Physical parameters for FeF2 and MnF2.
The nonlinearities of common optical materials result from the nonlinear response of their electric polarization to the electric field of electromagnetic waves (EMWs), or. From the Maxwell equations and related electromagnetic boundary conditions including this nonlinear polarization, one can present the origin of most nonlinear optical phenomena.
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. From these one can predict or explain various magnetic optical nonlinear features of magnetic materials. The magnetic mediums are optical dispersive, which originates from the magnetic permeability as a function of frequency. Since various nonlinear phenomena from ferromagnets and ferrimagnets almost exist in the microwave region, these phenomena are important for the microwave technology.
In the concept of ferromagnetism(Morrish, 2001), there is such a kind of magnetic ordering media, named antiferromagnets (AFs), such as NiO, MnF2, FeF2, and CoF2 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 H0 along the z axis. The sublattic magnetzation M0 is the absolute projection value of each total sublattice magnatization to the anisotropy axis. The driving magnetic field changes with time, according to.
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, each sublattice magnetization deviates the AF anisotropy axis and makes a precession with respect to the effective field acting on it. This precession is described by the Bloch’s equation with damping,
where is the gyromagnetic ratio and the damping coefficient, and are the total sublattice magnetizations and contain two parts, the static part and changing part with time, produced by the driving field,,
where is the AF anisotropy field, is the AF exchange field and indicates the driving field. Substituting (2-1b) and (2-1c) into (2-1a), we have
We shall use the perturbation expansion method to derive nonlinear magnetizations and susceptibilies of various orders. We take andas the 0-order magnetization and the 0-order field. is considered as the first-order field and we note that the complex conjugation of this field should be included in higher-order mathmetical procesures higher than the first-order one. In the third-order aproximation, the induced magnetizations are divided into the first-, second- and third-order parts, or
where c.c. indicates the complex conjugation. In practice, one needs the AF magnetization rather than the lattice magnetizations, so we define as the AF magnetization and as its supplemental quantity. In the linear case,，and considering that the linear magnetizations should change with time according to, Eqs.(2-2) can be simplified as
where the special frequencies are defined with, , and. The first-order z-components of the sublattice magnetizations are vanishing. From the definitions and, we have the nonzero elements of the first-order magnetic susceptibilty and supplementary susceptibility
where and with, and. The linear magnetic permeability often used in the past is, or and.
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. Here we first derive the dc susceptibility, which will appear in the third-order ones. Neglecting the linear, third-order terms and the second-harmonic terms in (2-2), reserving only the second-order 0-frequency terms, we obtain the following equations
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 and directly to represent and for simplicity. Substituting the linear results into (2-6) and (2-8), and using the definitions of and, we find the corresponding nonzero elements
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. Meanwhile the conservation of each sulattice magnetic moment results in
Applying the expressions of the first-order components and the expressions
2.3. The third-order approximation
The third-order magnetization also contains two part, or one varies with time according to and the orther is the third-harmonic part with. Because we do not consern with the third-harmonic (TH) generation, so the first-order, and second-order and TH terms in equations (2-2) all are ignored. Thus we have
Substituting the definitions of, , and into equations (2-13,2-14), and after some complicated algebra, we finally obtain
with the coefficients
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 is in the x-y plane, the third-order elements with only subscripts x and y, such as, , andet. al., are usefull. In addition, if the external magnetic field H0 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 is the AF dielectric constant and is the magnetic permeability tensor. It is interesting that the magnetic field of AF polaritons vibrates in the x-y plane since the field does not couple with the AF magnetization for it along the z axis. We take the magnetic field as with the amplitude. Thus applying equation (3-1) we find directly the dispersion relation of bulk polaritons
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 and are positive attenuation factors. From the magnetic field (3-3) and the Maxwell equation, we find the corresponding electric field
Here there are 4 amplitude components, but we know from equation that only two are independent. This bounding equation leads to,
The wave equation (3-1) shows that,
determining the two attenuation constants. The boundary conditions of and continuous at the interface (y=0) lead to the dispersion relation
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 much longer than the stack period (and are the AF and DE thicknesses). The main idea of this method is as follows. We assume that there are an effective relation between effective magnetic induction and magnetic field, and an effective relation between effective electric field and displacement, where these fields are considered as the wave fields in the structures. But and in any layer, where is given in section 2 for AF layers and for DE layers. These fields are local fields in the layers. For the components of magnetic induction and field continuous at the interface, one assumes
and for those components discontinuous at the interface, one assumes
where the AF ratio and the DE ratio. Thus the effective magnetic permeability is obtained from equations (3-8) and its definition,
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 lth stack period as follows.
where k1 and k2 are determined with and. Similar to Eq. (3-4) in subsection 3.1, the corresponding electric field in this period is written as
Here there is a relation between per pair of amplitude components, or
As a result, we can take and as 4 independent amplitude components. Next, according to the continuity of electromagnetic fields at that interface in the period, we find
At the interface between the lth and l+1th 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. From (3-15), the other relation also is obtained, or
Commonly, the matrix relation between the amplitude components in the lth and l+1th 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 with, and then the dispersion relation of bulk magnetic polaritons just is
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, where is the distance and is the attenuation coefficient and positive. As a result,
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 with the vacuum attenuation constant. The two components of the amplitude vector are related withand. The corresponding electric field is. The boundary conditions of field components Hx and Ez continuous at the surface lead to
with. These equations result in another relation,
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 style, but for the surface mode, and both are included dispersion equation (3-25).
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 and. Thus the reflection and transmission are determined with equation (3-27). In numerical calculations, the damping in the permeability cannot is ignored since it implies the existence of absorption. We have obtained the numerical results on the AF multilayer, and transmission spectra are consistent with the polariton spectra (Wang, J. J. et. al, 1999), as illustrated in Fig.5.
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 wavelenghtis much longer than the superlattice period L (), the superlattice behaves like an anisotropic bulk medium(Almeida & Mills,1988; Raj & Tilley,1987), and the effective-medium approch is reasonable. We have introduced a nonlinear effective-medium theory(Wang & Fu, 2004), to solve effective susceptibilities of magnetic superlattices or multilayers. This method has a key point that the effective second- and third-order magnetizations come from the contribution of AF layers or and.
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 and. In AF layers, the polariton wave equation is
where is the linear permeability of antiferromagnetic layers given in section 2, and the nonzeroelements,. The third-order magnetization is indicated by with the nonlinear susceptibility elements presened in section 2. As an approximation, we consider the field components in as linear ones to find the nonlinear solution of included in wave equanion (4-1). For the linear surface wave propagating along the x-axis and the linear bulk waves moving in the x-y plane,. Thus the wave equation is rewritten as
with. Eq.(4-2c) implies that is a third-order small quantity and equal to zero in the circumstance of linearity (TM waves). We begin from the linear wave solution that has been given section 3 to look for the nonlinear wave solution in AF layers. In the case of linearity, the relations among the wave amplitudes, with. The nonlinear terms in equations (4-2) should contain a factor with and is defined as the attenuation constant for the surface modes, and m=1 and with Q the Bloch’s wavwnumber for the bulk modes. and are nonlinear coefficients. After solving the derivation of equation (4-2b) with respect to y, substituting it into (4-2a) leads to the wave solutions
in which for the bulk modes and for the surface modes. The expressions of coefficients in Eqs.(4-3a) and (4-3b) are presented as follows:
When is a real number, the coefficients in Eq.(4-3a) are
where, ,.The field strength . From the boundary conditions of the linear field, one also can easily prove that included in the formulae is
for the surface modes and
for the bulk modes. The coefficients in Eq.(4-3) can written as
If is imaginary, i.e., these coefficients should be changed into
Note that all these coefficients contain implicitly the factor, so we say that they are of the second-order. For simplicity in the process of deriving dispersion equations, we introduce three second-order quantities,
Thus the nonlinear magnetic field can be rewritten as
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 and magnetic induction field continuous at the interfaces and surface (and). The results (4-3a) and (4-3b) related to the nth AF layer, as well as the solutions in the vacuum and in the nth NM layer will be used to determine the dispersion relations. In the following several paragraphs, we shall calculate the dispersion relations of the surface and bulk modes, respectively.
Bulk dispersion equation
For the bulk polaritons, there are 6 amplitude coefficients in the wave solutions, and. The magnetic induction.. in AF layers and in NM layers. The boundary conditions of and continuous at the interfaces (and) imply four equations, and in a NM layer leads to two additional relations and. Thus we have
From these four equations, we find the dispersion relation of the nonlinear bulk polaritons,
with the nonlinear factor N described by
Due to the nonlinear interaction, the nonlinear term appears in the dispersion equation of the polaritons and is directly proportional to. This term is a second-order quantity and makes a small correct to the dispersion properties of the linear bulk polaritons.
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, the bulk wave moves along the x-axis and the dispersion equation is a real equation for real. For such a dispersion equation, has a real solution, otherwise the solution of is a complex number with the real part, so-called the nonlinear mode frequency, and the imaginary part, the attenuation or gain coefficient. In addition, it is very interesting that the unreciprocity of the bulk modes, with, is seen, due to the existence of in the nonlinear term as a function of with the period.
Surface dispersion relations
For the surface modes, note and take the transformation in equations (4-10), we can find
in which can be obtained directly from Eq.(4-13) with the same transformation. This nonlinear term is directly proportional to the multiply of and, so in the same condition the nonlinearity makes larger contribution to the bulk modes than the surface modes. We can use the linear expression of to reduce the nonlinear term on the right-hand of Eq.(4-14), but have to derive its nonlinear expression to describe on the left-hand, since its nonlinear part may has the same numerical order as that of. So we need another equation to determine it. Applying the boundary conditions at the surface, and, we can find
Combining this with Eqs.(4-11a-c), the equation determining is found,
in Eq.(4-16) also can be considered as an linear quantity since it always appears in the multiply of it and. We also should note that there is a series of nonlinear surface eigen-modes as 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 and decreasing n have a similar effect in numerical calculation.
Because the nonlinear terms in Eqs.(4-12) and (4-14) all contain directly proportional to, the nonlinear effects may be too strong for us to use the third-order approximation for the nonlinear magnetization when is near to. In this situation we will take a smaller value of to assure of the availability of this approximation.
We take the FeF2/ZnF2 superlattice as an example for numerical calculations, the physical parameters of FeF2 are given in table 1. While the relative dielectric constant of ZnF2 are. We apply the SL period, and take for the surface modes. The nonlinear factor is the relative strength of the wave field. The nonlinear shift in frequency is defined as, where the nonlinear frequencyand attenuation or gain coefficient as the real and imagine parts of the frequency solution from the nonlinear dispersion equations both are solved numerically. is determined by the linear dispersion relations.
|FeF2||197kG||533kG||7.04 kG||5.5||rad s-1 kG|
|MnF2||7.87kG||550kG||5.65 kG||5.5||rad s-1 kG|
We illustrate the nonlinear shift in frequency as function of the component of wave vector in the three bulk-mode bands seperately in Fig. 6. is offered to illustrate the bottom band. (a) and ((b) show for and respectively. As shown in Fig.6(a), in the bottom band, when, the nonlinear shift is downward or negative in the region of smaller, but becomes positive from negative with the increase of. For a SL with thicker AF layers, for example, the shift is always positive. For the top bulk band, always is positive and possesses its maximum. In the middle band, Fig. 6(b) shows the positive frequency shift that increases basically with. In terms of the shape of a band, the second-order derivative of linear frequency with respect to, for a mode in it can be roughly estimated to be positive or negative. According to the Lighthill criterion for the existence of solitons(Lighthill,1965). One confirms from the figures that >0 for modes in the top band, <0 in the bottom band, but <0 or >0 in the middle band, depending on. The soliton solution may be found since the Lighthill criterion can be fulfilled in the two bands. In the middle bulk band, the mode attenuation is vanishing, the nonlinearity is very evident and the nonlinear shift is positive.
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, as a result, the nonlinear effect also is stronger. The attenuation as the dispersion equations are real. The shift is negative for 0.3, but positive for =0.1. For =0.2, it is positive and increases with in the range of small, but negative in the range of large and its absolute value decreases as is increased. Although there can be a series of surface eigen-modes in the nonlinear situation, the obvious nonlinear effect can be seen only for, so that we present only the corresponding mode. One should note that the Lighthill criterion is satisfied for =0.1 and 0.2, as a result, the surface soliton may form from the surface magnetic polariton.
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
in which the nonlinear terms are shown with
Finally the reflectivity and transmissivity are defined as and (Klingshirn, C. F.,1997). Here we should discuss a special situation. In the situation, from the expressions of to and to, we find. It is quite obvious that one finds no nonlinear effects on the reflection and transmission in the case of normal incidence. For, becomes imaginary as the incident angle exceeds a special value, then the transmission vanishes. The nonlinear effect can be seen only from the reflection. Due to the complicated expressions for the reflection and transmission coefficients, more properties of and can be obtained only by numerical calculation of Eq. (4-18).
We take a FeF2 film as an example for numerical calculations. with the physical parameters given in Table 1. The film thickness is fixed at and the incident wave intensity, implicitly included in the nonlinear coefficients, is fixed at, corresponding to a magnetic amplitude of 16G in the incident wave. In the figures for numerical results, we use dotted lines to show linear results and solid lines to show nonlinear results. We shall discuss transmission and reflection of the AF film put in a vacuum. The transmission and reflection versus frequency are illustrated in Fig.9 for the incident angle and are shown in Fig.10 versus incident angle for
First, the nonlinear modification is more evident in reflection for frequencies higher than. We see a very obvious discontinuity on the nonlinear R and T curves at, corresponding to the smallest value of whose real part changes in sign as the frequency moves cross this point. This causes the jump and obvious nonlinear modification, as the wave magnetic field is intense in the vicinity of this point. Secondly,and versus for a fixed frequency are shown in Fig.10. Here the discontinuity is also seen since the magnetic amplitude and the nonlinear terms vary with the wave vector. It is more interesting that when the incident angle the reflection and transmission are both lower than the linear ones, implying that the absorption is reinforced. However, in the range of they both are higher than the linear ones, and as a result the absorption is evidently restrained. The nonlinear influence disappears for normal incidence.
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 the SH magnetization is vanishing, as a result the SHG is absent. So the external magnetic field is necessary for the SHG. We take such an AF structure as example to describe the SHG theory, where the AF film is put two different dielectrics. In the coordinate system selected in Fig.11, the AF anisotropic field and dc magnetic field both parallel to the z-axis and the x-y plane as the incident plane. I is the incident wave, R the reflection wave and T the transmission wave, related to incident angle, reflection angle and transmission angle, respectively. If a subscript 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 in the film. In this case, only one component of the SH magnetization can be found easily
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 and show the amplitudes of the forward and backward waves in the film, respectively. The electric fields above and below the film are
The corresponding magnetic fields in different spaces are written as
where isand the vacuum magnetic permeability. and is the wave number in the above space, and and is the wave number below the film, but. Here c is the vacuum velocity of light. and. The boundary conditions of the fields at the surfaces first require that, and these wave-number components should be real since we assume that dielectric constants and magnetic permeabilities in nonmagnetic media all are real values. In addition, using the boundary conditions, we also find the pump-field amplitudes with the electric amplitude of, and
where d is the film thickness, and. The wave amplitudes and of R and T are not necessary for seeking the SHG, so they are given up here.To solve the output amplitudes of SHG, and, we should look for the solution of the SH wave equation in the film. In fact, there are three component equations, but only one contains a source term and this equation is
The other two are homogeneous and do not contain the field component. In addition, the other SH components cannot emerge voluntarily without source terms, so it is evident that the SH wave is a TM wave. Because the SH magnetization and pump field in the film both have been given, to find the solution of equation (5-8) is easy. Let
with. Substituting SH solution (5-9), expression (5-1) and solution (5-6a) into equation (5-8), we find the nonlinear amplitudes
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 and determined by and. The SH electric field in different spaces are found from to be
Considering the boundary conditions of these fields continuous at the surfaces, there must be and the these wave-number components all are real quantities, meaning the propagation angles of the SH outputs from the film
It is proven that the SH wave outputs and have the same propagation direction as reflection wave R and transmission wave T, respectively.
Finally we solve the amplitudes of the output SH wave. The continuity conditions of and at the interfaces lead to
After eliminating and from the above equations, we find the magnetic field-amplitudes of the output SH waves,
The numerical calculations are based on three examples, a single MnF2 film, SiO2/MnF2/air and ZnF2/MnF2/air, in which the MnF2 film is antiferromagnetic. The relative dielectric constants are 1.0 for air, 2.3 for SiO2 and 8.0 for ZnF2. The relative magnetic permeabilities of these media are 1.0. There are two resonance frequencies in the dc field of, and. We take the AF damping coefficient and the film thickness. The incident density is fixed at, which is much less than that in the previous papers (Almeida & Mills, 1987; Kahn, et. al., 1988; Costa, et. al., 1993; Wang & Li, 2005; Bai, et. al., 2007 ).
We first illustrate the output densities of a single film versus frequency and incident anglewith Fig.12 (a) for and (b) for. Evidently in terms of their respective maxima, is weaker than by about ten times. Their maxima both are situated at the second resonant frequency and correspond to the situation of normal incidence. The figure of is more complicated than that of since additional weaker peaks of are seen at large incident angles.
Next we discuss the SH outputs of SiO2/MnF2/air shown in Fig.13. Incident wave I and reflective wave R are in the SiO2 medium and transmission wave T in air. The maximum peak of is between the two resonant frequencies and in the region of. For the given parameters, this angle just satisfies and is related to, so it can be called a critical angle. When, is an imaginary number and transmission T vanishes. For, is very weak and numerically similar to that of the single film. However, the maximum of is about four times as large as that of, and decreases rapidly as the incident angle or frequency moves away from or the resonant frequency region. We find that the maxima of and are in intensity higher than those shown in Fig.12 by about 40 and 13 times, respectively.
Finally we discuss the SH outputs of ZnF2/MnF2/air, with the dielectric constant of ZnF2 larger than that of SiO2. The spectrum of SR 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 in the last paragraph and is equal to 20.1o. The second defined as corresponds to and is equal to. For, becomes an imaginary number
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 SR appear between the two critical angles, but the highest peak stands between the two resonant frequencies and is near to. Outside of the region between and, we almost cannot see SR. For ST, the pattern is more simple, as shown in Fig.14 (b). Only one main peak is seen clearly, which arises at and occupies a wider frequency range. Different from Fig.13, the maxima in Fig.14(a) and Fig.14(b) are about equal. Comparing Fig.14 with Fig.12, we find that the maximums of SR and ST 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 is equal to zero or an imaginary number. In this situation, the backward wave as the reflection wave is the most intense and equal in intensity to the forward wave. The interference of the two waves at the bottom surface makes the pump wave enlarged, and further leads to the appearance of the -peak in the vicinity of the critical angle. The intensity of, however, depends on that of the pump wave at the upper surface. When the phase difference between the forward and backward waves satisfies (k is an integer) at the surface, the interference results in the peaks of. Thus the interference effect in the film plays an important role in the enhancement of the SHG.
It is also interesting for us to examine the SH outputs versus the film thickness. We take the SiO2/MnF2/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, 0.09 W/cm2, and 0.012 W/cm2. If we enhance the incident wave density to 10.0kW/cm2, the two output densities are increased by 100 times, to 9.0W/cm2 and 1.2W/cm2, or if we focus SI 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.
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.
This work is financially supported by the National Natural Scienc Foundation of China with grant no.11074061 and the Natural Science Foundation of Heilongjiang Province with grant no.ZD200913.