the light passing through the fibers are different. An optical directional coupler is an optical device where the two wave guides are at close proximity over a length l [Fig. ]. The evanescent fields associated with the propagating modes in the two wave guides mutually interact among themselves and lead to a periodic exchange of energy between the two wave guides. 2,5 The modal field of the individual wave

3 Method of Developing All Optical Half Adder Based on Nonlinear Directional Coupler Input Fibre Output of Fibre ( I / ) L c Input Fibre 2 Fig.. A simple optical directional coupler. guide varies with z exponentially and if the amplitude of a mode at z in wave guide is denoted as aðzþ then daðzþ ¼ i dz aðzþ where signifies the propagation constant of the concerned mode in wave guide. In the same way bðzþ represents the amplitude at z of a mode in wave guide 2 with propagation constant 2. So, db dz ¼ i 2b: When the two wave guides are close together, the modes in the two wave guides must interact through the evanescent fields. Hence, in the presence of such an interaction, the variation of amplitudes of the modes in the two wave guides can be expressed as da dz ¼ ið þ k Þa ik 2 b; ð3þ db dz ¼ ið 2 þ k 22 Þb ik 2 a; ð4þ where k and k 22 represent the nonlinear correction terms to the propagation constants of each individual wave guide. Similarly k 2 and k 2 are the coupling constants from the first fiber to the second and from second to the first respectively. In order to solve the set of Eqs. (3) and (4) we can write aðzþ ¼a 0 expð izþ; bðzþ ¼b 0 expð izþ: Output of Fibre 2.( I / 2) ð5þ ð6þ

4 P. Mandal We consider here the existence of a wave in the system consisting of the two coupled wave guides, propagating with a phaseðþ, which is a superposition of the modes of wave guides and 2 with amplitudes a 0 and b 0. Using Eqs. (5) and (6) in Eqs. (3) and (4), we get aðzþ ¼a s expð i s zþþa a expð i a zþ ð7þ and bðzþ ¼ s k k 2 a s expð i s zþþ a k k 2 a a expð i a zþ: Here s and a represent the symmetric and anti-symmetric mode. In a coupled wave guide one has two independent set of modes, one propagating with propagation constant s and the other with a. Now we assume that at z ¼ 0, the mode in wave guide is launched with unit power and that there is no power in wave guide 2. Then we have the initial conditions at z ¼ 0 a s þ a a ¼ ð9þ and s k k 2 a s þ a k k 2 k 2 4 ð þ k Þ 2 þ k 2 sin2 a a ¼ 0: Power in wave guides and 2 is proportional to jaðzþj 2 and jbðzþj 2 respectively. Substituting from Eq. (9) into Eqs. (7), (8) it can be seen that 5 jaðzþj 2 k 2 =2 ¼ 4 ð þ k Þ 2 þ k 2 sin2 4 ð þ k Þ 2 þ k 2 z ; ðþ =2 jbðzþj 2 ¼ 4 ð þ k Þ 2 þ k 2 z : ð2þ where ¼ð 2 Þ; k ¼ðk k 22 Þ; and k ¼½=4k 2 k 2 Š =2 is the coupling constant: There is a periodic exchange of energy between the two wave guides with a period h ¼ ½ 4 ð þ k Þ 2 þ k 2 Š : =2 Therefore, the nonlinear coupling length of the directional coupler is L nc ¼ h 2 ¼ 2½ 4 ð þ k Þ 2 þ k 2 Š : =2 In absence of nonlinearity one can conclude that k ¼ k 22,ork ¼ 0 and ¼ 0. Hence, the linear coupling length is ð8þ ð0þ ð3þ L c ¼ 2k : ð4þ

5 Method of Developing All Optical Half Adder Based on Nonlinear Directional Coupler This is the conventional result of the coupling length in the directional coupler, when two wave guides are same and their nonlinearities are not considered at all. So, the change of length of the directional coupler from the conventional result is given by, L ¼ L nc L c ¼ 2½ 4 ð þ k Þ 2 þ k 2 Š =2 2k ¼ 2k 8k ð þ k Þ 2 : ð5þ 2 From equation (2) we see that the propagation constant for the wave in the media may be approximately written as p ¼ 0 þ k 0 n t 20 þ where 0 is the propagation constant for linear case and n 2 is the third order nonlinear correction term respectively. We may consider now as the propagation constant in one fiber and 2 as that in the other. Then ¼ 0 þ k 0 n 20 p t ; p 2 ¼ 0 þ k 0 n 2t 20 ; where p t and p 2t are the power terms in wave guide and 2 respectively. If input intensity of the second fiber is zero, then 2 ¼ 0. Then ¼ 2 ¼ k 0n 20 p t : ð6þ Putting the values of in Eq. (4) we get L nc ¼ 2 k 0 n 20 p : 2 =2 t þ k 4 A þ k 2 eff Again we may neglect k, as there is no input of the second fiber. So the equation becomes L nc ¼ 2 =2 : ð7þ 4 ðk 0n 20 I Þ 2 þ k 2 3. Estimation of Switching Energy and Efficiency of the Logic Operation In this method two laser beams are required. An ordinary continuous wave laser of 3.5 mw power and beam cross-section of 50 m 2 has the intensity W=m 2. If the above continuous wave beam is changed to a pulse beam of pulse duration 0 8 s, the pulse power

6 P. Mandal reaches a value of W=m 2. This pulsating beam can be obtained by the use of a suitable Q switching or mode locking technique. Considering the value of the coupling constant k ¼ 0:52354 mm, the coupling length L c becomes 3 mm for an ordinary fiber. For fused silica, n 0 ¼ :485 and n 2 ¼ 3: m 2 =W. If we consider the intensity of the pulse as I ¼ W=m 2, of wavelength.55 m, then the coupling length becomes 2.26 mm. If I ¼ W=m 2, the coupling length reaches.49 mm. If P ð0þ is the power launched into fiber at z ¼ 0, then at any value of z the powers propagating in the two fibers are given by P ðzþ P ð0þ ¼ K 2 2 sin2 z; P 2 ðzþ P ð0þ ¼ K 2 2 sin2 z; where 2 ¼ K 2 þ 4 ð þ kþ2 and ¼ 2, k ¼ k k 22. The maximum fractional power transfer from the input fiber to the coupler fiber is given by max ¼ P 2 max P ð0þ ¼ K 2 2 sin2 ¼ K 2 max ¼ 2 þð : ð8þ 2K Þ2 Putting the values of from Eq. (2) into Eq. (8), we get max ¼ þð k on 20 I : 2K Þ2 Putting the value of the coupling constant K ¼ 0:52354 mm, third order nonlinear correction term n 20 ¼ 3: m 2 =W, input power P ð0þ ¼3:5 mw (i.e. I ¼ W=m 2 ) and 0 ¼ :55 m, we get max ¼ 53. Therefore, the maximum fractional power transferred is 53%. When p ð0þ ¼7 mw (i.e I ¼ W=m 2 ), then max ¼ 36%. Hence we conclude that in this analytical treatment we should use a directional coupler with linear coupling length equal to 3 mm. When one input is present (3.5 mw) then the coupling changes to 0.74 mm (3 mm 2.26 mm), which is very small. So due to periodic exchange of light we get output at fiber 2 only (sum). The maximum fractional power transfer from the input fiber (fiber or fiber 3) to the coupled fiber (fiber 2) is 53%. When both the inputs are present then the coupling length changes to one half of its initial value (i.e.49 mm), so we get light output at fiber and fiber 3 only. Then maximum power transfer to the fiber 2 is 36%, due to periodic exchange of light power back to the fiber is 63%. 4. Construction of Half-adder Switch In the above discussion, it is clearly seen from Eq. (8) that the coupling length between the two wave guides changes as required by changing k and. This approach of reduction of coupling length may be used for developing an optical directional coupler based all optical

7 Method of Developing All Optical Half Adder Based on Nonlinear Directional Coupler 3.0 B Coupling Length L c mm Fig Power P Watt (0 9 ) Reduction of coupling length with intensity of the input light pulse. switch. Let two identical wave guides be used to make a directional coupler of length equal to L c. A major energy of incident wave at the input of wave guide comes out of the second wave guide at the exit. If we decrease the coupling length to L c =2 by controlling the intensity of the input light (i.e. changing k and ) then at the exit the energy will again emerge from the first wave guide. This can be achieved easily by proper choice of parameter of the wave guide and peak power of the input light pulse (from Eq. (2), propagation constant changes with peak power carried by an optical fiber). Figure 2 shows the variation of coupling length L c as a function of intensity. This is the basic principle behind the development of an optical directional coupler based optical switch. Now we propose the development of an optical half-adder based on this switching operation. To do this we first have to discuss the digital half-adder and then the optical half-adder. Figure 3 shows the digital half-adder. It has two inputs A and B that represent the bits to be added. A ¼ B ¼ 0 means there is no voltage at the input. So no voltage will come at the output. When A ¼ B ¼ at the input, some voltage is available in both the inputs. The A B EX-OR logic Sum (D) AND logic Carry (E) Fig. 3. A block diagram of a digital half-adder

8 P. Mandal figure consists of an EX-OR and an AND gate. The output of EX-OR gives the sum (D) of A and B. The output of the AND gate gives carry (E) bit. The truth table of the digital halfadder is given in Table. The truth Table shows that there is output voltage at D only when there is a voltage at either the A or B inputs. No voltage will be seen at the output D if there is no voltage in both the inputs A and B or voltage appears in both the inputs. The output of the carry (E) gives voltage if both the inputs received voltage. Now we consider three optical fibers coupled together (Fig. 4) (fiber, fiber 2, fiber 3). If light intensity is given only in fiber or fiber 3, the coupling length between the first and second fiber or the second and third fiber is L 0c. Now if input light is incident in fibers and 3, and no signal is used through the second fiber, output light exits at fibers and 2 or at fibers 3 and 2. The input light of fibers and 3 may be treated as inputs A and B. When A ¼ B ¼ 0, i.e. both the intensities are absent, no light will come at the output. When A ¼ B ¼, light intensity is available in both the inputs. The output of fiber 2 is called SUM (S) and the output of fiber and fiber 3 together is called CARRY (C). If the inputs and outputs are coded in this way then the truth table (shown in Table 2) for the half-adder operation can be realized. When light intensity (I ) is given at the input of fiber or fiber 3 (i.e. either A ¼, B ¼ 0 or A ¼ 0, B ¼ ) and no light is given in the input of fiber 2, due to periodic exchange of energy we can get a light signal at the outlet of fiber 2 (as coupling length L c arises due to this intensity). Fiber Fiber 2 Fiber 3 L c 2 Output of fiber Output of fiber 3 Output of fiber 2 (Sum) L c Output of fiber Output of the combined fibers and 3 (Carry output) Output of fiber 3 Fig. 4. Three optical directional couplers are coupled together with nonlinear wave guides (fiber, fiber 2, and fiber 3). When light is initially introduced in fiber or fiber 3 only but not in fiber 2, the coupling length is L c. Then we get output at fiber 2 only due to periodic exchange of light signal from fiber to 2 or fiber 3 to 2. On the other hand when light is initially introduced in fibers and 3 together but not in fiber 2, the coupling length is L c =2. We get output at fibers or 3 only because coupling length decreases and light goes back to the first fiber and to the third fiber. Output of fiber and fiber 3 are combined to give the Carry output while that of fiber 2 gives the Sum output

9 Method of Developing All Optical Half Adder Based on Nonlinear Directional Coupler Table. Truth table of digital half-adder logic operation. Here 0 (zero) means no voltage. means some voltage is available. INPUT OUTPUT A B Sum D Carry E Table 2. Truth table of optical half-adder logic operation. 0 (zero) means no light, means light is present there. INPUT OUTPUT A B Sum Y Carry C If light intensities are sent through first and third fiber, (i.e. A ¼ B ¼ ) due to coupling of high intensity light the refractive index of the second fiber is increased such that the coupling length decreases to L c =2. As the length is already set at L c the energy of the middle fiber will emerge from the first and third fiber. Hence, we get light output () at fiber or fiber 3. No light is in the second fiber due to periodic exchange of energy. Hence, the halfadder optical directional coupler switch is achieved. The truth table can be easily explained if fibers and 3 are combined together to get the Carry output of the half-adder and the output of the second fiber are kept separated. Ultimately to discuss the principle of operation, it can be said that we get light at the output of fiber 2 (sum output) only when there is some light either at input fiber or input fiber 3. No light will be seen at the output of fiber 2 if no light appears at both fibers and 3 or light appears in both of them. Similarly at carry output (combination of fibers and 2) light will come only when both the fibers (fibers and 2) receive light at their input. The coupling is done in such a way that the system will not encourage any back reflection. All the transitions of light in these three fibers are in the forward direction. The back reflection should be avoided strictly, otherwise some optical noise may be generated which may prevent the desired result. 5. Conclusion It is thus seen that if the coupling length of three nonlinear fibers are suitably adjusted and if the light intensity at the input side is properly selected for generating the data bit or 0,

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