**5. Programming the optical properties of a network**

Because the underlying physics of RGWNs is based on the interference of local waves, it allows for layouts that are inhomogeneous and non-periodic across the network. Unlike photonic crystals, which are restricted to Bragg wave effects in periodic structures, the flexibility of RGWNs open up design possibilities where the wave properties are varied across the structure. With respect to metamaterials, which could inherently be nonhomogeneous due to the local nature of the interaction between light and the meta-atoms, RGWNs have the advantage of having interference effects within the network, which allows for frequency spectrum reshaping designs through these effects.

An additional unique feature of RGWNs relates to the constraints on wave propagation within the structure. Unlike other photonic designs, RGWNs have a limited number of modes that are allowed to propagate within the structure (e.g. only the TM0 mode for the case of the plasmonic implementation described previously). Furthermore, the waves can propagate only inside the waveguides connecting the splitting elements. The different waveguides are coupled only by X-junctions, which each have only a limited number of terminals. This level of control is beneficial for several reasons. First, the interference pattern in the network can be controlled more directly. Second, it allows for a comprehensive mathematical representation of the RGWN by scattering matrix (S-matrix) formalism that greatly reduces the computational complexity of programming the network. Third, since the waveguides are isolated from each other, their only contribution to the network is to serve as phase retardation elements between the splitting elements. As a result, the waveguide length is the only effective parameter in its contour, as long as the bending is not too severe. This waveguide feature allows for the network to maintain its engineered function even when distorted. Additionally, the ability to utilize curved or bent waveguides to accommodate long contours is useful when designing the interference pattern of RGWNs.

Resonant Guided Wave Networks 53

The network S-matrix is assembled from the mathematical representation of its components according to the network layout. As a first step, a 'function library' of mathematical representations is generated for all the possible network components (i.e., waveguides and X-junctions) using finite difference time domain (FDTD) full wave electromagnetic simulations. Once this library is established, the RGWN S-matrix can be assembled according to the network layout. It is worth pointing out that the S-matrix calculation scheme is almost always found to be much faster than resolving the RGWN behavior from full wave electromagnetic simulations, yet reproduces the same information about the network. This becomes significant for optimization tasks and

To carry out this formalism, the two basic RGWN components (waveguides and Xjunctions) need to first be represented mathematically. The waveguides are mathematically represented by their complex phase retardation, determined by the complex propagation constant of the wave and the waveguide length. The propagation constants are extracted from FDTD simulations for waveguides with various widths at different frequencies. The X-junctions, which are comprised of two intersecting waveguides with four terminals, are mathematically represented by a (4x4) S-matrix. For a given set of waveguide widths, the complex transmission coefficients of the X-junction ports are extracted from FDTD simulations by measuring the amplitude and phase of the wave transmitted to the different ports when excited from one of the terminals at a given

The S-matrix of the 2x2 RGWN is then assembled from the mathematical representation of its constituent components according to the network layout. The phasor representation of the local wave H-fields in the network is represented by three column vectors (transposed

*tr oooo oo oo*

*tr iiiiiiii*

*A aaaaaaaa*

*A aaaaaaaa*

*A aaaaaaaa*

vectors by the network connectivity can then be represented by the system:

0

*tr iiiiiiii*

where *Aout* and *Ain* hold the values of the local input and output waves of the RGWN at its ports, and *Anet* represents the input wave on the X-junctions from the internal terminals of the RGWN. The i/o superscripts denote input/output waves with respect to the X-junction, the number subscripts corresponds to the junction number as defined in Fig. 7a, and the bracketed number subscripts label the ports as defined in Fig. 7c. The coupling of the H-field

*out net in FS RS*

*M KA M A*

*A MA MA*

where *MFS* and *MRS* are diagonal 8-by-8 matrices that originate from the splitting relations in the X-junctions and *K* is a sparse 8-by-8 matrix that stands for the wave propagation in the

*net in RS FS*

1(1) 1(2) 2(3) 2(2) 4(1) 4(4) 3(3) 3(4)

,,,,,,, ;

,,,,,,, ;

(1)

(2)

,,,,,,, ;

1(1) 1(2) 2(3) 2(2) 4(1) 4(4) 3(3) 3(4)

1(3) 1(4) 2(1) 2(4) 4(3) 4(2) 3(1) 3(2)

especially as the network size increases.

*out*

*in*

*net*

waveguides. These matrices are defined as:

wavelength.

for brevity):

These distinctive RGWN characteristics open up new opportunities for designing photonic devices by programming the entire network rather than by assembling interconnected discrete components with traceable functions. The usual way of designing photonic devices is to target the desired subsystem functions, map them logically into sub-functions, and then assembling components that carry out these sub-functions in the desired system. For example, a wavelength router could be designed using add/drop ports where the input and output waveguides are coupled by wavelength sensitive ring resonators (Little 1997) or by defects in a photonic crystal (Fan 1998). Similarly, in free space optics, this function could be achieved through the use of collinear beam splitters, each designed to deflect a desired wavelength band. In these schemes, the couplers and waveguides are discrete components that are associated with a specific function, and are combined in a logical way to carry out the overall system function. An alternative approach is to use a network of components that carries out the desired function but, unlike traditional designs, there is no specific logical sub-function associated with any individual component. While the inner connectivity of the device will be less intuitive, it has the potential to result in more efficient designs of complex and compact devices.

One possible way of representing a system function in a RGWN is through the use of a scattering operator that maps the set of local waves entering the device terminals to the set of the waves exiting from the same terminals (Feigenbaum 2010-2). Since a RGWN is composed of a discrete set of components (waveguides and X-junctions) and terminals, the system function is represented by a scattering matrix (S-matrix) connecting the vectors of the waves entering and emerging from the RGWN via the external ports (see Fig. 7). Designing the system function of the RGWN is then mathematically equivalent to designing the S-matrix to yield a desired output, given a set of inputs.

Fig. 7. Mathematical representation scheme of (a) a 2x2 RGWN system and its components, (b) a waveguide component, and (c) an X-junction component (Feigenbaum 2010-2).

Programming an optical function onto a network, according to the design principle described above, will first be demonstrated for a plasmonic 2x2 RGWN, in which the constituent MIM waveguides are allowed to differ in width, length, and contour. The device has eight terminals, numbered from '1' to '8', as illustrated in Fig. 7a. The input vector lists the complex amplitudes of the magnetic fields (H-fields) entering the network in the eight terminals, and similarly the output vector describes the complex H-field amplitudes of the waves exiting the network through these same terminals.

These distinctive RGWN characteristics open up new opportunities for designing photonic devices by programming the entire network rather than by assembling interconnected discrete components with traceable functions. The usual way of designing photonic devices is to target the desired subsystem functions, map them logically into sub-functions, and then assembling components that carry out these sub-functions in the desired system. For example, a wavelength router could be designed using add/drop ports where the input and output waveguides are coupled by wavelength sensitive ring resonators (Little 1997) or by defects in a photonic crystal (Fan 1998). Similarly, in free space optics, this function could be achieved through the use of collinear beam splitters, each designed to deflect a desired wavelength band. In these schemes, the couplers and waveguides are discrete components that are associated with a specific function, and are combined in a logical way to carry out the overall system function. An alternative approach is to use a network of components that carries out the desired function but, unlike traditional designs, there is no specific logical sub-function associated with any individual component. While the inner connectivity of the device will be less intuitive, it has the potential to result in more efficient designs of complex

One possible way of representing a system function in a RGWN is through the use of a scattering operator that maps the set of local waves entering the device terminals to the set of the waves exiting from the same terminals (Feigenbaum 2010-2). Since a RGWN is composed of a discrete set of components (waveguides and X-junctions) and terminals, the system function is represented by a scattering matrix (S-matrix) connecting the vectors of the waves entering and emerging from the RGWN via the external ports (see Fig. 7). Designing the system function of the RGWN is then mathematically equivalent to designing

Fig. 7. Mathematical representation scheme of (a) a 2x2 RGWN system and its components, (b) a waveguide component, and (c) an X-junction component (Feigenbaum 2010-2).

Programming an optical function onto a network, according to the design principle described above, will first be demonstrated for a plasmonic 2x2 RGWN, in which the constituent MIM waveguides are allowed to differ in width, length, and contour. The device has eight terminals, numbered from '1' to '8', as illustrated in Fig. 7a. The input vector lists the complex amplitudes of the magnetic fields (H-fields) entering the network in the eight terminals, and similarly the output vector describes the complex H-field amplitudes of the

the S-matrix to yield a desired output, given a set of inputs.

waves exiting the network through these same terminals.

and compact devices.

The network S-matrix is assembled from the mathematical representation of its components according to the network layout. As a first step, a 'function library' of mathematical representations is generated for all the possible network components (i.e., waveguides and X-junctions) using finite difference time domain (FDTD) full wave electromagnetic simulations. Once this library is established, the RGWN S-matrix can be assembled according to the network layout. It is worth pointing out that the S-matrix calculation scheme is almost always found to be much faster than resolving the RGWN behavior from full wave electromagnetic simulations, yet reproduces the same information about the network. This becomes significant for optimization tasks and especially as the network size increases.

To carry out this formalism, the two basic RGWN components (waveguides and Xjunctions) need to first be represented mathematically. The waveguides are mathematically represented by their complex phase retardation, determined by the complex propagation constant of the wave and the waveguide length. The propagation constants are extracted from FDTD simulations for waveguides with various widths at different frequencies. The X-junctions, which are comprised of two intersecting waveguides with four terminals, are mathematically represented by a (4x4) S-matrix. For a given set of waveguide widths, the complex transmission coefficients of the X-junction ports are extracted from FDTD simulations by measuring the amplitude and phase of the wave transmitted to the different ports when excited from one of the terminals at a given wavelength.

The S-matrix of the 2x2 RGWN is then assembled from the mathematical representation of its constituent components according to the network layout. The phasor representation of the local wave H-fields in the network is represented by three column vectors (transposed for brevity):

$$\underline{A}\_{out}^{\text{tr}} = \left\langle a\_{1(1)}^{o}, a\_{1(2)}^{o}, a\_{2(3)}^{o}, a\_{2(2)}^{o}, a\_{4(1)}^{o}, a\_{4(4)}^{o}, a\_{3(3)}^{o}, a\_{3(4)}^{o} \right\rangle;$$

$$\underline{A}\_{in}^{\text{tr}} = \left\langle a\_{1(1)}^{i}, a\_{1(2)}^{i}, a\_{2(3)}^{i}, a\_{2(2)}^{i}, a\_{4(1)}^{i}, a\_{4(4)}^{i}, a\_{3(3)}^{i}, a\_{3(4)}^{i} \right\rangle;\tag{1}$$

$$\underline{A}\_{net}^{\text{tr}} = \left\langle a\_{1(3)}^{i}, a\_{1(4)}^{i}, a\_{2(1)}^{i}, a\_{2(4)}^{i}, a\_{4(3)}^{i}, a\_{4(2)}^{i}, a\_{3(1)}^{i}, a\_{3(2)}^{i} \right\rangle;$$

where *Aout* and *Ain* hold the values of the local input and output waves of the RGWN at its ports, and *Anet* represents the input wave on the X-junctions from the internal terminals of the RGWN. The i/o superscripts denote input/output waves with respect to the X-junction, the number subscripts corresponds to the junction number as defined in Fig. 7a, and the bracketed number subscripts label the ports as defined in Fig. 7c. The coupling of the H-field vectors by the network connectivity can then be represented by the system:

$$\begin{cases} \underline{A}\_{out} = \underline{\underline{M}}\_{FS} \underline{A}\_{net} + \underline{\underline{M}}\_{RS} \underline{A}\_{in} \\ \underline{0} = \left( \underline{\underline{M}}\_{RS} - \underline{\underline{K}} \right) \underline{A}\_{net} + \underline{\underline{M}}\_{FS} \underline{A}\_{in} \end{cases} \tag{2}$$

where *MFS* and *MRS* are diagonal 8-by-8 matrices that originate from the splitting relations in the X-junctions and *K* is a sparse 8-by-8 matrix that stands for the wave propagation in the waveguides. These matrices are defined as:

Resonant Guided Wave Networks 55

do not have enough degrees of freedom in this small 2x2 network to exactly attain the desired outputs, we instead optimize the ratio of power going to the two sets of ports at the different

Fig. 8. 2x2 RGWN programmed to function as a dichroic router: (a) schematic drawing, and (b, c) time snapshots of the H-field at the two operation frequencies (Feigenbaum 2010-2).

The optimization procedure is implemented in Matlab using the pre-calculated mathematical representation data set of the RGWN components obtained from full-field electromagnetic

FDTD simulations excited with continuous wave sources (see illustration in Fig. 9).

Fig. 9. Flow chart of the RGWN S-matrix optimization procedure (Feigenbaum 2010-2).

61 52 1 2

 

 

> 

<sup>1</sup> () () 1

function for the dichroic router defined as follows:

 

1 61 51 2 52 62

*O Out Out O Out Out*

( )/ ( ) ( )/ ( )

The dichroic router network is defined by eight parameters: the length and width of the upper, lower, and side waveguides and the two wavelengths. The waveguide widths determine the effective index in the waveguides as well as the transmission coefficients of the X-junctions. The optimization procedure is conducted in Matlab with the optimization

*O O Out Out O O f O O*

where *O1* and *O2* represent the two terminal output ratios that need to be maximized at the two different wavelengths, *λ1* and *λ2*. The function *f* is used to merge the two ratios together

1 2

(5)

1 2

wavelengths.

 1324 1324 3 1 2 1 4 3 4 2 ,,, , ,,, (1,3) (3,1) (2,6) (6,2) (4,8) (8,4) (5,7) (7,5) *FS FS FS FS FS RS RS RS RS RS ii ii VH VH F S R S i i FS ii ii RS VH VH S F S R M Diag S S S S M Diag S S S S tt tt S S tt tt K K K K K K K K K other matrix el* , exp 0 *m <sup>i</sup> i m j L ements* (3)

where the V/H superscript index denotes if the transmission coefficient is for excitation of the vertical or the horizontal waveguide of that X-junction.

Algebra of equation set 3 gives the matrix representation of the 2x2 RGWN S-matrix:

$$\underline{S}^{2x2RGV\mathcal{W}} = \underline{\underline{M}}\_{RS} - \underline{\underline{M}}\_{FS} \left(\underline{\underline{M}}\_{RS} - \mathcal{K}\right)^{-1} \underline{\underline{M}}\_{FS} \tag{4}$$

When validating the field amplitude predictions of the S-matrix representation with FDTD simulations, less than 5% difference is found for various test cases. The two major contributions to this small deviation result from the interpolation between the parameter space points, where the library components were calculated, and from the error added when the waveguides are bent. For cases where no interpolation or waveguide bending occurs, the FDTD results differ by only 1% from the S-matrix predictions. The ability to accurately predict the RGWN interference using S-matrix representation reduces the complicated task of programing a desired optical function into a RGWN into an efficient optimization of its Smatrix.

For example, the RGWN can be programmed by minimizing the difference between the actual network output and the desired one (for a given input), as the network parameter space is swept across the various waveguide widths and lengths. The optimization process then results in a set of network parameters that can be translated to a network layout and then validated with FDTD simulations.

#### **6. Multi-chroic filters using RGWNs**

The S-matrix programming method can be exemplified by designing a 2x2 RGWN to function as a dichroic router (Fig. 8a). Although simple in concept, the exercise of setting a passive device to have different functions at different wavelengths is quite instructive. Explicitly, the required function is to route two different wavelengths (*λ1* and *λ2*) to a different set of ports ('1' and '6' for *λ1* and '2' and '5' for *λ2*) when the two bottom ports ('7' and '8') are simultaneously excited with equal power. Mathematically, we can represent the device as an 8x8 S-matrix **S**(*λ1*, *λ2*) connecting the input and the output vectors. For both wavelengths, the input vector is nonzero for the bottom ports (i.e. **In**=(0,0,0,0,0,0,1,1)) and the desired output vectors would be **Out**(*λ1*)=(1,0,0,0,0,1,0,0) for *λ1* and **Out**(*λ2*)=(0,1,0,0,1,0,0,0) for *λ2*. Because we

*FS FS FS FS FS RS RS RS RS RS*

*M Diag S S S S M Diag S S S S*

*S S*

*K K K K*

*K K other matrix el*

*K K K*

 

 

then validated with FDTD simulations.

**6. Multi-chroic filters using RGWNs** 

matrix.

(1,3) (3,1) (2,6) (6,2) (4,8) (8,4) (5,7) (7,5)

the vertical or the horizontal waveguide of that X-junction.

, exp

*RS FS RS FS*

*<sup>i</sup> i m j L*

 

(4)

(3)

,,, , ,,,

1324 1324

*m*

where the V/H superscript index denotes if the transmission coefficient is for excitation of

*ements*

0

Algebra of equation set 3 gives the matrix representation of the 2x2 RGWN S-matrix:

<sup>1</sup> 2 2*x RGWN*

*S M MM KM*

When validating the field amplitude predictions of the S-matrix representation with FDTD simulations, less than 5% difference is found for various test cases. The two major contributions to this small deviation result from the interpolation between the parameter space points, where the library components were calculated, and from the error added when the waveguides are bent. For cases where no interpolation or waveguide bending occurs, the FDTD results differ by only 1% from the S-matrix predictions. The ability to accurately predict the RGWN interference using S-matrix representation reduces the complicated task of programing a desired optical function into a RGWN into an efficient optimization of its S-

For example, the RGWN can be programmed by minimizing the difference between the actual network output and the desired one (for a given input), as the network parameter space is swept across the various waveguide widths and lengths. The optimization process then results in a set of network parameters that can be translated to a network layout and

The S-matrix programming method can be exemplified by designing a 2x2 RGWN to function as a dichroic router (Fig. 8a). Although simple in concept, the exercise of setting a passive device to have different functions at different wavelengths is quite instructive. Explicitly, the required function is to route two different wavelengths (*λ1* and *λ2*) to a different set of ports ('1' and '6' for *λ1* and '2' and '5' for *λ2*) when the two bottom ports ('7' and '8') are simultaneously excited with equal power. Mathematically, we can represent the device as an 8x8 S-matrix **S**(*λ1*, *λ2*) connecting the input and the output vectors. For both wavelengths, the input vector is nonzero for the bottom ports (i.e. **In**=(0,0,0,0,0,0,1,1)) and the desired output vectors would be **Out**(*λ1*)=(1,0,0,0,0,1,0,0) for *λ1* and **Out**(*λ2*)=(0,1,0,0,1,0,0,0) for *λ2*. Because we

*ii ii VH VH*

 

*tt tt*

*tt tt*

*F S R S i i FS ii ii RS VH VH S F S R*

> 

do not have enough degrees of freedom in this small 2x2 network to exactly attain the desired outputs, we instead optimize the ratio of power going to the two sets of ports at the different wavelengths.

Fig. 8. 2x2 RGWN programmed to function as a dichroic router: (a) schematic drawing, and (b, c) time snapshots of the H-field at the two operation frequencies (Feigenbaum 2010-2).

The optimization procedure is implemented in Matlab using the pre-calculated mathematical representation data set of the RGWN components obtained from full-field electromagnetic FDTD simulations excited with continuous wave sources (see illustration in Fig. 9).

Fig. 9. Flow chart of the RGWN S-matrix optimization procedure (Feigenbaum 2010-2).

The dichroic router network is defined by eight parameters: the length and width of the upper, lower, and side waveguides and the two wavelengths. The waveguide widths determine the effective index in the waveguides as well as the transmission coefficients of the X-junctions. The optimization procedure is conducted in Matlab with the optimization function for the dichroic router defined as follows:

$$\begin{aligned} O\_1 &= \left| Out\_6(\lambda\_1) \right| / \left| Out\_5(\lambda\_1) \right| \\ O\_2 &= \left| Out\_5(\lambda\_2) \right| / \left| Out\_6(\lambda\_2) \right| \\ \frac{1}{f} &= \left[ \sqrt{\left| Out\_6(\lambda\_1) \right| \cdot \left| Out\_5(\lambda\_2) \right|} \right] \cdot \left[ O\_1 \cdot O\_2 \right] \cdot \left[ 1 - \left| \frac{O\_1 - O\_2}{O\_1 + O\_2} \right] \right] \end{aligned} \tag{5}$$

where *O1* and *O2* represent the two terminal output ratios that need to be maximized at the two different wavelengths, *λ1* and *λ2*. The function *f* is used to merge the two ratios together

Resonant Guided Wave Networks 57

is the excitation conditions of the remaining two junction terminals that null the output in terminal '5.' Indeed, the excitation amplitudes of junction '3' obtained from the S-matrix representation are 0.23exp(-j0.21π) and 0.34exp(j0.64π), which are close in amplitude and ~ π phase-shifted. This is consistent with the results from section 3, which show that when an 'X-junction' is simultaneously excited π phase-shifted from two adjacent terminals, the two other terminals will be filtered out (Fig. 3a). The fact that the excitations are not exactly the same in amplitude and π phase-shifted is attributed to the additional constraints the design has on the other wavelength as well as the limitations imposed on the parameter space.

Similarly, the excitation conditions necessary for filtering out terminals '1' and '6' at *λ2* (Fig. 8b) are examined by focusing on the S-matrix amplitudes of X-junction '4.' In this case there are three terminals being excited: the lower terminal of the X-junction (port '7') is given by the network excitation, so the excitation of the other two internal ports will determine the filtering out of port '6.' Intuitively, the condition to filter out terminal '6' will be simply a Π phase-shifted excitation of the upper and lower terminals of junction '4', with zero excitation

cause a residual wave emerging from terminal '6', which could be compensated by a small amplitude excitation at the other side terminal of the junction '4.' Indeed, the excitation amplitudes of junction '4' in the S-matrix representation are 1 in lower terminal,

To further exemplify the programmability of RGWNs via S-matrix formalism, we consider a 3x3 RGWN programmed to function as a trichroic router. In order to implement the more complex task of routing three wavelengths we allow for more degrees of freedom in the network by increasing the number of components, effectively increasing the amount of data contained. The function is defined as an extension of the dichroic router, but here when the three bottom terminals are simultaneously excited at three different frequencies, the frequencies are filtered out to three different sets of side terminals as illustrated in Fig. 10.

Fig. 10. 3x3 RGWN programmed to function as a trichroic router. Time snapshots of the steady state H-field at the three operation frequencies (Feigenbaum 2010-2): a) λ1, b) λ2, c) λ3.

be the usual case for devices relying on material dispersion, such as a glass prism.

It is interesting to note that the wavelengths are not mapped monotonically to the output terminals (i.e. from bottom/top ports as the wavelength increases/decreases), which would

*<sup>1</sup>* we also know that additional constraints might

λ

0.9exp(j0.82π) in upper terminal, and 0.3exp(-j0.32π) in the side terminal.

The analysis results in the optimal RGWN parameters shown in Table 2.

from the side port. From the case of

into one weighted optimization parameter, where the first term in squared brackets maximizes the total power routed into the selected terminals, the second term maximizes the two ratios, and the third term is a weighting factor that ensures that the two ratios are maximized equally. The target function is defined as the inverse of these three terms in multiplication for the minimization Matlab function. At each point in the parameter space, the network output vector is calculated as the multiplication of the 2x2 RGWN S-matrix evaluated at the parameter values times the input vector representing excitation only from the two bottom terminals ('7' and '8').

After defining the optimization function, we constrain the parameter space based on practical considerations. The parameter space includes the width and length of the upper, lower, and side waveguides as well as the two wavelengths of operation (*λ1* and *λ2*). We decrease the number of parameters to optimize by restricting the device to have left-right symmetry based on the desired operation. We restrain the design to operate in the infrared frequency range (λ0 = 1.2-2μm) where the material dispersion and loss are less pronounced than in the visible. Furthermore, the waveguide thickness is constrained to be small enough to only support the lowest order plasmonic mode (air gap widths 100-500nm).

The optimization procedure yields the network parameters given in Table 1, which reveal that the required RGWN for color routing is distributed inhomogeneously.


Table 1. Set of optimized parameters for 2x2 RGWN dichroic router operating at *λ1*=2µm and *λ2*=1.26µm.

When translating the optimized network parameters into the network layout, we learn that the upper waveguide is longer than the lower one, and therefore needs to be bent. Importing the resulting layout into FDTD, we obtain the steady state H-field distribution shown in Fig. 8b and 8c which show time snap shots at the two operation wavelengths. The FDTD simulation results validate the S-matrix design, with *λ1* and *λ2* clearly routed to a different set of sideways ports as illustrated in Fig. 8b and 8c, respectively. From these FDTD results, it is also possible to observe the build-up of local resonance inside the network, which results in the filtering out of the desired output ports. We note that the transmission ('3' and '4') and reflection ('7' and '8') ports from the device are not identically zero since the device does not have enough degrees of freedom and were therefore not included in the optimization function.

The matrix representation can also be used to understand the interference conditions through which the RGWN accomplishes its desired function. From the known input vector and the network S-matrix, the wave complex amplitudes can be identified at any point in the network. For each wavelength, we resolve the excitation conditions of the X-junctions that have the ports that are to be filtered out. For example, for *λ1* to be filtered out from terminals '2' and '5', we examine the excitation conditions in X-junction '3,' which has four terminals. Two of the terminals are external device ports ('4' and '5') and the other two are internal network terminals. There is no input signal incident on the two external ports, so it

into one weighted optimization parameter, where the first term in squared brackets maximizes the total power routed into the selected terminals, the second term maximizes the two ratios, and the third term is a weighting factor that ensures that the two ratios are maximized equally. The target function is defined as the inverse of these three terms in multiplication for the minimization Matlab function. At each point in the parameter space, the network output vector is calculated as the multiplication of the 2x2 RGWN S-matrix evaluated at the parameter values times the input vector representing excitation only from

After defining the optimization function, we constrain the parameter space based on practical considerations. The parameter space includes the width and length of the upper, lower, and side waveguides as well as the two wavelengths of operation (*λ1* and *λ2*). We decrease the number of parameters to optimize by restricting the device to have left-right symmetry based on the desired operation. We restrain the design to operate in the infrared frequency range (λ0 = 1.2-2μm) where the material dispersion and loss are less pronounced than in the visible. Furthermore, the waveguide thickness is constrained to be small enough

The optimization procedure yields the network parameters given in Table 1, which reveal

Waveguides Width (µm) Length (µm)

Table 1. Set of optimized parameters for 2x2 RGWN dichroic router operating at *λ1*=2µm

When translating the optimized network parameters into the network layout, we learn that the upper waveguide is longer than the lower one, and therefore needs to be bent. Importing the resulting layout into FDTD, we obtain the steady state H-field distribution shown in Fig. 8b and 8c which show time snap shots at the two operation wavelengths. The FDTD simulation results validate the S-matrix design, with *λ1* and *λ2* clearly routed to a different set of sideways ports as illustrated in Fig. 8b and 8c, respectively. From these FDTD results, it is also possible to observe the build-up of local resonance inside the network, which results in the filtering out of the desired output ports. We note that the transmission ('3' and '4') and reflection ('7' and '8') ports from the device are not identically zero since the device does not have enough degrees of freedom and were therefore not

The matrix representation can also be used to understand the interference conditions through which the RGWN accomplishes its desired function. From the known input vector and the network S-matrix, the wave complex amplitudes can be identified at any point in the network. For each wavelength, we resolve the excitation conditions of the X-junctions that have the ports that are to be filtered out. For example, for *λ1* to be filtered out from terminals '2' and '5', we examine the excitation conditions in X-junction '3,' which has four terminals. Two of the terminals are external device ports ('4' and '5') and the other two are internal network terminals. There is no input signal incident on the two external ports, so it

to only support the lowest order plasmonic mode (air gap widths 100-500nm).

Lower 0.47 5.4 Side 0.31 1.34 Upper 0.38 6.6

that the required RGWN for color routing is distributed inhomogeneously.

the two bottom terminals ('7' and '8').

included in the optimization function.

and *λ2*=1.26µm.

is the excitation conditions of the remaining two junction terminals that null the output in terminal '5.' Indeed, the excitation amplitudes of junction '3' obtained from the S-matrix representation are 0.23exp(-j0.21π) and 0.34exp(j0.64π), which are close in amplitude and ~ π phase-shifted. This is consistent with the results from section 3, which show that when an 'X-junction' is simultaneously excited π phase-shifted from two adjacent terminals, the two other terminals will be filtered out (Fig. 3a). The fact that the excitations are not exactly the same in amplitude and π phase-shifted is attributed to the additional constraints the design has on the other wavelength as well as the limitations imposed on the parameter space.

Similarly, the excitation conditions necessary for filtering out terminals '1' and '6' at *λ2* (Fig. 8b) are examined by focusing on the S-matrix amplitudes of X-junction '4.' In this case there are three terminals being excited: the lower terminal of the X-junction (port '7') is given by the network excitation, so the excitation of the other two internal ports will determine the filtering out of port '6.' Intuitively, the condition to filter out terminal '6' will be simply a Π phase-shifted excitation of the upper and lower terminals of junction '4', with zero excitation from the side port. From the case of λ*<sup>1</sup>* we also know that additional constraints might cause a residual wave emerging from terminal '6', which could be compensated by a small amplitude excitation at the other side terminal of the junction '4.' Indeed, the excitation amplitudes of junction '4' in the S-matrix representation are 1 in lower terminal, 0.9exp(j0.82π) in upper terminal, and 0.3exp(-j0.32π) in the side terminal.

To further exemplify the programmability of RGWNs via S-matrix formalism, we consider a 3x3 RGWN programmed to function as a trichroic router. In order to implement the more complex task of routing three wavelengths we allow for more degrees of freedom in the network by increasing the number of components, effectively increasing the amount of data contained. The function is defined as an extension of the dichroic router, but here when the three bottom terminals are simultaneously excited at three different frequencies, the frequencies are filtered out to three different sets of side terminals as illustrated in Fig. 10. The analysis results in the optimal RGWN parameters shown in Table 2.

Fig. 10. 3x3 RGWN programmed to function as a trichroic router. Time snapshots of the steady state H-field at the three operation frequencies (Feigenbaum 2010-2): a) λ1, b) λ2, c) λ3.

It is interesting to note that the wavelengths are not mapped monotonically to the output terminals (i.e. from bottom/top ports as the wavelength increases/decreases), which would be the usual case for devices relying on material dispersion, such as a glass prism.

Resonant Guided Wave Networks 59

element of the 2D network is replaced by a six-arm 3D junction element. Using 3D FDTD, we have verified that six-way equal power splitting occurs for pulsed excitation in a coaxial Au-air waveguide junction. Like for the 2D-RGWN, the dispersion of the infinitely large periodic 3D-RGWN is predominantly determined by the network parameters rather than the waveguide dispersion. This is demonstrated by the noticeable difference in the band diagrams (Fig. 5b and 5c) obtained for two networks comprised of the same waveguides but

RGWNs offer a different approach for designing dispersive photonic materials. Whereas photonic crystals rely on the formation of Bloch wave states by interference of waves diffracted from an array of periodic elements, a truly non-local phenomenon; RGWNs rely on the coherent superposition of power flowing along isolated waveguides and splitting at X-junctions. Furthermore, in photonic crystals, the interference pattern of the diffracted waves depends on the nonlocal periodic spatial arrangement of the diffracting elements; and in RGWNs it is the local network topology that determines the dispersion and resonance features. For example, in a RGWN, the coherent wave propagation through the network is determined only by the total path length along the waveguide and the phase shift added at a power splitting event, having no restriction on whether the waveguides are straight or curved. Metamaterials also feature a design approach based on the attributes of localized resonances, but their dispersive properties do not depend on any length scale between resonant elements – thus differing substantially from RGWNs. Arrays of coupled resonator optical waveguides (CROWs) feature discrete identifiable resonators that act as the energy storage elements, and dispersion occurs as modes of adjacent resonators are evanescently coupled. By contrast, in RGWNs, energy is not stored resonantly in discrete resonators, but rather in the network of waveguides that are designed to exhibit a collective

The operation of RGWNs was demonstrated in this chapter using plasmonics, which allowed for a simple layout and broadband range of operation; however, this implementation also brought about substantial attenuation due to the fundamental loss of plasmonic modes. As indicated above, the plasmonic MIM modes used here have typical propagation lengths of about 50 microns due to metal loss. Since the RGWN scope is broader than the field of plasmonics, it calls for an all-dielectric implementation to mitigate the losses brought on by plasmonics. Implementing RGWNs using photonic circuitry would also address the coupling loss associated with the difference in the modal overlaps between

This new design paradigm is based on different underlying physics and thus opens up new directions for the design of artificial optical materials and devices. Since the RGWN design relies on the interference of local waves, we can use these accessible design parameters to program optical functions directly onto the network. Furthermore, the constraints on the propagation and coupling of the local waves in RGWNs allow for the device operation to reduce to a simple mathematical representation using S-matrix formalism. This allows for the network programming to take the form of an optimization procedure over a relatively small parameter space. The RGWN S-matrix representation was demonstrated here where the inputs were given and the S-matrix of device was designed to give a desired output (e.g.,

the plasmonic modes in the RGWN and the interfacing dielectric optics.

with different inter-node spacing.

resonant behaviour.

**8. Conclusions and future directions** 


Table 2. Set of optimized parameters for a 3x3 RGWN trichroic router operating at *λ1*=1.59µm, *λ2*=1.97µm, and *λ3*=1.23µm.
