## 1. Introduction

The nonlinearities of common optical materials result from the nonlinear response of their electric polarization to the electric field of electromagnetic waves (EMWs), or

However, the magnetically optical nonlinearities of magnetic materials come from the nonlinear response of their dynamical magnetization to the magnetic field of EWMs, or the magnetization

In the concept of ferromagnetism(Morrish, 2001), there is such a kind of magnetic ordering media, named antiferromagnets (AFs), such as NiO, MnF_{2}, FeF_{2}, and CoF_{2}
*et. al*. This kind of materials may possess two or more magnetic sublattices and all lattice points on any sublattice have the same magnetic moment, but the moments on adjacent sublattices are opposite in direction and counteract to each other. We here present an example in Fig.1, a bi-sublattice AF structure. In contrast to the ferromagnets or ferrimagnets, it is very difficult to magnetize AFs by a magnetic field of ordinary intensity since very intense AF exchange interaction exists in them, so they are almost not useful in the fields of electronic and electric engineering. But the dynamical properties of AFs should be paid a greater attention to. The resonant frequencies of the AFs usually fall in millimeter or far infrared (IR) frequency regime. Therefore the experimental methods to study AFs optical properties are optical or quasi-optical ones. In addition, these frequency regions also are the working frequency regions of the THz technology, so the AFs may be available to make new elements in the field of THz technology.

The propagation of electromagnetic waves in AFs can be divided into two cases. In the first case, the frequency of an EMW is far to the AF resonant frequency and then the AF can be optically considered as an ordinary dielectric. The second case means that the wave frequency is situated in the vicinity of the AF resonant frequency and the dynamical magnetization of the AF then couples with the magnetic field of the EMW. Consequently, modes of EMW propagation in this frequency region are some AF polaritons. In the linear case, the AF polaritons in AF films, multilayers and superlattices had been extendedly discussed before the year 2000 (Stamps & Camley, 1996; Camley & Mills, 1982; Zhu & Cao, 1987; Oliveros, et. al., 1992; Camley, 1992; Raj & Tilley, 1987; Wang & Tilley, 1987; Almeida & Tilley, 1990).

The magnetically nonlinear investigation of AF systems was not given great attention until the 1990s. In the recent years, many progresses have been made in understanding the magnetic dynamics of AF systems (Costa, et. al.,1993; Balakrishnan, et. al.,1990, 1992; Daniel & Bishop,1992; Daniel & Amuda,1994; Balakrishnan & Blumenfeld,1997). Many investigations have been carried out on nonlinear guided and surface waves (Wang & Awai,1998; Almeida & Mills, 1987; Kahn, et. al., 1988; Wright & Stegeman, 1992; Boardman & Egan,1986), second-harmonic generation (Lim, 2002, 2006; Fiebig et. al, 1994, 2001, 2005), bistability (Vukovic, 1992) and dispersion properties (Wang,Q, 2000). Almeida and Mills first discussed the nonlinear infrared responses of the AFs and explore the field-dependent of transmission through thin AF films and superlattices, where the third-order approximation of dynamical magnetization was used, but no analytical expressions of nonlinear magnetic susceptibilities in the AF films or layers were obtained (Almeida & Mills, 1987; Kahn, et. al., 1988). Lim first obtained the expressions of the susceptibilities in the third-order approximation, in a special situation where a circularly polarized magnetic field and the cylindrical coordinate system were applied in the derivation process (Lim, et. al., 2000). It is obvious that those expressions cannot be conveniently used in various geometries and boundaries of different shape. In analogue to what done in the ordinary nonlinear optics, the nonlinear magnetic susceptibilities were presented in the Cartesian coordinate system by Wang et. al. (Wang & Fu, 2004; Zhou, et. al., 2009), and were used to discuss the nonlinear polaritons of AF superlattices and the second-harmonic generation (SHG) of AF films (Wang & Li, 2005; Zhou & Wang, 2008), as well as transmission and reflection bi-stability (Bai, et. al., 2007; Zhou, 2010).

## 2. Nonlinear susceptibilities of antiferromagnets

AF susceptibility is considered as one important physical quantity to describe the response of magnetization in AFs to the driving magnetic filed. It is also a basis of investigating dynamic properties and magneto-optical properties. In this section, the main steps and results of deriving nonlinear magnetic susceptibilities of AFs will be presented in the right-angled coordinate system, or the Cartesian system. The detail mathmetical procedure can be found from our previous works (Wang & Fu, 2004; Zhou, et. al., 2009). The used bi-sublattice AF structure and coordinate system are shown in Fig.1, where we take the AF anisotropy axis and the external magnetic field H_{0} along the z axis. The sublattic magnetzation M_{0} is the absolute projection value of each total sublattice magnatization to the anisotropy axis. The driving magnetic field

### 2.1. Basical assumptions, definitions and the first-order susceptibilities

We begin with the assumption that this AF crystal is at a low temperature, or the temperature is much lower than its Neel temperature and the magnetic ordering is properly preserved. Then the magnetization on each sublattice is regarded as saturated without a driving field. In the alternating driving field

where

where

(4) |

(6) |

We shall use the perturbation expansion method to derive nonlinear magnetizations and susceptibilies of various orders. We take

where *c.c.* indicates the complex conjugation. In practice, one needs the AF magnetization rather than the lattice magnetizations, so we define

where the special frequencies are defined with

where

### 2.2. The second-order approximation

Similar to the second-order electric polarization in the nonlinear optics, the second-order magneizations also are divided into the dc part unchanging with time and the second-harmonic part varying with time according to

In addition, the z component of the dc magnetization can be obtained from the conservation of each sulattice magnetic moment, and we see

These lead directly to the z component to be

Here we have used

Next, we are going to derive the second-harmonic (SH) magnetization and susceptibility. They will not be used only in the third-order susceptibility, but also be applied to describe the SH generation in various AF systems. In equations (2-2), reserving only the SH terms, we obtain the following equations

with

Applying the expressions of the first-order components and the expressions

one finds

(35) |

(36) |

(38) |

(39) |

where

.### 2.3. The third-order approximation

The third-order magnetization also contains two part, or one varies with time according to

(46) |

where

(47) |

(48) |

(50) |

(51) |

Substituting the definitions of

(62) |

(63) |

(64) |

(65) |

(66) |

(67) |

with the coefficients

(68) |

(70) |

(72) |

(73) |

(75) |

(78) |

(79) |

(81) |

(82) |

(83) |

(85) |

The symmetry relations among the third-order elements are found to be

Although there are 81 elements of the third-order susceptibility tensor and their expressions are very complicated, but many among them may not be applied due to the plane or line polarization of used electromagnetic waves. for example when the magnetic field *x* and *y*, such as*et. al.*, are usefull. In addition, if the external magnetic field H_{0} is removed, many the first- second- and third-order elements will disappear, or become 0. In the following sections, when one discusses AF polaritons the damping is neglected, but when investigating transmission and reflection the damping is considered.

## 3. Linear polaritons in antiferromagnetic systems

The linear AF polaritons of AF systems (AF bulk, AF films and superlattices) are eigen modes of electromagnetic waves propagating in the systems. The features of these modes can predicate many optical and electromagnetic properties of the systems. There are two kinds of the AF polaritons, the surface modes and bulk modes. The surface modes propagate along a surface of the systems and exponentially attenuate with the increase of distance to this surface. For these AF systems, an optical technology was applied to measure the AF polariton spectra (Jensen, 1995). The experimental results are completely consistent with the theoretical predications. In this section, we take the Voigt geometry usually used in the experiment and theoretical works, where the waves propagate in the plane normal to the AF anisotropy axis and the external magnetic field is pointed along this anisotropy axis.

### 3.1. Polaritons in AF bulk and film

Bulk AF polaritons can be directly described by the wave equation of EMWs in an AF crystal,

where

The best and simplest example available to describe the surface AF polariton is a semi-infinite AF. We assume the semi-infinite AF occupies the lower semi-space and the upper semi-space is of vacuum. The y axis is normal to the surface. The surface polariton moves along the x axis. The wave field in different spaces can be shown by

where

Here there are 4 amplitude components, but we know from equation

The wave equation (3-1) shows that

determining the two attenuation constants. The boundary conditions of

where the permeability components and dielectric constants all are their relative values. Equation (3-7) describes the surface AF polariton under the condition that the attenuation factors both are positive. In practice, Eq.(3-6) also shows the dispersion relation of bulk modes as that attenuation factor is vanishing.

We illustrate the features of surface and bulk AF polaritons in Fig.2. There are three bulk continua where electromagnetic waves can propagate. Outside these regions, one sees the surface modes, or the surface polariton. The surface polariton is non-reciprocal, or the polariton exhibits completely different properties as it moves in two mutually opposite directions, respectively. This non-reciprocity is attributed to the applied external field that breaks the magnetic symmetry of the AF. If we take an AF film as example to discuss this subject, we are easy to see that the surface mode is changed only in quantity, but the bulk modes become so-called guided modes, which no longer form continua and are some separated modes (Cao & Caillé, 1982).

### 3.2. Polaritons in antiferromagnetic multilayers and superlattices

There have been many works on the magnetic polaritons in AF multilayers or superlattices. This AF structure is the one-dimension stack, commonly composed of alternative AF layers and dielectric (DE) layers, as illustrated in Fig.3.

In the limit case of small stack period, the effective-medium method was developed (Oliveros, et. al., 1992; Camley, 1992; Raj & Tilley, 1987; Almeida & Tilley, 1990; Cao & Caillé, 1982; Almeida & Mills,1988; Dumelow & Tilley,1993; Elmzughi, 1995a, 1995b). According to this method, one can consider these structures as some homogeneous films or bulk media with effective magnetic permeability and dielectric constant. This method and its results are very simple in mathematics. Of course, this is an approximate method. The other method is called as the transfer-matrix method (Born & Wolf, 1964; Raj & Tilley, 1989), where the electromagnetic boundary conditions at one interface set up a matrix relation between field amplitudes in the two adjacent layers, or adjacent media. Thus amplitudes in any layer can be related to those in another layer by the product of a series of matrixes. For an infinite AF superlattice, the Bloch’s theorem is available and can give an additional relation between the corresponding amplitudes in two adjacent periods. Using these matrix relations, bulk AF polaritons in the superlattices can be determined. For one semi-finite structure with one surface, the surface mode can exist and also will be discussed with the method.

#### 3.2.1. The limit case of short period, effective-medium method

Now we introduce the effective-medium method, with the condition of the wavelength

and for those components discontinuous at the interface, one assumes

(3-8b)

where the AF ratio

with the elements

On the similar principle, we can find that the effective dielectric permittivity tensor is diagonal and its elements are

On the base of these effective permeability and permittivity, one can consider the AF multilayers or superlattices as homogeneous and anisotropical AF films or bulk media, so the same theory as that in section 3.1 can be used. Magnetic polaritons of AF multilayers (Oliveros, et.al., 1992; Raj & Tilley, 1987), AF superlattices with parallel or transverse surfaces (Camley, et. al., 1992; Barnas, 1988) and one-dimension AF photonic crystals (Song, et.al., 2009; Ta, et. al.,2010) have been discussed with this method.

#### 3.2.2. Polaritons and transmission of AF multilayers: transfer-matrix method

If the wavelength is comparable to the stack period, the effective-medium method is no longer available so that a strict method is necessary. The transfer-matrix method is such a method. In this subsection, we shall present magnetic polaritons of AF multilayers or superlattices with this method. We introduce the wave magnetic field in two layers in the *l*th stack period as follows.

where k_{1} and k_{2} are determined with

(103) |

Here there is a relation between per pair of amplitude components, or

As a result, we can take

At the interface between the *l*th and *l+1*th periods, one see

Thus the matrix relation between the amplitude components in the same period is introduced as

where the matrix elements are given by

with

with

Commonly, the matrix relation between the amplitude components in the *l*th and *l+1*th periods is written as

In order to discuss bulk AF polaritons, an infinite AF superlattice should be considered. Then the Bloch’s theorem is available so that

It can be reduced into a more clearly formula, or

When one wants to discuss the surface polariton, the semi-infinite system is the best and simplest example. In this situation, the Bloch’s theorem is not available and the polariton wave attenuates with the distance to the surface, according to

It should remind that equation (3-23) cannot independently determine the dispersion of the surface polariton since the attenuation coefficient is unknown, so an additional equation is necessary. We take the wave function outside this semi-infinite structure as _{x} and E_{z} continuous at the surface lead to

with

Eqs. (3-23) and (3-25) jointly determine the dispersion properties of the surface polariton under the conditions of

We present a figure example to show features of bulk and surface polaritons, as shown in Fig.4. Because of the symmetry of dispersion curves with respective to *k=0,* we present only the dispersion pattern in the range of *k>0.* The bulk polaritons form several separated continuums, and the surface mode exists in the bulk-polariton stop-bands. The bulk polaritons are symmetrical in the propagation direction, or possess the reciprocity, but is not the surface mode. These properties also can be found from the dispersion relations. For the bulk polaritons, the wave vector appears in dispersion equation (3-22) in its

#### 3.2.3. Transmission of AF multilayers

In practice, infinite AF superlattices do not exist, so the conclusions from them are approximate results. For example, if the incident-wave frequency falls in a bulk-polariton stop-band of infinite AF superlattice, the transmission of the corresponding AF multilayer must be very weak, but not vanishing. Of course, it is more intensive in the case of frequency in a bulk-polariton continuum. Based on the above results, we derive the transmission ratio of an AF multilayer, where this structure has two surfaces, the upper surface and lower surface. We take a TE wave as the incident wave, with its electric component normal to the incident plane (the x-y plane) and along the z axis. The incident wave illuminates the upper surface and the transmission wave comes out from the lower surface. We set up the wave function above and below the multilayer as

The wave function in the multilayer has been given by (3-12) and (3-13). By the mathematical process similar to that in subsection 3.2.2, we can obtain the transmission and reflection of the multilayer with N periods from the following matrix relation,

in which two new matrixes are shown with

with

## 4. Nonlinear surface and bulk polaritons in AF superlattices

In the previous section, we have discussed the linear propagation of electromagnetic waves in various AF systems, including the transmission and reflection of finite thickness multilayer. The results are available to the situation of lower intensity of electromagnetic waves. If the intensity is very high, the nonlinear response of magnetzation in AF media to the magnetic component of electromagnetic waves cannot be neglected. Under the present laser technology, this case is practical. Because we have found the second- and third-order magnetic susceptibilities of AF media, we can directly derive and solve nonlinear dispersion equations of electromagnetic waves in various AF systems. There also are two situations to be discussed. First，if the wavelenght

### 4.1. Polaritons in AF superlattice

In this section we shall use a stricter method to deal with nonlinear propagation of AF polaritons in AF superlattices. In section 2, we have obtained various nonlinear susceptibilities of AF media, which means that one has obtained the expressions of

where

with*m*=1 and

(129) |

and

(130) |

in which

where

for the surface modes and

for the bulk modes. The coefficients in Eq.(4-3) can written as

(136) |

(137) |

If

Note that all these coefficients contain implicitly the factor

and

Thus the nonlinear magnetic field can be rewritten as

(148) |

and the third-order magnetization is equal to

The two formulae will be applied for solving the dispersion equations of the nonlinear surface and bulk polaritons from the boundary conditions satisfied by the wave fields.

Seeking the dispersion relations of AF polaritons should begin from the boundary conditions of the magnetic field *n*th AF layer, as well as the solutions in the vacuum *n*th NM layer

Bulk dispersion equation

For the bulk polaritons, there are 6 amplitude coefficients in the wave solutions,

From these four equations, we find the dispersion relation of the nonlinear bulk polaritons,

with the nonlinear factor *N* described by

(155) |

Due to the nonlinear interaction, the nonlinear term

Generally speaking, this nonlinear dispersion equation is a complex relation. However in some special circumstances it may be a real one. Here we illustrate it with an example. If

Surface dispersion relations

For the surface modes, note

in which

Combining this with Eqs.(4-11a-c), the equation determining

(158) |

with

*n* can be any integer value equal to or larger than 1. Actually the nonlinear contribution decreases rapidly as *n* is increased, so only for small *n*, the nonlinear effect is important. In addition, increasing *n* have a similar effect in numerical calculation.

Because the nonlinear terms in Eqs.(4-12) and (4-14) all contain

We take the FeF_{2}/ZnF_{2} superlattice as an example for numerical calculations, the physical parameters of FeF_{2} are given in table 1. While the relative dielectric constant of ZnF_{2} are

H_{a} | H_{e} | ||||

FeF2 | 197kG | 533kG | 7.04 kG | 5.5 | ^{-1} kG |

MnF2 | 7.87kG | 550kG | 5.65 kG | 5.5 | ^{-1} kG |

We illustrate the nonlinear shift in frequency as function of the component of wave vector

We examine the surface magnetic polariton in the case of nonlinearity, which is shown in Fig.7. Similar to those in the middle bulk band, the surface-mode frequency also is very closed to

### 4.2. Nonlinear infrared ransmission through and reflection off AF films

Finally, we discuss nonlinear transmission through the AF film. We assume that the media above and below the nonlinear AF film are both linear, but the film is nonlinear. Our geometry is shown in Fig. 8, where the anisotropy axis (the z axis) is parallel to the film surfaces and normal to the incident plane (the x–y plane). A linearly polarized radiation (TE wave) is obliquely incident on the upper surface.

Because we have known the nonlinear wave solution in the AF film and those above and below the film, to solve nonlinear transmission and reflection is a simple algebraical precess. Thus we directly present the finall results, the nonlinear refection and transmission coefficients

(160) |

in which the nonlinear terms

(162) |

Finally the reflectivity and transmissivity are defined as

We take a FeF_{2} film as an example for numerical calculations. with the physical parameters given in Table 1. The film thickness is fixed at

First, the nonlinear modification is more evident in reflection for frequencies higher than*R* and *T* curves at

we see the discontinuities on the reflection and transmission curves and the nonlinear effect is very obvious in the regions near to the jump points. The discontinuities are related to the bi-stable states. The nonlinear interaction also play an important role in decreasing or increasing the absorption in the AF film.

## 5. Second harmonic generation in antiferromagnetic films

In this section, the most fundamental nonlinear effect, second harmonic generation (SHG) of an AF film between two dielectrics (Zhou & Wang, 2008) and in one-dimensional photonic crystals (Zhou, et. al., 2009) have been analyzed based on the second-harmonic tensor elements obtained in section 2. We know from the expression of SH magnetization that if *I* is the incident wave, *R* the reflection wave and *T* the transmission wave, related to incident angle*s* is added to the above quantities, they represent the corresponding quantities of second harmonic (SH) waves. The pump wave in the film is not indicated in this figure. The dielectric constants and magnetic permeabilities are shown in corresponding spaces.

Although we have obtained all elements of the SH susceptibility in section 2, but only two will be used in this geometry. It is because that a plane EMW of incidence can be decomposed into two waves, or a TE wave with the electric field normal to the incident plane and a TM wave with the magnetic field transverse to this plane. Due to no coupling between magnetic moments in the film and the TM wave (Lim, 2002, 2006; Wang & Li, 2005; Bai, et. al., 2007), the incident TM wave does not excite the linear and SH magnetizations, so can be ignored. Thus we take the TE wave as the incident wave *I* which produces the TE pump wave

(164) |

The SHG magnetization arises as a source term in the harmonic wave equation and is excited by the pump wave, and in turn the pumping wave is induced by the incident wave. When the energy-flux density of the excited SH wave is much less than that of the incident wave, the assumption that the depletion of pump waves can be neglected(Shen, 1984) is commonly accepted. This assumption allows us to solve the pump wave in the film within the linear electromagnetic theory or with the linear optical method.

Based on the above assumption, to solve the pump wave is a linear problem. The method is well-known and just one usual optical process, so we give a simpler description for solving the pump wave in the film. Because the pump wave is a TE wave, we take its electric field to be

where

The corresponding magnetic fields in different spaces are written as

(168) |

where is*c* is the vacuum velocity of light.

where *d* is the film thickness, *R* and *T* are not necessary for seeking the SHG, so they are given up here.To solve the output amplitudes of SHG,

The other two are homogeneous and do not contain the field component

with

Solution (5-9) shows that the SH wave in the film also propagates in the incident plane and it will radiate out from the film. We use

to indicate the magnetic field of SH wave generated above the film and

to represent the SH field below, with

(180) |

Considering the boundary conditions of these fields continuous at the surfaces, there must be

It is proven that the SH wave outputs *R* and transmission wave *T*, respectively.

Finally we solve the amplitudes of the output SH wave. The continuity conditions of

After eliminating

(190) |

where

*a*,

*b*and

*c*that SH amplitudes

The numerical calculations are based on three examples, a single MnF_{2} film, SiO_{2}/MnF_{2}/air and ZnF_{2}/MnF_{2}/air, in which the MnF_{2} film is antiferromagnetic. The relative dielectric constants are 1.0 for air, 2.3 for SiO_{2} and 8.0 for ZnF_{2}. The relative magnetic permeabilities of these media are 1.0. There are two resonance frequencies in the dc field of

We first illustrate the output densities of a single film versus frequency

Next we discuss the SH outputs of SiO_{2}/MnF_{2}/air shown in Fig.13. Incident wave *I* and reflective wave *R* are in the SiO_{2} medium and transmission wave *T* in air. The maximum peak of *T* vanishes. For

Finally we discuss the SH outputs of ZnF_{2}/MnF_{2}/air, with the dielectric constant of ZnF_{2} larger than that of SiO_{2}. The spectrum of *S*_{R} is the most complicated and interesting, as shown in Fig.14 (a). First we see two special angles of incidence. The first angle has the same definition as ^{o}. The second defined as

and the incident wave *I* is completely reflected, so the SH wave is not excited. On this point, Fig.13(a) is completely different from Fig.12(a). More peaks of *S*_{R} appear between the two critical angles, but the highest peak stands between the two resonant frequencies and is near to*S*_{R}. For *S*_{T}, the pattern is more simple, as shown in Fig.14 (b). Only one main peak is seen clearly, which arises at *S*_{R} and *S*_{T} are larger than those shown in Fig.12 by about 240 times and 20 times, respectively.

For the SH output peaks in Fig.13 and Fig.14, we present the explanations as follows. The pump wave in the film is composed of two parts, the forward and backward waves corresponding to the signs + and－ in Eq.(5-3), respectively. The transmission (*T*) vanishes and the forward wave is completely reflected from the bottom surface of the film as *k* is an integer) at the surface, the interference results in the peaks of

It is also interesting for us to examine the SH outputs versus the film thickness. We take the SiO_{2}/MnF_{2}/air as an example and show the result in Fig.15. We think that the SH fringes result from the change of optical thickness of the film, and the SH outputs reache their individual saturation values about at^{2}, and 0.012 W/cm^{2}. If we enhance the incident wave density to 10.0kW/cm^{2}, the two output densities are increased by 100 times, to 9.0W/cm^{2} and 1.2W/cm^{2}, or if we focus *S*_{I} on a smaller area, higher SH outputs are also obtained, so it is not difficult to observe the SH outputs.

If we put this AF film into one-dimension Photonic crystals (PCs), the SHG has a higher efficiency(Zhou, et. al., 2009). It is because that when some AF films as defect layers are introduced into a one-dimension PC, the defect modes may appear in the band gaps. Thus electromagnetic radiations corresponding to the defect modes can enter the PC and be greatly localized in the AF films. This localization effect has been applied to the SHG from a traditional nonlinear film embedded in one-dimension photonic crystals(Ren, et. Al., 2004 ; Si, et. al., 2001 ; Zhu, et.al., 2008, Wang, F., et. al. 2006), where a giant enhancement of the SHG was found.

## 6. Summary

In this chacter, we first presented various-order nonlinear magnetizations and magnetic susceptibilities of antiferromagnets within the perturbation theory in a special geometry, where the external magnetic field is pointed along the anisotropy axis. As a base of the nonlinear subject, linear magnetic polariton theory of AF systems were introduced, including the effective-medium method and transfer-matrix-method. Here nonlinear propagation of electromagnetic waves in the AF systems was composed of three subjects, nonlinear polaritons, nonlinear transmition and reflection, and second-harmonic generation. For each subject, we presented a theoretical method and gave main results. However, magnetically optical nonlinearity is a great field. For AF systems, due to their infrared and millimeter resonant-frequency feature, they may possess great potential applications in infrared and THz technology fields. Many subjects parallel to the those in the traditional nonlinear optics have not been discussed up to now. So the magnetically nonlinear optics is a opening field. We also hope that more experimental and theoretical works can appear in future.