In Figure 4(a), we represent the permittivities E∥ and E⊥ of our plasmonic crystals for a wide range of frequencies. Note that the metal-filling factor takes control on the dissipative effects in the metamaterial; accordingly low values of f are of great convenience. We set f ¼ 1=4 in our numerical simulations. For low frequencies, Ω ≪ 1, the following approximations can be used: E⊥ ≈ fE2 < 0 and 0 < E∥ ≈ E1=ð Þ 1 � f . Therefore, propagating TEx modes (Ex ¼ 0) cannot exist in the bulk crystal since it behaves like a metal in these circumstances. On the other hand, TMx waves propagate following the spatial dispersion curve of Eq. (37). This is a characteristic of Type II hyperbolic media. As mentioned earlier, Eq. (37) denotes a hyperboloid of one sheet

Figure 4. (a) Variation of relative permittivities E∥ (blue solid line) and E⊥ (magenta solid line) as a function of normalized frequency Ω, for the plasmonic crystal including a lossless Drude metal and fused silica (E1 ¼ 2:25) as dielectric material. Ω1 and Ω2 yield 0.359 and 0.759, respectively. (b) and (c) plot Eq. (37) in the kxky plane for extraordinary waves (TMx modes) for a plasmonic effective crystal, including a dielectric of permittivity E1 ¼ 2:25, in the range Ω < 1. Solid line corresponds to kz ¼ 0 and shaded regions are associated with harmonic waves with kz > 0 (nonevanescent fields).

Ω1 ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

for which E⊥ ¼ 0. For slightly higher frequencies, both E∥ and E⊥ are positive and Eq. (37)

simulates a uniaxial medium with positive birefringence. Raising the frequency even more, E∥

Ω2 ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

leading to the so-called canalization regime. In general, Ω1 < Ω2 provided that f < 1=2. Beyond Ω2, Eq. (37) turns to a hyperboloidal shape. In the range Ω2 < Ω < 1; however, the dispersion curve has two sheets (Type I hyperbolic medium). Figure 4(c) illustrates this case. Note that the upper limit of this hyperbolic band is determined by the condition E∥ ¼ 0 or in an

1 þ E1ð Þ 1 � f =f p , (38)

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E⊥

1 þ E1f =ð Þ 1 � f p , (39)

p , the periodic multilayer

(see Figure 4(b) for Ω ¼ 0:20).

diverges at

Furthermore, the hyperbolic dispersion exists up to a frequency

becomes an ellipsoid of revolution. Since its minor semi-axis is Ω ffiffiffiffiffi

equivalent way, E2 ¼ 0, occurring at the plasma frequency.

being now kx, the Bloch wavenumber KTM.

The validity of the EMA is related on the assumption that the period Λ is much shorter than the wavelength, that is, Λ ≪ λ0. Apparently, Eq. (33) is in good agreement with the EMA in the vicinity of kx ¼ 0 for TMx waves only. In contrast, propagation along the x-axis, where ky ¼ kz ¼ 0, results in large discrepancies. Even small-filling factors of the metallic composite lead to enormous birefringences. Such metamaterials enlarge the birefringence of the effectiveuniaxial crystal at least in one order of magnitude in comparison with values shown in Table 1. However, the size of birefringence displayed by extraordinary waves is reduced if w2 increases. On the other hand, the isotropy of the isofrequency curve is practically conserved for ordinary waves.

#### 4.2. Hyperbolic media

As we have seen in Section 3.3, nanolayered metal-dielectric compounds behave like plasmonic crystals enabling a simplified description of the medium by using the longwavelength approximation [10–12]. Under certain conditions, the permittivity of the medium set in the form of a second-rank tensor includes elements of opposite signs, leading to a metamaterial of extreme anisotropy [13, 14]. This class of nanostructured media with hyperbolic dispersion is promising metamaterials with a plethora of practical applications from biosensing to fluorescence engineering [15].

Type I hyperbolic media refers to a special kind of uniaxially anisotropic media, that can be described by a permittivity tensor where element E∥ is negative and E⊥ is positive. In this case, Eq. (37) leads to a two-sheet hyperboloid. Type II hyperbolic media lead to positive E∥ and negative E⊥, and Eq. (37) gives us a one-sheet hyperboloid [16]. The fulfillment of hyperbolic dispersion allows wave propagation over a wide spatial spectrum that would be evanescent in an ordinary isotropic dielectric. At the optical range, hyperbolic media can be manufactured with metal-dielectric multilayers or metallic nanowires. Multilayered hyperbolic metamaterials at an optical range take advantage of the wide frequency band in which metals exhibit negative permittivity and support plasmonic modes.

The system under analysis is a periodic binary medium, where we take as medium 1 a transparent dielectric medium that is ideally nondispersive. In our three numerical simulations we take a lossless Drude metal where its permittivity is E2 ¼ 1 � Ω�2 and dielectric media with permittivities: (a) E1 ¼ 1, (b) E1 ¼ 2:25 and (c) E1 ¼ 11:55. Note that frequencies can be expressed in units of the plasma frequency, Ω ¼ ω=ωp.

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Figure 4. (a) Variation of relative permittivities E∥ (blue solid line) and E⊥ (magenta solid line) as a function of normalized frequency Ω, for the plasmonic crystal including a lossless Drude metal and fused silica (E1 ¼ 2:25) as dielectric material. Ω1 and Ω2 yield 0.359 and 0.759, respectively. (b) and (c) plot Eq. (37) in the kxky plane for extraordinary waves (TMx modes) for a plasmonic effective crystal, including a dielectric of permittivity E1 ¼ 2:25, in the range Ω < 1. Solid line corresponds to kz ¼ 0 and shaded regions are associated with harmonic waves with kz > 0 (nonevanescent fields).

In Figure 4(a), we represent the permittivities E∥ and E⊥ of our plasmonic crystals for a wide range of frequencies. Note that the metal-filling factor takes control on the dissipative effects in the metamaterial; accordingly low values of f are of great convenience. We set f ¼ 1=4 in our numerical simulations. For low frequencies, Ω ≪ 1, the following approximations can be used: E⊥ ≈ fE2 < 0 and 0 < E∥ ≈ E1=ð Þ 1 � f . Therefore, propagating TEx modes (Ex ¼ 0) cannot exist in the bulk crystal since it behaves like a metal in these circumstances. On the other hand, TMx waves propagate following the spatial dispersion curve of Eq. (37). This is a characteristic of Type II hyperbolic media. As mentioned earlier, Eq. (37) denotes a hyperboloid of one sheet (see Figure 4(b) for Ω ¼ 0:20).

Furthermore, the hyperbolic dispersion exists up to a frequency

filling factor of medium "2" providing the metal rate in a unit cell. The dispersion equations

The validity of the EMA is related on the assumption that the period Λ is much shorter than the wavelength, that is, Λ ≪ λ0. Apparently, Eq. (33) is in good agreement with the EMA in the vicinity of kx ¼ 0 for TMx waves only. In contrast, propagation along the x-axis, where ky ¼ kz ¼ 0, results in large discrepancies. Even small-filling factors of the metallic composite lead to enormous birefringences. Such metamaterials enlarge the birefringence of the effectiveuniaxial crystal at least in one order of magnitude in comparison with values shown in Table 1. However, the size of birefringence displayed by extraordinary waves is reduced if w2 increases. On the other hand, the isotropy of the isofrequency curve is practically conserved

As we have seen in Section 3.3, nanolayered metal-dielectric compounds behave like plasmonic crystals enabling a simplified description of the medium by using the longwavelength approximation [10–12]. Under certain conditions, the permittivity of the medium set in the form of a second-rank tensor includes elements of opposite signs, leading to a metamaterial of extreme anisotropy [13, 14]. This class of nanostructured media with hyperbolic dispersion is promising metamaterials with a plethora of practical applications from

Type I hyperbolic media refers to a special kind of uniaxially anisotropic media, that can be described by a permittivity tensor where element E∥ is negative and E⊥ is positive. In this case, Eq. (37) leads to a two-sheet hyperboloid. Type II hyperbolic media lead to positive E∥ and negative E⊥, and Eq. (37) gives us a one-sheet hyperboloid [16]. The fulfillment of hyperbolic dispersion allows wave propagation over a wide spatial spectrum that would be evanescent in an ordinary isotropic dielectric. At the optical range, hyperbolic media can be manufactured with metal-dielectric multilayers or metallic nanowires. Multilayered hyperbolic metamaterials at an optical range take advantage of the wide frequency band in which metals exhibit negative

The system under analysis is a periodic binary medium, where we take as medium 1 a transparent dielectric medium that is ideally nondispersive. In our three numerical simulations we take a lossless Drude metal where its permittivity is E2 ¼ 1 � Ω�2 and dielectric media with permittivities: (a) E1 ¼ 1, (b) E1 ¼ 2:25 and (c) E1 ¼ 11:55. Note that frequencies can be

k 2 x E⊥ þ k2 t E∥ ¼ k2

0, corresponding to TE (o-) waves, where kx represents the

0, (37)

given by the EMA are k

56 Surface Waves - New Trends and Developments

for ordinary waves.

4.2. Hyperbolic media

2 x þ k 2 t ¼ E⊥k 2

being now kx, the Bloch wavenumber KTM.

biosensing to fluorescence engineering [15].

permittivity and support plasmonic modes.

expressed in units of the plasma frequency, Ω ¼ ω=ωp.

Bloch wavenumber KTE, and for TM (e-) waves, we have

$$
\Omega\_1 = \frac{1}{\sqrt{1 + \epsilon\_1 (1 - f)/f}},\tag{38}
$$

for which E⊥ ¼ 0. For slightly higher frequencies, both E∥ and E⊥ are positive and Eq. (37) becomes an ellipsoid of revolution. Since its minor semi-axis is Ω ffiffiffiffiffi E⊥ p , the periodic multilayer simulates a uniaxial medium with positive birefringence. Raising the frequency even more, E∥ diverges at

$$
\Omega\_2 = \frac{1}{\sqrt{1 + \epsilon\_1 f/(1 - f)}},
\tag{39}
$$

leading to the so-called canalization regime. In general, Ω1 < Ω2 provided that f < 1=2. Beyond Ω2, Eq. (37) turns to a hyperboloidal shape. In the range Ω2 < Ω < 1; however, the dispersion curve has two sheets (Type I hyperbolic medium). Figure 4(c) illustrates this case. Note that the upper limit of this hyperbolic band is determined by the condition E∥ ¼ 0 or in an equivalent way, E2 ¼ 0, occurring at the plasma frequency.

#### 5. Dyakonov surface waves

Dyakonov surface waves (DSWs) are another kind of surface waves, supported at the interface between an optically isotropic medium and a uniaxial-birefringent material. In the original work by Dyakonov (English version was reported in 1988 [3]), the optical axis of the uniaxial medium was assumed in-plane with respect to the interface. This is the case we deal with here. Since we treat the plasmonic lattice as a uniaxial crystal, we may establish analytically the

follow Dyakonov [3] by considering hybrid-polarized surface modes. In the isotropic medium we consider TEx (Ex ¼ 0) and TMx (Hx ¼ 0) waves whose wave vectors have the same real components ky and kz in the plane x ¼ 0. Therefore the electric field in both media may be set as

Moreover, these fields are evanescent in the isotropic medium and in the superlattice. In the

where the ordinary and extraordinary waves in the effective uniaxial medium decay exponentially with rates given by κo ¼ �iko1 and κe ¼ �ike1, respectively. Taking the formulation given in Section 2.3.1, the amplitudes Ao1 and Ae1 are identically zero. In the isotropic medium

where the evanescent decay for TE and TM modes is κ ¼ �ikTE2 ¼ �ikTM2. Now the ampli-

Once we have the amplitudes in both sides of the interface, we apply the boundary conditions

A0 TE2 0 A0 TM2 0

of hybrid polarization modes. Using the elements Mij of the matrix Mh, and defining Mi and Ma as

� ikzð Þ κo þ κ 2 k2 z � k2 0E⊥ � �

� ð Þ κe þ κ ky 2κe k 2 z � k 2 0E⊥

ikzð Þ κo � κ

ð Þ κ � κe ky

2 k2 z � k 2 0E⊥ � �

2κe k 2 z � k 2 0E⊥

anisotropic medium (x < 0) the evanescent electric amplitude can be written as

TE2 ^aTE2 exp ð Þþ ikTE2x A0

0 Bo1 0 Be1

� � in x ¼ 0. For that purpose, we

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TM2 ^aTM2 exp ð Þ ikTM2x , (42)

, (43)

o

o

, (44a)

: (44b)

Etot ¼ Eð Þx exp ikyy þ ikzz � iωt � �: (40)

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

Eð Þ¼ x Bo1 ^bo1 exp ð Þþ �iko1x Be1 ^be1 exp ð Þ �ike1x , (41)

2, where D1 is provided by Eq. (16), D2 from Eq. (26), v1 from Eq. (17),

1 � D2 establishes a relationship between the amplitudes

0kyð Þ Eκo þ κE⊥

0kyð Þ Eκo � κE⊥

� �

z � k 2 0E⊥ � �

� �

k2

� � � ikz κκeE⊥ þ Eκ2

k2

� � � ikz κκeE⊥ � Eκ2

2κo k 2 z � k 2 0E⊥ � �

2κeE⊥ k2

2κo k 2 z � k 2 0E⊥ � �

2κeE⊥ k 2 z � k2 0E⊥ � �

diffraction equation that gives the 2D wave vector kD ¼ 0; ky; kz

(x > 0) the amplitude of the electric field is

tudes B0

and v0

TE2 and B0

at x ¼ 0, D1 � v1 ¼ D2 � v0

Eð Þ¼ x A0

TM2 are zero.

where the transmission matrix Mh ¼ D�1

Mi ¼ M11 M13

Ma ¼ M21 M23

M31 M33 � � ¼

M41 M43 � � ¼

2 from Eq. (27). This equation reduces to:

The importance of DSWs for integrated optical applications, such as sensing and nanowaveguiding, was appreciated in a series of papers [17, 18]. Indeed Dyakonov-like surface waves also emerge in the case that a biaxial crystal [19] or a structurally chiral material [20] takes the place of the uniaxial medium. The case of metal-dielectric (MD) multilayers as structurally anisotropic media is especially convenient since small-filling fractions of the metallic inclusions enable metamaterials with an enormous birefringence, thus enhancing density of DSWs and relaxing their prominent directivity [21–23].

#### 5.1. Dispersion equation of DSWs

The system under study is the plotted in Figure 5, where we have two semi-infinite media, one of them is isotropic and the second one is an MD lattice. In our case, the indices "1" and "2" make reference to the plasmonic lattice and the isotropic medium, respectively. We have previously reported a comprehensive analysis of this case in [4]. As we have seen earlier, the plasmonic lattice can be taken as an effective uniaxial crystal. In this case, the permittivity along its optical axis, Ez1 ¼ E∥, is given by Eq. (36); also, the permittivity in the transverse direction Ex1 ¼ Ey1 ¼ E⊥ may be appropriately averaged. From hereon, the permittivity E2 of the isotropic medium in x > 0 will be denoted by E. Note that our analysis serves for natural birefringent materials characterized by permittivities E∥ and E⊥.

Figure 5. Schematic setup under study, consisting in a semi-infinite dielectric-metal superlattice ð Þ x < 0 and an isotropic substrate ð Þ x > 0 .

Since we treat the plasmonic lattice as a uniaxial crystal, we may establish analytically the diffraction equation that gives the 2D wave vector kD ¼ 0; ky; kz � � in x ¼ 0. For that purpose, we follow Dyakonov [3] by considering hybrid-polarized surface modes. In the isotropic medium we consider TEx (Ex ¼ 0) and TMx (Hx ¼ 0) waves whose wave vectors have the same real components ky and kz in the plane x ¼ 0. Therefore the electric field in both media may be set as

5. Dyakonov surface waves

58 Surface Waves - New Trends and Developments

5.1. Dispersion equation of DSWs

substrate ð Þ x > 0 .

Dyakonov surface waves (DSWs) are another kind of surface waves, supported at the interface between an optically isotropic medium and a uniaxial-birefringent material. In the original work by Dyakonov (English version was reported in 1988 [3]), the optical axis of the uniaxial medium was assumed in-plane with respect to the interface. This is the case we deal with here. The importance of DSWs for integrated optical applications, such as sensing and nanowaveguiding, was appreciated in a series of papers [17, 18]. Indeed Dyakonov-like surface waves also emerge in the case that a biaxial crystal [19] or a structurally chiral material [20] takes the place of the uniaxial medium. The case of metal-dielectric (MD) multilayers as structurally anisotropic media is especially convenient since small-filling fractions of the metallic inclusions enable metamaterials with an enormous birefringence, thus enhancing

The system under study is the plotted in Figure 5, where we have two semi-infinite media, one of them is isotropic and the second one is an MD lattice. In our case, the indices "1" and "2" make reference to the plasmonic lattice and the isotropic medium, respectively. We have previously reported a comprehensive analysis of this case in [4]. As we have seen earlier, the plasmonic lattice can be taken as an effective uniaxial crystal. In this case, the permittivity along its optical axis, Ez1 ¼ E∥, is given by Eq. (36); also, the permittivity in the transverse direction Ex1 ¼ Ey1 ¼ E⊥ may be appropriately averaged. From hereon, the permittivity E2 of the isotropic medium in x > 0 will be denoted by E. Note that our analysis serves for natural

Figure 5. Schematic setup under study, consisting in a semi-infinite dielectric-metal superlattice ð Þ x < 0 and an isotropic

density of DSWs and relaxing their prominent directivity [21–23].

birefringent materials characterized by permittivities E∥ and E⊥.

$$\mathbf{E}\_{\text{tot}} = \mathbf{E}(\mathbf{x}) \exp\left(i\mathbf{k}\_y y + i\mathbf{k}\_z z - i\omega t\right). \tag{40}$$

Moreover, these fields are evanescent in the isotropic medium and in the superlattice. In the anisotropic medium (x < 0) the evanescent electric amplitude can be written as

$$\mathbf{E}(\mathbf{x}) = B\_{o1} \ddot{b}\_{o1} \exp\left(-i\mathbf{k}\_{o1}\mathbf{x}\right) + B\_{e1} \ddot{b}\_{e1} \exp\left(-i\mathbf{k}\_{e1}\mathbf{x}\right),\tag{41}$$

where the ordinary and extraordinary waves in the effective uniaxial medium decay exponentially with rates given by κo ¼ �iko1 and κe ¼ �ike1, respectively. Taking the formulation given in Section 2.3.1, the amplitudes Ao1 and Ae1 are identically zero. In the isotropic medium (x > 0) the amplitude of the electric field is

$$\mathbf{E(x)} = A'\_{T\mathbf{E2}} \lor\_{T\mathbf{E2}} \exp\left(i\mathbf{k}\_{T\mathbf{E2}}\mathbf{x}\right) + A'\_{T\mathbf{M2}} \lor\_{T\mathbf{M2}} \exp\left(i\mathbf{k}\_{T\mathbf{M2}}\mathbf{x}\right),\tag{42}$$

where the evanescent decay for TE and TM modes is κ ¼ �ikTE2 ¼ �ikTM2. Now the amplitudes B0 TE2 and B0 TM2 are zero.

Once we have the amplitudes in both sides of the interface, we apply the boundary conditions at x ¼ 0, D1 � v1 ¼ D2 � v0 2, where D1 is provided by Eq. (16), D2 from Eq. (26), v1 from Eq. (17), and v0 2 from Eq. (27). This equation reduces to:

$$
\begin{bmatrix} 0\\ B\_{o1} \\ 0 \\ B\_{e1} \end{bmatrix} = \mathbf{M}\_{\mathbb{H}} \begin{bmatrix} A'\_{T\mathbb{E}2} \\ 0 \\ A'\_{T\mathbb{M}2} \\ 0 \end{bmatrix}' \tag{43}
$$

where the transmission matrix Mh ¼ D�1 1 � D2 establishes a relationship between the amplitudes of hybrid polarization modes. Using the elements Mij of the matrix Mh, and defining Mi and Ma as

$$\mathbf{M}\_{i} = \begin{bmatrix} M\_{11} & M\_{13} \\ M\_{31} & M\_{33} \end{bmatrix} = \begin{bmatrix} -\frac{i\mathbf{k}\_{z}(\kappa\_{o} + \kappa)}{2\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} & \frac{k\_{0}^{2}k\_{y}(\kappa\epsilon\_{o} + \kappa\epsilon\_{\perp})}{2\kappa\_{o}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} \\ -\frac{(\kappa\_{e} + \kappa)k\_{y}}{2\kappa\_{e}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} & -\frac{i\mathbf{k}\_{z}(\kappa\kappa\_{e}\epsilon\_{\perp} + \epsilon\kappa\_{o}^{2})}{2\kappa\_{e}\epsilon\_{\perp}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} \end{bmatrix},\tag{44a}$$

$$\mathbf{M}\_{4} = \begin{bmatrix} M\_{21} & M\_{23} \\ M\_{41} & M\_{43} \end{bmatrix} = \begin{bmatrix} \frac{i\mathbf{k}\_{\varepsilon}(\kappa\_{o} - \kappa)}{2\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} & \frac{k\_{0}^{2}k\_{y}(\epsilon\kappa\_{o} - \kappa\epsilon\_{\perp})}{2\kappa\_{o}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} \\ \frac{(\kappa - \kappa\_{e})k\_{y}}{2\kappa\_{\varepsilon}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} & -\frac{i\mathbf{k}\_{z}(\kappa\kappa\_{e}\epsilon\_{\perp} - \kappa\kappa\_{o}^{2})}{2\kappa\_{\varepsilon}\epsilon\_{\perp}\left(k\_{z}^{2} - k\_{0}^{2}\epsilon\_{\perp}\right)} \end{bmatrix}. \tag{44b}$$

Eq. (43) can be rewritten as a set of two independent matrix equations, namely

$$
\begin{bmatrix} 0 \\ 0 \end{bmatrix} = \mathbf{M}\_i \cdot \begin{bmatrix} A'\_{\text{TE2}} \\ A'\_{\text{TM2}} \end{bmatrix} \prime \tag{45a}
$$

$$
\begin{bmatrix} B\_{o1} \\ B\_{\epsilon 1} \end{bmatrix} = \mathbf{M}\_{\mathfrak{a}} \cdot \begin{bmatrix} A'\_{T\mathbf{E}2} \\ A'\_{T\mathbf{M}2} \end{bmatrix}. \tag{45b}
$$

the substrate x > 0. Looking at the other side of the dispersion curve, θmax is established by ke ¼ 0, shown as a black solid line, for which the extraordinary wave will not decay spatially at

Figure 6. Dispersion Eq. (47) for DSWs (dotted-dashed line) propagating on the interface of a semi-infinite dielectricmetal lattice with metal filling factor f = 0.1. The solid elliptical line and the black dashed line are associated with homogeneous extraordinary waves ð Þ κe ¼ 0 and homogeneous ordinary waves ð Þ κo ¼ 0 , respectively. The isofrequency curve ð Þ κ ¼ 0 of isotropic N-BAK1 are represented in by the red solid line, which applies for TEx and TMx waves.

Consequently, the solution for Eq. (47) can be traced near the curves κ ¼ 0 and κe ¼ 0; thus,

As we discussed in Section 4.1, the EMA is limited to metallic slabs' width wm ≪ λ0. However, this condition must be taken into account with care, since the skin depth of noble metals is extremely short, δ ≈ c=ωp. For instance, we estimate δ ¼ 24 nm in the case of silver. If the metal thickness is comparable to its skin depth, the EMA will substantially deviate from exact calculations. Note that experimental studies from multilayer optics rarely incorporate metallic

We emphasize that moderate changes in the birefringence of the plasmonic crystal will substantially affect the existence of DSWs. More specifically, an enlargement of E⊥ driven by increasing wm, provided f is fixed (see Section 4.1), will lead to a significant modification of the DSW dispersion curves. Ultimately, this phenomenon is clearly attributed to nonlocal effects in the effective-medium response of nanolayered metamaterials [24], which is associ-

We conclude that, in order to excite DSWs, one may counterbalance the decrease of birefringence in the plasmonic lattice by means of a dielectric substrate of higher index of refraction. To illustrate this matter, the dispersion Eq. (47) for DSWs is represented in Figure 7, in addition to using the values of E∥ and E⊥ from nonlocal estimators [4]. When wm grows but f is kept fixed, the dispersion curve of the Dyakonov surface waves tends to approach the optic axis.

ated with a strong variation of the fields on the scale of a single layer.

of both curves.

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DSWs are always found close to the crosspoint P0 ky0; kz0

Adapted with permission from [4] of copyright ©2013 IEEE photonics society.

x ! �∞.

5.2.1. Nonlocal effects

slabs with a thickness below 10 nm.

Note that Mi governs the amplitudes A0 TE2 and A0 TM2 of the isotropic medium, and Ma (also Mh) may be used to determine the amplitudes Bo1 and Be1 of the anisotropic medium.

Dyakonov equation is obtained by means of letting the determinant of Mi equal to zero, giving

$$(k\_0^2 k\_y^2 \epsilon\_\perp(\kappa + \kappa\_\epsilon)(\epsilon \kappa\_o + \epsilon\_\perp \kappa) = \kappa\_o k\_z^2 (\kappa + \kappa\_o)(\epsilon \kappa\_o^2 + \epsilon\_\perp \kappa \kappa\_\epsilon),\tag{46}$$

which provides a spectral map of allowed values ky; kz � �. After fairly tedious algebraic transformations we can reduce Eq. (46) to a more convenient form [3].

$$(\kappa + \kappa\_{\epsilon})(\kappa + \kappa\_{o})(\epsilon \kappa\_{o} + \epsilon\_{\bot}\kappa\_{\epsilon}) = (\epsilon\_{\parallel} - \epsilon)(\epsilon - \epsilon\_{\bot})k\_{0}^{2}\kappa\_{o}.\tag{47}$$

Assuming that E∥, E⊥ and all decay rates are positive, the additional restriction E⊥ < E < E∥ can be deduced for the existence of surface waves. As a consequence, positive birefringence is mandatory to ensure a stationary solution of Maxwell's equations. Therefore, layered superlattices supporting Dyakonov-like surface waves cannot be formed by all dielectric materials [21].

#### 5.2. DSWs in nano-engineered materials

To illustrate the difference between using conventional birefringent materials and plasmonic crystals, we solve Eq. (47) for liquid crystal E7 with E∥ ¼ 2:98 i E⊥ ¼ 2:31 at a wavelength of λ0 ¼ 1:55μ m and N-BAK1 substrate of dielectric constant E ¼ 2:42. In this case, DSWs propagate in a narrow angular region Δθ ¼ θmax � θmin, where θ stands for the angle between the inplane vector ky; kz � � and the optical axis. More specifically, the angular range yields Δθ ¼ 0:92∘ around the mean angle θ ¼ 26:6∘ . It appears that the resulting angular range Δθ is short. But this range would become much smaller when using other optical crystals like quartz, exhibiting a common birefringence. In order to gain in angular extent Δθ, we consider a GaAs-Ag crystal (E1 ¼ 12:5 and E2 ¼ �103:3, where we neglect losses) that leads to values of E∥ ¼ 14:08, derived from Eq. (36), and E⊥ ¼ 0:92 with a metal-filling factor f ¼ 0:10. Form birefringence now yields Δn ¼ 2:79. Moreover, solutions to Eq. (47) can be found in the region of angles comprised between θmin ¼ 39:0∘ and θmax ¼ 71:3∘ . It is important to point out that the total angular range of existence of DSWs, Δθ ¼ 32:3∘ , grows by more than an order of magnitude. Figure 6 shows the dispersion curve for DSWs; one can observe that θmin is attained under the condition κ ¼ 0 (red solid line), where TEx and TMx waves are uniform in

Figure 6. Dispersion Eq. (47) for DSWs (dotted-dashed line) propagating on the interface of a semi-infinite dielectricmetal lattice with metal filling factor f = 0.1. The solid elliptical line and the black dashed line are associated with homogeneous extraordinary waves ð Þ κe ¼ 0 and homogeneous ordinary waves ð Þ κo ¼ 0 , respectively. The isofrequency curve ð Þ κ ¼ 0 of isotropic N-BAK1 are represented in by the red solid line, which applies for TEx and TMx waves. Adapted with permission from [4] of copyright ©2013 IEEE photonics society.

the substrate x > 0. Looking at the other side of the dispersion curve, θmax is established by ke ¼ 0, shown as a black solid line, for which the extraordinary wave will not decay spatially at x ! �∞.

Consequently, the solution for Eq. (47) can be traced near the curves κ ¼ 0 and κe ¼ 0; thus, DSWs are always found close to the crosspoint P0 ky0; kz0 of both curves.

#### 5.2.1. Nonlocal effects

Eq. (43) can be rewritten as a set of two independent matrix equations, namely

0 0 � �

Bo1 Be1 � �

yE⊥ð Þ κ þ κe ð Þ¼ Eκo þ E⊥κ κok2

Note that Mi governs the amplitudes A0

60 Surface Waves - New Trends and Developments

k 2 0k 2

5.2. DSWs in nano-engineered materials

rials [21].

plane vector ky; kz

around the mean angle θ ¼ 26:6∘

which provides a spectral map of allowed values ky; kz

formations we can reduce Eq. (46) to a more convenient form [3].

of angles comprised between θmin ¼ 39:0∘ and θmax ¼ 71:3∘

¼ Mi � A0

¼ Ma � A0

TE2 and A0

Dyakonov equation is obtained by means of letting the determinant of Mi equal to zero, giving

ð Þ κ þ κe ð Þ κ þ κo ð Þ¼ Eκo þ E⊥κe E∥ � E � �ð Þ E � E⊥ k

Assuming that E∥, E⊥ and all decay rates are positive, the additional restriction E⊥ < E < E∥ can be deduced for the existence of surface waves. As a consequence, positive birefringence is mandatory to ensure a stationary solution of Maxwell's equations. Therefore, layered superlattices supporting Dyakonov-like surface waves cannot be formed by all dielectric mate-

To illustrate the difference between using conventional birefringent materials and plasmonic crystals, we solve Eq. (47) for liquid crystal E7 with E∥ ¼ 2:98 i E⊥ ¼ 2:31 at a wavelength of λ0 ¼ 1:55μ m and N-BAK1 substrate of dielectric constant E ¼ 2:42. In this case, DSWs propagate in a narrow angular region Δθ ¼ θmax � θmin, where θ stands for the angle between the in-

this range would become much smaller when using other optical crystals like quartz, exhibiting a common birefringence. In order to gain in angular extent Δθ, we consider a GaAs-Ag crystal (E1 ¼ 12:5 and E2 ¼ �103:3, where we neglect losses) that leads to values of E∥ ¼ 14:08, derived from Eq. (36), and E⊥ ¼ 0:92 with a metal-filling factor f ¼ 0:10. Form birefringence now yields Δn ¼ 2:79. Moreover, solutions to Eq. (47) can be found in the region

the total angular range of existence of DSWs, Δθ ¼ 32:3∘ , grows by more than an order of magnitude. Figure 6 shows the dispersion curve for DSWs; one can observe that θmin is attained under the condition κ ¼ 0 (red solid line), where TEx and TMx waves are uniform in

� � and the optical axis. More specifically, the angular range yields Δθ ¼ 0:92∘

. It appears that the resulting angular range Δθ is short. But

Mh) may be used to determine the amplitudes Bo1 and Be1 of the anisotropic medium.

TE2 A0 TM2

> TE2 A0 TM2

z ð Þ κ þ κo Eκ2

" #

, (45a)

: (45b)

� �, (46)

. It is important to point out that

0κo: (47)

TM2 of the isotropic medium, and Ma (also

� �. After fairly tedious algebraic trans-

o þ E⊥κκe

2

" #

As we discussed in Section 4.1, the EMA is limited to metallic slabs' width wm ≪ λ0. However, this condition must be taken into account with care, since the skin depth of noble metals is extremely short, δ ≈ c=ωp. For instance, we estimate δ ¼ 24 nm in the case of silver. If the metal thickness is comparable to its skin depth, the EMA will substantially deviate from exact calculations. Note that experimental studies from multilayer optics rarely incorporate metallic slabs with a thickness below 10 nm.

We emphasize that moderate changes in the birefringence of the plasmonic crystal will substantially affect the existence of DSWs. More specifically, an enlargement of E⊥ driven by increasing wm, provided f is fixed (see Section 4.1), will lead to a significant modification of the DSW dispersion curves. Ultimately, this phenomenon is clearly attributed to nonlocal effects in the effective-medium response of nanolayered metamaterials [24], which is associated with a strong variation of the fields on the scale of a single layer.

We conclude that, in order to excite DSWs, one may counterbalance the decrease of birefringence in the plasmonic lattice by means of a dielectric substrate of higher index of refraction. To illustrate this matter, the dispersion Eq. (47) for DSWs is represented in Figure 7, in addition to using the values of E∥ and E⊥ from nonlocal estimators [4]. When wm grows but f is kept fixed, the dispersion curve of the Dyakonov surface waves tends to approach the optic axis.

Figure 7. Solutions to Dyakonov equation, drawn in dotted-dashed lines, for a MD lattice with the same filling factor f ¼ 0:10 but different wm, using estimates from the nonlocal birefringence approach described in [4]. Note that the red solid line designates the isofrequency curve of isotropic substrate (a) N-BAK1 and (b) P-SF68. Adapted with permission from [4] of copyright ©2013 IEEE photonics society.

For an N-BAK1 substrate, as shown in Figure 7(a), θmax ¼ 68:2∘ and 58:7∘ for wm ¼ 3 nm and 6 nm, respectively. Also, θmin ¼ 37:6∘ and 32:1∘ for these two cases. As a consequence the angular range Δθ shrinks when wm increases. In the limit E⊥ ! E, which occurs for wm ¼ 10:3 nm using a N-BAK1 substrate, DSWs are not supported at the interface of the MD lattice and the isotropic dielectric. A substrate with greater relative permittivity E would be necessary. For example, if we use a substrate with greater relative permittivity E as P-SF68 [see Figure 7(b)], DSWs exist for wm ¼ 12 nm with an angular range Δθ ¼ 12:5∘ .

#### 5.2.2. Dissipative effects

Up to now, we have avoided another important aspect of plasmonic devices namely dissipation in metallic elements. In this regard, effective permittivities are fundamentally complex, and consequently the Dyakonov Eq. (47) is expected to give complex values of ky; kz . This procedure has been discussed by Sorni et al. [25] recently. In order to tackle this problem, we evaluate numerically the value of the Bloch wavenumber kz for a given real value ky. The spatial frequency kz becomes complex since Im½ �¼ Em 8:1. As a consequence, the surface wave cannot propagate indefinitely, undergoing an energy attenuation given by l ¼ ð Þ 2Im½ � kz �1 . Furthermore, we naturally assume that the real part of the parameters κ, κo, and κe are all positive. These positive values correlate with a decay at ∣x∣ ! ∞ and thus with a confinement of the wave near x ¼ 0.

We observe that the dispersion curve for dissipative DSWs is flatter and larger than the curve

Figure 8. (a) Isofrequency curve that corresponds to hybrid surface waves existing at the boundary between a semiinfinite P-SF68 substrate and a lossy MD superlattice of f ¼ 0:10 and wm ¼ 12 nm. (b) Ratio of Im(kz) over re(kz) representing dissipation effects in the propagation of DSWs. (c) Three contour plots of the magnetic field ∣Hx∣ computed using the finite-element method. The superlattice is set on the left, for which only one period is represented. Capital letters a, B, and C, designate the transverse spatial frequencies ky ¼ 0:8k0, 1:2k0, and 1:6 k0, respectively. Adapted with permis-

waves. In these two figures, capital letters A, B, and C designate the transverse spatial frequencies ky ¼ 0:8k0, 1:2k0, and 1:6k0, respectively. Figure 8(c) shows the magnetic field ∣Hx∣ for the three different cases denoted by capital letters A, B, and C. Note that in the case of paraxial surface waves, for which ky reaches a minimum value (case A), one achieves Imð Þ kz ≪ Reð Þ kz as depicted in Figure 8(b). This is induced by an enormous shift undergone by the field maximum in the direction to the isotropic medium, as shown in Figure 8(c), where dissipation effects are barely disadvantageous on surface-wave propagation. Moreover, this would be consistent with a condition Reð Þ κ ≪ Reð Þ κe . On the other hand, for nonparaxial waves, having the largest values of ky, the fields show slow energy decay inside the plasmonic superlattice. In case C, the magnetic field ∣Hx∣ is localized around the metallic layer and takes significant values far from the boundary of the substrate. As a consequence, losses in the metal translate

. Figure 8(b) shows Im(kz)/Re(kz) in the range of existence of the surface

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

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63

, giving an angular

obtained by neglecting losses. Specifically θmax ¼ 49:9∘ and θmin ¼ 23:7∘

range Δθ ¼ 26:2∘

into a significant rise in the values of Im(kz).

sion from [4] of copyright ©2013 IEEE photonics society.

Figure 8(a) depicts the dispersion curve corresponding to dissipative DSWs, for the case of a plasmonic MD lattice with f ¼ 0:10 and wm ¼ 12 nm. We used a commercial software (COMSOL Multiphysics) based on the finite-element method (FEM) in order to perform our numerical simulations. We cannot observe surface waves by setting an N-BAK1 substrate with n ¼ 1:56, suggesting that this is a retardation effect. More specifically, Figure 8(a) shows the isofrequency curves when n ¼ 1:95, corresponding to P-SF68.

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge http://dx.doi.org/10.5772/intechopen.74126 63

For an N-BAK1 substrate, as shown in Figure 7(a), θmax ¼ 68:2∘ and 58:7∘ for wm ¼ 3 nm and 6 nm, respectively. Also, θmin ¼ 37:6∘ and 32:1∘ for these two cases. As a consequence the angular range Δθ shrinks when wm increases. In the limit E⊥ ! E, which occurs for wm ¼ 10:3 nm using a N-BAK1 substrate, DSWs are not supported at the interface of the MD lattice and the isotropic dielectric. A substrate with greater relative permittivity E would be necessary. For example, if we use a substrate with greater relative permittivity E as P-SF68 [see Figure 7(b)],

Figure 7. Solutions to Dyakonov equation, drawn in dotted-dashed lines, for a MD lattice with the same filling factor f ¼ 0:10 but different wm, using estimates from the nonlocal birefringence approach described in [4]. Note that the red solid line designates the isofrequency curve of isotropic substrate (a) N-BAK1 and (b) P-SF68. Adapted with permission

Up to now, we have avoided another important aspect of plasmonic devices namely dissipation in metallic elements. In this regard, effective permittivities are fundamentally complex, and consequently the Dyakonov Eq. (47) is expected to give complex values of ky; kz

procedure has been discussed by Sorni et al. [25] recently. In order to tackle this problem, we evaluate numerically the value of the Bloch wavenumber kz for a given real value ky. The spatial frequency kz becomes complex since Im½ �¼ Em 8:1. As a consequence, the surface wave cannot propagate indefinitely, undergoing an energy attenuation given by l ¼ ð Þ 2Im½ � kz �1

Furthermore, we naturally assume that the real part of the parameters κ, κo, and κe are all positive. These positive values correlate with a decay at ∣x∣ ! ∞ and thus with a confinement

Figure 8(a) depicts the dispersion curve corresponding to dissipative DSWs, for the case of a plasmonic MD lattice with f ¼ 0:10 and wm ¼ 12 nm. We used a commercial software (COMSOL Multiphysics) based on the finite-element method (FEM) in order to perform our numerical simulations. We cannot observe surface waves by setting an N-BAK1 substrate with n ¼ 1:56, suggesting that this is a retardation effect. More specifically, Figure 8(a) shows the

.

. This

.

DSWs exist for wm ¼ 12 nm with an angular range Δθ ¼ 12:5∘

isofrequency curves when n ¼ 1:95, corresponding to P-SF68.

5.2.2. Dissipative effects

from [4] of copyright ©2013 IEEE photonics society.

62 Surface Waves - New Trends and Developments

of the wave near x ¼ 0.

Figure 8. (a) Isofrequency curve that corresponds to hybrid surface waves existing at the boundary between a semiinfinite P-SF68 substrate and a lossy MD superlattice of f ¼ 0:10 and wm ¼ 12 nm. (b) Ratio of Im(kz) over re(kz) representing dissipation effects in the propagation of DSWs. (c) Three contour plots of the magnetic field ∣Hx∣ computed using the finite-element method. The superlattice is set on the left, for which only one period is represented. Capital letters a, B, and C, designate the transverse spatial frequencies ky ¼ 0:8k0, 1:2k0, and 1:6 k0, respectively. Adapted with permission from [4] of copyright ©2013 IEEE photonics society.

We observe that the dispersion curve for dissipative DSWs is flatter and larger than the curve obtained by neglecting losses. Specifically θmax ¼ 49:9∘ and θmin ¼ 23:7∘ , giving an angular range Δθ ¼ 26:2∘ . Figure 8(b) shows Im(kz)/Re(kz) in the range of existence of the surface waves. In these two figures, capital letters A, B, and C designate the transverse spatial frequencies ky ¼ 0:8k0, 1:2k0, and 1:6k0, respectively. Figure 8(c) shows the magnetic field ∣Hx∣ for the three different cases denoted by capital letters A, B, and C. Note that in the case of paraxial surface waves, for which ky reaches a minimum value (case A), one achieves Imð Þ kz ≪ Reð Þ kz as depicted in Figure 8(b). This is induced by an enormous shift undergone by the field maximum in the direction to the isotropic medium, as shown in Figure 8(c), where dissipation effects are barely disadvantageous on surface-wave propagation. Moreover, this would be consistent with a condition Reð Þ κ ≪ Reð Þ κe . On the other hand, for nonparaxial waves, having the largest values of ky, the fields show slow energy decay inside the plasmonic superlattice. In case C, the magnetic field ∣Hx∣ is localized around the metallic layer and takes significant values far from the boundary of the substrate. As a consequence, losses in the metal translate into a significant rise in the values of Im(kz).

#### 5.3. New families of DSWs in lossy media

In this section we carry out a thorough analysis of DSWs that takes place in lossy uniaxial metamaterials. Special emphasis is put when the effective-medium approach induces satisfactory results. The introduction of losses leads to a transformation of the isofrequency curves, which deviates from spheres and ellipsoids, as commonly considered by ordinary and extraordinary waves, respectively. As a consequence, one can find two different families of surface waves as reported by Sorni et al. [25]. One family of surface waves is directly related with the well-known solutions derived by Dyakonov [3]. Importantly, the existence of a new family of surface waves is revealed, closely connected to the presence of losses in the uniaxial effective crystal. We point out that the solutions to Dyakonov equation presented earlier are partial ones insofar as the z-component of the wavevector is kept real valued. Nevertheless, the whole set of solutions includes all possible wavevectors that feature a complex-valued kx.

#### 6. Dyakonov surface waves in hyperbolic media

In this section we perform a thorough analysis of DSWs taking place in semi-infinite MD lattices exhibiting hyperbolic dispersion. Part of this section was previously reported by Zapata-Rodríguez et al. [26]; we point out that recently further studies on DSW in hyperbolic metamaterials have been reported by other authors [27]. Our approach puts emphasis on the EMA. Under these conditions, different regimes can be found including DSWs with nonhyperbolic dispersion. The system under analysis is again as depicted in Figure 5. For simplicity, we assume that dielectric materials are nondispersive; indeed, we set E ¼ 1 and Ed ¼ 2:25 in our numerical simulations. Furthermore, Drude metals are included, and frequencies will be expressed in units of its plasma frequency, Ω ¼ ω=ωp. Again, Dyakonov Eq. (47) provides the spectral map of wave vectors kD ¼ 0; ky; kz � �. Note that in this section, spatial frequencies will be expressed in units of kp.

In the special case of the surface wave propagation perpendicular to the optical axis (kz ¼ 0), Eq. (47) reveals the following solution: Eκo þ E⊥κ ¼ 0. In the case: E⊥ < 0 and E < ∣E⊥∣, this equation has the well-known solution

$$k\_y = \Omega \sqrt{\frac{\epsilon \epsilon\_\perp}{\epsilon + \epsilon\_\perp}}\tag{48}$$

other cases are treated elsewhere [26]. In the effective-uniaxial medium, it is easy to realize that κ < κo and also κe < κo. Under these circumstances, all brackets in Dyakonov Eq. (47) are positive provided Eκo þ E⊥κe > 0. This happens within the spectral band Ω0 < Ω < Ω1, where

Ω0 ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

In Figure 9(a) and (b), we illustrate the dispersion equation of DSWs for two different frequencies within the spectral range 0 < Ω < Ω0. In these cases, the dispersion curve approaches a hyperbola. We find a bandgap around kz ¼ 0 in Figure 9(a), unlike what occurs in (b). Note that hybrid solutions near kz ¼ 0 are additionally constrained to the condition ky ≥ Ω ffiffiffiffi

also Eq. (48)], which is a necessary condition for κe to exhibit real and positive values. We consider the quasi-static regime (Ω ! 0) where ∣kD∣ ¼ kD ≫ Ω to determine the asymptotes of the hyperbolic-like DSW dispersion curve. Under this approximation, κ ¼ kD, κo ¼ kD and

being ky ¼ kD cos θ and kz ¼ kD sin θ. These asymptotes establish a canalization regime leading to a collective directional propagation of DSW beams [4, 28]. At this point it is necessary to remind that the asymptotes of the e-waves dispersion curve, in the kykz plane, have slopes

In the high-frequency band Ω2 < Ω < 1 we find that E∥ < 0 < E⊥, as occurs in Figure 9(c). Note the relevant proximity of DSW dispersion curve to κe ¼ 0, the same way we also find in Figure 9(a) and (b). Conversely it crosses the e-wave hyperbolic curve at two different points,

Figure 9. Solutions to Eq. (47), drawn in solid line, providing the spatial dispersion of DSWs which can exist in the arrangement of Figure 5, at different frequencies: (a) Ω ¼ 0:20, (b) Ω ¼ 0:28, and (c) Ω ¼ 0:85. Here, the isotropic medium is air and the multi-layered metamaterial has a filling factor f ¼ 0:25 and. Also, we include equations κ ¼ 0 (dotted line)

and κe ¼ 0 (dashed line). Adapted with permission from [26] of copyright ©2013 Optical Society of America.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos 2θ þ E∥=E⊥

� � sin 2θ

Note that Ω0 ¼ 0:292 in our numerical simulation.

Θ ¼

satisfying the condition θD < θe, as illustrated in Figure 9(b).

q

κe ¼ ΘkD, where

1 þ E=f þ Edð Þ 1 � f =f p : (49)

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

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E∥ p [see 65

, (50)

which resembles the dispersion equation of conventional SPPs [see Eq. (31)]. Here we have purely TMx polarized waves, as expected. Note that no solutions to Eq. (47) can be found, in the form of surface waves, considering wave propagation parallel to the optical axis (ky ¼ 0) for hyperbolic metamaterials: E⊥E∥ < 0. That means that a threshold value of ky can be found for the existence of surface waves.

Next we describe a specific configuration governing DSWs, subject to a low value of the refractive index n ¼ ffiffi E p , namely E < E∥ (E < E⊥) occurring at low and moderate frequencies; other cases are treated elsewhere [26]. In the effective-uniaxial medium, it is easy to realize that κ < κo and also κe < κo. Under these circumstances, all brackets in Dyakonov Eq. (47) are positive provided Eκo þ E⊥κe > 0. This happens within the spectral band Ω0 < Ω < Ω1, where

$$\Omega\_0 = \frac{1}{\sqrt{1 + \epsilon/f + \epsilon\_d(1 - f)/f}}.\tag{49}$$

Note that Ω0 ¼ 0:292 in our numerical simulation.

5.3. New families of DSWs in lossy media

64 Surface Waves - New Trends and Developments

In this section we carry out a thorough analysis of DSWs that takes place in lossy uniaxial metamaterials. Special emphasis is put when the effective-medium approach induces satisfactory results. The introduction of losses leads to a transformation of the isofrequency curves, which deviates from spheres and ellipsoids, as commonly considered by ordinary and extraordinary waves, respectively. As a consequence, one can find two different families of surface waves as reported by Sorni et al. [25]. One family of surface waves is directly related with the well-known solutions derived by Dyakonov [3]. Importantly, the existence of a new family of surface waves is revealed, closely connected to the presence of losses in the uniaxial effective crystal. We point out that the solutions to Dyakonov equation presented earlier are partial ones insofar as the z-component of the wavevector is kept real valued. Nevertheless, the whole set

In this section we perform a thorough analysis of DSWs taking place in semi-infinite MD lattices exhibiting hyperbolic dispersion. Part of this section was previously reported by Zapata-Rodríguez et al. [26]; we point out that recently further studies on DSW in hyperbolic metamaterials have been reported by other authors [27]. Our approach puts emphasis on the EMA. Under these conditions, different regimes can be found including DSWs with nonhyperbolic dispersion. The system under analysis is again as depicted in Figure 5. For simplicity, we assume that dielectric materials are nondispersive; indeed, we set E ¼ 1 and Ed ¼ 2:25 in our numerical simulations. Furthermore, Drude metals are included, and frequencies will be expressed in units of its plasma frequency, Ω ¼ ω=ωp. Again, Dyakonov Eq. (47)

In the special case of the surface wave propagation perpendicular to the optical axis (kz ¼ 0), Eq. (47) reveals the following solution: Eκo þ E⊥κ ¼ 0. In the case: E⊥ < 0 and E < ∣E⊥∣, this

r

which resembles the dispersion equation of conventional SPPs [see Eq. (31)]. Here we have purely TMx polarized waves, as expected. Note that no solutions to Eq. (47) can be found, in the form of surface waves, considering wave propagation parallel to the optical axis (ky ¼ 0) for hyperbolic metamaterials: E⊥E∥ < 0. That means that a threshold value of ky can be found

Next we describe a specific configuration governing DSWs, subject to a low value of the

E p , namely E < E∥ (E < E⊥) occurring at low and moderate frequencies;

ffiffiffiffiffiffiffiffiffiffiffiffi EE⊥ E þ E⊥

ky ¼ Ω

� �. Note that in this section, spatial

, (48)

of solutions includes all possible wavevectors that feature a complex-valued kx.

6. Dyakonov surface waves in hyperbolic media

provides the spectral map of wave vectors kD ¼ 0; ky; kz

frequencies will be expressed in units of kp.

equation has the well-known solution

for the existence of surface waves.

refractive index n ¼ ffiffi

In Figure 9(a) and (b), we illustrate the dispersion equation of DSWs for two different frequencies within the spectral range 0 < Ω < Ω0. In these cases, the dispersion curve approaches a hyperbola. We find a bandgap around kz ¼ 0 in Figure 9(a), unlike what occurs in (b). Note that hybrid solutions near kz ¼ 0 are additionally constrained to the condition ky ≥ Ω ffiffiffiffi E∥ p [see also Eq. (48)], which is a necessary condition for κe to exhibit real and positive values. We consider the quasi-static regime (Ω ! 0) where ∣kD∣ ¼ kD ≫ Ω to determine the asymptotes of the hyperbolic-like DSW dispersion curve. Under this approximation, κ ¼ kD, κo ¼ kD and κe ¼ ΘkD, where

$$\Theta = \sqrt{\cos^2 \theta + \left(\epsilon\_{\parallel}/\epsilon\_{\perp}\right) \sin^2 \theta},\tag{50}$$

being ky ¼ kD cos θ and kz ¼ kD sin θ. These asymptotes establish a canalization regime leading to a collective directional propagation of DSW beams [4, 28]. At this point it is necessary to remind that the asymptotes of the e-waves dispersion curve, in the kykz plane, have slopes satisfying the condition θD < θe, as illustrated in Figure 9(b).

In the high-frequency band Ω2 < Ω < 1 we find that E∥ < 0 < E⊥, as occurs in Figure 9(c). Note the relevant proximity of DSW dispersion curve to κe ¼ 0, the same way we also find in Figure 9(a) and (b). Conversely it crosses the e-wave hyperbolic curve at two different points,

Figure 9. Solutions to Eq. (47), drawn in solid line, providing the spatial dispersion of DSWs which can exist in the arrangement of Figure 5, at different frequencies: (a) Ω ¼ 0:20, (b) Ω ¼ 0:28, and (c) Ω ¼ 0:85. Here, the isotropic medium is air and the multi-layered metamaterial has a filling factor f ¼ 0:25 and. Also, we include equations κ ¼ 0 (dotted line) and κe ¼ 0 (dashed line). Adapted with permission from [26] of copyright ©2013 Optical Society of America.

where solutions to the Dyakonov equation begin and end, respectively. It is clear that the angular range of DSWs now turns to be significantly low.

showed that hybrid-polarized surface waves may propagate obliquely at the boundary between a plasmonic bilayer superlattice and an isotropic loss-free material. We revealed that realistic widths of the slabs might lead to solutions which deviate significantly from the results derived directly from the EMA and Dyakonov analysis. Finally, we showed that excitation of DSWs at the boundary of an isotropic dielectric and a hyperbolic metamaterial enables a distinct regime of propagation. It is important to note that the properties of the resulting bound states change drastically with the index of refraction of the surrounding medium, suggesting potential applications in chemical and biological sensing and nanoimaging.

This chapter was supported by the Qatar National Research Fund (Grant No. NPRP 8-028-1- 001) and the Spanish Ministry of Economy and Competitiveness (Grants No. TEC2014-53727-

1 Departament of Optics and Optometry and Vision Science, University of Valencia, Burjassot,

2 Center of Microelectronic Technologies, Institute of Chemistry, Technology and Metallurgy

4 Department of Optics, Pharmacology and Anatomy, University of Alicante, Alicante, Spain

[1] Zenneck J. Über die Fortpflanzung ebener elektromagnetischer wellen längs einer ebenen Lieterfläche und ihre Beziehung zur drahtlosen Telegraphie. Annalen der Physik (Leip-

[3] D'yakonov MI. New type of electromagnetic wave propagating at an interface. Sov. Phys.

[4] Zapata-Rodríguez CJ, Miret JJ, Sorni JA, Vuković SM. Propagation of dyakonon wavepackets at the boundary of metallodielectric lattices. IEEE Journal of Selected Topics in

[2] Maier SA. Plasmonics: Fundamentals and Applications. New York: Springer; 2007

2,3, Juan J. Miret4

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

http://dx.doi.org/10.5772/intechopen.74126

67

, Mahin Naserpour1,5 and

\*, Slobodan Vuković

5 Department of Physics, College of Sciences, Shiraz University, Shiraz, Iran

C2-1-R and TEC2017-86102-C2-1R).

Author details

Milivoj R. Belić

References

zig). 1907;23:846-866

JETP. 1988;67:714-716

Quantum Electronics. 2013;19:4601408

Spain

Carlos J. Zapata-Rodríguez1

3

\*Address all correspondence to<EMAIL_ADDRESS>
(IHTM), University of Belgrade, Belgrade, Serbia 3 Texas A&M University at Qatar, Doha, Qatar

#### 6.1. DSWs in band-gap hyperbolic media

In previous sections we demonstrated that the presence of metallic nano-elements leads to nonlocal effects and dissipation effects which reshape the propagation dynamics of the surface signal. Here, we briefly discuss the extraordinary favorable conditions which may appear in band-gap metal–insulator-layered media for the existence of DSWs. As reported thoroughly by Miret et al. [29], engineering secondary bands by tuning the plasmonic-crystal geometry may lead to a controlled optical anisotropy, which is markedly dissimilar to the prescribed hyperbolic regime that is derived by the EMA, however, assisting the presence of DSWs on the interface between such hyperbolic metamaterial and an insulator.

In particular, a surface wave propagating on an Ag-Ge grating was considered, where the environment medium that is set above the metallic grating is formed by SiO2. If the metalfilling factor was f ¼ 0:25, the effective permittivities of the anisotropic metamaterial would be estimated as E⊥ ¼ �11:48 þ i2:05 and Ek ¼ 25:96 þ i0:14 at a wavelength of λ0 ¼ 1550 nm. Disregarding losses, the DSW dispersion curve describes an incomplete hyperbolic curve, finding an endpoint under the condition κe ¼ 0, where the extraordinary wave breaks its confinement in the vicinities of the isotropic-uniaxial interface [26].

Considering now a realistic nanostructure consisting of Ag layers of w2 ¼ 40 nm interspersed between Ge layers of w1 ¼ 120 nm, thus maintaining a metal-filling factor of f ¼ 0:25 as analyzed earlier, a first TM band with hyperbolic-like characteristics dominates at high inplane frequencies kt. Additionally, a second band emerges for TM Bloch modes, which exhibits a moderate anisotropy, demonstrating near-elliptical dispersion curves (with positive effective permittivities) and positive birefringence. Furthermore, TE modal dispersion is roughly isotropic. Therefore, satisfactory conditions are found near the second TM band for the existence of Dyakonov-like surface waves. Finally, in order to numerically obtain the dispersion curves and wave fields associated with DSWs, one may follow the same computational procedure followed by Zapata-Rodríguez et al. and [4, 26].

#### 7. Summary

In this chapter we provide several methods to analytically calculate and numerically simulate modal propagation of DSWs governed by material anisotropy. We focused on the spatial properties of DSWs at optical and telecom wavelengths, particularly using uniaxial metamaterials formed of dielectric and metallic nanolayers. We developed an electromagnetic matrix procedure enabling different aspects reviewed in this chapter, specially adapted to complex multilayered configurations. The EMA results are particularly appropriate for the characterization of the form birefringence of a multilayered nanostructure, though limitations driven by the layers width have been discussed. Through a rigorous full-wave analysis, we showed that hybrid-polarized surface waves may propagate obliquely at the boundary between a plasmonic bilayer superlattice and an isotropic loss-free material. We revealed that realistic widths of the slabs might lead to solutions which deviate significantly from the results derived directly from the EMA and Dyakonov analysis. Finally, we showed that excitation of DSWs at the boundary of an isotropic dielectric and a hyperbolic metamaterial enables a distinct regime of propagation. It is important to note that the properties of the resulting bound states change drastically with the index of refraction of the surrounding medium, suggesting potential applications in chemical and biological sensing and nanoimaging.

This chapter was supported by the Qatar National Research Fund (Grant No. NPRP 8-028-1- 001) and the Spanish Ministry of Economy and Competitiveness (Grants No. TEC2014-53727- C2-1-R and TEC2017-86102-C2-1R).

## Author details

where solutions to the Dyakonov equation begin and end, respectively. It is clear that the

In previous sections we demonstrated that the presence of metallic nano-elements leads to nonlocal effects and dissipation effects which reshape the propagation dynamics of the surface signal. Here, we briefly discuss the extraordinary favorable conditions which may appear in band-gap metal–insulator-layered media for the existence of DSWs. As reported thoroughly by Miret et al. [29], engineering secondary bands by tuning the plasmonic-crystal geometry may lead to a controlled optical anisotropy, which is markedly dissimilar to the prescribed hyperbolic regime that is derived by the EMA, however, assisting the presence of DSWs on the

In particular, a surface wave propagating on an Ag-Ge grating was considered, where the environment medium that is set above the metallic grating is formed by SiO2. If the metalfilling factor was f ¼ 0:25, the effective permittivities of the anisotropic metamaterial would be estimated as E⊥ ¼ �11:48 þ i2:05 and Ek ¼ 25:96 þ i0:14 at a wavelength of λ0 ¼ 1550 nm. Disregarding losses, the DSW dispersion curve describes an incomplete hyperbolic curve, finding an endpoint under the condition κe ¼ 0, where the extraordinary wave breaks its

Considering now a realistic nanostructure consisting of Ag layers of w2 ¼ 40 nm interspersed between Ge layers of w1 ¼ 120 nm, thus maintaining a metal-filling factor of f ¼ 0:25 as analyzed earlier, a first TM band with hyperbolic-like characteristics dominates at high inplane frequencies kt. Additionally, a second band emerges for TM Bloch modes, which exhibits a moderate anisotropy, demonstrating near-elliptical dispersion curves (with positive effective permittivities) and positive birefringence. Furthermore, TE modal dispersion is roughly isotropic. Therefore, satisfactory conditions are found near the second TM band for the existence of Dyakonov-like surface waves. Finally, in order to numerically obtain the dispersion curves and wave fields associated with DSWs, one may follow the same computational procedure

In this chapter we provide several methods to analytically calculate and numerically simulate modal propagation of DSWs governed by material anisotropy. We focused on the spatial properties of DSWs at optical and telecom wavelengths, particularly using uniaxial metamaterials formed of dielectric and metallic nanolayers. We developed an electromagnetic matrix procedure enabling different aspects reviewed in this chapter, specially adapted to complex multilayered configurations. The EMA results are particularly appropriate for the characterization of the form birefringence of a multilayered nanostructure, though limitations driven by the layers width have been discussed. Through a rigorous full-wave analysis, we

angular range of DSWs now turns to be significantly low.

interface between such hyperbolic metamaterial and an insulator.

confinement in the vicinities of the isotropic-uniaxial interface [26].

followed by Zapata-Rodríguez et al. and [4, 26].

7. Summary

6.1. DSWs in band-gap hyperbolic media

66 Surface Waves - New Trends and Developments

Carlos J. Zapata-Rodríguez1 \*, Slobodan Vuković 2,3, Juan J. Miret4 , Mahin Naserpour1,5 and Milivoj R. Belić 3

\*Address all correspondence to<EMAIL_ADDRESS>
1 Departament of Optics and Optometry and Vision Science, University of Valencia, Burjassot, Spain

2 Center of Microelectronic Technologies, Institute of Chemistry, Technology and Metallurgy (IHTM), University of Belgrade, Belgrade, Serbia

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[21] Vuković SM, Miret JJ, Zapata-Rodríguez CJ, Jaksić Z. Oblique surface waves at an interface of metal-dielectric superlattice and isotropic dielectric. Physica Scripta. 2012;T149:

Dyakonov Surface Waves: Anisotropy-Enabling Confinement on the Edge

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[22] Ghasempour Ardakani A, Naserpour M, Zapata Rodríguez CJ. Dyakonov-like surface waves in the THz regime. Photonics and Nanostructures - Fundamentals and Applica-

[23] Miret JJ, Zapata-Rodrïguez CJ, Jaksić Z, Vuković SM, Belić MR. Substantial enlargement of angular existence range for Dyakonov-like surface waves at semi-infinite metal-dielectric

[24] Elser J, Podolskiy VA, Salakhutdinov I, Avrutsky I. Nonlocal effects in effective-medium response of nanolayered metamaterials. Applied Physics Letters. 2007;90:191109

[25] Sorni JA, Naserpour M, Zapata-Rodríguez CJ, Miret JJ. Dyakonov surface waves in lossy

[26] Zapata-Rodríguez CJ, Miret JJ, Vuković S, Belić MR. Engineered surface waves in hyper-

[27] Xiang Y, Guo J, Dai X, Wen S, Tang D. Engineered surface Bloch waves in graphene-based

[28] Jacob Z, Narimanov EE. Optical hyperspace for plasmons: Dyakonov states in metama-

[29] Miret JJ, Sorní JA, Naserpour M, Ghasempour Ardakani A, Zapata-Rodríguez CJ. Nonlocal dispersion anomalies of Dyakonov-like surface waves at hyperbolic media interfaces. Photonics and Nanostructures - Fundamentals and Applications. 2016;18:16-22

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[8] Barnes WL, Dereux A, Ebbesen TW. Surface plasmon subwavelength optics. Nature. 2003;

[9] Zayats AV, Smolyaninov II, Maradudin AA. Nano-optics of surface plasmon polaritons.

[10] Yariv A, Yeh P. Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers. Journal of the Optical Society of America. 1977;

[11] Vuković SM, Shadrivov IV, Kivshar YS. Surface Bloch waves in metamaterial and metal-

[12] Rytov SM. Electromagnetic properties of a finely stratified medium. Soviet Physics - JETP.

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[14] Smolyaninov II, Hwang E, Narimanov E. Hyperbolic metamaterial interfaces: Hawking radiation from Rindler horizons and spacetime signature transitions. Physical Review B.

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[16] Cortes CL, Newman W, Molesky S, Jacob Z. Quantum nanophotonics using hyperbolic

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[18] Takayama O, Artigas D, Torner L. Lossless directional guiding of light in dielectric nanosheets using Dyakonov surface waves. Nature Nanotechnology. 2014;9:419-424 [19] Walker DB, Glytsis EN, Gaylord TK. Surface mode at isotropic uniaxial and isotropicbiaxial interfaces. Journal of the Optical Society of America. A. 1998;15:248-260

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Physics Reports. 2005;408:131-314

**Chapter 4**

Provisional chapter

**Measurement of Sea Wave Spatial Spectra from High-**

DOI: 10.5772/intechopen.71834

The chapter is devoted to the development of methods for remote measurement of spatial spectra of waves arising on marine and ocean surface. It is shown that in most natural conditions of optical image formation, a nonlinear modulation of the brightness field occurs by slopes of water surface elements. Methods for reconstructing the spectra of surface waves from optical image spectra with allowance for such modulation are proposed. The methods are based on the numerical simulation of water surface taking into account wave formation conditions and conditions of light entering the sea surface from the upper and lower hemispheres. Using the results of numerical simulation, special operators are built to retrieve wave spectra from the spectra of aerospace images. These retrieving operators are presented in the form of analytical expressions, depending on the sets of parameters, which are determined by the conditions for the formation of images. The results of experimental studies of the sea wave spectra in various water areas using satellite optical images of high spatial resolution are presented. In the experimental studies, the spatial spectral characteristics of sea waves estimated from remote sensing data were compared with the corresponding characteristics measured by contact assets under

Keywords: wave spectra, surface waves, remote sensing, image processing

Registration of spatial spectra of surface waves is actual in solving many fundamental and applied problems of modern oceanology [1–3]. Obtaining information about such spectra is important for studying various physical processes occurring near the ocean-atmosphere interface, detecting water pollution, and monitoring anthropogenic impacts on the marine areas [1, 4–12].

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Measurement of Sea Wave Spatial Spectra from

High-Resolution Optical Aerospace Imagery

**Resolution Optical Aerospace Imagery**

Valery G. Bondur and Alexander B. Murynin

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Valery G. Bondur and Alexander B. Murynin

http://dx.doi.org/10.5772/intechopen.71834

Abstract

controlled conditions.

1. Introduction

Provisional chapter

#### **Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery** Measurement of Sea Wave Spatial Spectra from

DOI: 10.5772/intechopen.71834

Valery G. Bondur and Alexander B. Murynin

High-Resolution Optical Aerospace Imagery

Additional information is available at the end of the chapter Valery G. Bondur and Alexander B. Murynin

http://dx.doi.org/10.5772/intechopen.71834 Additional information is available at the end of the chapter

#### Abstract

The chapter is devoted to the development of methods for remote measurement of spatial spectra of waves arising on marine and ocean surface. It is shown that in most natural conditions of optical image formation, a nonlinear modulation of the brightness field occurs by slopes of water surface elements. Methods for reconstructing the spectra of surface waves from optical image spectra with allowance for such modulation are proposed. The methods are based on the numerical simulation of water surface taking into account wave formation conditions and conditions of light entering the sea surface from the upper and lower hemispheres. Using the results of numerical simulation, special operators are built to retrieve wave spectra from the spectra of aerospace images. These retrieving operators are presented in the form of analytical expressions, depending on the sets of parameters, which are determined by the conditions for the formation of images. The results of experimental studies of the sea wave spectra in various water areas using satellite optical images of high spatial resolution are presented. In the experimental studies, the spatial spectral characteristics of sea waves estimated from remote sensing data were compared with the corresponding characteristics measured by contact assets under controlled conditions.

Keywords: wave spectra, surface waves, remote sensing, image processing

#### 1. Introduction

Registration of spatial spectra of surface waves is actual in solving many fundamental and applied problems of modern oceanology [1–3]. Obtaining information about such spectra is important for studying various physical processes occurring near the ocean-atmosphere interface, detecting water pollution, and monitoring anthropogenic impacts on the marine areas [1, 4–12].

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

To obtain two-dimensional spectra of surface waves in large water areas, including hard-to-reach ones, the use of remote sensing methods based on the processing of various images obtained from air carriers and also space images of high spatial resolution is promising [4, 13–15].

An adequate estimation of such spectra formed by image processing obtained in the process of aerospace monitoring of marine areas of the seas and oceans requires the use of reconstructing operators that are functions that allow the spatial spectra of brightness fields recorded in optical aerospace images to be transformed into sea wave spectra [4, 16–18, 24]. These operators are built using numerical simulation methods based on the various conditions for the formation of aerospace images and the characteristics of remote sensing equipment [4, 16–21]. Initially, these methods were used to construct retrieving operators, which allow us to obtain wavelet spectra in the equilibrium interval [16–18]. At the present time, the development of modified retrieving operators, which are a superposition of the high-frequency and lowfrequency components, is being developed and is suitable for use in the low-frequency region, including near the spectral maximum [17, 18].

This chapter describes a method for retrieving sea wave spectra from the spectra of aerospace optical images over a wide range of spatial frequencies, including the equilibrium interval, the spectral maximum, and the low-frequency region. To calibrate and verify the adequacy of the developed method, the contact data obtained in synchronous measurements with the help of an array of string wave recorders are used. The results of experimental studies carried out using the developed method are presented.

#### 2. An approach to retrieve marine surface spectra

Rough sea surface is a random field of elevations (wave applications).

$$z = \zeta \left( x, y, t \right) \tag{1}$$

Ψð Þ¼ k S½ � ξ ð Þ k (3)

http://dx.doi.org/10.5772/intechopen.71834

ξx x; y ¼ ∂ξ x; y =∂x, ξy x; y ¼ ∂ξ x; y =∂y (4)

Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery

ξϕ x; y ¼ cosϕξx x; y þ sinϕξy x; y (5)

Ψð Þ k (6)

Φð Þ¼ k RSð Þ k (8)

,

73

where S is the spectral density operator, which is proportional to the square of the modulus of

Since optical images of the sea surface are formed as a result of reflection and refraction of light according to the laws of geometrical optics, for their analysis, the structure of the sea surface along with the field of elevations (x, y) is conveniently characterized by fields of slopes (or

The gradient of the sea surface in an arbitrary direction ϕ, with allowance for (3), can be

Taking into account the properties of the Fourier transform, one can associate the spectrum of

The brightness field, recorded by the remote sensing equipment at a fixed time, can be expanded in a power series along the surface slopes and is represented in the form [1, 11, 12]:

where N is the nonlinear component of the signal, containing terms proportional to (ξx(x,y))2

The contribution to the detected signal of the nonlinear component N (x, y, ξx, ξy) is determined by a number of parameters: lighting conditions, wave state, and recording equipment

In order to change from the spectrum of the optical image S(k), obtained under known conditions, to the slope spectrum of the sea surface Φ(k) in the direction determined by these

As a rule, analytical estimates of the contribution of the nonlinear component of N to the spatial spectrum of the luminosity field are difficult; therefore, the numerical simulation

The method of constructing the retrieving operator developed and described in this chapter is an extension of the method proposed in [1, 5, 6, 10–14]. To construct a retrieving operator corresponding to certain image acquisition conditions, direct numerical modeling of optical

method is used to solve the problem of constructing the recovery operator [1, 11–13].

and ξy(x,y) are fields of slopes (gradients of the elevation field) of the sea surface.

conditions, the definition of the retrieving operator R is introduced as:

L x; y ¼ C0 þ Cxξx x; y þ Cyξy x; y þ N x; y; ξx x; y ; ξy x; y (7)

, and so on. C0, Cx, and Cy are coefficients of the linear part of the expansion and ξx(x,y)

Φð Þ¼ k cosϕ kx þ sinϕ ky

the Fourier transform of the field ξ(x, y); k = (kx, ky) is the wave vector.

such a field of slopes with the spectrum of the field of elevations:

gradients) along the axes [2, 4, 17].

expressed as follows:

(ξy(x,y))2

characteristics [4, 17].

where ζ (x, y, t) is random function of sea surface elevations (elevation field); (x, y, z) is a rectangular Cartesian coordinate system in which the (x, y) plane coincides with the level of a calm (undisturbed) water surface; t is time.

Fixing the time instant t = t0 in (1), we obtain a two-dimensional random function of spatial coordinates:

$$z = \zeta(\mathbf{x}, y, t)|\_{t=t0} = \xi(\mathbf{x}, y) \tag{2}$$

Aerospace images, which are recorded by remote methods, are used to study the characteristics of the sea surface elevation field at a fixed time z = ξ (x, y). Two-dimensional signal fields that are represented in aerospace images are associated with the sea surface elevation field and can be used to estimate significant characteristics of this surface.

Since the sea surface elevation field (x, y) is a Gaussian quasistationary field, it is described quite adequately by the spectral density [2]:

Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery http://dx.doi.org/10.5772/intechopen.71834 73

$$\Psi(\mathbf{k}) = \mathbf{S}[\xi](\mathbf{k}) \tag{3}$$

where S is the spectral density operator, which is proportional to the square of the modulus of the Fourier transform of the field ξ(x, y); k = (kx, ky) is the wave vector.

To obtain two-dimensional spectra of surface waves in large water areas, including hard-to-reach ones, the use of remote sensing methods based on the processing of various images obtained

An adequate estimation of such spectra formed by image processing obtained in the process of aerospace monitoring of marine areas of the seas and oceans requires the use of reconstructing operators that are functions that allow the spatial spectra of brightness fields recorded in optical aerospace images to be transformed into sea wave spectra [4, 16–18, 24]. These operators are built using numerical simulation methods based on the various conditions for the formation of aerospace images and the characteristics of remote sensing equipment [4, 16–21]. Initially, these methods were used to construct retrieving operators, which allow us to obtain wavelet spectra in the equilibrium interval [16–18]. At the present time, the development of modified retrieving operators, which are a superposition of the high-frequency and lowfrequency components, is being developed and is suitable for use in the low-frequency region,

This chapter describes a method for retrieving sea wave spectra from the spectra of aerospace optical images over a wide range of spatial frequencies, including the equilibrium interval, the spectral maximum, and the low-frequency region. To calibrate and verify the adequacy of the developed method, the contact data obtained in synchronous measurements with the help of an array of string wave recorders are used. The results of experimental studies carried out

where ζ (x, y, t) is random function of sea surface elevations (elevation field); (x, y, z) is a rectangular Cartesian coordinate system in which the (x, y) plane coincides with the level of a

Fixing the time instant t = t0 in (1), we obtain a two-dimensional random function of spatial

Aerospace images, which are recorded by remote methods, are used to study the characteristics of the sea surface elevation field at a fixed time z = ξ (x, y). Two-dimensional signal fields that are represented in aerospace images are associated with the sea surface elevation field and

Since the sea surface elevation field (x, y) is a Gaussian quasistationary field, it is described

z ¼ ζ ð Þ x; y; t (1)

z ¼ ζ ð Þj x; y; t t¼t0 ¼ ξð Þ x; y (2)

from air carriers and also space images of high spatial resolution is promising [4, 13–15].

including near the spectral maximum [17, 18].

72 Surface Waves - New Trends and Developments

using the developed method are presented.

calm (undisturbed) water surface; t is time.

quite adequately by the spectral density [2]:

coordinates:

2. An approach to retrieve marine surface spectra

can be used to estimate significant characteristics of this surface.

Rough sea surface is a random field of elevations (wave applications).

Since optical images of the sea surface are formed as a result of reflection and refraction of light according to the laws of geometrical optics, for their analysis, the structure of the sea surface along with the field of elevations (x, y) is conveniently characterized by fields of slopes (or gradients) along the axes [2, 4, 17].

$$
\mathfrak{E}\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}) = \mathfrak{d}\xi(\mathbf{x}, \mathbf{y}) / \mathfrak{d}\mathfrak{x}, \mathfrak{E}\_{\mathbf{y}}(\mathbf{x}, \mathbf{y}) = \mathfrak{d}\xi(\mathbf{x}, \mathbf{y}) / \mathfrak{d}\mathfrak{y} \tag{4}
$$

The gradient of the sea surface in an arbitrary direction ϕ, with allowance for (3), can be expressed as follows:

$$\mathfrak{E}\_{\phi}(\mathbf{x}, \mathbf{y}) = \cos \phi \mathfrak{E}\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}) + \sin \phi \mathfrak{E}\_{\mathbf{y}}(\mathbf{x}, \mathbf{y}) \tag{5}$$

Taking into account the properties of the Fourier transform, one can associate the spectrum of such a field of slopes with the spectrum of the field of elevations:

$$\boldsymbol{\Phi}(\mathbf{k}) = \left(\cos\phi \,\mathbf{k}\_{\mathbf{x}} + \sin\phi \,\mathbf{k}\_{\mathbf{y}}\right) \,\Psi(\mathbf{k})\tag{6}$$

The brightness field, recorded by the remote sensing equipment at a fixed time, can be expanded in a power series along the surface slopes and is represented in the form [1, 11, 12]:

$$\mathbf{L}(\mathbf{x}, \mathbf{y}) = \mathbf{C}\_0 + \mathbf{C}\_\mathbf{x} \xi\_\mathbf{x}(\mathbf{x}, \mathbf{y}) + \mathbf{C}\_\mathbf{y} \xi\_\mathbf{y}(\mathbf{x}, \mathbf{y}) + \mathbf{N}(\mathbf{x}, \mathbf{y}, \xi\_\mathbf{x}(\mathbf{x}, \mathbf{y}), \xi\_\mathbf{y}(\mathbf{x}, \mathbf{y})) \tag{7}$$

where N is the nonlinear component of the signal, containing terms proportional to (ξx(x,y))2 , (ξy(x,y))2 , and so on. C0, Cx, and Cy are coefficients of the linear part of the expansion and ξx(x,y) and ξy(x,y) are fields of slopes (gradients of the elevation field) of the sea surface.

The contribution to the detected signal of the nonlinear component N (x, y, ξx, ξy) is determined by a number of parameters: lighting conditions, wave state, and recording equipment characteristics [4, 17].

In order to change from the spectrum of the optical image S(k), obtained under known conditions, to the slope spectrum of the sea surface Φ(k) in the direction determined by these conditions, the definition of the retrieving operator R is introduced as:

$$\boldsymbol{\Phi}(\mathbf{k}) = \mathbf{R}\mathbf{S}(\mathbf{k})\tag{8}$$

As a rule, analytical estimates of the contribution of the nonlinear component of N to the spatial spectrum of the luminosity field are difficult; therefore, the numerical simulation method is used to solve the problem of constructing the recovery operator [1, 11–13].

The method of constructing the retrieving operator developed and described in this chapter is an extension of the method proposed in [1, 5, 6, 10–14]. To construct a retrieving operator corresponding to certain image acquisition conditions, direct numerical modeling of optical images is performed under a given set of conditions [8–11], after which an approximation of the spatial frequency filter (transfer function) [11–14] is constructed, which allows obtaining a spatial spectrum of sea surface slopes from the aerospace image spectrum.

Parametrization of the spatial-frequency filter using a set of parameters that depends on the experimental conditions was proposed in [21]. The adequacy of the parametrization was experimentally verified in the equilibrium interval using contact data obtained by wave recorders and also by stereoscopic photography from the oceanographic platform [22].

However, the operator obtained in [21] does not allow for reconstructing the wave spectra in the region of low spatial frequencies and near the spectral maximum. To eliminate this shortcoming, an improvement was made to the method of constructing the retrieving operator and its approbation in various conditions [20, 19].

The modified retrieving operator Rmod is represented as the product of two operators (transfer functions):

$$\mathbf{R}\_{\text{mod}}(\mathbf{k}) = \mathbf{R}\_{\text{low}}(\mathbf{k}) \mathbf{R}\_{\text{high}}(\mathbf{k}) \tag{9}$$

and temporal characteristics of sea waves based on the corresponding hydrodynamic models, it is possible to calibrate remote sensing assets for measuring sea surface characteristics.

Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery

http://dx.doi.org/10.5772/intechopen.71834

75

As sources of information for comparing the results of remote and direct contact measure-

• Measurements in which temporal sequences of elevations of the sea surface are formed

• Stereo-photogrammetric measurements performed from close distances and allowing us to directly measure the two-dimensional realizations of the sea surface elevation field with

For direct determination of characteristics of sea waves, contact methods usually employ arrays of sensors (wave recorders) measuring the parameters of sea surface (gradient, elevation, and acceleration) at one or several points spaced from each other by some distance. To estimate spatiotemporal spectra from the arrays of wave recorders, indirect iterative calculation methods are widely used, in which some hypotheses about the statistical properties of

The stereo-photogrammetric processing of synchronously registered optical images of the sea surface from two different points of view (stereopairs) makes it possible to directly measure the realizations of two-dimensional fields of sea surface elevations, and calculate spatial spec-

The stereo-photogrammetric method allows measuring the shape of rough sea surface. The drawbacks of this method include the greater complexity of processing primary data and high

To compare the frequency and frequency-directed wave spectra recorded by wave recorders with two-dimensional and one-dimensional spatial spectra of sea surface elevations recorded by remote data, it is necessary to develop a special approach that takes into account the

Taking into account the features considered, the present chapter proposes an approach to

• Conducting experiments under controlled conditions, including satellite imagery of sea surface test areas and synchronous measurements using contact equipment and/or stereo-

• Retrieving sea surface spectra from space images under fixed conditions for obtaining

• Calculation of frequency spectra of waves from contact data obtained by wave recorders. • Formation of wave spectra from the satellite data of stereo-photogrammetric image

comparing satellite and contact measurements, which consists of the following:

photogrammetric imagery of the sea surface from a low altitude.

ments of sea surface characteristics, we will use the following types of measurements:

• Measurements using wave buoys drifting in the space survey area.

using stationary contact sensors.

tra of elevations on the base of these realizations.

features of gravitational and gravitational-capillary waves.

demands on the computing resources used.

high spatial resolution.

wave are postulated.

these images.

processing.

where Rhigh is a recovery operator (transfer function) in the high-frequency region; Rlow is a recovery operator (transfer function) in the low-frequency range.

An approximation of the retrieving operator in the form

$$\mathbf{R(k)} = a\_0 \left( \exp \left( a\_4 k^{a\_5} \right) \right) \left( \cos \left( \wp - \wp\_c \right) \right)^{a\_3} k^{a\_1 + a\_2 \cos \left( \wp - \wp\_c \right)} \tag{10}$$

where the parameter vector a = (a0, a1, a2, a3, a4, a5) is formed on the basis of experimental data obtained in complex experiments, including remote and in situ measurements of wave spectra.

#### 3. Calibration and verification of the adequacy of remote methods

When verifying the adequacy of remote methods of measuring wave spectra, sea truth data obtained in specially conducted experiments under controlled conditions using reference techniques and means ensuring the measurement of the sea surface spectra with sufficient accuracy should be used. As such methods and means in this chapter, measurements of sea surface elevations made using a grid of string wave recorders, stereo photography from low altitudes, and measurements using wave buoys will be used.

In an experimental verification of the reliability of methods for recording wave spectra from the spectra of satellite images, we will make a quantitative comparison of the results of remote measurements with data obtained by direct measurements by contact methods.

Having obtained under the same conditions the results of remote measurements of instantaneous two-dimensional fields of slopes and elevations, as well as data of direct (contact) measurements of local time series of heights, and also using the method of comparing spatial and temporal characteristics of sea waves based on the corresponding hydrodynamic models, it is possible to calibrate remote sensing assets for measuring sea surface characteristics.

As sources of information for comparing the results of remote and direct contact measurements of sea surface characteristics, we will use the following types of measurements:

images is performed under a given set of conditions [8–11], after which an approximation of the spatial frequency filter (transfer function) [11–14] is constructed, which allows obtaining a

Parametrization of the spatial-frequency filter using a set of parameters that depends on the experimental conditions was proposed in [21]. The adequacy of the parametrization was experimentally verified in the equilibrium interval using contact data obtained by wave

However, the operator obtained in [21] does not allow for reconstructing the wave spectra in the region of low spatial frequencies and near the spectral maximum. To eliminate this shortcoming, an improvement was made to the method of constructing the retrieving operator and

The modified retrieving operator Rmod is represented as the product of two operators (transfer

where Rhigh is a recovery operator (transfer function) in the high-frequency region; Rlow is a

where the parameter vector a = (a0, a1, a2, a3, a4, a5) is formed on the basis of experimental data obtained in complex experiments, including remote and in situ measurements of wave spectra.

When verifying the adequacy of remote methods of measuring wave spectra, sea truth data obtained in specially conducted experiments under controlled conditions using reference techniques and means ensuring the measurement of the sea surface spectra with sufficient accuracy should be used. As such methods and means in this chapter, measurements of sea surface elevations made using a grid of string wave recorders, stereo photography from low

In an experimental verification of the reliability of methods for recording wave spectra from the spectra of satellite images, we will make a quantitative comparison of the results of remote

Having obtained under the same conditions the results of remote measurements of instantaneous two-dimensional fields of slopes and elevations, as well as data of direct (contact) measurements of local time series of heights, and also using the method of comparing spatial

measurements with data obtained by direct measurements by contact methods.

a3

a5 ð Þ cos φ � φc

3. Calibration and verification of the adequacy of remote methods

Rmodð Þ¼ k Rlowð Þ k Rhighð Þ k (9)

a1þa2 cos ð Þ φ�φc (10)

k

recorders and also by stereoscopic photography from the oceanographic platform [22].

spatial spectrum of sea surface slopes from the aerospace image spectrum.

its approbation in various conditions [20, 19].

74 Surface Waves - New Trends and Developments

recovery operator (transfer function) in the low-frequency range.

An approximation of the retrieving operator in the form

R kð Þ¼ a0 exp a4k

altitudes, and measurements using wave buoys will be used.

functions):

• Stereo-photogrammetric measurements performed from close distances and allowing us to directly measure the two-dimensional realizations of the sea surface elevation field with high spatial resolution.

For direct determination of characteristics of sea waves, contact methods usually employ arrays of sensors (wave recorders) measuring the parameters of sea surface (gradient, elevation, and acceleration) at one or several points spaced from each other by some distance. To estimate spatiotemporal spectra from the arrays of wave recorders, indirect iterative calculation methods are widely used, in which some hypotheses about the statistical properties of wave are postulated.

The stereo-photogrammetric processing of synchronously registered optical images of the sea surface from two different points of view (stereopairs) makes it possible to directly measure the realizations of two-dimensional fields of sea surface elevations, and calculate spatial spectra of elevations on the base of these realizations.

The stereo-photogrammetric method allows measuring the shape of rough sea surface. The drawbacks of this method include the greater complexity of processing primary data and high demands on the computing resources used.

To compare the frequency and frequency-directed wave spectra recorded by wave recorders with two-dimensional and one-dimensional spatial spectra of sea surface elevations recorded by remote data, it is necessary to develop a special approach that takes into account the features of gravitational and gravitational-capillary waves.

Taking into account the features considered, the present chapter proposes an approach to comparing satellite and contact measurements, which consists of the following:

• Comparison of wave spectra obtained remotely with contact and/or stereo-photogrammetric measurements.

Consider the relationship between the spatial spectra of the sea surface, reconstructed from optical images, as well as frequency and frequency-directed wave spectra, measured by contact sensors, and characterize the fluctuations in sea level over time at a fixed point.

The frequency spectrum of sea surface elevations Sξ(ω) characterizes the distribution of wave oscillations at a given point along the cyclic frequencies ω, ω = 2π/τ, where τ is the period of the wave oscillation.

The spatial wave spectrum characterizes the energy distribution at a fixed time instant with respect to wave numbers k (or spatial frequencies ν = 1/Λ), k = 2π/Λ, where Λ is the length of the surface wave.

Spatial and frequency spectra of waves are functions of different arguments. They have different physical meanings and require the use of different methods of measurement. Therefore, to compare the spectra measured by different methods, a dispersion relation is used that describes the relationship between the time and spatial frequencies of waves, depending on the physical mechanisms that form the surface waves of the range in question. Within the framework of the linear hydrodynamic model, the components of the wave spectrum can be considered as elementary plane waves, for which the dispersion relation of the theory of potential waves of small amplitude that relates the cyclic frequency of the wave (ω) to the wave number k is valid. In deep water (for kh ≫ 1, where h is the depth), the dispersion relation taking into account the contribution of gravitational and capillary forces to wave formation has the form

$$
\omega(k) = \sqrt{\text{gk} + (\text{T}/\rho)k^3} \tag{11}
$$

4. The results of experimental studies

types of experiments were used [19, 22]:

drifting wave buoys.

top of the pentagon, were used.

ment zone.

• Resolution—no worse than 3 mm • Number of measuring channels—6

To solve the problems of validation of the developed methods, the results obtained in three

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• The first type of experiments consisted of carrying out space surveys and synchronous sea truth measurements of wave spectra near the stationary oceanographic platform.

• The second type of experiments consisted of carrying out a space survey of the investigated water area and simultaneous measurements of wave spectra with the help of

• The third type of experiments consisted of carrying out studies in the short-wave region of wave spectra (Λ = 0.04–1.0 m) with the help of string wave recorders, object photography

Experimental work of the first type was carried out in the area of the oceanographic platform

For the measurement of space-time wave spectra, contact data obtained using a wave measuring unit based on an array of string wave recorders, which were a set of six resistive wave recorders measuring the elevations of the sea surface at points located in the center and at the

The distance from the central string to each of the external strings was 25 cm. The main

During the first type of experiments, a special space imagery was conducted in the vicinity of the oceanographic platform using high-spatial resolution (0.5 m) optic-electronic equipment installed on board the GEOEYE spacecraft. A brief overview of the complex experiments conducted on September 24, 2015 at 11:52 (LT) and on September 12, 2011 at 12:06 is presented below. The presented experiments were characterized by different conditions of wave formation. During the first complex experiment, developing wind waves were observed at a nearsurface wind speed of about 4 m/s. During the second complex experiment, the velocity of the near-surface wind varied from 0 to 2 m/s; however, swell waves were present in the experi-

Figures 1 and 2 show the results of a joint analysis of sea wave spectra obtained from satellite and contact data under various wave formation conditions. Figure 1 shows the results of an experiment conducted on September 24, 2015 under the conditions of developing wind waves, and Figure 2 shows the results experiment, conducted on September 12, 2012 with weak wind

(Katsiveli settlement, Crimea), installed at a distance of about 600 m from the shore.

and stereo survey from the deck of the oceanographic platform.

Let us briefly summarize some experimental results of these types.

technical characteristics of this complex were as follows:

• Frequency of interrogation of channels—10, 20, 50, and 100 Hz • The maximum height of the measured waves was up to 4 m.

For gravitational waves (for Λ > 10�1 m), the relation (11) is simplified and takes the form

$$
\omega(k) = \sqrt{gk} \tag{12}
$$

The dispersion relation makes it possible to establish a relationship between the spatial spectrum of χ(k) characterizing the total energy of waves with a wave number k propagating in all possible directions and a frequency spectrum S(ω) characterizing the total energy of waves with a cyclic frequency ω propagating in all possible directions

$$
\chi(k) = \mathcal{S}\left(\omega(k)\right)d\omega(k)\,dk.\tag{13}
$$

On the other hand, a one-dimensional spatial spectrum can be obtained from the twodimensional spectrum of sea surface elevations Ψ(k) by integrating over the azimuth

$$\chi(k) = \int\_{-\pi/2}^{\pi/2} \Psi(k\cos\varphi, k\sin\varphi) k d\varphi \tag{14}$$

Thus, the spectrum χð Þk can be used as a base function and recalculate the spectra obtained by various measurement methods in this spectrum by means of appropriate transformations.

## 4. The results of experimental studies

• Comparison of wave spectra obtained remotely with contact and/or stereo-photogrammetric

Consider the relationship between the spatial spectra of the sea surface, reconstructed from optical images, as well as frequency and frequency-directed wave spectra, measured by con-

The frequency spectrum of sea surface elevations Sξ(ω) characterizes the distribution of wave oscillations at a given point along the cyclic frequencies ω, ω = 2π/τ, where τ is the period of

The spatial wave spectrum characterizes the energy distribution at a fixed time instant with respect to wave numbers k (or spatial frequencies ν = 1/Λ), k = 2π/Λ, where Λ is the length of

Spatial and frequency spectra of waves are functions of different arguments. They have different physical meanings and require the use of different methods of measurement. Therefore, to compare the spectra measured by different methods, a dispersion relation is used that describes the relationship between the time and spatial frequencies of waves, depending on the physical mechanisms that form the surface waves of the range in question. Within the framework of the linear hydrodynamic model, the components of the wave spectrum can be considered as elementary plane waves, for which the dispersion relation of the theory of potential waves of small amplitude that relates the cyclic frequency of the wave (ω) to the wave number k is valid. In deep water (for kh ≫ 1, where h is the depth), the dispersion relation taking into account the contribution of gravitational and capillary forces to wave formation has the form

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gk þ ð Þ Τ=r k3

> > gk p (12)

χð Þ¼ k S ð Þ ωð Þk dωð Þk dk: (13)

Ψð Þ kcosφ; ksinφ kdφ (14)

(11)

ωð Þ¼ k

with a cyclic frequency ω propagating in all possible directions

χð Þ¼ k

q

For gravitational waves (for Λ > 10�1 m), the relation (11) is simplified and takes the form

ωð Þ¼ k ffiffiffiffiffi

The dispersion relation makes it possible to establish a relationship between the spatial spectrum of χ(k) characterizing the total energy of waves with a wave number k propagating in all possible directions and a frequency spectrum S(ω) characterizing the total energy of waves

On the other hand, a one-dimensional spatial spectrum can be obtained from the two-

Thus, the spectrum χð Þk can be used as a base function and recalculate the spectra obtained by various measurement methods in this spectrum by means of appropriate transformations.

dimensional spectrum of sea surface elevations Ψ(k) by integrating over the azimuth

ðπ=2 �π=2

tact sensors, and characterize the fluctuations in sea level over time at a fixed point.

measurements.

76 Surface Waves - New Trends and Developments

the wave oscillation.

the surface wave.

To solve the problems of validation of the developed methods, the results obtained in three types of experiments were used [19, 22]:

Let us briefly summarize some experimental results of these types.

Experimental work of the first type was carried out in the area of the oceanographic platform (Katsiveli settlement, Crimea), installed at a distance of about 600 m from the shore.

For the measurement of space-time wave spectra, contact data obtained using a wave measuring unit based on an array of string wave recorders, which were a set of six resistive wave recorders measuring the elevations of the sea surface at points located in the center and at the top of the pentagon, were used.

The distance from the central string to each of the external strings was 25 cm. The main technical characteristics of this complex were as follows:

During the first type of experiments, a special space imagery was conducted in the vicinity of the oceanographic platform using high-spatial resolution (0.5 m) optic-electronic equipment installed on board the GEOEYE spacecraft. A brief overview of the complex experiments conducted on September 24, 2015 at 11:52 (LT) and on September 12, 2011 at 12:06 is presented below. The presented experiments were characterized by different conditions of wave formation. During the first complex experiment, developing wind waves were observed at a nearsurface wind speed of about 4 m/s. During the second complex experiment, the velocity of the near-surface wind varied from 0 to 2 m/s; however, swell waves were present in the experiment zone.

Figures 1 and 2 show the results of a joint analysis of sea wave spectra obtained from satellite and contact data under various wave formation conditions. Figure 1 shows the results of an experiment conducted on September 24, 2015 under the conditions of developing wind waves, and Figure 2 shows the results experiment, conducted on September 12, 2012 with weak wind

Figure 1. Comparison of the spectra of developing wind waves measured from remote and contact data of the complex experiment. September 24, 2015: (a) and (b) two-dimensional slope spectrum recovered in two intervals of spatial frequencies; (c) comparison of the one-dimensional frequency spectrum of the elevations obtained from the twodimensional spectrum of slopes by the retrieving operator R, with the spectrum obtained from the data of the string wave recorder. The dashed lines denote the Toba approximation [23].

waves in the presence of swell waves [19, 20]. Figures 1(a) and 2(a) show the reconstructed spatial spectra of sea surface slopes in a wide range of spatial frequencies, and Figures 1(b) and 2(b) show the slope spectra in the region of the spectral maximum and low spatial frequencies are enlarged.

Figures 1(c) and 2(c) show a comparison of the one-dimensional frequency spectrum of the elevations obtained from the two-dimensional spectrum of slopes with the frequency spectrum of the elevations obtained from the data of the array of string wave recorders. The conjugated spectra conditionally show the boundary ω0 between the frequency domains corresponding to the regions of action of the high-frequency and low-frequency reconstructing operators included in the formula (9) [20].

The comparison of the graphs of one-dimensional frequency spectra of waves, shown in Figures 1 and 2, allows us to visually evaluate the good correspondence between the spectra measured by contact data and spectra reconstructed from remote data.

For the quantitative analysis of the correspondence between the results of remote and contact measurements, a measure of discrepancy was used, calculated from formula

$$\Delta = \sqrt{\left(1/N\right)\sum\_{n=1}^{N} \left(1 - \Psi\_{cn}(\omega\_n)/\Psi\_{contact}(\omega\_n)\right)^2} \tag{15}$$

2012, this measure of divergence amounted to Δ ≈ 0.08. The obtained results indicate the adequacy of the determination of wave spectra from satellite images of high spatial resolution. During the experiments of the second type, a space survey was conducted in the vicinity of Oahu Island (Hawaii, USA) from the Ikonos and QuickBird satellites, which provided space images of high spatial resolution (0.65–1.0 m) and synchronous measurements of the frequency-angular

Figure 2. Comparison of the spectra of weak wind waves in the presence of swell waves measured from remote and contact experimental data. September 12, 2012: (a) and (b) two-dimensional slope spectrum recovered in two intervals of spatial frequencies; (c) comparison of the one-dimensional frequency spectrum of the elevations obtained from the twodimensional spectrum of slopes by the retrieving operator R, with the spectrum obtained from the data of the string wave

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The experiments were carried out to assess the anthropogenic impacts on the water area of Mamala bay, using satellite and contact data [19]. In the course of complex experiments, space surveys and synchronous sea truth measurements were carried out, including with the help of drifting wave buoys, which ensure the registration of frequency-angle spectra of surface waves. The following circumstance was taken into account when processing the data of the complex experiment. IKONOS and QuickBird satellites were photographed in these experiments far from the sunlight, and the resolution of the survey cameras of these satellites allowed us to fix surface waves of the submeter range (0.6–1.0 m). Under such conditions, the nonlinear components of

wave spectra with the help of two wave buoys, drifting in the space survey area.

recorder.

For the complex experiment carried out on September 24, 2010, the measure of discrepancy estimated by formula (15) was Δ ≈ 0.07, and for the experiment conducted on September 12, Measurement of Sea Wave Spatial Spectra from High-Resolution Optical Aerospace Imagery http://dx.doi.org/10.5772/intechopen.71834 79

Figure 2. Comparison of the spectra of weak wind waves in the presence of swell waves measured from remote and contact experimental data. September 12, 2012: (a) and (b) two-dimensional slope spectrum recovered in two intervals of spatial frequencies; (c) comparison of the one-dimensional frequency spectrum of the elevations obtained from the twodimensional spectrum of slopes by the retrieving operator R, with the spectrum obtained from the data of the string wave recorder.

waves in the presence of swell waves [19, 20]. Figures 1(a) and 2(a) show the reconstructed spatial spectra of sea surface slopes in a wide range of spatial frequencies, and Figures 1(b) and 2(b) show the slope spectra in the region of the spectral maximum and low spatial frequencies

Figure 1. Comparison of the spectra of developing wind waves measured from remote and contact data of the complex experiment. September 24, 2015: (a) and (b) two-dimensional slope spectrum recovered in two intervals of spatial frequencies; (c) comparison of the one-dimensional frequency spectrum of the elevations obtained from the twodimensional spectrum of slopes by the retrieving operator R, with the spectrum obtained from the data of the string wave

Figures 1(c) and 2(c) show a comparison of the one-dimensional frequency spectrum of the elevations obtained from the two-dimensional spectrum of slopes with the frequency spectrum of the elevations obtained from the data of the array of string wave recorders. The conjugated spectra conditionally show the boundary ω0 between the frequency domains corresponding to the regions of action of the high-frequency and low-frequency reconstructing operators

The comparison of the graphs of one-dimensional frequency spectra of waves, shown in Figures 1 and 2, allows us to visually evaluate the good correspondence between the spectra

For the quantitative analysis of the correspondence between the results of remote and contact

For the complex experiment carried out on September 24, 2010, the measure of discrepancy estimated by formula (15) was Δ ≈ 0.07, and for the experiment conducted on September 12,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ 1 � Ψспð Þ ωn =Ψcontactð Þ ωn 2

vuut (15)

measured by contact data and spectra reconstructed from remote data.

Δ ¼

measurements, a measure of discrepancy was used, calculated from formula

ð Þ 1=N X N

n¼1

are enlarged.

included in the formula (9) [20].

recorder. The dashed lines denote the Toba approximation [23].

78 Surface Waves - New Trends and Developments

2012, this measure of divergence amounted to Δ ≈ 0.08. The obtained results indicate the adequacy of the determination of wave spectra from satellite images of high spatial resolution.

During the experiments of the second type, a space survey was conducted in the vicinity of Oahu Island (Hawaii, USA) from the Ikonos and QuickBird satellites, which provided space images of high spatial resolution (0.65–1.0 m) and synchronous measurements of the frequency-angular wave spectra with the help of two wave buoys, drifting in the space survey area.

The experiments were carried out to assess the anthropogenic impacts on the water area of Mamala bay, using satellite and contact data [19]. In the course of complex experiments, space surveys and synchronous sea truth measurements were carried out, including with the help of drifting wave buoys, which ensure the registration of frequency-angle spectra of surface waves.

The following circumstance was taken into account when processing the data of the complex experiment. IKONOS and QuickBird satellites were photographed in these experiments far from the sunlight, and the resolution of the survey cameras of these satellites allowed us to fix surface waves of the submeter range (0.6–1.0 m). Under such conditions, the nonlinear components of the brightness field of the sea surface make an insignificant contribution to the spectra of satellite images. In this connection, in the analysis of complex experiment data, it is permissible to use the linear model, according to which the two-dimensional spatial spectra of satellite images S (kx, ky) are linearly related to the spatial spectra of sea wave inclinations Ψφ (νx, νy).

the time of shooting (see Figure 3a). The data obtained by measurements with a wave buoy (see Figure 3c) were selected in such a way that the moment of the survey was as close as possible to the end of the half-hour accumulation of data obtained by the buoy. For the selected image fragments, two-dimensional spatial spectra S (kx, ky) were formed, examples of which are shown in Figure 3b, and according to measurements by wave buoys, the frequency-angle

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Two-dimensional spectra of satellite images S (kx, ky) (see Figure 3b) also yielded one-dimensional spatial spectra S (k) (see Figure 3d), and from the frequency-angle spectra of the elevations Ψ (ω, θ) obtained by wave buoys, the frequency spectra of Ψ (ω) (see Figure 3e) were

From the one-dimensional spatial spectra obtained from two-dimensional spectra of satellite images using the dispersion relation for gravitational waves, the frequency spectra S (ω) were

An important stage in the comparison of wave spectra obtained from space images and data from wave buoys is the detection of additional spectral harmonics due to anthropogenic effects on the surface waves caused by buried wastewater outfalls in the region of the experiments [7–12]. Such spectral harmonics arise under the action of high-frequency internal waves generated by turbulent jets of nonsalty water discharged into the saline marine environment through the diffusers of deep outfalls. Similar effects are caused by various physical mechanisms, analyzed in [14, 15].

The conducted experiments' results and comparison of additional spectral harmonics detected on the base of satellite image spectra and wave buoy data have confirmed the efficiency of remote methods for detection of manifestations of anthropogenic impacts on the water envi-

Experimental work of the third type was performed completely on the oceanographic platform

To verify the adequacy of the methods for retrieving sea wave spectra in the 0.1–1 m shortwave range, the results of complex field experiments of the third type were analyzed. These experiments were carried out at an oceanographic platform, including conventional and stereo survey, as well as contact measurements of wave spectra using a wave recorder array. Stereo surveying of the sea surface was carried out from the working platform, which was located at a height of 16 m above sea level with a 10.2 m basis using stereocameras with a focal length of 99 mm. From the measured elevation field samples, the spectra Ψϕ (kϕ) were calculated using the interpolation and discrete Fourier transform procedures. Simultaneously, with the stereoscopic shooting on the platform, contact measurements of the frequency spectra of waves were performed using a wave spectrograph providing registration in the frequency range of 0.1– 15 Hz with a tunable filter bandwidth of 0.1 Hz. In the wave spectrograph, a method for sequential analysis of the frequency spectrum was realized by automatically tuning the trans-

Images of the sea surface obtained during stereo photography were also used to retrieve the wave spectra by a nonlinear multiposition method [22]. At the same time, fragments of two

spectra of the elevations Ψε (ω, θ) are constructed.

ronment taking into account wave spectra changes.

mission frequency of a narrowband filter in a given interval [22].

constructed in the chosen directions.

determined from the formula (12).

in Katsiveli settlement (Crimea).

The results of comparison of the wave spectra obtained on the basis of remote data and drifting buoy data are shown in Figure 3.

During the processing, fragments of satellite images with dimensions of 1024 1024 or 2048 2048 pixels were selected in such a way that the fragment overlapped the wave buoy path at

Figure 3. The results of a joint analysis of sea wave spectra obtained from space images and from drifting wave buoys: (a) the initial space image and its fragment used for processing; (b) two-dimensional spatial spectrum of the selected fragment and its enlarged central part (right); (c) installation of a wave buoy; (d) a one-dimensional spatial spectrum obtained from the two-dimensional spectrum of the satellite image; (e) the frequency spectrum obtained by a wave buoy; (f) comparison of frequency spectra obtained by different methods.

the time of shooting (see Figure 3a). The data obtained by measurements with a wave buoy (see Figure 3c) were selected in such a way that the moment of the survey was as close as possible to the end of the half-hour accumulation of data obtained by the buoy. For the selected image fragments, two-dimensional spatial spectra S (kx, ky) were formed, examples of which are shown in Figure 3b, and according to measurements by wave buoys, the frequency-angle spectra of the elevations Ψε (ω, θ) are constructed.

the brightness field of the sea surface make an insignificant contribution to the spectra of satellite images. In this connection, in the analysis of complex experiment data, it is permissible to use the linear model, according to which the two-dimensional spatial spectra of satellite images S (kx, ky)

The results of comparison of the wave spectra obtained on the basis of remote data and drifting

During the processing, fragments of satellite images with dimensions of 1024 1024 or 2048 2048 pixels were selected in such a way that the fragment overlapped the wave buoy path at

Figure 3. The results of a joint analysis of sea wave spectra obtained from space images and from drifting wave buoys: (a) the initial space image and its fragment used for processing; (b) two-dimensional spatial spectrum of the selected fragment and its enlarged central part (right); (c) installation of a wave buoy; (d) a one-dimensional spatial spectrum obtained from the two-dimensional spectrum of the satellite image; (e) the frequency spectrum obtained by a wave buoy;

(f) comparison of frequency spectra obtained by different methods.

are linearly related to the spatial spectra of sea wave inclinations Ψφ (νx, νy).