Modeling the Device

Here we discuss the means for calculating the power triples P0max, P1min, and Pmax at the outputs of a given device, given the values of the triples at each of its inputs. ...

...The power triple for the jth output of a device is computed from the input triples and the coupling terms as follows:

P1min(out)j =
min
s∈states
 { $\displaystyle \begin{array}[t]{c}{\rm min} \\ [-15pt] {\rm inputs}\, i \end{array}$  [P1min(in)i - Lij(s)]}, Lij(s)  ∈  loss,		 (2)
P0max(out)j =
max
s∈states
 $\displaystyle \sum_{{{\rm inputs}\, i}}^{}$$\displaystyle \left\{\vphantom{ \begin{array}{l}P_{\rm max}{\rm (in)}_i - L_{ij... ...ax}({\rm in})_i - L_{ij}(s)\; ,\; L_{ij}(s)\in {\rm loss}, \end{array} }\right.$$\displaystyle \begin{array}{l}P_{\rm max}{\rm (in)}_i - L_{ij}(s)\; ,\; L_{ij}(... ...P_{0\rm max}({\rm in})_i - L_{ij}(s)\; ,\; L_{ij}(s)\in {\rm loss}, \end{array}$		 (3)
Pmax(out)j =
max
s∈states
 $\displaystyle \sum_{{{\rm inputs}\, i}}^{}$Pmax(in)i - Lij(s).		 (4)

Equation ([*]) states that the power of the minimum 1 emerging from the jth output of the device will be the minimum over all possible states of the minimum over all possible inputs having loss terms of the minimum 1's arriving at those inputs minus the loss terms. Equation ([*]) states that the power of the maximum 0 emerging from the jth output of the device will be the maximum over all possible states of the sum of the inputs of ( Pmax minus the cross talk term) for those inputs that have cross talk terms in that state, plus ( P0max minus the loss term) for those inputs that have loss terms in the particular state. Equation ([*]) states that Pmax emerging from the jth output of the device will be the maximum over all the possible states of the sum over all the i inputs of Pmax minus the loss or cross talk between each of those inputs and the output j. These equations are in representational format, as the subtraction of the loss parameters implies logarithmic units for power; so in practice the summations require conversion to linear units. ...

The loss and cross talk terms Lij are the edge weights mentioned above. Note that circuit heuristics are ignored because the extremes are taken over all device states. That is, the power triple is guaranteed to be a bound on the worst case; but in a circuit, the worst case may not be as poor as the triple owing to the exclusion of some combinations of states and inputs. Equation ([*]) shows the most important reason for tracking Pmax: the greatest power produces the largest possible cross talk term in this model. Thus Pmax is essential for calculating subsequent P0max terms.

As example system components, consider lithium niobate directional couplers and passive 3-dB couplers as logic devices and optical fiber and 3-dB splitters for interconnection. Figure 3 shows a lithium niobate directional coupler configured as a five-terminal optical device.[#!6!#] Of the three device inputs, a, b, and c, only the first two are transmitting inputs; inputs that couple power directly to the outputs. Input c, a detection point in our terminology, functions as a device control. As the logic equation shows, when sufficient power is applied to c, the switch is placed in the bar state; otherwise it is in the cross state. The graph model on the right of the figure makes c into a detection point that is independent of the two-state coupling between the other inputs and outputs. Figure 4 illustrates at a more functional level how the transmission coupling occurs.

3-dB couplers and splitters are modeled as devices with two inputs and two outputs, with 3-dB of loss from each input to each output, no cross talk, a single device state, and no detection points. Lossy interconnections, such as optical fibers, are modeled as devices with one input, one output, a single loss term, no cross talk, a single device state, and no detection points. There is no need to model loss-free interconnections, since they add nothing to the analysis. However, if it is desired to model them for clarity, or if a graphical system model already exists that contains them, they may be modeled exactly like lossy interconnections, but with zero loss and cross talk.