where / *a a f dD* is the AF filling ratio, and / *i i f dD* is IC filling ratio with *Dd d i a* as a bi-layer thickness.

Let us consider an incident plane electromagnetic wave propagating in the x-y plane as shown in Fig.11. Such a wave can be divided into two polarizations, a TE mode with its electric field parallel to axis z and a TM mode with its magnetic field parallel to axis z. According to Maxwell's equations, these wave vectors and frequencies of the two modes inside the film satisfy the following expressions

$$\frac{k\_y^2}{\varepsilon\_{zz}^\varepsilon \mu\_{xx}^\varepsilon} + \frac{k\_x^2}{\varepsilon\_{zz}^\varepsilon \mu\_{yy}^\varepsilon} = (o \,/\, c)^2 \quad \left(\text{for TE mode}\right). \tag{50}$$

$$k\_y^2 + k\_x^2 = \left(\alpha/c\right)^2 \varepsilon\_{xx}^\varepsilon \mu\_{zz}^\varepsilon \left(for \text{ TE mode}\right),\tag{51}$$

Since 1 *e zz* is a positive real number, relation (51) corresponds to the case of an ordinary optical (when 0 *e xx* ) or an opaque (when 0 *e xx* , contributing to an imaginary *y k* ) film, and so, we no longer consider the TM case, but deal with only the case of TE incident mode, and found the left-handed feature and negative refraction behavior. For the TE mode, the presence of left-handed feature (or negative refraction) needs the satisfaction of the prerequisite condition that 0 *e zz* and *e xx* and *e yy* can not be simultaneously positive [i.e., at least one of *e xx* and *e yy* is negative, see (50)]. According to expressions (48) and (49),

$$
\rho\_{\Gamma} \le \alpha \le \alpha\_{\Gamma} \sqrt{1 + f\_i(\varepsilon\_0 - \varepsilon\_a) / (f\_a \varepsilon\_a + f\_i \varepsilon\_a)} \text{ (for } \varepsilon\_{zz}^{\varepsilon} < 0\text{)}.\tag{52}
$$

$$
\Delta o\_r \le o \le \sqrt{o\_r^2 + 2f\_a o\_a o\_m} \text{ (for } \mu\_{\text{xx}}^\epsilon < 0\text{)}\,\tag{53}
$$

$$\sqrt{\alpha\_r^2 + 2f\_i \alpha\_a \alpha\_m} \le \alpha \le \sqrt{\alpha\_r^2 + 2\alpha\_a \alpha\_m} \text{ (for } \mu\_{yy}^\varepsilon < 0\text{)}\,\tag{54}$$

where *a a H* and 0 4 *m M* . Frequency region (52) is very large and covers regions (53) and (54) for the selected physical parameters. It is noted that both 0 *e xx* and 0 *e yy*

can occur simultaneously only when AF layers are thicker than IC layers, which corresponds to spectral domain 2 1/2 2 1/2 (2 ) (2 ) *r iam r aam f f* .The wave electric field in the film can be written at

$$E\_z = \left[ A\_0 \exp(ik\_y y) + B\_0 \exp(-ik\_y y) \right] \exp(ik\_x x - i\alpha t),\tag{55}$$

**H** (56)

and the corresponding magnetic field can be given by

226 Ferromagnetic Resonance – Theory and Applications

permeability *e*

where / *a a*

Since 1 *e zz*

optical (when 0 *e*

at least one of *e*

where

*a a*

*xx*

prerequisite condition that 0 *e*

*xx* and *e*

bi-layer thickness.

and dielectric permittivity *e*

*f dD* is the AF filling ratio, and / *i i*

2 2

*k k*

*ee ee zz xx zz yy*

) or an opaque (when 0 *e*

and *e*

>

(53) and (54) for the selected physical parameters. It is noted that both 0 *e*

*zz*

>

*yy*

 *H* and 0 4

*y x xx zz k k*

inside the film satisfy the following expressions

**3. Presence of left-handedness and negative refraction of AF/IC PCs**

In the previous section, we have discussed MPPs in AF/IC PCs with the TMM and EMM for one- and two-dimension. Based on FeF2/TIBr, there are a number of surface and bulk polaritons in which the negative refraction and left-handedness can appear. In order to investigate the formation mechanism of LHM in AF/IC PCs, the external magnetic field and magnetic damping is set to be zero. In this case, according Eqs.(7) and (8), the effective

( ), / ( ), 1, *ee e xx a i yy a i zz*

/ ( ), ( ), *e e e xx yy a i a i i a zz a a i i*

Let us consider an incident plane electromagnetic wave propagating in the x-y plane as shown in Fig.11. Such a wave can be divided into two polarizations, a TE mode with its electric field parallel to axis z and a TM mode with its magnetic field parallel to axis z. According to Maxwell's equations, these wave vectors and frequencies of the two modes

2 ( /) , *y x*

22 2 ( ) / , *e e*

*xx*

and so, we no longer consider the TM case, but deal with only the case of TE incident mode, and found the left-handed feature and negative refraction behavior. For the TE mode, the presence of left-handed feature (or negative refraction) needs the satisfaction of the

is a positive real number, relation (51) corresponds to the case of an ordinary

*yy*

>

0 1 ( )/( ) ( , 0) *e*

2 2 ( 0), *e*

is negative, see (50)]. According to expressions (48) and (49),

*f f f for* (52)

*f for* (53)

 *f for* (54)

*m M* . Frequency region (52) is very large and covers regions

*xx* and *e*

*T Ti aa i zz*

*r r a a m xx*

2 2 2 2 ( 0), *e r i a m r a m yy*

will be described as

 *f f ff* (48)

>

(50)

*c for TE m do e* (51)

, contributing to an imaginary *y k* ) film,

can not be simultaneously positive [i.e.,

*xx*

and 0 *e*

*yy*

*for T*

*c E mode*

 *ff ff* (49)

*f dD* is IC filling ratio with *Dd d i a* as a

1 . *z z e e xx yy*

*iyx*

 **x y e e**

  *E E*

**Figure 11.** Draft for incidence, reflection, refraction and transmission rays. After Wang & Song, 2009.

The radiation in the film consists of two parts, one is the forward light (refraction light) related to amplitude *A*0 and the other is the backward light (reflection light) related to amplitude 0 *B* . Here, *y k* is defined as a negative number, otherwise the refraction wave corresponds to amplitude 0 *B* . These two situations are equivalent in essence. According to the definition of energy flow density of electromagnetic wave \* **S EH** Re( ) / 2 , the flow densities of the two lights can be given by

$$\mathbf{S}\_{\mathbf{A}} = \text{Re}\left[\frac{\left|A\_{0}\right|^{2}}{2\alpha\nu}(\frac{k\_{\text{x}}}{\mu\_{yy}^{\varepsilon}}\mathbf{e}\_{\mathbf{x}} + \frac{k\_{\text{y}}}{\mu\_{\text{xx}}^{\varepsilon}}\mathbf{e}\_{\mathbf{y}})\right], \mathbf{S}\_{\mathbf{B}} = \text{Re}\left[\frac{\left|B\_{0}\right|^{2}}{2\alpha\nu}(\frac{k\_{\text{x}}}{\mu\_{yy}^{\varepsilon}}\mathbf{e}\_{\mathbf{x}} - \frac{k\_{\text{y}}}{\mu\_{\text{xx}}^{\varepsilon}}\mathbf{e}\_{\mathbf{y}})\right].\tag{57}$$

The inner product between a wave vector ( [ , ,0] *x y k k* **KA** or [ , ,0] *x y k k* **KB** ) and its corresponding energy flow is given by expression 2 Re( ) / 2 0 *e A zz I A* , or 2 Re( ) / 2 0 *e B zz I B* . Thus the angle between energy flow and wave vector can be expressed as

$$\alpha\_{j} = \arccos[I\_{j}/(\left|\mathbf{K}\_{j}\right| \left|\mathbf{S}\_{j}\right|)].\tag{58}$$

where *j*=*A* or *B*. It is obvious that *A* is equal to *B* and larger than / 2 in the range (52). It can be seen from the expression (57) of **AS** that its *x* component is negative and *y*  component is positive when both 0 *e xx* and 0 *e yy* , since *x k* is positive and identical to the equivalent component of the incident wave vector. Thus an important condition is found for the existence of negative refraction, or AF layers must be thicker than IC layers. The refraction angle can be expressed as

$$\theta' = \arctan(k\_x \mu\_{xx}^\varepsilon / k\_y \mu\_{yy}^\varepsilon). \tag{59}$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 229

[43]. The matrix

. According to the boundary conditions of *z E*

(61)

*kd i kd* (62)

*k d* (63)

*k d* (64)

**Figure 13.** Refraction angle versus frequency for various incident angles, and for filling ratios (a)

**4. Transmission, refraction and absorption properties of AF/IC PCs**

In this section, we shall examine transmission, refraction and absorption of AF/IC PCs, where the condition of the period much smaller than the wavelength is not necessary. The transmission spectra based on FeF2/TIBr PCs reveal that there exist two intriguing guided modes in a wide stop band [11]. Additionally, FeF2/TIBr PCs possess either the negative refraction or the quasi left-handedness, or even simultaneously hold them at certain frequencies of two guided modes, which require both negative magnetic permeability of AF layers and negative permittivity of IC layers. The handedness and refraction properties of the system can be manipulated by modifying the external magnetic field which will

The geometry is shown in Fig. 1. We assume the electric field solutions in AF and IC layers

[ exp( ) exp( )]exp( ), *j zj j j j x E e A ik y B ik y ik x i t*

where *j a i*, signify AF or IC layers, respectively. The corresponding magnetic field

and *Hx* continuous at interfaces, the relation between wave amplitudes in the two same

11 [( )cos( ) (1 )sin( )]/ ( ), *a i i i i T*

12 (1 )sin( ) / ( ), *a i i T i*

21 ( 1)sin( ) / ( ), *a i i T i*

1 2

2

layers of the two adjacent periods can be shown as a transfer matrix T

*f* . After Wang & Song, 2009.

determine the frequency regimes of the guided modes.

solutions are also achieved via *j j E iB*

elements are expressed by the following equations:

0.8 *a*

as

*f* and (b) 0.6 *a*

FeF2 and TlBr are used as constituent materials where the AF resonant frequency *r* is closer to the phonon resonant frequency *T* and located in the far infrared regime. Fig.12 shows this angle as a function of frequency for 0.8 *a f* and 0.6. It can be seen from Fig.12 that for most of the frequency range occupied by the curves, angle is at least bigger than 160*o* for various incident angles. So the wave vector, electric and magnetic fields form an approximate left-handed triplet. The operation frequency range becomes narrow as *a f* decreases, as shown in Fig.12b.

As shown in Fig.13(a) for 0.8 *a f* , the refraction angle is positive on the left side and negative on the right side of the intersection point of the curves. This corresponds to the following critical frequency obtained under the condition of 0 *e xx* and 0 *e yy* :

$$
\alpha o\_c = \sqrt{o\_r^2 + 2f\_i o\_m o\_a} \,. \tag{60}
$$

It can be seen from Fig.13(b) in comparing with Fig.13(a) that the frequency region of negative refraction is obviously narrower and the negative refraction angle becomes smaller. Numerical simulations also show both positive and negative refraction angles are in the spectral range of approximate left-handed feature shown in Fig.12.

**Figure 12.** Angle beteen refraction energy flow and corresponding wave vector versus frequency for different incident angles and for filling ratios (a) 0.8 *a f* and (b) 0.6 *a f* . After Wang & Song, 2009.

component is positive when both 0 *e*

closer to the phonon resonant frequency

decreases, as shown in Fig.12b.

**Figure 12.** Angle

different incident angles and for filling ratios (a) 0.8 *a*

As shown in Fig.13(a) for 0.8 *a*

refraction angle can be expressed as

It can be seen from the expression (57) of **AS** that its *x* component is negative and *y*

the equivalent component of the incident wave vector. Thus an important condition is found for the existence of negative refraction, or AF layers must be thicker than IC layers. The

>

160*o* for various incident angles. So the wave vector, electric and magnetic fields form an approximate left-handed triplet. The operation frequency range becomes narrow as *a*

negative on the right side of the intersection point of the curves. This corresponds to the

2 2 . *c r ima*

It can be seen from Fig.13(b) in comparing with Fig.13(a) that the frequency region of negative refraction is obviously narrower and the negative refraction angle becomes smaller. Numerical simulations also show both positive and negative refraction angles are in the

beteen refraction energy flow and corresponding wave vector versus frequency for

*f* and (b) 0.6 *a*

FeF2 and TlBr are used as constituent materials where the AF resonant frequency

*yy*

*x xx y yy*

, since *x k* is positive and identical to

*r* is

*f*

is at least bigger than

*k k* (59)

*f* , the refraction angle is positive on the left side and

*xx*

*T* and located in the far infrared regime. Fig.12

and 0 *e*

*f* (60)

*yy* :

*f* . After Wang & Song, 2009.

*f* and 0.6. It can be seen from Fig.12

and 0 *e*

*xx*

arctan( / ). *e e*

shows this angle as a function of frequency for 0.8 *a*

that for most of the frequency range occupied by the curves, angle

following critical frequency obtained under the condition of 0 *e*

spectral range of approximate left-handed feature shown in Fig.12.

**Figure 13.** Refraction angle versus frequency for various incident angles, and for filling ratios (a) 0.8 *a f* and (b) 0.6 *a f* . After Wang & Song, 2009.

### **4. Transmission, refraction and absorption properties of AF/IC PCs**

In this section, we shall examine transmission, refraction and absorption of AF/IC PCs, where the condition of the period much smaller than the wavelength is not necessary. The transmission spectra based on FeF2/TIBr PCs reveal that there exist two intriguing guided modes in a wide stop band [11]. Additionally, FeF2/TIBr PCs possess either the negative refraction or the quasi left-handedness, or even simultaneously hold them at certain frequencies of two guided modes, which require both negative magnetic permeability of AF layers and negative permittivity of IC layers. The handedness and refraction properties of the system can be manipulated by modifying the external magnetic field which will determine the frequency regimes of the guided modes.

The geometry is shown in Fig. 1. We assume the electric field solutions in AF and IC layers as

$$\vec{E}\_{j} = \vec{e}\_{z} \{ A\_{j} \exp(ik\_{j}y) + B\_{j} \exp(-ik\_{j}y) \} \exp(ik\_{x}x - i\alpha t),\tag{61}$$

where *j a i*, signify AF or IC layers, respectively. The corresponding magnetic field solutions are also achieved via *j j E iB* . According to the boundary conditions of *z E* and *Hx* continuous at interfaces, the relation between wave amplitudes in the two same layers of the two adjacent periods can be shown as a transfer matrix T [43]. The matrix elements are expressed by the following equations:

$$T\_{11} = \delta\_a \left[ (\Delta - \Delta') \cos(k\_i d\_j) + i(1 - \Delta \Delta') \sin(k\_i d\_j) \right] / (\Delta - \Delta'),\tag{62}$$

$$T\_{12} = \mathrm{i}\,\delta\_a^{-1} (1 - \Delta'^2) \sin(k\_i d\_i) / (\Delta - \Delta'),\tag{63}$$

$$T\_{21} = \mathrm{i}\mathcal{S}\_{\underline{a}}(\Delta^2 - 1)\sin(k\_{\underline{i}}d\_{\underline{i}})/(\Delta - \Delta'),\tag{64}$$

$$T\_{22} = \delta\_a^{-1} [ (\Delta - \Delta') \cos(k\_i d\_i) + i (\Delta \Delta' - 1) \sin(k\_i d\_i) ] / (\Delta - \Delta') \tag{65}$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 231

(74)

, where volume fractions are

. The stacking number is *N* 9 .  Based on wave solutions of the electric field and Maxwell equations, the magnetic field components of the forward-going wave in AF layers

> ( /) exp( ). *a x ax a ax k ik H A ik y ik x i t*

> (/ ) exp( ). *a x ay a ax ik k H A ik y ik x i t*

exp( ), exp( ). *i i x i*

1 2

(1 ) (1 ) (1 ) (1 ) *i a aa i a a a A A B B*

1 2

According to boundary conditions, the electric and magnetic fields of every layer are acquired when the incident wave is known. Then the expressions of refraction energy flow

What needs to be emphasized is that we here concentrate only on the refraction, so only the forward-going wave corresponding to the first term in Eq.(61) is considered and the backward-going wave is ignored. Owing to refraction angles being different in various layers, the refraction angle of the AF/IC PCs should be effective one. The angle between the energy flow and wave vector, and the refraction angle of the AF/IC PCs are defined as

arccos[( ) / ( )], *KS KS*

12 12 ' [( ) / ( )], *ax ix ay iy*

Numerical calculations based on FeF2/TlBr PCs. We take the AF layer thickness 4 *a d m*

transmission spectra with specific angles of the incidence in an external magnetic field

0

0

*k A k A H ik y ik x i t H*

0 *H T* 3 . As illustrated in Fig. 14(a), the forbidden band ranges from 0.9

1 *aai f ddd* and 2 1 *f f* 1 , respectively.

and the thickness of IC layers 1 *i d m*

and 1 2 *a i S fS fS*

in all layers are written as

with *K ke fk fk e xx a i y* 1 2

The amplitudes of two neighboring layers satisfy

0 0

The magnetic field components of the forward-going wave in the adjacent IC layers are

*ix i x iy i x*

are shown as

with / *x ai ik k k* and / *x ai ik k k* . In AF layers, there are the relation 2 22 2 *av x* a *k ck* with 2 2 *v* , 2 22 2 sin *x k c* and *a* the dielectric constant. The magnetic permeability tensor components of AF layers are represented by

$$\mu = 1 + \alpha\_m \alpha\_a \{ 1/\left[ \alpha\_r^2 - (\alpha\_0 - \alpha - i\sigma)^2 \right] + 1/\left[ \alpha\_r^2 - (\alpha\_0 + \alpha + i\sigma)^2 \right] \}.\tag{66}$$

$$\mu\_{\perp} = \alpha\_m \alpha\_a \{ 1/\left[ \alpha\_r^2 - (\alpha\_0 - \alpha - i\sigma)^2 \right] - 1/\left[ \alpha\_r^2 - (\alpha\_0 + \alpha + i\sigma)^2 \right] \}. \tag{67}$$

with 0 4 *m M* , *a a H* , 1 2 [ (2 )] *r aea* ,*e e H* and 0 0 *H* . *M*0 represents the sublattice magnetization, and *H*0 , *Ha* and *H*e indicate the external magnetic field, anisotropy and the exchange fields, respectively. *r* denotes the zero-field resonant frequency, and are the gyromagnetic ratio and the magnetic damping. In IC layers, we have the relation 2 22 2 *ii x k ck* with the dielectric function 2 22 ( ) *i h l hT T i* , where *h* and *l* are the high- and low-frequency dielectric constants, but *T* indicates the frequency of the TO vibrating mode in the longwavelength limition and denotes the phonon damping. The magnetic permeability of IC layers is considered as 1 *i* . We assume here that the stacking number included in the magnetic superlattices is *N*. Then transmission and reflection coefficients of the AISL can be written as

$$
\begin{pmatrix} 1 \\ r \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 1+\Lambda\_1 & 1+\Lambda\_1' \\ 1-\Lambda\_1 & 1-\Lambda\_1' \end{pmatrix} \overleftrightarrow{T}^{N-1} \begin{pmatrix} k\_i \delta\_a \delta\_i \left(1+\Lambda\right)/\left(k\_1+k\_i\right) & k\_i \delta\_a^{-1} \delta\_i \left(1+\Lambda'\right)/\left(k\_1+k\_i\right) \\ k\_i \delta\_a \delta\_i^{-1} \left(1-\Lambda\right)/\left(k\_i-k\_1\right) & k\_i \delta\_a^{-1} \delta\_i^{-1} \left(1-\Lambda'\right)/\left(k\_i-k\_1\right) \end{pmatrix} \begin{pmatrix} t \\ t \end{pmatrix}. \tag{68}
$$

Note that incident wave amplitude is taken as 1. Therefore, the transmission and reflection coefficients can be determined with Eq. (68), and then the transmission ratio is 2 *t* and reflection ratio is 2 *r* . Additionally, absorption ratio is represented with 2 2 *A* 1 *r t* . Other quantities in Eq. (68) are 1 1 / *x a ik k k* , 1 1 / *x a ik k k* , exp( ) *i ii ik d* , 2 22 2 1*k c* cos .

As described in Ref. [8], magnetic superlattices possess two mini-bands with negative group velocity. When the incident wave is located in the frequency regions corresponding to the two mini-bands, what are the optical properties of the AF/IC PCs? In the preceding section, the expressions of transmission and absorption to be used have been derived. To grasp handedness and refraction properties of the AF/IC PCs, the refraction angle and propagation direction need to be determined. Therefore, subsequently we give the expression of the refraction angle. However, this structure possibly possesses a negative refraction, and generally the directions of the energy flow of electromagnetic wave and the wave vector misalign. We start with the definition of energy flow ( \* *S EH* Re[ ] / 2 ) with a view to achieving refraction properties. Based on wave solutions of the electric field and Maxwell equations, the magnetic field components of the forward-going wave in AF layers are shown as

$$H\_{ax} = \frac{\left(k\_a + ik\_x \mu\_\perp \mid \mu\right)}{o\mu\_0 \mu\_\nu} A\_a \exp(ik\_a y + ik\_x \mathbf{x} - i\alpha t). \tag{69}$$

$$H\_{ay} = \frac{(ik\_a \mu\_\perp / \mu - k\_x)}{\alpha \mu\_0 \mu\_\nu} A\_a \exp(ik\_a y + ik\_x x - i\alpha t). \tag{70}$$

The magnetic field components of the forward-going wave in the adjacent IC layers are

$$\frac{k\_i H\_i}{\alpha \mu\_0} = \frac{k\_i A\_i}{\alpha \mu\_0} \exp(ik\_i y + ik\_x x - i\alpha t),\\ H\_{ij} = -\frac{k\_x A\_i}{\alpha \mu\_0} \exp(ik\_i y + ik\_x x - i\alpha t). \tag{71}$$

The amplitudes of two neighboring layers satisfy

230 Ferromagnetic Resonance – Theory and Applications

*av x* a *k ck*

2 22 ( ) *i h l hT T*

*a a*

have the relation 2 22 2

anisotropy and the exchange fields, respectively.

*T*

Other quantities in Eq. (68) are 1 1 / *x a ik k k*

.

with / *x ai ik k k*

relation 2 22 2

with 0 4

frequency,

written as

 *m M* ,

> and

dielectric constants, but

wavelength limition and

reflection ratio is 2

*ik d* , 2 22 2

1*k c* cos

exp( ) *i ii*

layers is considered as 1 *i*

  1

>

 and / *x ai ik k k*

22 [( )cos( ) ( 1)sin( )] / ( ) *a i i i i T*

0 0 1 {1 / [ ( ) ] 1 / [ ( ) ]}, *ma r r*

the sublattice magnetization, and *H*0 , *Ha* and *H*e indicate the external magnetic field,

 and *l*

magnetic superlattices is *N*. Then transmission and reflection coefficients of the AISL can be

1 1 1 1 1

*r k kk k k k t*

Note that incident wave amplitude is taken as 1. Therefore, the transmission and reflection

As described in Ref. [8], magnetic superlattices possess two mini-bands with negative group velocity. When the incident wave is located in the frequency regions corresponding to the two mini-bands, what are the optical properties of the AF/IC PCs? In the preceding section, the expressions of transmission and absorption to be used have been derived. To grasp handedness and refraction properties of the AF/IC PCs, the refraction angle and propagation direction need to be determined. Therefore, subsequently we give the expression of the refraction angle. However, this structure possibly possesses a negative refraction, and generally the directions of the energy flow of electromagnetic wave and the wave vector misalign. We start with the definition of energy flow ( \* *S EH* Re[ ] / 2 ) with a

coefficients can be determined with Eq. (68), and then the transmission ratio is 2

1 1 1 1 1 1 1 1 1 1 1 1 /( ) 1 / . 2 1 1 1 / 1 /

*N iai i ia i i iai i ia i i*

*r* . Additionally, absorption ratio is represented with 2 2 *A* 1 *r t* .

 , 2 22 2 sin *x k c*

2 22 2

2 22 2 0 0 {1 / [ ( ) ] 1 / [ ( ) ]}. *ma r r*

*e e H* and

*i i* (67)

are the gyromagnetic ratio and the magnetic damping. In IC layers, we

*T* indicates the frequency of the TO vibrating mode in the long-

. We assume here that the stacking number included in the

denotes the phonon damping. The magnetic permeability of IC

(68)

*k kk k k k t*

1

 , 1 1 / *x a ik k k*

*i i* (66)

constant. The magnetic permeability tensor components of AF layers are represented by

 with 2 2 *v*

*H* , 1 2 [ (2 )]

*ii x k ck*

>

*i* , where *h*

*r aea* ,

*kd i k d* (65)

. In AF layers, there are the

 and *a*

with the dielectric function

*r* denotes the zero-field resonant

are the high- and low-frequency

 ,

the dielectric

0 0 *H* . *M*0 represents

*t* and

$$
\begin{pmatrix} A\_i \\ B\_i \end{pmatrix} = \begin{pmatrix} (1 + \Delta\_1)\delta\_a & (1 - \Delta\_2)\delta\_a^{-1} \\ (1 - \Delta\_1)\delta\_a & (1 + \Delta\_2)\delta\_a^{-1} \end{pmatrix} \begin{pmatrix} A\_a \\ B\_a \end{pmatrix} \tag{72}
$$

According to boundary conditions, the electric and magnetic fields of every layer are acquired when the incident wave is known. Then the expressions of refraction energy flow in all layers are written as

$$\vec{S}\_{j} = \text{Re}\left(\vec{E}\_{j} \times \vec{H}\_{j}^{\ast}\right) \Big/ \mathbf{2} = \text{Re}\left(-H\_{\dagger y}^{\ast} E\_{\dagger z} \vec{e}\_{x} + E\_{\dagger z} H\_{\dagger x}^{\ast} \vec{e}\_{y}\right) \Big/ \mathbf{2} \,, \tag{73} \tag{73}$$

What needs to be emphasized is that we here concentrate only on the refraction, so only the forward-going wave corresponding to the first term in Eq.(61) is considered and the backward-going wave is ignored. Owing to refraction angles being different in various layers, the refraction angle of the AF/IC PCs should be effective one. The angle between the energy flow and wave vector, and the refraction angle of the AF/IC PCs are defined as

$$\alpha = \arccos[\left(\bar{K} \cdot \bar{S}\right) / \left(\left|\bar{K}\right| \left|\bar{S}\right|\right)].\tag{74}$$

$$\theta' = \operatorname{arctg} [(f\_1 S\_{ax} + f\_2 S\_{ix}) / (f\_1 S\_{ay} + f\_2 S\_{iy})] \,\text{.}\tag{75}$$

with *K ke fk fk e xx a i y* 1 2 and 1 2 *a i S fS fS* , where volume fractions are 1 *aai f ddd* and 2 1 *f f* 1 , respectively.

Numerical calculations based on FeF2/TlBr PCs. We take the AF layer thickness 4 *a d m* and the thickness of IC layers 1 *i d m* . The stacking number is *N* 9 . Figure 14 shows the transmission spectra with specific angles of the incidence in an external magnetic field 0 *H T* 3 . As illustrated in Fig. 14(a), the forbidden band ranges from 0.9*r* to 1.2*r*

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 233

*r* , the absorption is obviously

*f* 0.8 , *N* 9 , 0 *H T* 3 and 0 *H T* 1 . (a)

maximum transmission. In the absorbing band at 1.024

strengthened with enhancing the incident angle.

**Figure 15.** Partially enlarged absorption spectra with 1

AF/IC PCs possesses a quasi left-handedness in these frequency regions.

16. We find the refraction angles are negative in the neighborhood of 0.943

handedness in Fig.17. However, the result is opposite in the vicinity of 1.064

Therefore, we here reckon the left-handedness is not always accompanied by negative refraction. FeF2/TlBrsuperlattices have the natures of either negative refraction or quasi lefthandedness, or even simultaneously bear them at the certain frequencies of two guided

To have a deeper understanding of the negative refraction and quasi- left-handedness of the

*r* . The two frequency ranges for negative angle do not completely coincide with those of the quasi-left-handedness in Fig. 17. Namely, the frequency regime of negative

>

AF/IC PCs, subsequently the expressions of the dielectric function

corresponding to Fig. 15(d). After Wang & Ta, 2012.

frequency for 0 *H T* 3 and o

Figure 17 shows the refraction angle

angle

modes.

0.943

1.064

refraction near to 0.943

 *r* and 1.064

corresponding to Fig. 14(b); (b) corresponding to Fig.14(d); (c) corresponding to Fig. 15(b); (d)

To capture the handedness and refraction behaviors of the AF/IC PCs, the angle of refraction and the angle between the energy flow and wave vector are illustrated. Fig. 16 shows the

between the energy flow and wave vector of forward-going wave varies with

45 . As illustrated in Fig. 16, the angles in the vicinities of

' versus frequency under the same condition as Fig.

 *r* and

> *r* .

*r* are greater than o 90 , but less than 180 . It indicates that the

*r* is strikingly greater than that occupied by the quasi-left-

**Figure 14.** Transmission spectra for the fraction 1 *f* 0.8 , external magnetic field 0 *H T* 3 and stacking number *N* 9 . (a) the incident angle 0 0 ; (b) a zoomed view of guided modes in (a); (c) the incident angle 0 45 ; (d) a zoomed view of guided modes in (c). After Wang & Ta, 2012.

corresponding to the band gap of magneto-phonon polariton in Ref. [8]. Here the most interesting may be that guided modes arise in the forbidden band. The two guided modes lie in the proximity of 0.943 *r* and 1.064 *r* , corresponding to the mini-bands with negative group velocity in Ref. [8]. At the same time, the magnetic permeability of AF layers and the dielectric function for IC layers are both negative. To distinctly observe two guided modes, the partially enlarged Fig. 14(b) corresponding to Fig. 14(a) is exhibited. Seen from Fig. 14(b), the maximum transmission of the guided mode with lower-frequency is 40% and that of higher mode is 28.4%. As is well known, the optical thicknesses of films are determined by the frequency-dependent magnetic permeability and the dielectric function. Then the optical thicknesses of thin films are varied with the frequency of incident wave. The optical path of wave in media can also be altered by changing the incident angle. Fig. 14(c) shows the transmission spectrum with incident angle o 45 and other parameters are the same as those in 14(a). The partially enlarged Fig. 14(d) corresponding to Fig. 14(c) is given. Compared with the normal incidence case, for 0 the forbidden band becomes wide and their maximum transmissions are reduced to 27.1% and 21.9%, but two guided modes keep their frequency positions unaltered.

As already noted, the damping is included and then the absorption appears. We are more interest in the two guided modes, so only the absorption corresponding to two guided modes will be considered in Fig. 15 (a) and (c) display the absorption spectra in the case of right incidence, but (b) and (d) illustrate the absorption spectra for incident angle o 45 . We see that the absorption has a great influence on the transmission spectra. In the absorbing band at 0.9426 *r* , relative tiny absorption corresponds exactly to the maximum transmission. In the absorbing band at 1.024 *r* , the absorption is obviously strengthened with enhancing the incident angle.

232 Ferromagnetic Resonance – Theory and Applications

**Figure 14.** Transmission spectra for the fraction 1

the incident angle 0

lie in the proximity of 0.943

stacking number *N* 9 . (a) the incident angle 0

given. Compared with the normal incidence case, for

modes keep their frequency positions unaltered.

 absorbing band at 0.9426

*r* and 1.064

45 ; (d) a zoomed view of guided modes in (c). After Wang & Ta, 2012.

*r* , relative tiny absorption corresponds exactly to the

corresponding to the band gap of magneto-phonon polariton in Ref. [8]. Here the most interesting may be that guided modes arise in the forbidden band. The two guided modes

negative group velocity in Ref. [8]. At the same time, the magnetic permeability of AF layers and the dielectric function for IC layers are both negative. To distinctly observe two guided modes, the partially enlarged Fig. 14(b) corresponding to Fig. 14(a) is exhibited. Seen from Fig. 14(b), the maximum transmission of the guided mode with lower-frequency is 40% and that of higher mode is 28.4%. As is well known, the optical thicknesses of films are determined by the frequency-dependent magnetic permeability and the dielectric function. Then the optical thicknesses of thin films are varied with the frequency of incident wave. The optical path of wave in media can also be altered by changing the incident angle. Fig. 14(c) shows the transmission spectrum with incident angle o 45 and other parameters are the same as those in 14(a). The partially enlarged Fig. 14(d) corresponding to Fig. 14(c) is

wide and their maximum transmissions are reduced to 27.1% and 21.9%, but two guided

As already noted, the damping is included and then the absorption appears. We are more interest in the two guided modes, so only the absorption corresponding to two guided modes will be considered in Fig. 15 (a) and (c) display the absorption spectra in the case of right incidence, but (b) and (d) illustrate the absorption spectra for incident angle o

We see that the absorption has a great influence on the transmission spectra. In the

*f* 0.8 , external magnetic field 0 *H T* 3 and

0 ; (b) a zoomed view of guided modes in (a); (c)

*r* , corresponding to the mini-bands with

0 the forbidden band becomes

45 .

**Figure 15.** Partially enlarged absorption spectra with 1 *f* 0.8 , *N* 9 , 0 *H T* 3 and 0 *H T* 1 . (a) corresponding to Fig. 14(b); (b) corresponding to Fig.14(d); (c) corresponding to Fig. 15(b); (d) corresponding to Fig. 15(d). After Wang & Ta, 2012.

To capture the handedness and refraction behaviors of the AF/IC PCs, the angle of refraction and the angle between the energy flow and wave vector are illustrated. Fig. 16 shows the angle between the energy flow and wave vector of forward-going wave varies with frequency for 0 *H T* 3 and o 45 . As illustrated in Fig. 16, the angles in the vicinities of 0.943 *r* and 1.064 *r* are greater than o 90 , but less than 180 . It indicates that the AF/IC PCs possesses a quasi left-handedness in these frequency regions.

Figure 17 shows the refraction angle ' versus frequency under the same condition as Fig. 16. We find the refraction angles are negative in the neighborhood of 0.943 *r* and 1.064 *r* . The two frequency ranges for negative angle do not completely coincide with those of the quasi-left-handedness in Fig. 17. Namely, the frequency regime of negative refraction near to 0.943 *r* is strikingly greater than that occupied by the quasi-lefthandedness in Fig.17. However, the result is opposite in the vicinity of 1.064 *r* . Therefore, we here reckon the left-handedness is not always accompanied by negative refraction. FeF2/TlBrsuperlattices have the natures of either negative refraction or quasi lefthandedness, or even simultaneously bear them at the certain frequencies of two guided modes.

To have a deeper understanding of the negative refraction and quasi- left-handedness of the AF/IC PCs, subsequently the expressions of the dielectric function

**Figure 16.** Angle between energy flow and wave vector of down going wave versus the change of frequencies at 0 *H T* 3 , 0 45 and 1 *f* 0.8 . After Wang & Ta, 2012.

**Figure 17.** Refraction angle ' versus the alteration of frequencies at 0 *H T* 3 , 0 45 and 1 *f* 0.8 . After Wang & Ta, 2012.

 2 22 ( ) *i h l hT T i* of IC layers and the magnetic permeability of AF layers are analyzed. We find that when frequency lies in

$$
\alpha \alpha\_{\Gamma} < \alpha < \left(\varepsilon \sqrt{\varepsilon\_h \beta\_h}\right)^{1/2} \alpha\_{\Gamma} \tag{76}
$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 235

versus the different frequencies at 0 *H T* 3 , 0

45 and

refraction in the limit of long wavelength. Regarding the results arising from the two methods mentioned above, we conclude that the necessary condition of negative refraction

are both negative in this PCs.

or left-handedness is that *v*

**Figure 18.** Voigt magnetic permeability *v*

Shu-Fang Fu and Xuan-Zhang Wang

1

2012.

**5. Summary**

**Author details**

**Acknowledgement**

**6. References**

Province, with no. ZD200913.

 and *i*

*f* 0.8 (solid line denotes real parts and broken line indicates imaginary parts). After Wang & Ta,

This chapter aims to discover optical properties of AF/IC PCs in the presence of external static magnetic field. First, within the effective-medium theory, we investigated dispersion properties of MPPs in one- and two-dimension AF/IC PCs. The ATR (attenuated total reflection) technique should be powerful in probing these MPPs. Second, there is a frequency region where the negative refraction and the quasi left-handedness appear when the AF/IC PCs period is much shorter than the incident wavelength. Finally, an external

magnetic field can be used to modulate the optical properties of the AF/IC PCs.

*Physics and Electronic Engineering, Harbin Normal University, Harbin, China*

*Key Laboratory for Photonic and Electronic Bandgap Materials, Ministry of Education, School of*

This work was financially supported by the National Natural Science Foundation of China with Grant no.11084061, 11104050, and the Natural Science Foundation of Heilongjiang

[1] T. Goto, A .V. Baryshev, M. Inoue, A. V. Dorofeenko, A. M. Merzlikin, A. P. Vinogradov, A. A. Lisyansky, A. B. Granovsky, Tailoring surfaces of one-dimensional

the dielectric function *i* is negative, namely the range 0.9152 2.1835 *r r* . This completely covers the frequency range of AF resonance, so the dielectric function must be negative in the region of negative magnetic permeability *v* . It is found from Fig.18 that the magnetic permeability *v* is negative in certain regions, where the dielectric function *i* is also negative. In our previous work [8], utilizing the effective medium theory, we verified the effective dielectric function and magnetic permeability are both negative in the long wavelength limit when *v* and *i* are negative. In other words, the AF/IC PCs is of negative refraction in the limit of long wavelength. Regarding the results arising from the two methods mentioned above, we conclude that the necessary condition of negative refraction or left-handedness is that *v* and *i* are both negative in this PCs.

**Figure 18.** Voigt magnetic permeability *v* versus the different frequencies at 0 *H T* 3 , 0 45 and 1 *f* 0.8 (solid line denotes real parts and broken line indicates imaginary parts). After Wang & Ta, 2012.

### **5. Summary**

234 Ferromagnetic Resonance – Theory and Applications

**Figure 16.** Angle

layers are analyzed. We find that when frequency lies in

negative in the region of negative magnetic permeability *v*

 and *i*

45 and 1

frequencies at 0 *H T* 3 , 0

**Figure 17.** Refraction angle

the dielectric function *i*

magnetic permeability *v*

wavelength limit when *v*

2 22 ( ) *i h l hT T*

After Wang & Ta, 2012.

between energy flow and wave vector of down going wave versus the change of

*f* 0.8 . After Wang & Ta, 2012.

' versus the alteration of frequencies at 0 *H T* 3 , 0

1 2 ,

is negative, namely the range 0.9152 2.1835

is negative in certain regions, where the dielectric function *i*

are negative. In other words, the AF/IC PCs is of negative

*T lh T*

completely covers the frequency range of AF resonance, so the dielectric function must be

also negative. In our previous work [8], utilizing the effective medium theory, we verified the effective dielectric function and magnetic permeability are both negative in the long

*i* of IC layers and the magnetic permeability of AF

(76)

. It is found from Fig.18 that the

*r r*

 . This

45 and 1

*f* 0.8 .

is This chapter aims to discover optical properties of AF/IC PCs in the presence of external static magnetic field. First, within the effective-medium theory, we investigated dispersion properties of MPPs in one- and two-dimension AF/IC PCs. The ATR (attenuated total reflection) technique should be powerful in probing these MPPs. Second, there is a frequency region where the negative refraction and the quasi left-handedness appear when the AF/IC PCs period is much shorter than the incident wavelength. Finally, an external magnetic field can be used to modulate the optical properties of the AF/IC PCs.

### **Author details**

Shu-Fang Fu and Xuan-Zhang Wang

*Key Laboratory for Photonic and Electronic Bandgap Materials, Ministry of Education, School of Physics and Electronic Engineering, Harbin Normal University, Harbin, China*

### **Acknowledgement**

This work was financially supported by the National Natural Science Foundation of China with Grant no.11084061, 11104050, and the Natural Science Foundation of Heilongjiang Province, with no. ZD200913.

### **6. References**

[1] T. Goto, A .V. Baryshev, M. Inoue, A. V. Dorofeenko, A. M. Merzlikin, A. P. Vinogradov, A. A. Lisyansky, A. B. Granovsky, Tailoring surfaces of one-dimensional magnetophotonic crystals: optical Tamm state and Faraday rotation, Phys. Rev. B. 79 (2009) 125103/1-5.

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 237

[18] C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's

[19] A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed

[20] Zhaolin Lu, Janusz A. Murakowski, Christopher A. Schuetz, Shouyuan Shi, Garrett J. Schneider, and Dennis W. Prather, "Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies," Phys.

[21] J. Lezec Henri, A. Dionne Jennifer, and A. Atwater Harry, "Negative refraction at

[22] A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, "Negative refraction at infrared wavelengths in a two-dimensional photonic crystal," Phys. Rev.

[23] W. T. Lu and S. Sridhar, "Superlens imaging theory for anisotropic nanostructured metamaterials with broadband all-angle negative refraction," Phys. Rev. B 77, (2008)

[24] F. M. Wang, H. Liu, T. Li, Z. G. Dong, S. N. Zhu, and X. Zhang, "Metamaterial of rod pairs standing on gold plate and its negative refraction property in the far-infrared

[25] Andrea Alu and Nader Engheta, "Three-dimensional nanotransmission lines at optical frequencies: A recipe for broadband negative-refraction optical metamaterials," Phys.

[26] R.H. Tarkhanyana and D.G. Niarchos, "Wave refraction and backward magnon– Plasmon, polaritons in left-handed antiferromagnet/semiconductor superlattices," J.

[29] R. T. Tagiyeva, M. Saglam,"Magnetic polaritons at the junctions of magnetic superlattice and magnetic material." Solid State Commun. 122 (2002) 413-417. [30] N.Raj, D.R.Tilley, "Polariton and effective-medium theory of magnetic Superlattices ,"

[31] Na Liu, Xiao-Dong Hou, Xiangdong Zhang, J. "Retarded modes in layered magnetic structures containing left-handed materials," Phys. Condens. Matter 20, (2008)

[32] C.A.A.Araújo, E.L.Albuquerque, D.H.A.L.Anselmo, M.S. Vasconcelos,"Magnetic polaritons in metamagnet layered structures: Spectra and localization properties," Phys.

[33] Xuan-Zhang Wang, Tilley. D. R, "Retarded modes of a lateral antiferromagnetic

[34] M. C. Oliveros, N. S. Almeida, D. R. Tilley, J. Thomas, R. E. Camley, "Magnetostatic modes and polaritons in antiferromagnetic-non-magnetic superlattices," J. Phys.

material that obeys Snell's law," Phys. Rev. Lett. 90, (2003)137401-1-4.

law," Phys. Rev. Lett. 90, (2003)107401-1-4.

visible frequencies," Science 316, (2007)430-432.

frequency regime," Phys. Rev. E 75, (2007) 016604-1-4.

[27] C. F. Klingshirn, Semiconductor Optics (Springer, Berlin 1997) Ch.13.

/nonmagnetic superlattice," Phys. Rev. B 52, (1995) 13353-13357.

[28] A.D. Boardman, Electromagnetic Surface Modes, John Wiley, New York, 1982.

Rev. Lett. 95, (2005)153901-1-4.

Lett. 93, (2004) 073902-1-4.

Rev. B 75, (2007) 024304-1-20.

Phys.Rev. B 36, (1987) 7003-07.

Lett. A 372, (2008) 1135-1140.

Condens. Matter 4, (1992) 8497-8510.

Magn. Magn. Materials, 312, (2007) 6-15.

233101-1-4.

335210/1-5.

[18] C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, (2003)107401-1-4.

236 Ferromagnetic Resonance – Theory and Applications

layer, J. Phys.:Condens. Matter. 17 (2005) 5447-5452.

opal/Fe3O4," J. Magn. Magn. Mater. 321 (2009) 840-842.

on photonic band gaps," Phys. Rev. B 56, (1997 ) 959-962.

crystal superlattices," J. Appl. Phys. 108 (2010) 013520/1-4.

superlattices," J. Magn. Magn. Mater. 324 (2012) 72-74.

Fundamentals and Applications 10(2012)/ 1-8.

(2009) 125103/1-5.

072502/1-3.

1005.

3969.

065601-1-4.

84, (2000) 4184-4187.

(2004) 026604-1-5.

(2009) 063119/1-4.

Rev. Lett. 85, (2000) 2933-2936.

of refraction," Science 292, (2001)77-79.

magnetophotonic crystals: optical Tamm state and Faraday rotation, Phys. Rev. B. 79

[2] R. H. Zhu, S. N. Fu, H. Y. Peng, Far infrared Faraday rotation effect in one-dimensional microcavity type magnetic photonic crystals, J. Magn. Magn. Mater. 323 (2011) 145-149. [3] X. Z. Wang, The Faraday effect of an antiferromagnetic photonic crystal with a defect

[4] V. V. Pavlov, P. A. Usachev, R. V. Pisarev, D. A. Kurdyukov, S. F. Kaplan, A. V. Kimel, A. Kirilyuk, Th. Rasing, "Enhancement of optical and magneto-optical effects in threedimensional opal/Fe3O4 magnetic photonic crystals," Appl. Phys. Lett. 93 (2008)

[5] V. V. Pavlov, P. A. Usachev, R. V. Pisarev, D. A. Kurdyukov, S. F. Kaplan, A. V. Kimel, A. Kirilyuk, Th. Rasing, "Optical study of three-dimensional magnetic photonic crystals

[6] R. Atkinson, "Limits of enhancement of the Faraday effect in ultra-thin magnetic layers by one-dimensional magnetophotonic crystals, "J. Phys. D: Appl. Phys. 39 (2006) 999-

[7] M.M. Sigalas, C.M. Soukoulis, R. Biswas, K.M. Ho, "Effect of the magnetic permeability

[8] J. X. Ta, Y. L. Song, X. Z. Wang, "Magneto-phonon polaritons of antiferromagnetic/ion-

[9] J. X. Ta, Y. L. Song, X. Z. Wang, "Magneto-phonon polaritons in two-dimension antiferromagnetic/ion-crystalic photonic crystals," Photonics and Nanostructures -

[10] Y. L. Song, J. X. Ta, H. Li, X. Z. Wang, "Presence of left-handedness and negative refraction in antiferromagnetic/ionic-crystal multilayered films, " J. Appl. Phys. 106

[11] J. X. Ta, Y. L. Song, X. Z. Wang, "Optical properties of antiferromagnetic/ion-crystal

[12] J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, (2000) 3966-

[13] D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials," Phys.

[14] D. Schurig and D. R. Smith, "Negative index lens aberrations," Phys. Rev. E 70, (2004)

[15] R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index

[16] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett.

[17] W. T. Lu, J. B. Sokoloff, and S. Sridhar, "Refraction of electromagnetic energy for wave packets incident on a negative-index medium is always negative," Phys. Rev. E 69,

Hefei, 2001, pp. 234 (in Chinese).

Amsterdam, 1982, pp. 5-67.

6808-6816.

[35] I. Balin, N. Dahan, V. Kleiner, E. Hasman, "Slow surface phonon polaritons for sensing

[36] A. J. Huber, B. Deutsch, L. Novotny, R. Hillenbrand, "Focusing of surface phonon

[37] K. Abraha, D.E. Brown, T. Dumelow, T.J. Parker, D.R. Tilley, "Obliqu-incidence farinfrared reflectivity study of the uniaxial antiferromagnet FeF2," Phys. Rev. B 50 ,(1994)

[38] E.D. Palik, Handbook of Optical Constants, Vol.3, Academic Press, San Diego, 1998, pp.926; R.C. Fang, Solid State Spectroscopy, Press., Univ. of Sci. and Tech. of China,

[39] Xuan-Zhang Wang and Hua Li, Nonlinear propagation of electromagnetic waves in antiferromagnets, in Propagation of Electromagnetic Waves in Complex Matter, edited

[40] X.Z. Wang, D.R. Tilley, "Magnetostatic surface and guided modes of lateral-magnetic-

[41] X.Z. Wang, X.R. Xu, "Nonlinear Magnetostatic surface waves of magnetic multilayers:

[42] D.N. Mirlin, in: V.M. Agranovich, D.L. Mills (Eds.), Surface phonon polaritons in dielectrics and semiconductors, in Surface Polaritons, North-Holland Press,

[43] X. F. Zhou, J. J. Wang, X .Z. Wang, D.R. Tilley, "Reflection and transmission by

in the midinfrared spectrum," Appl. Phys. Lett. 94 , (2009) 111112/1-3.

by Ahmed Kishk, (Intech-Open Access Publisher, 2011)pp.55-96, Ch.3

superlattice films," Phys. Rev. B 50, (1994) 13472-13479.

Effective-medium theory," Phys. Rev. B 63, (2001) 054415/1-11.

magnetic multilayers," J. Magn. Magn. Mater. 212, (2000) 82-90.

polaritons," Appl. Phys. Lett. 92 , (2008) 203104/1-3.

## *Edited by Orhan Yalçın*

The book Ferromagnetic Resonance - Theory and Applications highlights recent advances at the interface between the science and technology of nanostructures (bilayer-multilayers, nanowires, spinel type nanoparticles, photonic crystal, etc.). The electromagnetic resonance techniques have become a central field of modern scientific and technical activity. The modern technical applications of ferromagnetic resonance are in spintronics, electronics, space navigation, remote-control equipment, radio engineering, electronic computers, maritime, electrical engineering, instrumentmaking and geophysical methods of prospecting.

Ferromagnetic Resonance - Theory and Applications

Ferromagnetic Resonance

Theory and Applications

*Edited by Orhan Yalçın*

Photo by MADDRAT / iStock


**2. Methodology**

The analysis presented herein has been carried out using the methodology proposed by Miranda and Ferreira [1]. This particular methodology is adjusted so as to be implemented as a micro-scale assessment. In this case, the elements of interest are the historical buildings and the losses, which are mainly defined in terms of physical damage caused to the building fabric, namely, the direct tangible damage. For historic buildings, this damage is deemed the most severe. Moreover, to deal with historic sites, the method addresses direct intangible damage, in terms of losses to the cultural value of heritage buildings.

The vulnerability in this methodology refers to the intrinsic characteristics of the building. Other factors define the social, historical, and economic value of the asset and its interaction with the water. The results can be seen as a synthetic vulnerability function as they are not derived from damage data after flood events. The vulnerability index is a relative measure of the expected severity of the physical *Flood Risk Assessment in Urban Areas: The Historic City Centre of Aveiro as a Case Study DOI: http://dx.doi.org/10.5772/intechopen.109867*

impact of the flood on the building in comparison to other assets ranked. The value of the vulnerability index does not account for the hazard and only partially accounts for the exposure, following an approach that is becoming common in flood risk assessment [6, 7].

As shown in **Figure 1**, the methodology defines flood vulnerability (FV) as the product of the exposure component (EC) and the sensitivity component (SC). Five parameters are considered to define the sensitivity component, each one of them containing four attributes. The exposure parameter refers to the exposure of the building to the flood, which encompasses aspects like the orientation of the most exposed façade to the water flow or the location and dimension of the openings doors and windows [1].

### **2.1 Flood vulnerability**

The goal of vulnerability assessment is to understand how a system will be affected by floods. Even though there are different classes of vulnerability, physical vulnerability is the most commonly assessed because it is the easiest way to characterize it. Furthermore, the nonphysical vulnerability is usually evaluated in terms of economic loss, which is related to the interruption of various activities and is often much more challenging to estimate.

According to Balica [8], flood vulnerability can be determined by a physical approach, where hydrological models are used to estimate flood hazard and economic consequences. Moreover, Balica also mentions an empirical approach related to quantitative or qualitative indicators to rate the vulnerability without inputting the hazard intensity. Regarding the methodology adopted for the present study, no attribute has been assigned the value 0, since it is considered that every historic-building feature measured indicates some degree of vulnerability to the hazard. However, for all the descriptors to contribute equally, a lower boundary of 10 is applied to the rating scale for them, except for the land slope [9].

Based on the estimated individual parameters, the flood vulnerability is computed according to Eq. (1).

Flood Vulnerability FV ð Þ¼ Exposure Component EC ð Þ� Sensitivity Component SC ð Þ (1)

### **2.2 Exposure component**

Exposure analysis aims to examine the economic assets and activities influenced by the flood. Exposure can be determined by geospatial mapping to identify the location of the assets of interest related to the flood hazard. In addition to this, the exposure contemplates some intrinsic characteristics of the elements analyzed.

#### *2.2.1 Wall orientation*

Wall orientation assesses the position of the asset with respect to the expected water flow. This parameter considers that the wall in a plane perpendicular to the water stream direction (fully exposed) presents a bigger pressure load than the ones that are simply immersed. It also considers the presence of openings reachable by the water. The characterization of the buildings as fully or partially exposed without openings is attributed to the ones where the windows and doors are not facing the flow of the water [10]. The assessment is based on the orientation and the characteristics of the main façade. In the case of buildings located in low-lying areas, the class is increased to C or to D for partially and fully exposed buildings.

#### **2.3 Sensitivity component**

#### *2.3.1 Material*

This parameter considers the lateral capacity of the structure, the resilience to ground movement, and the absorption capacity, factors that contribute to the risk of collapse in the short term and degradation in the longer term. In the case of degradation, it depends on two factors: the consequences of improper drying and on the contaminants and salts present in the flow [11].

Among materials, earth is considered the most vulnerable. The materials' porosity determines their absorption capacity, but it can also promote their resilience, as it allows them to tolerate heavy wetting. It should be mentioned that the unbaked earthen structures are the most likely to present disintegration caused by the wetting and water pressure [12].

Masonry buildings are less prone to suffer collapse. However, they can present moisture in the elements and different types of decay such as spalling or exfoliation induced by the presence of salts and contaminants. Previous experiences demonstrated reduction up to 50% of the capacity in dry condition. Therefore, masonry structures should be properly inspected post-event [10].

Modern materials such as steel and concrete are less vulnerable. Moreover, the reparation and replacement of traditional materials may be more expensive. Timber elements are prone to biological attacks, such as fungi and insects. After the flood event, if the element is properly dried, serious damage is not expected. Moreover, saturated elements undergo significant deformation.

*Flood Risk Assessment in Urban Areas: The Historic City Centre of Aveiro as a Case Study DOI: http://dx.doi.org/10.5772/intechopen.109867*

#### *2.3.2 Number of stories*

It is argued that single-story buildings are the most vulnerable, since they do not possess a flood-free story to facilitate the evacuation [13]. Nonetheless, in terms of stability, higher buildings with the same footprint and superficial foundations are more susceptible to the effects of the flood on the subsoil, such as differential settlements [5]. Therefore, for the present study, the vulnerability is considered as increasing with increasing number of stories.

#### *2.3.3 Damage and cracking*

Damage can lead to higher water absorption and lower strength in the materials, such as in the case of wood. Indeed, water ingress can easily occur through cracks, and once inside the building, water can damage movable assets and valuable nonstructural elements, contributing significantly to the economic losses. Therefore, restoration and maintenance measures are essential to prevent water penetration. This indicator includes an assessment of the state and quality of the interventions that can be carried out [6].

#### *2.3.4 Age*

This parameter may represent more valuable assets. This is valid for countries where historic structures have a higher real-estate value, being thus inhabited by higher-income residents [14]. Furthermore, older buildings are likely more prone to suffer flood damage. It is worth noting that the four levels of the age indicator may require an adaptation to the local evolution over time of the investigated building stock.

#### *2.3.5 Heritage status*

The heritage status is directly related to the socioeconomic impact that the preservation of these assets could have. For this reason, a higher value implies higher exposure of the asset. Moreover, the buildings that are listed are often subjected to more requirements, which results in higher intervention costs. In these cases, it is also important to contemplate the nonstructural value of artworks and other movable assets that are hosted in the building, a fact that will increase their vulnerability [15].


**5. Discussion**

As previously mentioned, the goal of this study was to propose an efficient and easily implemented method to perform radiometric corrections of UAV-borne hyperspectral imagery—without adding embedded sensors and with minimum equipment (i.e., only two Spectralon panels and a field spectrometer)—which is adequate for "hard-to-access" ecosystems, such as the huge intertidal mudbanks along the Amazon-influenced, northeast coast of South America. Smith and Milton [55] reported that the empirical line method allows the calibration of remotely sensed data to reflectance with errors of only a few percent. This efficiency was confirmed for our UAV data by the obtained SAM distances and correlation scores. Here, we calculated a higher SAM distance for the pioneer mangrove forest than for the benthic biofilm. This result is coherent with the higher diversity observed among the mangroves spectra (both with the field spectrometer and with the UAV-borne camera—Figure 7c,d). Indeed, point-wise measurement (with the field spectrometer) and very high spatial resolution hyperspectral images (with UAV-borne camera) reflect the complex morphology induced by the mangroves canopy. On the contrary, a slightly lower correlation coefficient was obtained for the benthic biofilm. This could be due to the higher disparity, both spatially and temporally, of the biofilm concentration at the sediment surface, alongside local changes in sediment elevation and bioturbation activities [37].

This method of radiometric corrections, especially the initial calibration step, can also be used to monitor the drift of the manufacturer's calibration. As previously mentioned, Hyper-DRELIO data are noisier for wavelengths higher than 800 nm. Proctor and He [8] explained that this effect is common and is due to the combination of a sharp decrease of quantum efficiency in the NIR, the lower solar output in the NIR, and reduced sensitivity of the imager resulting in lower signal and greater noise. The proposed radiometric correction process relies on simplifying assumptions, which are likely to cause some errors. For instance, remaining noise can be due to DC, which has been considered steady here. Besides, as the "sun–target–sensor" geometry varies for all the pixels recorded by the camera, bidirectional reflectance effects can also influence the signal [33]. As mentioned by Garzonio et al. [34], light intensity could be optimized by placing neutral filters to avoid saturation for very high light intensities. Lastly, the proposed approach to take into account temporal variations of irradiance suits for the variations of ambient light but fails to take into account local effects, which are not captured by the field spectrometer above the 99% Spectralon panel. Therefore, some parts of the images can be affected by spatial changes in the illumination fields, due to isolated clouds or shadows originating from the geometrical configuration of the scene (e.g., presence of trees).

Moreover, as the Spectralons panels are of great importance in this method for radiometric corrections, the results would be sensitive to defects in Spectralon surfaces. For that matter, Bachmann et al. [48] mention that white Spectralon panels are subject to repeated handling and exposed to various environmental factors. Therefore, their calibration coefficients drift over time.

The methodology proposed can be extended to other types of areas, provided that topography-induced illumination differences are taken into account [59].

For now, we have not assessed the impact of platform vibrations on the radiometric and spectral stability of the hyperspectral camera (shift, band broadening, etc.). According to Garzonio et al. [34], however, the impact of platform mechanical vibrations would be almost insignificant in terms of band centre, width, and radiometric response.

Field access in the mangroves developing under Amazonian influence is particularly difficult. Thus, in this particular rapidly evolving ecosystem, future work should aim to assess the extent to which the very high spatial and spectral resolution provided by hyperspectral UAV can capture the biological complexity of the substrate. The good correlations found in this study between drone imagery after radiometric corrections and the in situ spectrometer measurements for the benthic biofilm and pioneer mangroves should allow simultaneous mapping of mangrove forests and benthic biofilm distribution at the mudbank scale, integrating small-scale heterogeneity, caused by the combined effects of geomorphology, tides, and biology (e.g., bioturbation). This method, tested in mangroves under limited human impacts, shows new possibilities for monitoring mangrove ecosystems facing different levels of pressure and subsequent alteration along the nearby coastline (Guyana and Surinam), as well as in other biogeographic regions with other mangrove species and dynamics.

#### **6. Conclusions**

Hyper-DRELIO allowed hyperspectral data to be collected above few hectares of mangrove forests and mudbanks in French Guiana, with both high spatial resolution and high spectral resolution in the VNIR domain. One of the main advantages of drones being their flexibility, the associated imagery calibration procedures have to be as simple as possible to keep the latter. This study proposes an easy, in situ radiometric calibration method, dedicated to drone-based hyperspectral surveys, without adding embedded sensors and with minimum equipment, using only two Spectralon (white and grey) and a ground spectrometer. The proposed procedure enables to calibrate the sensor, by correcting lining effects and transforming the raw relative DN generated by the hyperspectral camera into reflectance values standardized to in situ illumination conditions.

The radiometric corrections were applied to a small subset of a dataset collected above mudbanks colonised by benthic biofilm and a pioneer mangrove forest. Besides the fact that the shapes of spectra are globally consistent between the radiometrically corrected Hyper-DRELIO spectra and the in situ typical spectra, their degree of similarity was assessed using the SAM distance and correlation coefficient. SAM distance values of 0.039 above biofilm and 0.159 above pioneer mangrove forest, together with associated correlation coefficients (of 0.96 and 0.97, respectively), are greatly satisfying for substrate classification. Future work will consist of applying this method to the entire study area, in order to spatialize the results, and comparing hyperspectral and in situ data in order to obtain the finest possible classification of the various detectable elements.

**Author Contributions:** Conceptualization, M.J. and G.S.; data curation, M.J.; formal analysis, M.J., G.B. and P.L.; funding acquisition, E.M., N.L.D. and C.D.; investigation, M.J., G.B., E.M., J.A., P.G., G.T. and J.F.; methodology, M.J. and G.S.; project administration, E.M.; resources, E.M., G.T. and C.D.; software, M.J.; supervision, E.M., N.L.D., P.L., G.T. and C.D.; validation, M.J. and G.B.; visualization, M.J.; writing—original draft preparation, M.J., G.B. and E.M.; writing—review and editing, M.J., G.S., G.B., E.M., N.L.D., G.T. and C.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the French National Agency under the programs "Investissements d'Avenir" (LabexMER: ANR-10-LABX-19; EQUIPEX CRITEX: ANR-11-EQPX-0011) and by the CNRS MITI program ("Pépinière Interdisciplinaire de Guyane" through the BIOGEOMORPHO project). This work was supported by ISblue project, Interdisciplinary graduate school for the blue planet (ANR-17-EURE-0015) and co-funded by a grant from the French government under the program "Investissements d'Avenir".

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available on request due to privacy restrictions.

**Acknowledgments:** The authors are grateful to the Reserve Naturelle de l'Amana, Parc Naturel Régional de Guyane, DEAL Guyane for the use of their facilities and to I. Bihannic, A. Gardel, M. Jolivet, and S. Morvan for their help on the field. This paper is a GDR Liga's contribution.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Urban Ecosystem Services**

Editors

**Alessio Russo Giuseppe T. Cirella**

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

*Editors* Alessio Russo University of Gloucestershire UK

Giuseppe T. Cirella University of Gdansk Poland

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal *Land* (ISSN 2073-445X) (available at: https://www.mdpi.com/journal/land/special issues/Urban Ecosystem Services).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. *Journal Name* **Year**, *Volume Number*, Page Range.

**ISBN 978-3-0365-0582-4 (Hbk) ISBN 978-3-0365-0583-1 (PDF)**

Cover image courtesy of Alessio Russo.

© 2021 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications.

The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.