The transformation to simplify the circuit based on OSRR (Figs. 7b, 12 c and 12 d) is:

$$L\_{\rm s}^{\prime} = L\_{\rm s} + 2L$$

and the transformations to simplify the circuit based on OCSRR (Figs. 7a, 12 a and 12 b) are:

$$\mathbf{C}'\_{p} = \mathbf{2C}\_{p} + \mathbf{2C} \tag{17}$$

$$L\_p' = \frac{L\_p}{2} \tag{18}$$

#### **4. Parameter extraction technique for metamaterial transmission lines based on the resonant approach**

In previous sections we have presented different metamaterial transmission lines loaded with resonators and their equivalent circuit models. In this section we present a parameter extraction technique which, properly modified, can be applied to all of them. With this technique we can determinate the parameters of the circuit model of the structure. This represents an important assist in the design process, which becomes easier and faster.

The parameter extraction method consists in the imposition of several conditions obtained either from the simulated or measured response of the considered structure. The number of imposed conditions must be enough to obtain the values of all the parameters of the circuit.

#### **4.1 Parameter extraction technique metamaterial transmission lines based on Split-Ring Resonators (SRRs)**

The parameter extraction for the metamaterial transmission line based on SRRs is focused on the simplified equivalent circuit of the Fig. 8(b). This method was proposed in the reference (Aznar et al., 2008e). Since the number of parameters of the circuit model is five, we also need five conditions to univocally determine all parameters. From the representation of the reflection coefficient of a single unit cell, *S*11, in the Smith chart, two

Characterization of Metamaterial

circuit parameters of the circuit model in Fig. 8(b).

left handed cell based on a CPW structure.

expressions:

**Complementary Split-Ring Resonators (CSRRs)**

Transmission Lines with Coupled Resonators Through Parameter Extraction 315

reflection coefficient on a Smith chart and read the susceptance seen from the ports at that frequency where *S*11 intercepts the unit conductance circle. Since this is simply the susceptance corresponding the line capacitance (provided *Lp* has been removed), we can thus univocally determine *C*. Hence, this is the fifth condition that is required to extract the

Fig. 13. Reflection coefficient on the Smith chart (a); frequency response (reflection, *S11*, and transmission, *S21*, coefficients) depicted in a decibel scale and the dispersion relation (b) for a

The technique is based on the equivalent circuit model of a CSRR-loaded transmission line shown in Fig. 10(b). This method was proposed in 2006 (Bonache et al., 2006c). The considered structures are a negative permittivity as well as a left handed microstrip line (gap capacitors are required in the latter case). In view of the models, if losses are neglected (this is reasonable in a first order approximation), two characteristic frequencies can be identified: the frequency that nulls the shunt impedance (transmission zero frequency, *fz*) and the frequency that nulls the shunt admittance (which obviously coincides with the intrinsic resonance frequency of the CSRR, *f0*). These frequencies are given by the following

( )

(24)

*L C* = (25)

*C C*

1

1 2 *C C*

π

and they can be either experimentally determined, or obtained from the simulated response of the structure. At *fz* a notch in the transmission coefficient is expected and this frequency can be accurately measured. To obtain *f0*, a representation of the transmission coefficient on a Smith chart is required (see example in Fig. 14). At this frequency the shunt path to ground

2 *Z*

0

*f*

π*LCC* = +

*f*

**4.2 Parameter extraction technique metamaterial transmission lines based on**

conditions are obtained. Firstly, we can determine the frequency that nulls the series reactance, *fs*, from the intercept of *S*11 with the unit conductance circle (see Fig. 13 a). This is obvious since at this frequency, the real part of the admittance seen from the ports is simply the admittance of the opposite port, that is, *Y0*=(*Z0)*−1=(50Ω)−1=0.02 S. Hence, *S*11 must be allocated in the unit conductance circle at *fs*, as illustrated in the example provided in Fig. 13. This frequency is given by the following expression:

$$f\_s = \frac{1}{2\pi} \sqrt{\frac{1}{L\_s' C\_s'} + \frac{1}{L'C\_s'}}\tag{19}$$

Secondly, the susceptance of the unit cell seen from the ports at *fs*, whose value can be inferred from the Smith chart, is

$$B\left(a o\_{\rm s}\right) = \frac{CL\_{\rm p}'a o\_{\rm s}^2 - 2}{L\_{\rm p}'a o\_{\rm s}}\tag{20}$$

with 2 *S S* ω π = *f* . The next condition concerns the parallel resonator of the series branch. Namely, the resonance frequency of this resonator is given by

$$f\_Z = \frac{1}{2\pi} \sqrt{\frac{1}{L\_s' C\_s'}}\tag{21}$$

Notice that this frequency does not coincide with the intrinsic resonance frequency of the magnetically driven resonator, *fo (*which is the resonance frequency of the tank formed by *Ls* and *Cs).* The transmission zero frequency *fz (*Eq. 21) can be easily obtained from the transmission coefficient *S*21 of the unit cell since at this frequency the series branch is opened and the whole power injected from the input port is reflected back to the source. Thus, the transmission coefficient nulls (zero transmission) and *fz* can easily be identified from the representation of the transmission coefficient in a decibel scale (see example in Fig. 13 b).

Another condition can be deduced from the phase of the transmission coefficient, φ*S21*. At the frequency where φ*S21*=90º, *f*π*/2*, the electrical length of the unit cell, φ*=βl* (*β* being the phase constant and *l* the length of the unit cell), is ( ) 2 *f* 90º φ π = − . Since the dispersion relation of a periodic structure consisting of cascaded unit cells, as those in Fig. 8(b) is

$$\cos \phi = 1 + \frac{Z\_s(o)}{Z\_\rho(o)} \tag{22}$$

with *Zs* and *Zp* being the series and shunt impedances, respectively, of the π-circuit model, it follows that

$$Z\_S\left(\alpha\_{\pi\ell^2}\right) = -Z\_P\left(\alpha\_{\pi\ell^2}\right) \tag{23}$$

with 2 2 2 *f* ω π π π = . Expressions (19-21) and (23) are four of the five conditions needed to univocally determine the circuit parameters in Fig. 8(b). Now, by removing the shunt connected vias or strips in the layouts in Fig. 3, we can represent the corresponding

conditions are obtained. Firstly, we can determine the frequency that nulls the series reactance, *fs*, from the intercept of *S*11 with the unit conductance circle (see Fig. 13 a). This is obvious since at this frequency, the real part of the admittance seen from the ports is simply the admittance of the opposite port, that is, *Y0*=(*Z0)*−1=(50Ω)−1=0.02 S. Hence, *S*11 must be allocated in the unit conductance circle at *fs*, as illustrated in the example provided in Fig. 13.

11 1

*LC LC*

Secondly, the susceptance of the unit cell seen from the ports at *fs*, whose value can be

( ) 2 2 *P S*

*L* ω

1 1

π

Notice that this frequency does not coincide with the intrinsic resonance frequency of the magnetically driven resonator, *fo (*which is the resonance frequency of the tank formed by *Ls* and *Cs).* The transmission zero frequency *fz (*Eq. 21) can be easily obtained from the transmission coefficient *S*21 of the unit cell since at this frequency the series branch is opened and the whole power injected from the input port is reflected back to the source. Thus, the transmission coefficient nulls (zero transmission) and *fz* can easily be identified from the representation of the transmission coefficient in a decibel scale (see example in Fig. 13 b).

*SS S*

*P S*

ω

= *f* . The next condition concerns the parallel resonator of the series branch.

*S S*

*/2*, the electrical length of the unit cell,

( )

ω

ω

 ω

= . Expressions (19-21) and (23) are four of the five conditions needed to

 π

= + ′′ ′′ (19)

′ − = ′ (20)

*L C* = ′ ′ (21)

φ

= − . Since the dispersion relation of a

= + (22)

2 2 = − ( ) (23)

φ*S21*. At

*=βl* (*β* being the phase

2 *S*

π

*S*

ω

*CL B*

2 *Z*

Another condition can be deduced from the phase of the transmission coefficient,

φ

*Z Z S P* ( ) ω

π

φ π

( ) cos 1 *S P Z Z*

with *Zs* and *Zp* being the series and shunt impedances, respectively, of the π-circuit model, it

univocally determine the circuit parameters in Fig. 8(b). Now, by removing the shunt connected vias or strips in the layouts in Fig. 3, we can represent the corresponding

*f*

*f*

Namely, the resonance frequency of this resonator is given by

π

periodic structure consisting of cascaded unit cells, as those in Fig. 8(b) is

constant and *l* the length of the unit cell), is ( ) 2 *f* 90º

This frequency is given by the following expression:

inferred from the Smith chart, is

with 2 *S S* ω

 π

the frequency where

follows that

with 2 2 2 *f* ω

π

 π

 π φ*S21*=90º, *f* reflection coefficient on a Smith chart and read the susceptance seen from the ports at that frequency where *S*11 intercepts the unit conductance circle. Since this is simply the susceptance corresponding the line capacitance (provided *Lp* has been removed), we can thus univocally determine *C*. Hence, this is the fifth condition that is required to extract the circuit parameters of the circuit model in Fig. 8(b).

Fig. 13. Reflection coefficient on the Smith chart (a); frequency response (reflection, *S11*, and transmission, *S21*, coefficients) depicted in a decibel scale and the dispersion relation (b) for a left handed cell based on a CPW structure.

#### **4.2 Parameter extraction technique metamaterial transmission lines based on Complementary Split-Ring Resonators (CSRRs)**

The technique is based on the equivalent circuit model of a CSRR-loaded transmission line shown in Fig. 10(b). This method was proposed in 2006 (Bonache et al., 2006c). The considered structures are a negative permittivity as well as a left handed microstrip line (gap capacitors are required in the latter case). In view of the models, if losses are neglected (this is reasonable in a first order approximation), two characteristic frequencies can be identified: the frequency that nulls the shunt impedance (transmission zero frequency, *fz*) and the frequency that nulls the shunt admittance (which obviously coincides with the intrinsic resonance frequency of the CSRR, *f0*). These frequencies are given by the following expressions:

$$f\_{\chi} = \frac{1}{2\pi\sqrt{L\_{\subset}(\mathbb{C} + \mathbb{C}\_{\subset})}}\tag{24}$$

$$f\_0 = \frac{1}{2\pi\sqrt{L\_\odot C\_\odot}}\tag{25}$$

and they can be either experimentally determined, or obtained from the simulated response of the structure. At *fz* a notch in the transmission coefficient is expected and this frequency can be accurately measured. To obtain *f0*, a representation of the transmission coefficient on a Smith chart is required (see example in Fig. 14). At this frequency the shunt path to ground

Characterization of Metamaterial

**Split-Ring Resonators (OSRRs)**

resonance frequency of the series branch

characteristic impedance is given by

obtained:

that is,

Transmission Lines with Coupled Resonators Through Parameter Extraction 317

This subsection is focused on the parameter extraction for the structures represented in Fig. 7. The parameters of the circuit model of a CPW loaded with an OSRR (Fig. 12 d) can be extracted from the measurement or the electromagnetic simulation of the structure following a straightforward procedure (Durán-Sindreu et al., 2009a, 2009b). First of all, from the intercept of the return loss curve with the unit conductance circle in the Smith chart, we

> 0 2 *ZS B C* ω=

> > 1 *ZS C LS S*

where *B* is the susceptance in the intercept point. The frequency at this intercept point is the

To determine the other two element values of this branch, one more condition is needed.

transmission) the characteristic impedance of the structure is 50Ω. In this π-circuit, the

( ) () ()

Thus, by forcing this impedance to 50Ω, the second condition results. By inverting (27) and (28), we can determine the element values of the series branch. The following results are

> 2 2 2 0 0 1 1

> > 2 0

The parameters of the circuit model of a CPW loaded with an OCSRR (Fig. 12 b) can be extracted following a similar procedure. In this case, the intercept of the return loss curve with the unit resistance circle in the Smith chart gives the value of the series inductance

> 2 *ZP X L* ω→∞

where *X* is the reactance at the intercept point. The shunt branch resonates at this frequency,

ω *C*= ′ = ′

*Z Z C C*

=− +

1

*S*

*S Z*

() ()

 ω

2 2

*Z C*

ω  ω

2

*P S P S Z Z*

*Z Z* ω

ω

2 0

ω

This condition comes from the fact that at the reflection zero frequency

0 2

2

ω

ω

*S Z*

*S*

*L*

=

ω

*Z*

*S*

= (27)

= = ′ (28)

= + (29)

ω

(30)

(31)

= (32)

*z* (maximum

**4.3 Parameter extraction technique for metamaterial transmission lines based on open resonators: Open Split-Ring Resonators (OSRRs) and Open Complementary**

can directly infer the value of the shunt capacitance according to

is opened, and the input impedance seen from the ports is solely formed by the series elements of the structure (*L*, for the negative permittivity line, and *L* and *Cg* for the left handed line) and the resistance of the opposite port (50 Ω). Therefore, *f0* is given by the intersection between the measured (or simulated) *S11* curve and the unit normalized resistance circle. From this result we can also obtain the impedance of the series elements at that frequency. This gives directly the value of *L* for the negative permittivity line. For the left handed line, *L* can be independently estimated from a transmission line calculator, or from the value extracted for the negative permeability line, corrected by the presence of the gap, whereas *Cg* can be determined by adjusting its value to fit the impedance value read from the Smith chart at *f0* from the simulation or experiment.

Fig. 14. (a) Layouts of the structures employed in the parameter extraction method for the CSRR-based unit cell. (b) Frequency responses of the measurement, the electromagnetic simulation and the electric simulation employing the extracted parameters. Representation of the S11 parameter for the identification of the CSRR resonance frequency (c) for the complete (LH) structure (d) for the structure without gap (ε<0). Measurement and electric simulation.

Expressions (24) and (25) are dependent on three parameters. Therefore, we cannot directly obtain the element values of the CSRR (as desired) and the coupling capacitance. To this end, we need an additional condition, namely:

$$Z\_s\left(j\alpha\_{\pi\sharp^2}\right) = -Z\_p\left(j\alpha\_{\pi\sharp^2}\right) \tag{26}$$

where *ZS* ( ) *j*ω and *ZP* ( ) *j*ω are the series and shunt impedance of the T-circuit model of the structure, respectively, and ωπ 2 is the angular frequency where the phase of the transmission coefficient (which is a measurable quantity) is ( ) 21 φ *S* = π 2 . Thus, from (23)–(25) we can determine the three reactive element values that contribute to the shunt impedance.

is opened, and the input impedance seen from the ports is solely formed by the series elements of the structure (*L*, for the negative permittivity line, and *L* and *Cg* for the left handed line) and the resistance of the opposite port (50 Ω). Therefore, *f0* is given by the intersection between the measured (or simulated) *S11* curve and the unit normalized resistance circle. From this result we can also obtain the impedance of the series elements at that frequency. This gives directly the value of *L* for the negative permittivity line. For the left handed line, *L* can be independently estimated from a transmission line calculator, or from the value extracted for the negative permeability line, corrected by the presence of the gap, whereas *Cg* can be determined by adjusting its value to fit the impedance value read

Fig. 14. (a) Layouts of the structures employed in the parameter extraction method for the CSRR-based unit cell. (b) Frequency responses of the measurement, the electromagnetic simulation and the electric simulation employing the extracted parameters. Representation of the S11 parameter for the identification of the CSRR resonance frequency (c) for the complete (LH) structure (d) for the structure without gap (ε<0). Measurement and electric

Expressions (24) and (25) are dependent on three parameters. Therefore, we cannot directly obtain the element values of the CSRR (as desired) and the coupling capacitance. To this

> ω

φ*S* =

 π

are the series and shunt impedance of the T-circuit model of the

π

2 is the angular frequency where the phase of the transmission

2 2 = − ( ) (26)

2 . Thus, from (23)–(25) we can

*Zj Zj S P* ( ) ω

π

determine the three reactive element values that contribute to the shunt impedance.

from the Smith chart at *f0* from the simulation or experiment.

simulation.

where *ZS* ( ) *j*

ω

structure, respectively, and

end, we need an additional condition, namely:

ω

ωπ

coefficient (which is a measurable quantity) is ( ) 21

and *ZP* ( ) *j*

#### **4.3 Parameter extraction technique for metamaterial transmission lines based on open resonators: Open Split-Ring Resonators (OSRRs) and Open Complementary Split-Ring Resonators (OSRRs)**

This subsection is focused on the parameter extraction for the structures represented in Fig. 7. The parameters of the circuit model of a CPW loaded with an OSRR (Fig. 12 d) can be extracted from the measurement or the electromagnetic simulation of the structure following a straightforward procedure (Durán-Sindreu et al., 2009a, 2009b). First of all, from the intercept of the return loss curve with the unit conductance circle in the Smith chart, we can directly infer the value of the shunt capacitance according to

$$C = \frac{B}{2a\_{\mathbb{Z}\_8 \to 0}}\tag{27}$$

where *B* is the susceptance in the intercept point. The frequency at this intercept point is the resonance frequency of the series branch

$$\left. \phi^2 \right|\_{Z\_s=0} = \frac{1}{C\_s L\_s'} \tag{28}$$

To determine the other two element values of this branch, one more condition is needed. This condition comes from the fact that at the reflection zero frequency ω*z* (maximum transmission) the characteristic impedance of the structure is 50Ω. In this π-circuit, the characteristic impedance is given by

$$Z\_0\left(\left.\alpha\right>\right) = \sqrt{\frac{Z\_\rho\left(\left.\alpha\right>^2 Z\_s\left(\left.\alpha\right)\right>}{2Z\_\rho\left(\left.\alpha\right> + Z\_s\left(\left.\alpha\right)\right>}}}\tag{29}$$

Thus, by forcing this impedance to 50Ω, the second condition results. By inverting (27) and (28), we can determine the element values of the series branch. The following results are obtained:

$$C\_s = \left[\frac{\alpha\_\varkappa^2}{\alpha^2\Big|\_{Z\_s=0}} - 1\right] \left\{\frac{1}{2Z\_0^2\alpha\_\varkappa^2\mathcal{C}} + \frac{\mathcal{C}}{2}\right\} \tag{30}$$

$$L\_{\rm S}^{\prime} = \frac{1}{a\nu^2 \Big|\_{Z\_{\rm S} = 0} C\_{\rm S}^{\prime}} \tag{31}$$

The parameters of the circuit model of a CPW loaded with an OCSRR (Fig. 12 b) can be extracted following a similar procedure. In this case, the intercept of the return loss curve with the unit resistance circle in the Smith chart gives the value of the series inductance

$$L = \frac{X}{2a\Big|\_{Z\_{\mathbb{P}} \to \cdots}}\tag{32}$$

where *X* is the reactance at the intercept point. The shunt branch resonates at this frequency, that is,

Characterization of Metamaterial

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$$\left. \phi \phi^2 \right|\_{Z\_P \to =} = \frac{1}{L\_P' \mathcal{C}\_P'} \tag{33}$$

Finally, at the reflection zero frequency (ω*z*), the characteristic impedance, given by

$$Z\_o(\ \alpha) = \sqrt{Z\_s(\ \alpha) \left[Z\_s(\ \alpha) + 2Z\_p(\ \alpha)\right]} \tag{34}$$

must be forced to be 50Ω. From these two latter conditions, we finally obtain

$$L\_p' = \left[\frac{\alpha\_\mathcal{Z}^2}{\alpha^2\Big|\_{Z\_p \to \omega}} - 1\right] \left\{\frac{Z\_0^2}{2\alpha\_\mathcal{Z}^2 L} + \frac{L}{2}\right\} \tag{35}$$

$$C\_p' = \frac{1}{o\big|\_{Z\_p \to \dotsb} L\_p'} \tag{36}$$

and the element values are determined.

#### **5. Conclusion**

In this chapter, different kinds of resonant-type metamaterial transmission lines based on subwavelength resonators have been presented and studied. There are several types of resonators which allow their use in the implementation of this kind of artificial transmission lines. SRR-, CSRR- and open resonator-based structures have been presented and studied. The equivalent circuit models for each of the kinds of structures presented have been exposed, together with their parameter extraction methods. These methods provide the circuit model parameters, extracted from the frequency response of the structure, being a very useful design tool and allowing the corroboration of the proposed circuits as correct models for the considered structures. Furthermore, thanks to their accuracy, the parameter extraction methods have already been applied in automation procedures which allow the automatic generation of layouts for this kind of structures (Selga et al., 2010). After the imposition of certain conditions on the frequency response, which implies certain values for the circuit model parameters, the layout of the structure satisfying the required conditions can be automatically generated by means of space mapping techniques. This is a good example of the usefulness and importance of the equivalent circuit models and the parameter extraction methods.

#### **6. References**

2 1 *ZP L CP P*

0 () () () () 2 *Z ZZ Z*

2 2

2 2 1

*Z Z Z L L*

*P*

In this chapter, different kinds of resonant-type metamaterial transmission lines based on subwavelength resonators have been presented and studied. There are several types of resonators which allow their use in the implementation of this kind of artificial transmission lines. SRR-, CSRR- and open resonator-based structures have been presented and studied. The equivalent circuit models for each of the kinds of structures presented have been exposed, together with their parameter extraction methods. These methods provide the circuit model parameters, extracted from the frequency response of the structure, being a very useful design tool and allowing the corroboration of the proposed circuits as correct models for the considered structures. Furthermore, thanks to their accuracy, the parameter extraction methods have already been applied in automation procedures which allow the automatic generation of layouts for this kind of structures (Selga et al., 2010). After the imposition of certain conditions on the frequency response, which implies certain values for the circuit model parameters, the layout of the structure satisfying the required conditions can be automatically generated by means of space mapping techniques. This is a good example of the usefulness and importance of the equivalent circuit models and the

Aznar, F.; Bonache, J. & Martín, F., Improved circuit model for left-handed lines loaded with

Aznar, F.; García-García, J.; Gil, M.; Bonache J. & Martín, F. Strategies for the miniaturization

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*Symposium*, pp. 323-326, ISBN: 978-1-4244-1780-3 Atlanta (USA), June 2008.

split ring resonators, *Applied Physics Letters*, vol. 92, February 2008, pp. 043512-3,

of metamaterial resonators, *Microwave and Optical Technology Letters*, vol. 50, May

type metamaterial transmission lines, *Proceedings of the IEEE MTT-S Int'l Microwave*

ω*L* →∞

1

*Z P*

0

ω

2 2

*L*

→∞ = ′ ′ (33)

′ = ′ (36)

(35)

*z*), the characteristic impedance, given by

 ω= *SS P* + (34)

ω

must be forced to be 50Ω. From these two latter conditions, we finally obtain

ω

ω

*P*

ω

 ωω

*P Z*

*P*

*C*

→∞ ′ =−+

ω

Finally, at the reflection zero frequency (

and the element values are determined.

parameter extraction methods.

ISSN 0003-6951.

2008, pp. 1263-1270, ISSN 1098-2760.

**6. References**

**5. Conclusion**

**0**

**13**

*Italy*

**Genetic Programming**

*University of Cagliari*

Luisa Deias, Giuseppe Mazzarella and Nicola Sirena

**Synthesis of Planar EBG Structures Based on**

*Frequency selective surfaces* (FSS) consist of two-dimensional periodic arrays of metal patches patterned on a dielectric substrate or apertures etched on a metal screen (Hosseini et al., 2006;

These periodic structures resonate at certain frequencies, thus ensuring filtering characteristics, exploited both in the microwave and optical region of the electromagnetic spectrum. Frequency selective surfaces have been thoroughly studied over the years (Mittra et al., 1988; Munk, 2000), and they have found new life in the past decade when *electromagnetic bandgap* (EBG) (Sievenpiper et al., 1999; Yang et al., 1999) structures were introduced, firstly under the name of photonic bandgap (PBG) materials, in analogy to the bandgaps present in electric crystals, even though no photons were involved. Some well–known EBG structures are the Uniplanar Compact Photonic Band-Gap (UC–PBG)(Yang et al., 1999) (Fig. 1) and the

Sievenpiper "mushroom" high–impedance surface (Sievenpiper et al., 1999).

Fig. 1. Uniplanar Compact Photonic Band-Gap (UC–PBG)(Yang et al., 1999)

This entirely new class of structures, encompassing FSS as one of its subclasses (planar EBG) display some very interesting new electromagnetic properties. The presence of a stopband for this structure has been theoretically and experimentally verified and exploited

**1. Introduction**

Wu, 1995; Yang et al., 1999).

**1.1 Electronic Band–Gap (EBG) structures**

### **Synthesis of Planar EBG Structures Based on Genetic Programming**

Luisa Deias, Giuseppe Mazzarella and Nicola Sirena *University of Cagliari Italy*

#### **1. Introduction**

320 Metamaterial

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#### **1.1 Electronic Band–Gap (EBG) structures**

*Frequency selective surfaces* (FSS) consist of two-dimensional periodic arrays of metal patches patterned on a dielectric substrate or apertures etched on a metal screen (Hosseini et al., 2006; Wu, 1995; Yang et al., 1999).

These periodic structures resonate at certain frequencies, thus ensuring filtering characteristics, exploited both in the microwave and optical region of the electromagnetic spectrum. Frequency selective surfaces have been thoroughly studied over the years (Mittra et al., 1988; Munk, 2000), and they have found new life in the past decade when *electromagnetic bandgap* (EBG) (Sievenpiper et al., 1999; Yang et al., 1999) structures were introduced, firstly under the name of photonic bandgap (PBG) materials, in analogy to the bandgaps present in electric crystals, even though no photons were involved. Some well–known EBG structures are the Uniplanar Compact Photonic Band-Gap (UC–PBG)(Yang et al., 1999) (Fig. 1) and the Sievenpiper "mushroom" high–impedance surface (Sievenpiper et al., 1999).

Fig. 1. Uniplanar Compact Photonic Band-Gap (UC–PBG)(Yang et al., 1999)

This entirely new class of structures, encompassing FSS as one of its subclasses (planar EBG) display some very interesting new electromagnetic properties. The presence of a stopband for this structure has been theoretically and experimentally verified and exploited

Genetic Algorithms (GA), already widely used in antenna design (Jones & Joines, 2000; Lohn et al., 2001) and more recently also applied to EBG design and optimization (Bray et al., 2006; Ge et al., 2007; Yeo et al., 2002) iteratively transform populations of mathematical objects (typically fixed–length binary character strings), each with an associated fitness value, into

Synthesis of Planar EBG Structures Based on Genetic Programming 323

In GP, unlike GA chromosomes need not be represented by bit-strings and the alteration process includes other "genetic" operators appropriate for the given structure and the given problem. We can say that GA works on the "nucleotide" (i.e. bit) level, in the sense that the antenna or EBG structure is completely defined from the beginning and only a handful of parameters remains to be optimized. The approach proposed by Koza assumes no "a priori" structure. Instead, it builds up the structure of the individuals as the procedure evolves. As a consequence, its solution space has the power of the continuum, while the GA solution space is a discrete one, so it is a very small subspace of the former. The goal of genetic programming is not simply to evolve a bit–string representation of some problem but the computer code

A fixed–length coding such as GA is rather artificial. As it cannot provide a dynamic variability in length, such a coding often causes considerable redundancy and reduces the efficiency of genetic search. In contrast, GP uses high–level building blocks of variable length. Their size and complexity can change during breeding. Moreover, the typical evolution operators work on actual physical structures rather than on sequences of bits with no intuitive link to the EGB surface shape. The enormous power of this strategy fully allows

Getting machines to produce human–like results is the reason for the existence of the fields of artificial intelligence and machine learning. Genetic programming addresses this challenge by providing a method for automatically creating a working computer program from a high–level description of the problem. This is the reason why, unlike Genetic Algorithms, Genetic Programs often deliver elegant human–like solutions not anticipated by the programmer, providing only a minimum amount of pre–supplied human knowledge,

Yet, while allowing to explore and evaluate general configurations, this approach can lead to a severely ill–conditioned synthesis problem. A suitable stabilization is therefore obtained by imposing problem–specific requirements, in our case the periodicity of the the surface elements, which is directly related to the resonant frequency, and the physical parameters, which are not so relevant to the problem. We therefore let then the Genetic Programming

Representation is a key issue in genetic programming (Koza, 1992) because the representation scheme can severely limit the window through which the system "observes" its world.

Since GP manipulates programs by applying genetic operators, a programming language such as LISP was chosen, since it allows each individual, i.e. computer program, to be manipulated as data. Any LISP S–expression can be depicted as a rooted point-labelled tree with ordered branches, as shown in Fig.2. Therefore in GP each individual is a computer program, described through the set of instructions needed to "build" it, typically implemented

strategy evaluate every possible shape in the solution space we delimited.

new populations using the Darwinian principle of natural selection.

that solves that problem.

analysis and information.

in S–expressions.

the exploration of more general shapes.

in different realizations, i.e. TEM waveguide, slow–wave planar structure and low–loss conductor–backed coplanar waveguide.

By choosing the proper geometry of the periodic surface we can shape the electromagnetic behavior of these structures and they can be made to act as a so–called electromagnetic crystal, exhibiting frequency bands inside which the propagation of electromagnetic waves is not allowed or is highly attenuated. The concept of suppressing surfaces waves on metals is not new. Surface waves can be eliminated from a metal surface over a finite frequency band by applying a periodic texture. It has been done long before EGB structures were introduced using several geometries, such as a metal sheet covered with small bumps or a corrugated metal slab.

The novelty of EBGs is the application of an array of periodic patches or apertures, i.e. lumped circuit elements, to produce a thin two–dimensional structure that must generally be described by band structure concepts, even though the thickness and periodicity are both much smaller than the operating wavelength.

These periodic structures can also be designed to act as an *artificial magnetic conductor* (AMC) or high–impedance electromagnetic ground plane over a desired (quite) narrow frequency range, corresponding to the forbidden frequency band. Hence, the key feature of these structures is the reflection of an incident plane wave with no phase reversal, unlike normal metal surfaces (Sievenpiper et al., 1999).

High–impedance surfaces are widely studied now as promising antenna substrates (Feresidis et al., 2005; Gonzalo et al., 1999; Hosseini et al., 2006). A possibility of realizing a magnetic wall near the resonant frequency of a very thin structure is very attractive, since this allows one to design low–profile antennas and enhance the performance of printed antennas. The main drawback of this strategy is the reduced bandwidth of the complete antenna, since the frequency range over which these EBG surfaces behave as an AMC is usually narrowband and fixed by their geometrical configuration. The ultimate goal is then to design and incorporate such metamaterial–substrates in antenna structures in order to improve antenna performance.

A key issue in the research field of metamaterials is then represented by the design and optimization of EBG structures. Different techniques have been investigated, among which methods of global optimization such as genetic algorithm and particle swarm optimization (PSO),(Bray et al., 2006; Ge et al., 2007; Kovács et al., 2010; Tavallaee & Rahmat-Samii, 2007; Yeo et al., 2002).

#### **1.2 Genetic programming**

*Genetic Programming* (GP) (Koza, 1992) falls into the larger class of evolutionary computations (Fogel, 2006; Michalewicz, 1992), including genetic algorithms, evolution strategies and evolutionary programming, which can be described as highly parallel probabilistic search algorithms imitating the principles of natural evolution for optimization problems, based on the idea that most real–world problems cannot be handled with binary representations. Such algorithms allow task specific knowledge emerge while solving the problem. Genetic Programming, a variant of genetic algorithms yet with marked differences, is an especially interesting form of computational problem solving.

2 Will-be-set-by-IN-TECH

in different realizations, i.e. TEM waveguide, slow–wave planar structure and low–loss

By choosing the proper geometry of the periodic surface we can shape the electromagnetic behavior of these structures and they can be made to act as a so–called electromagnetic crystal, exhibiting frequency bands inside which the propagation of electromagnetic waves is not allowed or is highly attenuated. The concept of suppressing surfaces waves on metals is not new. Surface waves can be eliminated from a metal surface over a finite frequency band by applying a periodic texture. It has been done long before EGB structures were introduced using several geometries, such as a metal sheet covered with small bumps or a corrugated

The novelty of EBGs is the application of an array of periodic patches or apertures, i.e. lumped circuit elements, to produce a thin two–dimensional structure that must generally be described by band structure concepts, even though the thickness and periodicity are both

These periodic structures can also be designed to act as an *artificial magnetic conductor* (AMC) or high–impedance electromagnetic ground plane over a desired (quite) narrow frequency range, corresponding to the forbidden frequency band. Hence, the key feature of these structures is the reflection of an incident plane wave with no phase reversal, unlike normal

High–impedance surfaces are widely studied now as promising antenna substrates (Feresidis et al., 2005; Gonzalo et al., 1999; Hosseini et al., 2006). A possibility of realizing a magnetic wall near the resonant frequency of a very thin structure is very attractive, since this allows one to design low–profile antennas and enhance the performance of printed antennas. The main drawback of this strategy is the reduced bandwidth of the complete antenna, since the frequency range over which these EBG surfaces behave as an AMC is usually narrowband and fixed by their geometrical configuration. The ultimate goal is then to design and incorporate such metamaterial–substrates in antenna structures in order to improve antenna performance. A key issue in the research field of metamaterials is then represented by the design and optimization of EBG structures. Different techniques have been investigated, among which methods of global optimization such as genetic algorithm and particle swarm optimization (PSO),(Bray et al., 2006; Ge et al., 2007; Kovács et al., 2010; Tavallaee & Rahmat-Samii, 2007;

*Genetic Programming* (GP) (Koza, 1992) falls into the larger class of evolutionary computations (Fogel, 2006; Michalewicz, 1992), including genetic algorithms, evolution strategies and evolutionary programming, which can be described as highly parallel probabilistic search algorithms imitating the principles of natural evolution for optimization problems, based on the idea that most real–world problems cannot be handled with binary representations. Such algorithms allow task specific knowledge emerge while solving the problem. Genetic Programming, a variant of genetic algorithms yet with marked differences, is an especially

conductor–backed coplanar waveguide.

much smaller than the operating wavelength.

metal surfaces (Sievenpiper et al., 1999).

metal slab.

Yeo et al., 2002).

**1.2 Genetic programming**

interesting form of computational problem solving.

Genetic Algorithms (GA), already widely used in antenna design (Jones & Joines, 2000; Lohn et al., 2001) and more recently also applied to EBG design and optimization (Bray et al., 2006; Ge et al., 2007; Yeo et al., 2002) iteratively transform populations of mathematical objects (typically fixed–length binary character strings), each with an associated fitness value, into new populations using the Darwinian principle of natural selection.

In GP, unlike GA chromosomes need not be represented by bit-strings and the alteration process includes other "genetic" operators appropriate for the given structure and the given problem. We can say that GA works on the "nucleotide" (i.e. bit) level, in the sense that the antenna or EBG structure is completely defined from the beginning and only a handful of parameters remains to be optimized. The approach proposed by Koza assumes no "a priori" structure. Instead, it builds up the structure of the individuals as the procedure evolves. As a consequence, its solution space has the power of the continuum, while the GA solution space is a discrete one, so it is a very small subspace of the former. The goal of genetic programming is not simply to evolve a bit–string representation of some problem but the computer code that solves that problem.

A fixed–length coding such as GA is rather artificial. As it cannot provide a dynamic variability in length, such a coding often causes considerable redundancy and reduces the efficiency of genetic search. In contrast, GP uses high–level building blocks of variable length. Their size and complexity can change during breeding. Moreover, the typical evolution operators work on actual physical structures rather than on sequences of bits with no intuitive link to the EGB surface shape. The enormous power of this strategy fully allows the exploration of more general shapes.

Getting machines to produce human–like results is the reason for the existence of the fields of artificial intelligence and machine learning. Genetic programming addresses this challenge by providing a method for automatically creating a working computer program from a high–level description of the problem. This is the reason why, unlike Genetic Algorithms, Genetic Programs often deliver elegant human–like solutions not anticipated by the programmer, providing only a minimum amount of pre–supplied human knowledge, analysis and information.

Yet, while allowing to explore and evaluate general configurations, this approach can lead to a severely ill–conditioned synthesis problem. A suitable stabilization is therefore obtained by imposing problem–specific requirements, in our case the periodicity of the the surface elements, which is directly related to the resonant frequency, and the physical parameters, which are not so relevant to the problem. We therefore let then the Genetic Programming strategy evaluate every possible shape in the solution space we delimited.

Representation is a key issue in genetic programming (Koza, 1992) because the representation scheme can severely limit the window through which the system "observes" its world.

Since GP manipulates programs by applying genetic operators, a programming language such as LISP was chosen, since it allows each individual, i.e. computer program, to be manipulated as data. Any LISP S–expression can be depicted as a rooted point-labelled tree with ordered branches, as shown in Fig.2. Therefore in GP each individual is a computer program, described through the set of instructions needed to "build" it, typically implemented in S–expressions.

**2. EBG design usign genetic programming**

patch (d) circular loop (e) square loops (f) tripole.

(Branch (Rettangolo 0.16589776 0.2873323 (Ruota 90.0 (Ruota -90.0 END))) (Ruota 90.0

(Branch (Branch END END) (Ruota 90.0 END))

(Ruota 90.0 (Rettangolo 0.51570845 0.25133058

has been implemented in Fortran.

END))))))))

(Branch END END)

END)))))

Tree 0:

Tree 1:

Tree 2:

Tree 3:

In the design of frequency selective surfaces, the choice of the proper element may be of utmost importance. In fact some elements are intrinsically more broad–banded or more narrow–banded than others, while some can be more easily varied by design. In literature we can find a large variety of element types, the more common are illustrated in Fig. 3. It can be now devised the great potential of Genetic Programming, since we don't really want any possible geometry to be taken into account but only feasible geometries, just like a human being would design them on a piece of paper, yet faster and with much more "fantasy".

Synthesis of Planar EBG Structures Based on Genetic Programming 325

Fig. 3. Some typical FSS unit cell geometries: (a) cross dipole (b) Jerusalem cross (c) square

Therefore the goal of the design process is to obtain the unit cell aperture geometry of the periodic surface which fulfills the desired requirements on the resonant frequency of the periodic structure, analyzed with a full–wave technique. The GP approach has been implemented in Java, while the full–wave analysis of the periodic structure for each individual

The S–expression of an individual of the population can take for example the following form:

(Rettangolo 0.6837649 0.2757175 (Rettangolo 0.7254247 0.23175128

(Ruota 90.0 (Branch (Ruota -90.0 END) (Rettangolo 0.75231904 0.2833914

(Ruota 90.0 (Ruota 90.0 (Ruota -90.0 (Rettangolo 0.19471756 0.19593427

Fig. 2. S-expression: tree architecture.

While the evolutionary "strategy" is almost standard for a variety of problems (we used ECJ, a general purpose Java-based Evolutionary Computation research system and developed at George Mason University ECLab), three elements must be defined to evolve a design, i.e. "individual":

	- **–** CROSSOVER:creates new S–expressions by exchanging sub S–expressions between two S–expressions. The sub S-expressions exchanged are selected randomly.
	- **–** MUTATION: creates a new S–expression by replacing an existing argument symbol with the other possible symbol. The argument symbol replaced is selected randomly.

In other words, representation refers to a form of data structure. Variation operations are applied to existing solutions to create new solutions. Usually, variation is based on random perturbation.

The starting point is an initial population of randomly generated computer programs composed of functions and terminals appropriate to the specified problem domain. The evolutionary strategy works in order to find the best individual, in terms of their closeness to the constraints set in the design and evaluated as a "fitness" function. This strategy let to interesting and promising results in the synthesis of EBGs, even when considering more complex structures and requirements (Deias et al., 2009a;b; 2010).

#### **2. EBG design usign genetic programming**

4 Will-be-set-by-IN-TECH

While the evolutionary "strategy" is almost standard for a variety of problems (we used ECJ, a general purpose Java-based Evolutionary Computation research system and developed at George Mason University ECLab), three elements must be defined to evolve a design, i.e.

• VARIATION OPERATORS: code that takes one or more design representations as input and

S–expressions. The sub S-expressions exchanged are selected randomly.

**–** CROSSOVER:creates new S–expressions by exchanging sub S–expressions between two

**–** MUTATION: creates a new S–expression by replacing an existing argument symbol with the other possible symbol. The argument symbol replaced is selected randomly.

In other words, representation refers to a form of data structure. Variation operations are applied to existing solutions to create new solutions. Usually, variation is based on random

The starting point is an initial population of randomly generated computer programs composed of functions and terminals appropriate to the specified problem domain. The evolutionary strategy works in order to find the best individual, in terms of their closeness to the constraints set in the design and evaluated as a "fitness" function. This strategy let to interesting and promising results in the synthesis of EBGs, even when considering more

Fig. 2. S-expression: tree architecture.

outputs a design derived from them

• REPRESENTATION SCHEME:definition of the design space

• FITNESS FUNCTION: a function that evaluates the individuals

complex structures and requirements (Deias et al., 2009a;b; 2010).

"individual":

perturbation.

In the design of frequency selective surfaces, the choice of the proper element may be of utmost importance. In fact some elements are intrinsically more broad–banded or more narrow–banded than others, while some can be more easily varied by design. In literature we can find a large variety of element types, the more common are illustrated in Fig. 3. It can be now devised the great potential of Genetic Programming, since we don't really want any possible geometry to be taken into account but only feasible geometries, just like a human being would design them on a piece of paper, yet faster and with much more "fantasy".

Fig. 3. Some typical FSS unit cell geometries: (a) cross dipole (b) Jerusalem cross (c) square patch (d) circular loop (e) square loops (f) tripole.

Therefore the goal of the design process is to obtain the unit cell aperture geometry of the periodic surface which fulfills the desired requirements on the resonant frequency of the periodic structure, analyzed with a full–wave technique. The GP approach has been implemented in Java, while the full–wave analysis of the periodic structure for each individual has been implemented in Fortran.

The S–expression of an individual of the population can take for example the following form:

```
Tree 0:
 (Branch (Rettangolo 0.16589776 0.2873323
  (Ruota 90.0 (Ruota -90.0 END))) (Ruota 90.0
    (Rettangolo 0.6837649 0.2757175 (Rettangolo 0.7254247 0.23175128
     (Ruota 90.0 (Ruota 90.0 (Ruota -90.0 (Rettangolo 0.19471756 0.19593427
       END))))))))
Tree 1:
 (Branch (Branch END END) (Ruota 90.0 END))
Tree 2:
 (Branch END END)
Tree 3:
 (Ruota 90.0 (Rettangolo 0.51570845 0.25133058
   (Ruota 90.0 (Branch (Ruota -90.0 END) (Rettangolo 0.75231904 0.2833914
    END)))))
```
Fig. 5. Flowchart.

discretization step.

The fitness function employed by our GP optimization is:

*FF* = *ω*1 ∗ *ω*2 ∗ *e*

frequency points evaluation, individuals with larger bandwidth to prevail.

shown in (Collin, 1985) for a different problem, namely wire antennas.

where *ω*1 and *ω*2 are penalty coefficients developed in order to avoid geometries with high number of discretization elements within the unit cell and allowing, when using a three

Synthesis of Planar EBG Structures Based on Genetic Programming 327

A full–wave simulation using Method of Moments is carried out on a single design, i.e. unit cell of the infinite periodic surface and individual of a set population. The starting point is a random initial population of individuals. The external parameters of the structure (i.e. substrate thickness and dielectric constant) are fixed, altogether with the periodicity of the planar EBG. The geometry of the aperture is subject to the GP optimization, while the unit square cell within which each individual/design can evolve is fixed together with its

The set of admissible solution is then composed by every geometry/aperture that can be designed in this square, built as series of segments that can evolve in every direction, with no limit on the number of segments, on their width and length and number of subsequent ramifications. We have fixed the discretization step within the unit cell in order to pose a limit not only to the computational burden but also to the geometrical spatial bandwidth of the possible solutions. In such a way we obtain a stabilizing effect on the problem, as much as

The phase of the reflection coefficient, computed at one or more frequencies depending on the goal, is then incorporated into the GP strategy for the evaluation procedure, the fitness function determining the environment within which the solutions "live" and the best

*Phase*(*fc* )

20 (1)

The corresponding input to the Fortran executable is the following, i.e. a collection of rectangular patches that form the geometry of the aperture:

Patch 48 -0.0 , -0.0 , 0.4572 , 0.3048 -0.4572 , -0.0 , 0.4572 , 0.3048 -0.0 , -0.3048 , 0.4572 , 0.3048 -0.4572 , -0.3048 , 0.4572 , 0.3048 -0.3048 , -0.0 , 0.3048 , 0.4572 -0.0 , -0.0 , 0.3048 , 0.4572 -0.3048 , -0.4572 , 0.3048 , 0.4572 -0.0 , -0.4572 , 0.3048 , 0.4572 0.3048 , -0.0 , 0.15239999 , 0.762 -0.4572 , -0.0 , 0.15239999 , 0.762 ... -1.3716 , -0.0 , 0.762 , 0.3048 0.6096 , -0.0 , 0.762 , 0.3048 -0.0 , -0.0 , 0.1524 , 0.3048 -0.1524 , -0.0 , 0.1524 , 0.3048 -0.0 , -0.3048 , 0.1524 , 0.3048 -0.1524 , -0.3048 , 0.1524 , 0.3048 -0.3048 , -0.0 , 0.3048 , 0.1524 -0.0 , -0.0 , 0.3048 , 0.1524 -0.3048 , -0.1524 , 0.3048 , 0.1524 -0.0 , -0.1524 , 0.3048 , 0.1524

And the corresponding geometry of the aperture is shown in Fig.4.

Fig. 4. Aperture Geometry.

In our specific problem, we thus obtain a set of rectangular patches which describe the aperture geometry of the EBG unit cell.

The fitness criteria measure the quality of any given solution. The selection method uses the score obtained for each solution to determine which to save and which to eliminate from the population at each generation. Those solutions that survive are the "parents" of the next generation. The initialization of an evolutionary algorithm can be completely at random, or can incorporate human or other expertise about solutions that may work better than others.

Fig. 5. Flowchart.

6 Will-be-set-by-IN-TECH

The corresponding input to the Fortran executable is the following, i.e. a collection of

rectangular patches that form the geometry of the aperture:

And the corresponding geometry of the aperture is shown in Fig.4.

In our specific problem, we thus obtain a set of rectangular patches which describe the

The fitness criteria measure the quality of any given solution. The selection method uses the score obtained for each solution to determine which to save and which to eliminate from the population at each generation. Those solutions that survive are the "parents" of the next generation. The initialization of an evolutionary algorithm can be completely at random, or can incorporate human or other expertise about solutions that may work better than others.

Patch 48

...

Fig. 4. Aperture Geometry.

aperture geometry of the EBG unit cell.

The fitness function employed by our GP optimization is:

$$FF = \omega\_1 \* \omega\_2 \* e^{\frac{\text{Phase}(\mathcal{f}\_\mathcal{C})}{20}} \tag{1}$$

where *ω*1 and *ω*2 are penalty coefficients developed in order to avoid geometries with high number of discretization elements within the unit cell and allowing, when using a three frequency points evaluation, individuals with larger bandwidth to prevail.

A full–wave simulation using Method of Moments is carried out on a single design, i.e. unit cell of the infinite periodic surface and individual of a set population. The starting point is a random initial population of individuals. The external parameters of the structure (i.e. substrate thickness and dielectric constant) are fixed, altogether with the periodicity of the planar EBG. The geometry of the aperture is subject to the GP optimization, while the unit square cell within which each individual/design can evolve is fixed together with its discretization step.

The set of admissible solution is then composed by every geometry/aperture that can be designed in this square, built as series of segments that can evolve in every direction, with no limit on the number of segments, on their width and length and number of subsequent ramifications. We have fixed the discretization step within the unit cell in order to pose a limit not only to the computational burden but also to the geometrical spatial bandwidth of the possible solutions. In such a way we obtain a stabilizing effect on the problem, as much as shown in (Collin, 1985) for a different problem, namely wire antennas.

The phase of the reflection coefficient, computed at one or more frequencies depending on the goal, is then incorporated into the GP strategy for the evaluation procedure, the fitness function determining the environment within which the solutions "live" and the best

Fig. 7. Unit cell geometry of a Frequency Selective Surface with a dielectric substrate on ground plane, and magnetic currents equivalent to an aperture. The air–dielectric interface is

Synthesis of Planar EBG Structures Based on Genetic Programming 329

(a)

V*eq*

V*eq* V*eq*

Z1

h

Z1

h

MA1

MA2

MA3

(b)

Fig. 8. Equivalent circuits resulting from the equivalence applied to a FSS (in the spectral

the advantages of the aperture approach are far more greater (Asole et al., 2007).

of the method. The generalization to N-layer FSS is really straightforward.

PEC

A1

A2

A3

As a consequence of our aperture approach, the MoM matrix can be decoupled in the sum of "localized" admittance matrices, each one relevant to a single region. For multi–layered FSS

Let's consider the 3–layer AMC structure as shown in Fig.9, since it contains all the features

h1

Fig. 9. Geometry of a 3–layer structure and magnetic currents equivalent to the apertures

h2

h3

Z0

Z0

domain): (a) metallic patches; (b) apertures.

z

(central cell).

εr1

εr2

εr3

a PEC.

individual of a generation is selected. The evolutionary operators (reproduction, crossover and mutation) are then applied to the best individual leading to the subsequent evolution of the population and the next generation. In this way the geometrical properties are optimized to evolve the best solution (zero phase at the desired resonant frequency).

#### **3. Single and multi–layer EBG full–wave analysis**

Various techniques have been proposed in literature in order to analyze frequency selective surfaces (FSS) (Bardi et al., 2002; Bozzi & Perregrini, 1999; Harms et al., 1994; Mittra et al., 1988; Wu, 1995). The standard method of moments (MoM) approach is based on the induced electric currents on the FSS but it can be prohibitively costly from a computational point of view, if not even impractical in some cases, and this could explain why MoM is so unpopular.

Frequency Selective Surfaces (FSS) consist of two-dimensional periodic arrays of metallic patches patterned on a dielectric substrate or apertures etched on a metal screen, either entities can be isolated or connected in a rectangular grid. Hence, for any FSS, we can consider either the periodicity of the metallic patches or that of the apertures. By taking properly into account the continuity of electric or magnetic currents along adjacent cells we can conveniently adopt one of the two approaches, considering as unknowns of the problem either the electric or magnetic currents.

Fig. 6. Cross section view of a single layer FSS with dielectric substrate on ground plane.

The integral equation for the electric field (EFIE) applied to the periodic array of metal patches and the integral equation for the magnetic field (MFIE) applied to the periodic array of apertures thus represent two alternative formulations of the same problem, yielding to the same solution. The corresponding equivalent circuit for each approach is shown in Fig.8

Using a MoM approach based on the apertures, a very effective procedure can be devised (Deias & Mazzarella, 2006).

The aperture oriented approach (i.e. MFIE formulation) starts by applying the equivalence theorem (Balanis, 1996). The aperture is closed by a conductive sheet, and two unknown magnetic current densities **M***A* and −**M***A* on the opposite sides of the conductive sheet, as shown in Fig.7, are defined so that the continuity of the tangential electric field is guaranteed.

The MoM matrix is then obtained by imposing the continuity of the tangential component of the magnetic field across the aperture. The magnetic field is computed using a Green function in the spectral domain, too. But the relevant equivalent circuit, shown in Fig.8(b), allows to decouple the two regions below and above the metallization. This is a key feature not available in the conventional method based on the EFIE: in Fig.8(a) we can see that the equivalent circuit in this case bounds together the different regions.

8 Will-be-set-by-IN-TECH

individual of a generation is selected. The evolutionary operators (reproduction, crossover and mutation) are then applied to the best individual leading to the subsequent evolution of the population and the next generation. In this way the geometrical properties are optimized

Various techniques have been proposed in literature in order to analyze frequency selective surfaces (FSS) (Bardi et al., 2002; Bozzi & Perregrini, 1999; Harms et al., 1994; Mittra et al., 1988; Wu, 1995). The standard method of moments (MoM) approach is based on the induced electric currents on the FSS but it can be prohibitively costly from a computational point of view, if not even impractical in some cases, and this could explain why MoM is so unpopular. Frequency Selective Surfaces (FSS) consist of two-dimensional periodic arrays of metallic patches patterned on a dielectric substrate or apertures etched on a metal screen, either entities can be isolated or connected in a rectangular grid. Hence, for any FSS, we can consider either the periodicity of the metallic patches or that of the apertures. By taking properly into account the continuity of electric or magnetic currents along adjacent cells we can conveniently adopt one of the two approaches, considering as unknowns of the problem either the electric or

Fig. 6. Cross section view of a single layer FSS with dielectric substrate on ground plane.

The integral equation for the electric field (EFIE) applied to the periodic array of metal patches and the integral equation for the magnetic field (MFIE) applied to the periodic array of apertures thus represent two alternative formulations of the same problem, yielding to the same solution. The corresponding equivalent circuit for each approach is shown in Fig.8

Using a MoM approach based on the apertures, a very effective procedure can be devised

The aperture oriented approach (i.e. MFIE formulation) starts by applying the equivalence theorem (Balanis, 1996). The aperture is closed by a conductive sheet, and two unknown magnetic current densities **M***A* and −**M***A* on the opposite sides of the conductive sheet, as shown in Fig.7, are defined so that the continuity of the tangential electric field is guaranteed. The MoM matrix is then obtained by imposing the continuity of the tangential component of the magnetic field across the aperture. The magnetic field is computed using a Green function in the spectral domain, too. But the relevant equivalent circuit, shown in Fig.8(b), allows to decouple the two regions below and above the metallization. This is a key feature not available in the conventional method based on the EFIE: in Fig.8(a) we can see that the equivalent circuit

to evolve the best solution (zero phase at the desired resonant frequency).

**3. Single and multi–layer EBG full–wave analysis**

magnetic currents.

(Deias & Mazzarella, 2006).

in this case bounds together the different regions.

Fig. 7. Unit cell geometry of a Frequency Selective Surface with a dielectric substrate on ground plane, and magnetic currents equivalent to an aperture. The air–dielectric interface is a PEC.

Fig. 8. Equivalent circuits resulting from the equivalence applied to a FSS (in the spectral domain): (a) metallic patches; (b) apertures.

As a consequence of our aperture approach, the MoM matrix can be decoupled in the sum of "localized" admittance matrices, each one relevant to a single region. For multi–layered FSS the advantages of the aperture approach are far more greater (Asole et al., 2007).

Let's consider the 3–layer AMC structure as shown in Fig.9, since it contains all the features of the method. The generalization to N-layer FSS is really straightforward.

Fig. 9. Geometry of a 3–layer structure and magnetic currents equivalent to the apertures (central cell).

For the second discontinuity *A*2, as for any intermediate aperture, enforcing the boundary

Synthesis of Planar EBG Structures Based on Genetic Programming 331

2,*t* = **H***A*2*A*2

where *m* = 1, . . . , *Nb* and the l.h.s. term represents the self–term and the r.h.s. term the coupling terms, i.e. the effect of the magnetic currents **M***A*3 and **M***A*1 on the aperture *A*2.

3,*t* = **H***A*3*A*3

where *m* = 1, . . . , *Nb* and, on the l.h.s we have the self–term and the first r.h.s. term is the coupling term, i.e. the effect of the magnetic current **M***A*2 on the aperture *A*3. The second term

The presence of the coupling terms is the critical point in the multi–layer formulation. The key feature of the aperture approach, as previously illustrated, relies on the decoupling of the different regions. If we have a more general structure made up of N layers, we know that when dealing with the aperture *A*1, which lies in the bottom layer, we only have to consider the effect of the magnetic current on the aperture *A*2. When considering the aperture *AN*, the last one, we only have to consider the effect of the magnetic current on the aperture *AN*−1, and when considering any intermediate aperture *Ai*, for *i* = 2, ..., *N* − 1 we have to take into account only the effect of the apertures immediately below and above, i.e. *Ai*−1 and *Ai*+1.

where Y is the coefficient matrix, Mis the vector of the unknown coefficients, i.e. the magnetic current on the apertures, and T is the r.h.s, which represent the incident magnetic field on the

The solution of the above equation (12) yields the unknown induced magnetic current on the apertures. The reflection coefficient is then easily calculated from the magnetic current on the

It is important to stress out that inserting more layers will result in a larger matrix, affecting thus the overall computational time, yet for each additional layer only two blocks have to be

3,*t* + **H***A*3*A*2

**G**ˆ *A*2 *A*2 3 · ˆ **f** *A*2 *n* · **f** *A*2 *m* =

**G**ˆ *A*3 *A*3 4 · ˆ **f** *A*3 *n* · **f** *A*3 *m*

Y·M = T (12)

**G**ˆ *A*1 *A*2 2 · ˆ **f** *A*1 *n* · **f** *A*2 *m*

4,*t* + **H***inc*,*t* (10)

*A*3 *m* (*x*, *y*) *dS* (11)

3,*t* (8)

(9)

**H***A*2*A*2

**G**ˆ *A*3 *A*2 3 · ˆ **f** *A*3 *n* · **f** *A*2 *m* − *Nb* ∑ *n*=1 *mA*1 *n A*2 *A*2

For the last (third) discontinuity *A*3, the boundary condition leads to:

3,*t* + **H***A*2*A*3

**G**ˆ *A*3 *A*3 3 · ˆ **f** *A*3 *n* · **f** *A*3 *m* + *A*3 *A*3

**G**ˆ *A*2 *A*3 3 · ˆ **f** *A*2 *n* · **f** *A*3 *m* − *S* **H***inc* · **f**

**H***A*3*A*3

of the r.h.s represent the known terms of the MoM system.

Equations (7),(9),(11) can be written in matrix form:

2,*t* + **H***A*1*A*2

**G**ˆ *A*2 *A*2 2 · ˆ **f** *A*2 *n* · **f** *A*2 *m* + *A*2 *A*2

condition we obtain:

*Nb* ∑ *n*=1

*Nb* ∑ *n*=1 *mA*3 *n A*2 *A*2

*Nb* ∑ *n*=1

= − *Nb* ∑ *n*=1 *mA*2 *n A*3 *A*3

−*mA*3 *n A*3 *A*3

−*mA*2 *n A*2 *A*2

which leads to:

and we obtain:

last aperture *A*3.

upper layer.

computed.

The dielectric layers can be chosen with the desired physical parameter and a superstrate can also be used to cover the structure. The unit cell of the periodic printed surfaces on the different layers may be of different shape but they must have the same periodicity.

According to the equivalence theorem, each aperture is closed by a PEC, and two unknown magnetic current densities **M***Anm* and −**M***Anm* on the opposite sides are defined in order to guarantee the continuity of the tangential electric field. By the Floquet theorem, the equivalent currents on the (*n*, *m*)*th* cell of the periodic surface are connected to the central one by:

$$\mathbf{M}\_{A,nm} = e^{-j\beta nd\_x \chi} e^{-j\beta md\_y y} \mathbf{M}\_A \tag{2}$$

where **M***A* is the current in the central cell.

As a consequence, only the current in the central cell is the unknown of the problem, and need to be discretized as a linear combination of *Nb* basis functions:

$$\mathbf{M}\_A = \sum\_{n=1}^{N\_b} m\_n^A \mathbf{f}\_n \tag{3}$$

Then we obtain the MoM linear equation system by imposing the continuity of the tangential component of the magnetic field across each aperture. Again, by the Floquet theorem, this continuity needs to be forced only on the central cell (see Fig.**??**). It can be easily seen that, in the multilayered structure, we can distinguish three cases, namely an aperture lying on the first layer, on an intermediate layer or on the last layer.

For the first layer, i.e. for the discontinuity *A*1, the continuity of the transverse magnetic field can be written as:

$$\mathbf{H}\_{1,t}^{A\_1 A\_1} = \mathbf{H}\_{2,t}^{A\_1 A\_1} + \mathbf{H}\_{2,t}^{A\_2 A\_1} \tag{4}$$

which is then enforced in a weak form. A set of testing functions is selected and the boundary condition on the magnetic field is multiplied by each testing function and the result integrated over aperture. Choosing the Galerkin method, i.e. the testing functions used are the same basis functions used to express the unknown magnetic currents, we obtain for layer *A*1:

$$\int\_{A\_1} \mathbf{H}\_{1,t}^{A\_1 A\_1} \cdot \mathbf{f}\_{\mathfrak{m}} = \int\_{A\_1} \mathbf{H}\_{2,t}^{A\_1 A\_1} \cdot \mathbf{f}\_{\mathfrak{m}} + \int\_{A\_1} \mathbf{H}\_{2,t}^{A\_2 A\_1} \cdot \mathbf{f}\_{\mathfrak{m}\_{\ell}} \qquad m = 1, \dots, N\_b \tag{5}$$

which, introducing the Green Functions can be written as:

$$
\int\_{A\_1} \langle \mathbf{\hat{G}\_1^{A\_1 A\_1}}, -\mathbf{M}\_{A\_1} \rangle \cdot \mathbf{f}\_m = \int\_{A\_1} \langle \mathbf{\hat{G}\_2^{A\_1 A\_1}}, \mathbf{M}\_{A\_1} \rangle \cdot \mathbf{f}\_m + \\
$$

$$
\int\_{A\_1} \langle \mathbf{\hat{G}\_2^{A\_2 A\_1}}, \mathbf{M}\_{A\_2} \rangle \cdot \mathbf{f}\_m + \int\_{A\_1} \langle \mathbf{\hat{G}\_2^{A\_2 A\_1}}, \mathbf{M}\_{A\_2} \rangle \cdot \mathbf{f}\_m \qquad m = 1, \dots, N\_b \tag{6}
$$

Expressing the unknown magnetic currents in the form (3):

$$\sum\_{n=1}^{N\_b} -m\_n^{A\_1} \left( \int\_{A\_1} \left( \int\_{A\_1} \mathbf{G}\_1^{A\_1 A\_1} \cdot \mathbf{f}\_n^{A\_1} \right) \cdot \mathbf{f}\_m^{A\_1} + \int\_{A\_1} \left( \int\_{A\_1} \mathbf{G}\_2^{A\_1 A\_1} \cdot \mathbf{f}\_n^{A\_1} \right) \cdot \mathbf{f}\_m^{A\_1} \right)$$

$$= \sum\_{n=1}^{N\_b} m\_n^{A\_2} \int\_{A\_1} \left( \int\_{A\_1} \mathbf{\hat{G}}\_2^{A\_2 A\_1} \cdot \mathbf{f}\_n^{A\_2} \right) \cdot \mathbf{f}\_m^{A\_1}, \qquad m = 1, \dots, N\_b \tag{7}$$

where the l.h.s. term represents the self–term, i.e. the effect of the magnetic current **M***A*1 on the aperture *A*1 , and the r.h.s. term is the coupling term, i.e. the effect of the magnetic current **M***A*2 on the aperture *A*1.

For the second discontinuity *A*2, as for any intermediate aperture, enforcing the boundary condition we obtain:

$$\mathbf{H}\_{2,t}^{A\_2 A\_2} + \mathbf{H}\_{2,t}^{A\_1 A\_2} = \mathbf{H}\_{3,t}^{A\_2 A\_2} + \mathbf{H}\_{3,t}^{A\_3 A\_2} \tag{8}$$

which leads to:

10 Will-be-set-by-IN-TECH

The dielectric layers can be chosen with the desired physical parameter and a superstrate can also be used to cover the structure. The unit cell of the periodic printed surfaces on the

According to the equivalence theorem, each aperture is closed by a PEC, and two unknown magnetic current densities **M***Anm* and −**M***Anm* on the opposite sides are defined in order to guarantee the continuity of the tangential electric field. By the Floquet theorem, the equivalent currents on the (*n*, *m*)*th* cell of the periodic surface are connected to the central one by:

−*jβndx xe*

As a consequence, only the current in the central cell is the unknown of the problem, and need

*Nb* ∑ *n*=1

Then we obtain the MoM linear equation system by imposing the continuity of the tangential component of the magnetic field across each aperture. Again, by the Floquet theorem, this continuity needs to be forced only on the central cell (see Fig.**??**). It can be easily seen that, in the multilayered structure, we can distinguish three cases, namely an aperture lying on the

For the first layer, i.e. for the discontinuity *A*1, the continuity of the transverse magnetic field

which is then enforced in a weak form. A set of testing functions is selected and the boundary condition on the magnetic field is multiplied by each testing function and the result integrated over aperture. Choosing the Galerkin method, i.e. the testing functions used are the same basis functions used to express the unknown magnetic currents, we obtain for layer *A*1:

> *A*1

where the l.h.s. term represents the self–term, i.e. the effect of the magnetic current **M***A*1 on the aperture *A*1 , and the r.h.s. term is the coupling term, i.e. the effect of the magnetic current

2,*t* + **H***A*2*A*1

**H***A*2*A*1

2 , **M***A*1 � · **f***m* +

1,*t* = **H***A*1*A*1

 *A*1 �**G**ˆ *A*1 *A*1

 *A*1 �**G**ˆ *A*2 *A*1

*mA*

**M***A* =

−*jβmdy y***M***A* (2)

*n* **f***n* (3)

2,*t* (4)

2,*t* · **f***m*, *m* = 1, . . . , *Nb* (5)

2 , **M***A*2 � · **f***m*, *m* = 1, . . . , *Nb* (6)

*A*1 *m* , *m* = 1, . . . , *Nb* (7)

**G**ˆ *A*1 *A*1 2 · ˆ **f** *A*1 *n* · **f** *A*1 *m*

different layers may be of different shape but they must have the same periodicity.

**M***A*,*nm* = *e*

**H***A*1*A*1

**H***A*1*A*1 2,*t* · **f***m* +

**G**ˆ *A*1 *A*1 1 · ˆ **f** *A*1 *n* · **f** *A*1 *m* + *A*1 *A*1

**G**ˆ *A*2 *A*1 2 · ˆ **f** *A*2 *n* · **f**

to be discretized as a linear combination of *Nb* basis functions:

first layer, on an intermediate layer or on the last layer.

 *A*1

1 , −**M***A*1 � · **f***m* =

2 , **M***A*2 � · **f***m*, +

Expressing the unknown magnetic currents in the form (3):

which, introducing the Green Functions can be written as:

can be written as:

 *A*1

**H***A*1*A*1 1,*t* · **f***m* =

> *A*1 �**G**ˆ *A*1 *A*1

> *A*1 �**G**ˆ *A*2 *A*1

*Nb* ∑ *n*=1

= *Nb* ∑ *n*=1 *mA*2 *n A*1 *A*1

**M***A*2 on the aperture *A*1.

−*mA*1 *n A*1 *A*1

where **M***A* is the current in the central cell.

$$\sum\_{n=1}^{N\_b} -m\_n^{A\_2} \left( \int\_{A\_2} \left( \int\_{A\_2} \hat{\mathbf{G}}\_2^{A\_2 A\_2} \cdot \hat{\mathbf{f}}\_n^{A\_2} \right) \cdot \mathbf{f}\_m^{A\_2} + \int\_{A\_2} \left( \int\_{A\_2} \hat{\mathbf{G}}\_3^{A\_2 A\_2} \cdot \hat{\mathbf{f}}\_n^{A\_2} \right) \cdot \mathbf{f}\_m^{A\_2} \right) =$$

$$\sum\_{n=1}^{N\_b} m\_n^{A\_3} \int\_{A\_2} \left( \int\_{A\_2} \hat{\mathbf{G}}\_3^{A\_3 A\_2} \cdot \hat{\mathbf{f}}\_n^{A\_3} \right) \cdot \mathbf{f}\_m^{A\_2} - \sum\_{n=1}^{N\_b} m\_n^{A\_1} \int\_{A\_2} \left( \int\_{A\_2} \hat{\mathbf{G}}\_2^{A\_1 A\_2} \cdot \hat{\mathbf{f}}\_n^{A\_1} \right) \cdot \mathbf{f}\_m^{A\_2}$$

where *m* = 1, . . . , *Nb* and the l.h.s. term represents the self–term and the r.h.s. term the coupling terms, i.e. the effect of the magnetic currents **M***A*3 and **M***A*1 on the aperture *A*2.

For the last (third) discontinuity *A*3, the boundary condition leads to:

$$\mathbf{H}\_{3,t}^{A\_3A\_3} + \mathbf{H}\_{3,t}^{A\_2A\_3} = \mathbf{H}\_{4,t}^{A\_3A\_3} + \mathbf{H}\_{\text{inc},t} \tag{10}$$

and we obtain:

$$\sum\_{n=1}^{N\_b} -m\_n^{A\_3} \left( \int\_{A\_3} \left( \int\_{A\_3} \hat{\mathbf{G}}\_3^{A\_3 A\_3} \cdot \hat{\mathbf{f}}\_n^{A\_3} \right) \cdot \mathbf{f}\_m^{A\_3} + \int\_{A\_3} \left( \int\_{A\_3} \hat{\mathbf{G}}\_4^{A\_3 A\_3} \cdot \hat{\mathbf{f}}\_n^{A\_3} \right) \cdot \mathbf{f}\_m^{A\_3} \right)$$

$$= -\sum\_{n=1}^{N\_b} m\_n^{A\_2} \int\_{A\_3} \left( \int\_{A\_3} \hat{\mathbf{G}}\_3^{A\_2 A\_3} \cdot \hat{\mathbf{f}}\_n^{A\_2} \right) \cdot \mathbf{f}\_m^{A\_3} - \int\_S \mathbf{H}\_{\text{inc}} \cdot \mathbf{f}\_m^{A\_3} (\mathbf{x}, y) \, dS \tag{11}$$

where *m* = 1, . . . , *Nb* and, on the l.h.s we have the self–term and the first r.h.s. term is the coupling term, i.e. the effect of the magnetic current **M***A*2 on the aperture *A*3. The second term of the r.h.s represent the known terms of the MoM system.

The presence of the coupling terms is the critical point in the multi–layer formulation. The key feature of the aperture approach, as previously illustrated, relies on the decoupling of the different regions. If we have a more general structure made up of N layers, we know that when dealing with the aperture *A*1, which lies in the bottom layer, we only have to consider the effect of the magnetic current on the aperture *A*2. When considering the aperture *AN*, the last one, we only have to consider the effect of the magnetic current on the aperture *AN*−1, and when considering any intermediate aperture *Ai*, for *i* = 2, ..., *N* − 1 we have to take into account only the effect of the apertures immediately below and above, i.e. *Ai*−1 and *Ai*+1. Equations (7),(9),(11) can be written in matrix form:

$$
\mathcal{Y} \cdot \mathcal{M} = \mathcal{T} \tag{12}
$$

where Y is the coefficient matrix, Mis the vector of the unknown coefficients, i.e. the magnetic current on the apertures, and T is the r.h.s, which represent the incident magnetic field on the last aperture *A*3.

The solution of the above equation (12) yields the unknown induced magnetic current on the apertures. The reflection coefficient is then easily calculated from the magnetic current on the upper layer.

It is important to stress out that inserting more layers will result in a larger matrix, affecting thus the overall computational time, yet for each additional layer only two blocks have to be computed.

(a) Geometry

optimization stage cannot be known in advance.

Fig. 10. UC–EBG

Table 1. Data

Table 2. Results

the reflection coefficient is 2.2◦).

10 11 12 13 14 15 16 −200

(b) Reflection Coefficient

frequency

Synthesis of Planar EBG Structures Based on Genetic Programming 333

this approach is irrespective of the particular shape of the metallization, which in the GP

The GP optimization was launched to find the best structure, keeping as fixed parameters the periodicity, substrate dielectric constant and thickness, as shown in the following table.

**Fixed Parameters**

Periodicity (unit square cell) *dx* = 3.048 mm; *dy* = 3.048 mm

In Fig.11 and Fig.12 we can see two different runs of our optimization. And in Tab.2 it is

Itoh (lit.) 76.05◦ 20.52◦ -48.56◦ Case A 57.8◦ 2.2◦ -63.16◦ Case B 39.28◦ 0.83◦ -35.16◦ Case C 13.06◦ -6.13◦ -24.93◦

In Case A we used for the unit cell a discretization step of 0.1016mm, making the playground for the GP optimization a 30x30 grid. The penalty coefficients, both for the number of unknowns (i.e. borders of the discretized geometry) and for the bandwidth were initially not too strict, making the evolution process quickly evolve towards quite bulky structures. The resonant frequency was reached after few generations (for case A at gen. 13 the phase of

**Phase of Refl. Coeff. 14 GHz 14.2 GHz 14.4 GHz**

Discretization step 0.1016 mm (30 × 30 *grid*)

Center Frequency 14.2 GHz

Substrate dielectric constant *r* = 10.2 Substrate thickness 0.635 mm

shown how the goal is achieved, in comparison also to the reference structure.

phase of reflection coefficient [Deg]

The explicit evaluation of MoM system matrix Y requires the computation of the magnetic fields we previously introduced in (6),(8),(10). These are total magnetic fields,i.e. fields due to the magnetic currents of all the periodicity cells. Since the problem is linear, we start computing the magnetic field due to the magnetic current of a single cell. This field can be expressed in terms of a spectral domain (dyadic) Green function **G**ˆ (*u*, *v*) connecting the transverse magnetic current density on the aperture to the transverse magnetic field:

$$\mathbf{H}\_{t}^{\text{unit cell}}(\mathbf{x}, y) = \int\_{-\infty}^{+\infty} \int\_{-\infty}^{+\infty} \mathbf{\hat{G}}(u, v) \cdot \mathbf{\hat{M}}(u, v) \, e^{-j(ux + vy)} \, du \, dv \tag{13}$$

where **M** is the magnetic current on the aperture of the central cell.

We can then write the overall magnetic transverse field of the infinite planar array by summing (13) on the infinite cells of the AMC:

$$\mathbf{H}\_{l}(x,y) = \sum\_{r} \sum\_{s} e^{-jk\_{0}r d\_{x} \sin \theta \cos \phi} \ e^{-jk\_{0}s d\_{y} \sin \theta \sin \phi}.$$

$$\cdot \int\_{-\infty}^{+\infty} \int\_{-\infty}^{+\infty} \mathbf{\hat{G}}(u,v) \cdot \mathbf{\hat{M}}(u,v) \ e^{-j(ux+vy)} \, du \, dv \tag{14}$$

where *dx* and *dy* are the periodicity along the *x*–axis and the *y*–axis respectively, (*θ*, *φ*) is the direction of the incident field and *k*0 is the free–space propagation constant.

Equation (14) can be simplyfied taking into account the periodicity of the structure. The fields on either side of the AMC can be expanded in terms of the Floquet space harmonics (Munk, 2000). Therefore the cell summation in (14) can be rearranged (Pozar & Schaubert, 1984) using the Poisson formula. As a result, we obtain the following compact expression:

$$\mathbf{H}\_{l}(\mathbf{x},\mathbf{y}) = \frac{4\pi^{2}}{d\_{x}d\_{\mathcal{Y}}} \sum\_{r} \sum\_{s} \hat{\mathbf{G}}(\boldsymbol{u}\_{r\star}\boldsymbol{v}\_{s}) \cdot \mathbf{\hat{M}}(\boldsymbol{u}\_{r\star}\boldsymbol{v}\_{s}) \, e^{-j(\boldsymbol{u}\_{r\star}\boldsymbol{x}+\boldsymbol{v}\_{s}\boldsymbol{y})} \tag{15}$$

where the summation is now on the Floquet modes.

Equation (15) can be used for both fields (above and below the metallization) by using the pertinent Green function, which can be derived from the equivalent circuit shown in Fig.8(b) and setting **M** = ±**M***A* (current on the pertinent layer of the central cell).

#### **4. Single–layer EBG design results**

Significative and promising results have been found when this approach was applied to a single–layer EBG structure (Deias et al., 2009a).

One of the most interesting structures that can be found in literature is the Itoh's UC–EBG structure (Yang et al., 1999), shown in Fig.10(a) consisting of a Jerusalem cross aperture.

The reference structure has periodicity *dx* = *dy* = 3.048*mm*, *h* = 0.635*mm* and *ε* = 10.2, displaying a resonant frequency at 14.25GHz. The reflection coefficient is shown in Fig.10(b).

We decided to use these parameters as constraints for our test, in order to compare the results obtained using the GP approach with Itoh's UC–EBG.

We used for the full–wave analysis of the periodic structure our previously developed Fortran code, implementing the aperture approach described in the previous section. Furthermore

Fig. 10. UC–EBG

12 Will-be-set-by-IN-TECH

The explicit evaluation of MoM system matrix Y requires the computation of the magnetic fields we previously introduced in (6),(8),(10). These are total magnetic fields,i.e. fields due to the magnetic currents of all the periodicity cells. Since the problem is linear, we start computing the magnetic field due to the magnetic current of a single cell. This field can be expressed in terms of a spectral domain (dyadic) Green function **G**ˆ (*u*, *v*) connecting the

We can then write the overall magnetic transverse field of the infinite planar array by summing

−*jk*0*rdx* sin *θ* cos *φ e*

**G**ˆ (*u*, *v*) · **M**ˆ (*u*, *v*) *e*

where *dx* and *dy* are the periodicity along the *x*–axis and the *y*–axis respectively, (*θ*, *φ*) is the

Equation (14) can be simplyfied taking into account the periodicity of the structure. The fields on either side of the AMC can be expanded in terms of the Floquet space harmonics (Munk, 2000). Therefore the cell summation in (14) can be rearranged (Pozar & Schaubert, 1984) using

Equation (15) can be used for both fields (above and below the metallization) by using the pertinent Green function, which can be derived from the equivalent circuit shown in Fig.8(b)

Significative and promising results have been found when this approach was applied to a

One of the most interesting structures that can be found in literature is the Itoh's UC–EBG structure (Yang et al., 1999), shown in Fig.10(a) consisting of a Jerusalem cross aperture.

The reference structure has periodicity *dx* = *dy* = 3.048*mm*, *h* = 0.635*mm* and *ε* = 10.2, displaying a resonant frequency at 14.25GHz. The reflection coefficient is shown in Fig.10(b). We decided to use these parameters as constraints for our test, in order to compare the results

We used for the full–wave analysis of the periodic structure our previously developed Fortran code, implementing the aperture approach described in the previous section. Furthermore

**G**ˆ (*ur*, *vs*) · **M**ˆ (*ur*, *vs*) *e*

**G**ˆ (*u*, *v*) · **M**ˆ (*u*, *v*) *e*

−*jk*0*sdy* sin *θ* sin *φ* ·

−*j*(*ux*+*vy*) *du dv* (13)

−*j*(*ux*+*vy*) *du dv* (14)

−*j*(*urx*+*vsy*) (15)

transverse magnetic current density on the aperture to the transverse magnetic field:

 +∞ −∞

*e*

direction of the incident field and *k*0 is the free–space propagation constant.

the Poisson formula. As a result, we obtain the following compact expression:

∑*r* ∑*s*

and setting **M** = ±**M***A* (current on the pertinent layer of the central cell).

 +∞ −∞

where **M** is the magnetic current on the aperture of the central cell.

 +∞ −∞

**H***t*(*x*, *y*) = ∑*r* ∑*s*

**H***unit cell*

(13) on the infinite cells of the AMC:

*t* (*x*, *y*) =

· +∞ −∞

**H***t*(*x*, *y*) = 4*π*2

where the summation is now on the Floquet modes.

**4. Single–layer EBG design results**

single–layer EBG structure (Deias et al., 2009a).

obtained using the GP approach with Itoh's UC–EBG.

*dxdy*

this approach is irrespective of the particular shape of the metallization, which in the GP optimization stage cannot be known in advance.

The GP optimization was launched to find the best structure, keeping as fixed parameters the periodicity, substrate dielectric constant and thickness, as shown in the following table.

Table 1. Data

In Fig.11 and Fig.12 we can see two different runs of our optimization. And in Tab.2 it is shown how the goal is achieved, in comparison also to the reference structure.

Table 2. Results

In Case A we used for the unit cell a discretization step of 0.1016mm, making the playground for the GP optimization a 30x30 grid. The penalty coefficients, both for the number of unknowns (i.e. borders of the discretized geometry) and for the bandwidth were initially not too strict, making the evolution process quickly evolve towards quite bulky structures. The resonant frequency was reached after few generations (for case A at gen. 13 the phase of the reflection coefficient is 2.2◦).

13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6 14.8 15

Synthesis of Planar EBG Structures Based on Genetic Programming 335

[GHz]

(a) Case A (b) Case B (c) Case C

We can exploit this approach using a more complex fitness function in order to obtain, for example, larger bandwidth (Deias et al., 2009a),as shown in case B, or at more interesting

It is well known that there is a growing interest in antennas integrated with EBG surfaces for communication system applications, covering the 2.45 GHz and the 5 GHz wireless networking bands (Hung-Hsuan et al., 2007; Zhu & Langley, 2009). As proof of the effectiveness of our approach a simple EBG surface at this working frequencies was found

In Fig.16(b) we can see the reflection coefficient for the best individual shown Fig.16(a), as obtained after nine generations. The performance of our optimization is quite good leading

These results prove that the main advantage of the genetic approach is to be identified in the generalization of the "solution space", evolution being able to breed geometries that we could

to a simple and effective solution with a yet limited computational effort.

−150

Fig. 13. Reflection Coefficient.

Fig. 14. Best individual

frequencies (Deias et al., 2010).

with a yet limited computational effort.

Fixed parameters in this case are the following:

−100

case A case B case C Itoh

−50

0

Refl. Coeff. [°]

50

100

150

Fig. 12. Case B.

Fig. 11. Case A.

In Case B we then acted more stringently on the penalty coefficients. The structure remains spatially more limited, as the number of unknowns is kept reasonably low, and the bandwidth is largely improved. The number of generations in order to achieve a good result is still low.

As expected, the geometries resemble the reference one and other well–known configurations, since the only stringent requirement in this case is the resonant frequency. The fitness in this case is relatively simple, aimed almost uniquely to direct the evolution process towards a structure that resonates at the desired frequency. We then can insert an additional penalty coefficient in order to maximize the bandwidth, a secondary objective overlapped to the main goal.

Finally, as we can see in Fig.14(c), we allowed the geometry to touch the borders of the unit cell, thus considering a continuous aperture (while the geometry of the metallic patch will therefore be finite), and we found out a further improvement in the bandwidth. In Fig.15 the entire surface is shown.

Fig. 13. Reflection Coefficient.

14 Will-be-set-by-IN-TECH

(a) GEN.0 (b) GEN.4 (c) GEN.5

(d) GEN.8 (e) GEN.11 (f) GEN.12

(a) GEN.0 (b) GEN.2 (c) GEN.5

(d) GEN.6 (e) GEN.9 (f) GEN.11

In Case B we then acted more stringently on the penalty coefficients. The structure remains spatially more limited, as the number of unknowns is kept reasonably low, and the bandwidth is largely improved. The number of generations in order to achieve a good result is still low. As expected, the geometries resemble the reference one and other well–known configurations, since the only stringent requirement in this case is the resonant frequency. The fitness in this case is relatively simple, aimed almost uniquely to direct the evolution process towards a structure that resonates at the desired frequency. We then can insert an additional penalty coefficient in order to maximize the bandwidth, a secondary objective overlapped to the main

Finally, as we can see in Fig.14(c), we allowed the geometry to touch the borders of the unit cell, thus considering a continuous aperture (while the geometry of the metallic patch will therefore be finite), and we found out a further improvement in the bandwidth. In Fig.15 the

Fig. 11. Case A.

Fig. 12. Case B.

goal.

entire surface is shown.

Fig. 14. Best individual

We can exploit this approach using a more complex fitness function in order to obtain, for example, larger bandwidth (Deias et al., 2009a),as shown in case B, or at more interesting frequencies (Deias et al., 2010).

It is well known that there is a growing interest in antennas integrated with EBG surfaces for communication system applications, covering the 2.45 GHz and the 5 GHz wireless networking bands (Hung-Hsuan et al., 2007; Zhu & Langley, 2009). As proof of the effectiveness of our approach a simple EBG surface at this working frequencies was found with a yet limited computational effort.

Fixed parameters in this case are the following:

In Fig.16(b) we can see the reflection coefficient for the best individual shown Fig.16(a), as obtained after nine generations. The performance of our optimization is quite good leading to a simple and effective solution with a yet limited computational effort.

These results prove that the main advantage of the genetic approach is to be identified in the generalization of the "solution space", evolution being able to breed geometries that we could

(a) Geometry

Synthesis of Planar EBG Structures Based on Genetic Programming 337

2 2.2 2.4 2.6 2.8 3

[GHz]

(b) Reflection Coefficient

**Fixed Parameters**

Periodicity (unit square cell) *dx* = 3.048 mm; *dy* = 3.048 mm

This quite simple result proves that genetic programming can be efficiently adopted for this kind of design synthesis and optimization even in the case of complex structures consisting of multiple layers of EBG surfaces. Moreover the fitness function can be modified in order to

Discretization step 0.1524 mm (20 × 20 *grid*)

Center Frequency 2.45 GHz

Substrate dielectric constant *r* = 10.2 Substrate thickness 3.354772 mm

−180 −140 −100 −60 −20 20 60 100 140 180

take into consideration other requirements.

Fig. 16

Table 4. Data

Refl. Coeff. [°]

Fig. 15. EBG Surface (Case C)

Table 3. Data

never think of. Evolutionary computation on the other hand let us scope through geometries as though they were human–described.

#### **5. Multi–layer EBG design results**

The periodic surfaces in the single–layer configuration shown so far exhibit perfect reflection or transmission only at resonance. However, many applications, exploiting the filtering properties of frequency selective surfaces, seek for a resonant curve with a flat top and faster roll off. This goal can be achieved by using two or more periodic surfaces cascaded with dielectric slabs sandwiched in between . By using multiple layers of frequency selective surfaces as part of the substrate we operate in a manner similar to that used for designing broad–band microwave filters. With such a configuration typically we can obtain a bandwidth that is considerably larger than that of a single structure.

Genetic Programming strategy in conjunction with the flexible aperture approach previously described proved to be effective in the design and optimization of EBG surfaces in the case of a two–layer structure with the same geometry in both layers. When considering more layers or different geometries we only have to take into consideration an increased computational effort.

In the case of a two–layer configuration with the following fixed parameters:

We obtained the following geometry after two generations, each generation with a population of 150 individuals:

(a) Geometry

(b) Reflection Coefficient

#### Fig. 16

16 Will-be-set-by-IN-TECH

**Fixed Parameters**

Periodicity (unit square cell) *dx* = 40 mm; *dy* = 40 mm

Discretization step 0.8 mm (50 × 50 *grid*)

never think of. Evolutionary computation on the other hand let us scope through geometries

The periodic surfaces in the single–layer configuration shown so far exhibit perfect reflection or transmission only at resonance. However, many applications, exploiting the filtering properties of frequency selective surfaces, seek for a resonant curve with a flat top and faster roll off. This goal can be achieved by using two or more periodic surfaces cascaded with dielectric slabs sandwiched in between . By using multiple layers of frequency selective surfaces as part of the substrate we operate in a manner similar to that used for designing broad–band microwave filters. With such a configuration typically we can obtain a bandwidth

Genetic Programming strategy in conjunction with the flexible aperture approach previously described proved to be effective in the design and optimization of EBG surfaces in the case of a two–layer structure with the same geometry in both layers. When considering more layers or different geometries we only have to take into consideration an increased computational

We obtained the following geometry after two generations, each generation with a population

In the case of a two–layer configuration with the following fixed parameters:

Center Frequency 2.45 GHz

Substrate dielectric constant *r* = 1.38 Substrate thickness 2.2 mm

Fig. 15. EBG Surface (Case C)

as though they were human–described.

**5. Multi–layer EBG design results**

that is considerably larger than that of a single structure.

Table 3. Data

effort.

of 150 individuals:

#### Table 4. Data

This quite simple result proves that genetic programming can be efficiently adopted for this kind of design synthesis and optimization even in the case of complex structures consisting of multiple layers of EBG surfaces. Moreover the fitness function can be modified in order to take into consideration other requirements.

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#### Fig. 17

#### **6. Acknowledgements**

Work partially supported by Regione Autonoma della Sardegna, under contract CRP1\_511: "Valutazione e utilizzo della Genetic Programming nel progetto di strutture a radiofrequenza e microonde".

#### **7. References**

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18 Will-be-set-by-IN-TECH

(a) Geometry

9 10 11 12 13 14

[GHz]

(b) Reflection Coefficient

Work partially supported by Regione Autonoma della Sardegna, under contract CRP1\_511: "Valutazione e utilizzo della Genetic Programming nel progetto di strutture a radiofrequenza

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−200

−150

−100

−50

Refl. Coeff. [°]

Fig. 17

e microonde".

**7. References**

**6. Acknowledgements**

2): 641–644.

0

50

100

150

200

**1. Introduction**

importance.

Berkov8, E. Semenova8 and N. Gorn8

*Dipartimento di Fisica, Perugia, Italy*

*Garching bei Muenchen, Germany*

*8Innovent Technology Development, Jena, Germany*

*3CNISM, Unità di Perugia and Dipartimento di Fisica, Perugia, Italy 4Dipartimento di Fisica and CNISM, Università di Ferrara, Ferrara, Italy 5Faculty of Physics, Adam Mickiewicz University, Poznan, Poland 6 H. H. Wills Physics Laboratory, Bristol, United Kingdom*

**14**

**Magnonic Metamaterials**

A large proportion of the recent growth of the volume of electromagnetics research has been associated with the emergence of so called electromagnetic metamaterials1 and the discovered ability to design their unusual properties2,3 by tweaking the geometry and structure of the constituent "meta-atoms"4. For example, negative permittivity and negative permeability can be achieved, leading to negative refractive index metamaterials2. The negative permeability could be obtained via geometrical control of high frequency currents, e.g. in arrays of split ring resonators5, or alternatively one could rely on spin resonances in natural magnetic materials6,7, as was suggested by Veselago in Ref. 2. The age of nanotechnology therefore sets an intriguing quest for additional benefits to be gained by structuring natural magnetic materials into so called *magnonic* metamaterials, in which the frequency and strength of resonances based on spin waves (magnons)8 are determined by the geometry and magnetization configuration of meta-atoms. Spin waves can have frequencies of up to hundreds of GHz (in the exchange dominated regime)6-9 and have already been shown to play an important role in the high frequency magnetic response of composites10-14. Moreover, in view of the rapid advances in the field of magnonics9,15,16,, which in particular promises devices employing propagating spin waves, the appropriate design of magnonic metamaterials with properties defined with respect to propagating spin waves rather than electromagnetic waves acquires an independent and significant

\*M. Dvornik1, R.V. Mikhaylovskiy1, O. Dmytriiev1, G. Gubbiotti2,3, S. Tacchi3, M. Madami3, G. Carlotti3, F. Montoncello4, L. Giovannini4, R. Zivieri4, J.W. Klos5, M.L. Sokolovskyy5, S. Mamica5, M. Krawczyk5, M. Okuda6, J.C. Eloi6, S. Ward Jones6, W. Schwarzacher6, T. Schwarze7, F. Brandl7, D. Grundler7, D.V.

2*Istituto Officina dei Materiali del Consiglio Nazionale delle Ricerche (IOM-CNR), Sede di Perugia, c/o*

*7Lehrstuhl fuer Physik funktionaler Schichtsysteme, Physik Department, Technische Universität München,*

V.V. Kruglyak et al.\*

*University of Exeter, Exeter,*

*School of Physics,*

*United Kingdom*

### **Magnonic Metamaterials**

### V.V. Kruglyak et al.\*

*School of Physics, University of Exeter, Exeter, United Kingdom*

#### **1. Introduction**

20 Will-be-set-by-IN-TECH

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A large proportion of the recent growth of the volume of electromagnetics research has been associated with the emergence of so called electromagnetic metamaterials1 and the discovered ability to design their unusual properties2,3 by tweaking the geometry and structure of the constituent "meta-atoms"4. For example, negative permittivity and negative permeability can be achieved, leading to negative refractive index metamaterials2. The negative permeability could be obtained via geometrical control of high frequency currents, e.g. in arrays of split ring resonators5, or alternatively one could rely on spin resonances in natural magnetic materials6,7, as was suggested by Veselago in Ref. 2. The age of nanotechnology therefore sets an intriguing quest for additional benefits to be gained by structuring natural magnetic materials into so called *magnonic* metamaterials, in which the frequency and strength of resonances based on spin waves (magnons)8 are determined by the geometry and magnetization configuration of meta-atoms. Spin waves can have frequencies of up to hundreds of GHz (in the exchange dominated regime)6-9 and have already been shown to play an important role in the high frequency magnetic response of composites10-14. Moreover, in view of the rapid advances in the field of magnonics9,15,16,, which in particular promises devices employing propagating spin waves, the appropriate design of magnonic metamaterials with properties defined with respect to propagating spin waves rather than electromagnetic waves acquires an independent and significant importance.

\*M. Dvornik1, R.V. Mikhaylovskiy1, O. Dmytriiev1, G. Gubbiotti2,3, S. Tacchi3, M. Madami3, G. Carlotti3, F. Montoncello4, L. Giovannini4, R. Zivieri4, J.W. Klos5, M.L. Sokolovskyy5, S. Mamica5, M. Krawczyk5, M. Okuda6, J.C. Eloi6, S. Ward Jones6, W. Schwarzacher6, T. Schwarze7, F. Brandl7, D. Grundler7, D.V. Berkov8, E. Semenova8 and N. Gorn8

2*Istituto Officina dei Materiali del Consiglio Nazionale delle Ricerche (IOM-CNR), Sede di Perugia, c/o Dipartimento di Fisica, Perugia, Italy*

*7Lehrstuhl fuer Physik funktionaler Schichtsysteme, Physik Department, Technische Universität München, Garching bei Muenchen, Germany*

*8Innovent Technology Development, Jena, Germany*

Magnonic Metamaterials 343

Some metamaterials (e.g. arrays of voids) are simpler to consider as created by geometrical modification of previously continuous materials. This however does not exclude their treatment as if they were constructed from building blocks considered above. Also, one has to note that, from the point of view of electromagnetic waves, vacuum is a continuous material in that important sense that it supports propagation of electromagnetic waves. Therefore, even arrays of entirely disconnected blocks are metamaterials for electromagnetic

Generally, various dynamical properties of metamaterials can be considered relative to excitations with wavelength either comparable to or much greater than the characteristic size of the building blocks. The former case is associated with studies of artificial "band gap crystals", e.g. photonic17, plasmonic18, phononic19, and magnonic20 crystals. The latter case allows one to study the metamaterials as some effectively continuous media. The physical object under study is nonetheless the same in both cases. For example, metamaterials with artificial periodic modulation of the refractive index with periodicity comparable to the wavelength of electromagnetic waves in the visible range are known as photonic band gap structures, using which it is possible to enable propagation of light in particular directions, to localize it in chosen channels or zones, or even to completely prohibit its propagation. In the microwave frequency range, the same structures would behave as effectively continuous

In a particular frequency region of interest, the same metamaterials can behave differently with respect to excitations of different kinds, e.g. electromagnetic, sound or spin waves. Attractive opportunities arise from use of one excitation, with respect to which the sample behaves as a discrete structure (e. g. a band gap metamaterial), to design a resonance feature for another excitation, with respect to which the structure behaves as quasi-continuous metamaterial. For example, plasmonic resonances can be used to alter effective electromagnetic properties in THz to visible frequency range. Magnonic and LC resonances can be used for the same purpose in GHz-THz and GHz frequency ranges, respectively.

A thorough reader will probably find that, although term "metamaterial" emerged recently as a result of ever increasing pressure to spice up funding applications and publications in high profile journals, the underpinning research can be sometimes traced back to the middle or even beginning of the 20th century. Yet, one can also argue that it is the recent advances in nanofabrication and experimental tools that justify the boom by allowing researchers to revisit the established notions at reduced length scales and to apply results to modern

The main reason for electrically neutral atoms to be able to interact with electromagnetic radiation is that they themselves possess electrical structure. Indeed, each atom consists of positively charged core and negatively charged electrons. The core and electrons are not rigidly coupled, and so, the atom can be electrically polarised by and thereby interact with

However, the electric charge is not the only property responsible for the interaction of atoms or materials with electromagnetic radiation. In particular, spins and associated magnetic

**3. Spin waves and effectively continuous magnonic metamaterials**

waves. This distinguishes electromagnetic waves from the other excitations.

materials.

technology.

an external electric field.

In this Chapter, we review the recent advances in studies of magnonic metamaterials and challenges that have to be overcome in order for the rich opportunities of exploitation of such metamaterials to be realised. We start from a discussion of the notion of a metamaterial and its relevance to areas beyond that of common electromagnetics. We introduce band gap and effectively continuous magnonic metamaterials and demonstrate their properties and functionalities, presenting either experimental or theoretical evidence as appropriate for the illustration.

#### **2. What makes a metamaterial?**

To answer the question in the title in the most general sense (i.e. going beyond the topic of more common electromagnetic metamaterials), one has to learn first what makes standard (i.e., "nature-made") *materials*. One talks about a material when many atoms or ions are joined together to collectively create a novel quality with properties that are not observed in the constituent atoms in isolation. For example, isolated atoms have discrete electronic energy levels and associated discrete electromagnetic spectra. In a material consisting of many of such atoms, each discrete electronic level is split into a continuous "electronic band", and accordingly the electromagnetic spectrum of the material is also continuous.

As a result of joining into a material, atoms also collectively acquire new properties that are less trivially connected with those of isolated atoms. Firstly, different classes of materials are formed, including e.g. dielectrics, semiconductors, or metals, ferro-, piezo-, or segnetoelectrics, dia-, para-, antiferro- or ferro-magnets, superconductors, materials with a range of different mechanical properties, etc. Secondly, new classes of waves (excitations) and their quanta (so called quasi-particles) are observed, including plasmons, magnons, phonons, excitons etc. One can also distinguish many "hybrids" of the solid state quasi-particles among themselves as well as with photons, e.g. surface plasmon-polaritons, magnetoexcitons etc.

A material does not have to have a periodic arrangement of atoms and can be amorphous or quasi-crystalline. The classification of materials and their properties often does not have strict boundaries, while the differences are often quantitative. For example, dielectrics and semiconductors are different only by the size of the band gap; the same material can be ferromagnetic below and paramagnetic above a certain temperature, etc. Some electronic energy levels can remain discrete, therefore retaining their atomistic character. Materials can also consist of rather complex building blocks, e.g. molecules, crystal unit cells (often consisting of many atoms or ions), or crystallites. The most complex nature-made (truly functional!) materials – biological tissues – consist of living cells composed of millions of atoms.

This brings us to the idea of a *metamaterial* – an artificial assembly of "man-made" building blocks with tailored properties. Living aside static properties and biological tissues, the most practical ways to "tailor" properties of the building blocks are via their geometrical shaping and compositional modulation. The former leads to confinement of solid state excitations (e.g. those listed above) and hence to formation of a discrete spectrum of the allowed modes. Brought sufficiently close together, the blocks can form a metamaterial with the discrete spectrum split into a band structure.

In this Chapter, we review the recent advances in studies of magnonic metamaterials and challenges that have to be overcome in order for the rich opportunities of exploitation of such metamaterials to be realised. We start from a discussion of the notion of a metamaterial and its relevance to areas beyond that of common electromagnetics. We introduce band gap and effectively continuous magnonic metamaterials and demonstrate their properties and functionalities, presenting either experimental or theoretical evidence as appropriate for the

To answer the question in the title in the most general sense (i.e. going beyond the topic of more common electromagnetic metamaterials), one has to learn first what makes standard (i.e., "nature-made") *materials*. One talks about a material when many atoms or ions are joined together to collectively create a novel quality with properties that are not observed in the constituent atoms in isolation. For example, isolated atoms have discrete electronic energy levels and associated discrete electromagnetic spectra. In a material consisting of many of such atoms, each discrete electronic level is split into a continuous "electronic band", and accordingly the electromagnetic spectrum of the material is also continuous.

As a result of joining into a material, atoms also collectively acquire new properties that are less trivially connected with those of isolated atoms. Firstly, different classes of materials are formed, including e.g. dielectrics, semiconductors, or metals, ferro-, piezo-, or segnetoelectrics, dia-, para-, antiferro- or ferro-magnets, superconductors, materials with a range of different mechanical properties, etc. Secondly, new classes of waves (excitations) and their quanta (so called quasi-particles) are observed, including plasmons, magnons, phonons, excitons etc. One can also distinguish many "hybrids" of the solid state quasi-particles among themselves as well as with photons, e.g. surface plasmon-polaritons, magneto-

A material does not have to have a periodic arrangement of atoms and can be amorphous or quasi-crystalline. The classification of materials and their properties often does not have strict boundaries, while the differences are often quantitative. For example, dielectrics and semiconductors are different only by the size of the band gap; the same material can be ferromagnetic below and paramagnetic above a certain temperature, etc. Some electronic energy levels can remain discrete, therefore retaining their atomistic character. Materials can also consist of rather complex building blocks, e.g. molecules, crystal unit cells (often consisting of many atoms or ions), or crystallites. The most complex nature-made (truly functional!) materials – biological tissues – consist of living cells composed of millions of

This brings us to the idea of a *metamaterial* – an artificial assembly of "man-made" building blocks with tailored properties. Living aside static properties and biological tissues, the most practical ways to "tailor" properties of the building blocks are via their geometrical shaping and compositional modulation. The former leads to confinement of solid state excitations (e.g. those listed above) and hence to formation of a discrete spectrum of the allowed modes. Brought sufficiently close together, the blocks can form a metamaterial with the discrete

illustration.

excitons etc.

atoms.

spectrum split into a band structure.

**2. What makes a metamaterial?**

Some metamaterials (e.g. arrays of voids) are simpler to consider as created by geometrical modification of previously continuous materials. This however does not exclude their treatment as if they were constructed from building blocks considered above. Also, one has to note that, from the point of view of electromagnetic waves, vacuum is a continuous material in that important sense that it supports propagation of electromagnetic waves. Therefore, even arrays of entirely disconnected blocks are metamaterials for electromagnetic waves. This distinguishes electromagnetic waves from the other excitations.