FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 29 Integrated Optics Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 1
In this section we shall pursue a quantitative investigation of the phenomenon of Evanescent coupling which forms the basic principle of a directional coupler. For the convenience of discussion let us take the help of figure 28.6 of an integrated directional coupler which is shown below: Figure 29.1: Integrated Directional Coupler Let us assume the channel A and B to be exactly identical in all respects and we excite only channel A at z=0. According to the assumed geometry, a wave would now propagate in the positive z direction as a result of excitation by a source. However, as already discussed, the power of the wave will not remain confined only to channel A as the wave propagates. Due to the phenomenon of Evanescent coupling, there would be coupling of power into the channel B as well and if the length of the channels are sufficiently large, then at some z, the entire power launched into A may get coupled to B. Since we assume the channels A and B to be identical, the field distribution of the travelling wave in both the channels would also be exactly identical. If b is the isolated phase constant of each of the two channels in the absence of the other channel, then a forward propagating wave in channel A and channel B can be represented by the following two equations: (29.1) (29.2) Here, a and b are the signals in the two channels. Since we assume only one channel to be present, there would be no coupling of power and the wave propagation in the two channels is governed by the following differential equations: (29.3) Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 2
(29.4) Now, in presence of both the channels, the fields of the waves in the two channels would get coupled. Let us define a quantity κ known as the coupling coefficient which gives a measure of the amount of coupling between the fields in the two channels. In other words, κ is a measure of the amount of overlap between the modal fields in the two waveguides. More the overlap more is the coupling. Thus the quantity κ is governed by an overlap integral which indicates the behaviour of the coupling between the modal fields resulting in transfer of energy from one waveguide to the other. From our earlier discussion on filed distribution in an optical fiber, we can infer that the modal field distribution in the channel waveguide would depend on various parameters such as the width of the channel (d), the separation between the two channels (S), the refractive indices of the two waveguides (n 1 ) and the substrate material (n 2 ) and the wavelength of operation (λ). So, the coupling coefficient can be expressed as a function of all these parameters as shown below: κ ( λ) (29.5) In the presence of coupling, the differential equations 29.3 and 29.4 would now change to: κ (29.6) κ (29.7) The equations 29.6 and 29.7 are the coupled equations between a and b which govern the propagation of energy in the two channels in the presence of coupling. The analysis of these equations is, hence, known as coupled mode analysis. Analytical solutions of the two differential equations above w.r.t. appropriate boundary conditions give the field distribution in the two waveguides in the presence of coupling between the two channels. To solve the above equations, we need to de-couple them which can be done by taking a second derivative followed by a substitution. Following the same procedure for both the equations, we obtain the following decoupled equations: (κ ) (29.8) (κ ) (29.9) As already obvious, the decoupled equations are second order differential equations with constant coefficients and can be easily solved to obtain the following solutions: Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 3
( ) ( κ κ ) (29.10) ( ) ( κ κ ) (29.11) The quantities A 1, A 2, B 1 and B 2 are arbitrary constants that need to be determined with the help of appropriate boundary conditions. The analysis, so far, has been addressed to in a general manner, but for generating appropriate boundary conditions we need to consider some specific conditions. The initial condition is that at z=0, only the channel A has been excited with a unity source and there is no excitation applied to channel B. Therefore: ( ) ( ) (29.12) Applying the initial conditions to the equations 29.10 and 29.11 and solving for the arbitrary constants, we obtain the following solutions: ( ) κ (29.13) ( ) κ (29.14) The above equations suggest that the field propagation in the two channels occur in a sinusoidal fashion where the amplitudes of the waves undergo sinusoidal variations and the phase variation is indicated by the exponential term. If we plot these variations, we obtain the following plot: Figure 29.2: Field Variations in the two channels As seen from the plot of the field variation, at z=0, the entire power lies in the channel A and the amplitude of the field is 1(maximum) at z=0 in Channel A whereas in B it is zero indicating no power in channel B. However, as the fields propagate (as Z increases), the coupling begins and the power in the channel A starts coupling into channel B which is indicated by decreasing amplitude of the field in A and a corresponding increase field amplitude in channel B. As the wave moves further ahead, at a distance z=l c, then entire power that was launched into A gets coupled into B which is indicated by the field amplitude in B becoming maximum whereas in A it goes to zero. As the wave moves beyond L c, the power in B gets re-coupled to the channel A and at a distance 2L c Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 4
the entire power is available in A again. This cycle of exchange of power between the two channels goes on as the wave moves further ahead and this fact is indicated by the sinusoidal field amplitude variations shown in the above figure. The distance L c is, in fact, the point where κ π/. This length L c is called the coupling length of the channels. We can, now, relate the coupling coefficient κ and the coupling length L c as shown below: κ (29.15) This relation shows that the coupling coefficient is inversely proportional to the coupling length of the two channels. Stronger the coupling more is the value of the coupling coefficient and shorter is the coupling length. The important thing to note is the fact that however small the coupling coefficient may be, as long as the fields of the two channels interact, there would always be a length at which the entire power in one channel would get coupled to the other channel. If we take a channel length equal to L c and install a detector at the other end, we would observe that even if energy was launched into channel A initially, yet the detector at A shows no output detected energy whereas the detector at B would detect the entire energy initially launched channel A which has been coupled into B over the coupling length of the channel. The directional coupler is another basic integrated device in WDM networks and a variety of other complex devices are fundamentally based on the directional coupler. For example, with appropriate modifications to the basic design, a directional coupler can be made to act as an optical switch or even a power divider or a router. Let us now look into some of the applications that a directional coupler can be put into. For the convenience of discussion we re-consider figure 28.5 here: Figure 29.3: Directional Coupler Case 1: If L<< L c, In this case, a very small amount of power gets coupled to channel B ( in figure 29.2 this condition signifies a point much closer to 0 along the z axis) and most of the power will remain confined to channel A. That is if coupled power P 1 =a, then P 2 = 1-a for a launched power P=1. This observation is made use in tapping of optical signals. Case 2: If L=L c /2, This condition signifies a operating condition corresponding to point A In figure 29.2. In this case, output power at both the waveguides is equal to half of the total power Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 5
launched into A (3dB power). Thus the directional coupler, now, acts as a 3dB power divider for optical signals. Case 3: If L=L c, In this case Optical power launched into channel A would be available at the output of channel B and power launched into channel B would be available at the output of channel A due to evanescent coupling between the two channels. Thus making the length of the channels equal to the coupling length, the directional coupler may be made to function as a signal cross-over module. If a directional coupler is fabricated on a electro-optic substrate such as LiNbO 3, then the electro-optic effect of the substrate material can be combined with the above directional coupler principle and a new range of complex devices can be constructed. Applying an electric field to such kind of device changes the refractive index of the channel and as a result changes the coupling coefficient of the device and by equation 29.15, the coupling length of the two channels change too. Using this observation, a very basic directional coupler fabricated on an electro-optic substrate can now be dynamically converted onto a different module altogether. For example, application of appropriate electric field can result in directional couple being switched to operate in any one of the three cases mentioned above. Directional Coupler as an Amplitude Modulator A directional coupler fabricated on an electro-optic modulator can be made to function as an amplitude modulator. For our discussion we shall consider the following device: Figure 29.4: Amplitude Modulator Using a Directional Coupler Two channel waveguides A and B, each of length L, are fabricated on a substrate of LiNbO 3 and a provision of application of electric field to these channels have been made as shown in the above figure. Continuous wave (CW) optical signal is provided as an input to channel A. The data signal which is a stream of rectangular pulses, are fed to the electrode. This signal dynamically alters the nature if the channel and the optical outputs of the two channels are detected on the other end as shown. Adopting proper design methodology, the dimensions of the channels are such that when the data signal goes HIGH, the length L becomes equal to 2L c and when the data signal is low L=L c. With this configuration, the outputs at A and B corresponding to the data signal input are shown in the figure below: Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 6
Figure 29.5: Output of Amplitude Modulator of Figure 29.4 As seen from the above figure, when the data signal goes HIGH, the length L=2L c and the output at A corresponds to the input at A and hence it goes HIGH and the output at B goes low due to no power being coupled to B at that L. When the data signal goes LOW, the length L=L c and due to evanescent coupling entire input to A gets coupled to B. Consequently, B goes HIGH and A goes low. Thus, the output at B may be assumed to be the complement of the output at A. Directional coupler as an Optical Multiplexer The next device that a directional coupler on an electro-optic material can be made into is an optical multiplexer which is shown in the figure below: Figure 29.6: Optical Multiplexer Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 7
The presence or absence of the input voltage changes the coupling length of the two channels so as to route the optical input signal either to the output of channel A or channel B thus creating a multiplexing action in the optical domain. Directional Coupler as Cross-Connect The same principles may be used to construct another device known as the crossconnect which, basically, is a 2x2 optical switch which is shown below: Figure 29.7: Cross-Connect using a directional coupler The cross-connect has a similar operating principle as an optical multiplexer discussed above, the only difference being the presence of simultaneous inputs on both channels A and B. When the control signal goes HIGH, the length L of the device becomes equal to L c and the signal power in the input of A gets coupled to B and the signal power in the input of B gets coupled to A. Thus the device is in a cross-connected mode and this state of the device is said to be a cross-state. In the absence (or LOW) of the control signal, the length L=2L c and the input of A is detected at the output of A and the input to B is detected at the output of B. This state of the device is said to be Bar-state. The two states of operation the cross-connect is illustratively shown below: Figure 29.8: CROSS and BAR states of a cross-connect Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 8
2x2 cross-connects may be intelligently connected in layered fashion to construct higher order (nxn) cross-connects. For example, a 3x3 cross-connect can be constructed using 2x2 cross-connects as shown below: Figure 29.9: 3x3 cross-connect using 2x2 cross-connects Depending on the control signals C 1, C 2 and C 3 the 2x2 cross-connects operate in the CROSS (x) and BAR (=) states and the corresponding outputs are produced. The output table corresponding to the different configurations of the control signals is given below: Table 29.1: Output table of 3x3 cross-connect of figure 29.9 Control Signal Configuration Outputs C 1 C 2 C 3 P 2 Q 2 R 2 BAR BAR BAR P 1 Q 1 R 1 BAR BAR CROSS Q 1 P 1 R 1 BAR CROSS BAR P 1 R 1 Q 1 BAR CROSS CROSS R 1 P 1 Q 1 CROSS BAR BAR Q 1 P 1 R 1 CROSS BAR CROSS P 1 Q 1 R 1 CROSS CROSS BAR Q 1 R 1 P 1 CROSS CROSS CROSS R 1 Q 1 P 1 Directional Couplers as Wavelength Dependent Switches Another device which may be constructed based on evanescent coupling, is the wavelength dependent switch. The fundamental principle behind the design of this device is the dependence of the coupling coefficient κ on the wavelength of operation. Consequently, the coupling length is also dependent on the wavelength of operation. The structure of a wavelength dependent switch has been shown in the following figure: Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 9
Figure 29.10: Wavelength Dependent Switch As obvious from the figure, the wavelength dependent switch is, in fact a directional coupler put to a different fundamental use on the basis of the notion of wavelength dependence of the coupling length. Signals of two different optical wavelengths are input to one of the channels of length L. Depending on the value of the particular wavelength (say red) it may so happen that the length L may be an even multiple of the coupling length corresponding to that wavelength and as a result that particular wavelength travels uncoupled through the device. But if the length L is an odd multiple of the coupling length corresponding to that wavelength, the entire energy in that wavelength gets coupled to the other channel and is available at its output. Thus, from number of wavelengths supplied as an input to the channel, the one whose coupling length is such that L is its odd multiple, gets separated out from the pool and is detected separately at the output of the other channel. This operation is similar to the operation of a wavelength selective filter or a wavelength selective dropping switch because it drops one particular wavelength out of many others. This is particularly useful in networks where multiple wavelengths travel together and at a user node such a switch can be connected so that the wavelength corresponding to that user may be dropped at his node and data is thus received by the intended user. The above integrated devices enable light energy to be switched and routed from the source to the destination in the optical domain itself without creating the need for a conversion into the electrical domain. This has resulted in reduction of costs of optical networks and also increased the speed of communication in the optical domain by not having to convert the signal into equivalent electrical signal for routing, switching and other such signal processing purposes. Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 10