8.2. WDM COMPONENTS 353
difference∆Lbetween two neighboring waveguides remains constant from one wave-
guide to next. The phase difference for a signal of wavelengthλ, traveling from the
pth input port to theqth output port through themth waveguide (compared to the path
connecting central ports), can be written as [13]
φpqm=( 2 πm/λ)(n 1 δp+n 2 ∆L+n 1 δq′), (8.2.4)wheren 1 andn 2 are the refractive indices in the regions occupied by the star couplers
and waveguides, respectively. The lengthsδpandδq′depend on the location of the
input and output ports. When the condition
n 1 (δp+δq′)+n 2 ∆L=Qλ (8.2.5)is satisfied for some integerQ, the channel at the wavelengthλacquires phase shifts
that are multiples of 2πwhile passing through different waveguides. As a result, all
fields coming out of theMwaveguides will interfere constructively at theqth port.
Other wavelengths entering from thepth port will be directed to other output ports de-
termined by the condition (8.2.5). Clearly, the device acts as a demultiplexer since a
WDM signal entering from thepth port is distributed to different output ports depend-
ing on the channel wavelengths.
The routing function of a WGR results from the periodicity of the transmission
spectrum. This property is also easily understood from Eq. (8.2.5). The phase condition
for constructive interference can be satisfied for many integer values ofQ. Thus, ifQ
is changed toQ+1, a different wavelength will satisfy Eq. (8.2.5) and will be directed
toward the same port. The frequency difference between these two wavelengths is the
free spectral range (FSR), analogous to that of FP filters. For a WGR, it is given by
FSR=c
n 1 (δp+δq′)+n 2 ∆L. (8.2.6)
Strictly speaking, FSR is not the same for all ports, an undesirable feature from a
practical standpoint. However, whenδpandδq′are designed to be relatively small
compared with∆L, FSR becomes nearly constant for all ports. In that case, a WGR
can be viewed asNdemultiplexers working in parallel with the following property.
If the WDM signal from the first input port is distributed toNoutput ports in the
orderλ 1 ,λ 2 ,...,λN, the WDM signal from the second input port will be distributed as
λN,λ 1 ,...,λN− 1 , and the same cyclic pattern is followed for other input ports.
The optimization of a WGR requires precise control of many design parameters
for reducing the crosstalk and maximizing the coupling efficiency. Despite the com-
plexity of the design, WGRs are routinely fabricated in the form of a compact com-
mercial device (each dimension∼1 cm) using either silica-on-silicon technology or
InGaAsP/InP technology [74]–[81]. WGRs with 128 input and output ports were avail-
able by 1996 in the form of a planar lightwave circuit and were able to operate on WDM
signals with a channel spacing as small as 0.2 nm while maintaining crosstalk below
16 dB. WGRs with 256 input and output ports have been fabricated using this tech-
nology [102]. WGRs can also be used for applications other than wavelength routing
such as multichannel transmitters and receivers (discussed later in this section), tunable
add–drop optical filters, and add–drop multiplexers.