Abstract
Propagation of transverse electric electromagnetic waves in a homogeneous plane two-layered dielectric waveguide filled with a nonlinear medium is considered. The original wave propagation problem is reduced to a nonlinear eigenvalue problem for an equation with discontinuous coefficients. The eigenvalues are propagation constants (PCs) of the guided waves that the waveguide supports. The existence of PCs that do not have linear counterparts and therefore cannot be found with any perturbation method is proven. PCs without linear counterparts correspond to a novel propagation regime that arises due to the nonlinearity. Numerical results are also presented; the comparison between linear and nonlinear cases is made.
1. Introduction
Theory of electromagnetic wave propagation in regular (planar, cylindrical, etc.) waveguides filled with linear dielectrics traditionally attracts attention [1–4]. This theory is interesting due to several reasons: first, such problems describe real physical processes that are of importance for applications; second, from the mathematical point of view, this theory is an affluent source of sophisticated and interesting mathematical problems.
Theory of electromagnetic waves in nonlinear media has also attracted attention for decades [5–12]. There are a lot of topics in this field, for example, electromagnetic wave propagation in self-focusing and self-defocusing media, higher harmonic generation (especially second and third), and Raman scattering [6, 8, 10, 13].
In the theory of nonlinear electromagnetic wave propagation, the most advanced results can be found for the case of monochromatic polarised (TE and TM) waves in planar layered dielectric waveguides. From the mathematical standpoint, similar problems for circle cylindrical waveguides are much more complicated. To the best of our knowledge, the first rigorous formulation of TE and TM wave propagation in plane and circle cylindrical waveguides with nonlinear filling had been proposed in [14], and since then these and similar problems have been studied very intensively [7, 10, 15–25]. Nevertheless, key results in the cases of TE and TM wave propagation in a layer with Kerr nonlinearity have been found only recently [23–25].
It is worth noting that the development of the wave propagation theory in a single layer is a first step towards studying stratified (or multilayered) waveguide structures. Layered and periodic waveguides are of special interest for optical guiding industry. Since such structures play an important role in a number of applications in optics, then they compel attention of researchers [26–35].
This paper focuses on the problem of monochromatic TE wave propagation in a plane two-layered dielectric waveguide filled with Kerr media. The guided wave harmonically depends on one of the longitudinal coordinates and decays along the transverse coordinate. Perfectly conducted wall is located on one of the waveguide boundaries; on the opposite side, the waveguide is open and the half-space is filled with a homogeneous isotropic nonmagnetic medium having constant permittivity. We apply the approach developed in [36]. From the mathematical point of view, the problem under investigation is a nonlinear eigenvalue problem for an ordinary nonlinear autonomous differential equation of the second order with discontinuous coefficients and boundary and transmission conditions followed from electromagnetic theory. Eigenvalues of the problem are propagation constants (PCs) of eigenwaves of the waveguide. The PCs are solutions to the so-called dispersion equation (DE). We derive the DE in the general case. If one of the layers is nonlinear and the other one is linear, then the DE can be studied in detail [23, 36].
2. Statement of the Problem
We consider the propagation of a monochromatic TE wave , where is the circular frequency, in a lossless two-layered plane dielectric waveguide , where
The TE wave is described as follows:where and are the complex amplitudes [14] and is an unknown real propagation constant (spectral parameter).
In the half-space , the permittivity is constant and is equal to , where and is the permittivity of free space. There are no sources in the entire space. Everywhere, , where is the permeability of free space.
The waveguide is characterised by the permittivity , where and (see Figure 1). In what follows, we assume that are real constants. There is a perfectly conducted wall at the boundary .
Complex amplitudes (2) satisfy Maxwell’s equations,and decay as when ; tangential components of the fields are continuous on the boundaries and ; tangential component of the electric field vanishes on the boundary . It is assumed that the value is prescribed.
If it does not lead to misunderstanding, the explicit dependence on and is omitted.
Substituting (2) into (4), one getsLet . Expressing and from the second and third equations in (5) and substituting the results into the first equation, one obtains
Denoting by and in the layers and , respectively, one obtains the equationswhere and ; are not necessarily positive.
Equation (6) is linear in the half-space . Taking into account conditions at infinity, one obtains its solution in the formwhere . This solution results in the condition .
The continuity condition for the tangential field components results in the continuity of and at and . Using solution (9) and the continuity of and , one obtains and . Since corresponds to the tangential component of the electric field, then it vanishes at . In view of this, one gets the following conditions:where is supposed to be known (without loss of generality, ).
Thus, the original wave propagation problem is reduced to the problem , which is to determine PCs , such that there exists a nontrivial function which satisfies (7)-(8), conditions (10)-(11), and
The problem can be treated as an eigenvalue problem for a nonlinear differential equation of the second order with discontinuous coefficients on a segment with mixed boundary conditions and transmission conditions at the point .
3. Linear Problem
Here we consider the case , which corresponds to the linear problem denoted by . In this case, (7) and (8) are linear:
Using (10), solutions to (14) are written in the following form:
(i) For ,where and .
(ii) For ,where .
(iii) For ,
Using solutions (15)–(17) and (11), the DE for the problem is written in the following form:
(i) For ,where and .
(ii) For ,where .
(iii) For ,
It is clear that (18) has no solutions. Indeed, the left-hand side is positive and right-hand side is negative. Thus, solutions to the linear problem satisfy the condition
Let us consider (19). Introduce the function
Let ; therefore , where is an integer, such that [see formula (19)]. Thus, one obtains Choosing the lowest possible , one gets . The inequality must be fulfilled.
Now let ; therefore , where is an integer, such that [see formula (19)]. Thus, one obtains Choosing the lowest possible , one gets . The inequality must be fulfilled.
Since is continuous for and , then there is such that . The inequalityis a sufficient condition of existence of solutions to (19).
Let us pass to (20). Introduce the function
Let ; therefore , where is an integer, such that [see formula (20)]. Thus, one obtains where and . Choosing the lowest possible , one gets subject to . Since the inequality must be fulfilled, then .
Now let ; therefore , where is an integer, such that [see formula (20)]. Thus, one obtains Choosing the lowest possible , one gets subject to . Since the inequality must be fulfilled, then .
It follows from the above that the inequalitiesgive a sufficient condition of existence of solutions to (20).
Thus, we formulate the following.
Statement 1. The problem does not have more than a finite number of PCs , . For any , it is true that .
Proof. The existence of solutions to (19) and (20) subject to conditions (25) and (29), respectively, results from the analysis given above. These conditions are only sufficient. It is also clear how conditions (25) and (29) change when the number of solutions increases.
The functions depend analytically on . Since belongs to a finite interval [see formula (21)], then each of (19) and (20) does not have more than a finite number of isolated solutions inside the interval.
4. Nonlinear Problem: Dispersion Equations and Theorem of Equivalence
In the following, we need auxiliary results given below by Statements 2 and 3.
Statement 2. The Cauchy problem for (7) with initial datawhere and are constants, has a unique continuous solution defined globally on , where is an arbitrary real point. This solution depends continuously on for all .
Proof. First integral of (7) takes the formwhere, using conditions (30), one findsIt is clear that does not depend on , and if .
Introduce new variables:Equation (7) can be rewritten as a system:First integral (31) takes the formSolving (35) with respect to , taking into account the fact that , and substituting the result into the right-hand side of the second equation in (34), one obtainswhere and the radicand is positive for all real and .
Using conditions (30), one findsSince , monotonically decreases for . In the general case, can have zeros at some points on the interval . Suppose that has zeros . Then has break points . If , then does not become zero for any and, therefore, is continuous for . It is clear that for all . Formula (36) implies thatThereby, solutions to (36) are sought on each of the intervals , :Substituting , , and into the first, second, and third equations, respectively, in (39), one determines Using the found , one can rewrite (39) in the following form: By substituting , , and into the first, second, and third equations, respectively, of previous equation, one obtainsTaking into account (37) and (38), one finds, from (42),Formulas (43) give explicit expressions for distances between zeros of . Moreover, since the left-hand sides in (43) are finite, the right-hand sides are also finite. Therefore, the improper integrals on the right-hand sides converge.
Summing up all the terms in (43), one getsFrom the last formula, one findsFormula (45) shows that the solution of the Cauchy problem to (7) with initial conditions (30) exists and is defined globally at any segment . The uniqueness of this solution and its continuity with respect to follows from smoothness of the right-hand side of (7) with respect to and [37].
Let us consider the function . Since the right-hand side of (7) depends on analytically, the solution of the considered Cauchy problem depends on analytically as well [38] and therefore and depend analytically on . Since and do not vanish simultaneously, is an analytical function that can have only poles of the first order.
Passing to the limit in (45), one getswhere . We notice that if , then necessarily .
Statement 3. The Cauchy problem for (8) with initial datawhere is a real constant, has a unique continuous solution defined globally on , where is an arbitrary real point. This solution depends continuously on for all .
Proof. Now let us consider (8). First integral of (8) has the formwhere, using condition (47), one calculateswhere depends on . The positivity of is essential.
Introduce new variables:Equation (8) can be rewritten as a system:First integral (48) takes the formSolving (52) with respect to , taking into account the fact that , and substituting the result into the right-hand side of the second equation in (51), one obtainswhere and the radicand is positive for all real and .
Using condition (47) and the fact that , one findsSince , monotonically decreases for . However, is continuous if and only if does not vanish for all . In the general case, can have zeros at some points on the interval . Suppose that has zeros . Then has break points . It is clear that for all . Formula (53) implies thatThereby, solutions to (53) are sought on each of the intervals , .
Using the same reasoning as in the proof of Statement 2, one obtains the expressionFormula (56) shows that the solution to the Cauchy problem for (8) with initial conditions (47) exists and is defined globally at any segment . The uniqueness of this solution and its continuity with respect to follows from smoothness of the right-hand side of (8) with respect to and [37].
Passing to the limit in (56), one getswhere .
The following theorem takes place.
Theorem 1 (of equivalence). The value is a PC of the problem if and only if there are integers and such that is a solution to the DE:with certain .
Proof. It follows from the derivation of expressions (46) and (57) that if is an eigenvalue of the problem , then it is a solution to system (58) with , , and . Let us prove that each solution to system (58) is an eigenvalue.
Let system (58) have a solution with , , , and .
Consider the Cauchy problem for (7) with initial data (30), where . In accordance with Statement 2, its solution exists, is unique, and is defined for . At this step, we do not claim that . Using the found solution and formula (33), one determines the functions and . It is clear that and .
Assuming that and using the found and , one obtains the expression which corresponds to the first line in (58). We rewrite it in the formSince satisfies the first line in (58) with , then in the first line in (58) and in (60) the integrands coincide. Subtracting one from another, one obtainsDue to the obvious estimates , one obtains that (61) is fulfilled only if . Therefore, the condition is false. In the same way, it can be shown that the condition is also false. Thus, .
Now let us pass to the Cauchy problem for (8) with initial data (47), where . In accordance with Statement 3, its solution exists, is unique, and is defined globally for . In this case, using the continuity of and , first integral (48), and (49), one finds that the quantity is determined from the equation where and are calculated at with ( and are already found).
At this step, we do not claim that . Using the found solution and formula (50), one determines the functions and .
We assume that . Using the found and , one obtains the expression which corresponds to the second line in (58). We rewrite it in the following form:Since satisfies the second line in (58) with , then in the second line in (58) and in (64) the integrands coincide. Subtracting one from another, one obtainsDue to the obvious estimates , one obtains that (65) is fulfilled only if . Thus, the condition is false. In the same way, it can be shown that the condition is also false. Thus, .
In other words, we have shown that the functions and satisfy (7) and (8) and conditions (10) and (11), and, therefore, the function is an eigenfunction of the problem corresponding to the eigenvalue .
5. Solvability of the Nonlinear Problem
Theorem 1 is derived for the general case, that is, . The DE (58) can be studied theoretically and numerically. However, the existence of infinitely many PCs for the general case can be proven only if a special restriction is imposed on . This restriction establishes the behaviour of for big . To be more precise, must not decrease “too” rapid. As a matter of fact, such a restriction does not result from the physical formulation of the problem. For this reason below we study two simplified problems, where either or vanishes.
In this section, we use the following notation for the eigenvalues of problems and : means that all the eigenvalues are arranged in the ascending order; means that this eigenvalue is a solution of (57) with [for the problem ] and (46) with [for the problem ].
5.1. Case and
If (or ), solutions to (7) are found elementarily. This allows one to explicitly compute the quantity , which isThis expression (due to its analytical dependence on ) can be used for as well as for . Note that is a real value for all . Indeed, for , one has , where , and then and ; then
It can be checked that the substitution of the explicit expression for into the first equation of system (58) leads to the identity. Thus, in this case, system (58) reduces to the only equationwhere and are given by (57) and (67), respectively; .
The solvability of the problem is established by the following.
Theorem 2. For , any , and any fixed , the problem has an infinite number of PCs with the following properties:(1)If is the solution to , then and .(2)If the linear problem has solutions , then there exists a constant such that, for any , it is true that and , where are first solutions to .(3)If , then .(4)For big and arbitrary small , the asymptotic two-sided inequalityis valid.
Proof. Now let us study the behaviour of with respect to . Explicit solution to (7) with initial data (30) has the form . At the point , one getsSubstituting (71) into (48), one obtainsIf , then and and, therefore, .
If , then and . As before, we denote , where . ThenIf , then and . As before, we denote and , where and . ThenGrouping terms in the last expression for , one gets where and .
ThenSince and are bounded above, then it is clear that for sufficiently small the constant for all . It is also clear that for any fixed there exists such that for all .
Using the expansion , where and , for sufficiently large , one gets the asymptoticsSince both integrands in (69) are positive, the estimatetakes place, where and . Thus we need to evaluate . For further analysis, we use the easy checked inequalities , where and . These inequalities givewhere . Integral is calculated explicitly.
For , the value and, therefore, one has Let ; thenObviously, the denominator in is always positive.
Calculating , one gets Two cases are possible for . If , that is, , then If , that is, , then Combining the found results, one arrives at the formulawhere , , and .
Taking into account formula (77), it is clear that the inequality with fixed holds only if belongs to a finite interval.
Taking into account the fact that for large the value is asymptotically equal to , one finds that the second formula in (85) must be used for sufficiently big .
Further, we need the following expansions: and , which are valid for . Using these expansions for the second line in (85), one findsTaking into account the fact that one finally obtainsfor sufficiently large .
It follows from (88) that . This means that for any there is an integer such that, for any integer , the DE (69) has at least one solution. This implies the existence of infinitely many with an accumulation point at infinity.
Passing to the limit in the first line of (85), one obtains either (20) or (19) depending on or . This implies item of the theorem.
Multiplying (8) by and integrating over , one gets the following for : This givesTaking into account the fact that where and , formula (90) results in If , then, obviously, the right-hand side of this equality tends to and so does the left-hand side. This implies that .
For using (48), one can derive an exact formula provided that has at least two zeros inside the segment.
In order to derive the asymptotic estimates for , let us get back to formula (88). It follows from (78) and (79) that Since bounds from below and bounds from above, then, solving the equations and , with respect to , one finds (70).
5.2. Case and
If (or ), solutions to (8) are found elementarily. This allows one to explicitly compute the quantity , which isThis expression (due to its analytical dependence on ) can be used for as well as for . Note that is a real value for all . Indeed, for , one has , where and then and ; then
It is clear that the substitution of the explicit expression for into the second equation of system (58) leads to the identity. Thus, in this case, system (58) reduces to the only equationwhere and are given by (46) and (94), respectively; .
The solvability of the problem is established by the following.
Theorem 3. For , any , and any fixed , the problem has an infinite number of PCs with the following properties:(1)If is the solution to , then and .(2)If the linear problem has solutions , then there exists a constant such that, for any , it is true that and , where are first solutions to .(3)If , then .(4)If , then .(5)For big and arbitrary small , the asymptotic two-sided inequality is valid, where and is the inversion of .
Proof. The constant in (96) is defined explicitly by (32).
Using (16) at the point , one obtains Substituting these values into (31) and using (32), one obtains the biquadratic equation Using the same estimations as in the proof of Theorem 2 and evaluating the integral contained in (96), one findswhere .
Integral is computed in the same way as in the proof of Theorem 2; the result has the formwhere , , and .
In order to derive the asymptotic estimates for the eigenvalues , let us get back to formula (101). Since is a (fixed) constant [it does not depend on ; see formula (32)], the third line in (101) corresponds to sufficiently big . Thus, for sufficiently big , the third line of (101) gives the asymptoticsIt follows from (102) that . This implies that for any prescribed there is an integer such that (96) has at least one solution for any integer . Thus, there exists an infinite number of positive eigenvalues with accumulation point at infinity.
Passing to the limit in the first line of (101) gives either (20) or (19) depending on or . This implies item of the theorem.
Item of the theorem results from the behaviour of the second and third lines in (101) as .
Multiplying (7) by and integrating over , one obtainsIt follows from (103) that as . This implies item of the theorem.
The asymptotic estimates in the theorem result from formula (102).
6. Numerical Results
Figures 2–13 show results of numerical experiments. The following parameters are used: [E] and ; and (Figures 2–9); and (Figures 10 and 11); and (Figures 12 and 13); other parameters are specified in the captions; , where is the speed of light.
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In Figures 2, 4, 6, 8, 10, and 12(a), the dispersion curves (DCs) are plotted ( versus ). The DCs for the linear and nonlinear cases are given in red and blue colours, respectively. Dashed lines in Figure 2 correspond to the boundaries for . Points of intersections of the vertical dashed line with DCs are PCs.
Dashed lines in Figures 3, 5, 7, and 9 correspond to the boundary between the layers.
Figures 2 and 3 correspond to the linear problem (). For chosen frequency, there are 2 PCs; eigenmodes for them are shown in Figure 3.
The dependence versus , where , is shown in Figures 2, 4, 6, 8, 10, and 12(b). The linear and nonlinear cases are given in red and blue colours, respectively. Points of intersections of the vertical dashed line (corresponding to ) with the curves are PCs. Obviously, in the linear case, the PCs do not depend on and therefore linear solutions are horizontal (red) lines.
There are two types of PCs for the nonlinear case. The first type corresponds to the PCs, which are close to the corresponding linear solutions (in the linear limit, they reduce to the linear solutions; see item in Theorems 2 and 3). The second type corresponds to the PCs, which present a novel guiding regime (they do not reduce to any linear solutions in the linear limit; see item in Theorem 3). In the latter case, we call them “purely nonlinear” PCs; see also [23–25]. In Figures 4, 6, and 8, the PC marked with a blue dot corresponds to the first type; the PCs and marked with green and brown dots, respectively, correspond to the second type.
In Figures 6 and 8, for any frequency, there are infinitely many PCs in the nonlinear cases (few of them are shown and marked in the figures) and only 2 PCs in the linear case.
DCs for the linear cases (red DCs) presented in Figures 6 and 8 coincide with the DCs for the linear case presented in Figure 2.
Figures 5, 7, and 9 allow one to compare linear and nonlinear modes. It can be seen from these pictures that, for a “linear” PC and for a “nonlinear” one (which reduces to the “linear” one in the linear limit), eigenmodes are also close. Obviously, for such a nonlinear PC, perturbation methods can be applied. However, two other nonlinear eigenmodes (shown in green and brown colours) are not close to any linear solution and they do not reduce to linear solutions in the linear limit; these very eigenmodes we call purely nonlinear.
In Figures 8–13, the nonlinear case for different sets of is shown, where both coefficients and are nonzero. In this case, we did not prove the existence of infinitely many PCs; however, these figures are similar to corresponding figures, where or equals zero.
7. Conclusion
The paper focuses on the problem of wave propagation in a plane layered dielectric waveguide filled with Kerr medium. The existence of guided modes that have linear counterparts and guided modes that do not have linear counterparts is proven. The latter waves correspond to a novel guided regime. Since the Kerr nonlinearity is widely studied in nonlinear optics (see, e.g., [7–14]), the results found here can be interesting and important from both theoretical and applied points of view.
It is worth noting that similar results have been found for some other cases. Indeed, it was proven lately that, even in a simpler case of a one-layer waveguide, the Kerr nonlinearity results in the existence of novel guided regimes as well [23–25]. Moreover, similar results have recently been found in the case of polynomial nonlinearity [39]. Thus, the existence of infinitely many nonperturbative PCs is a general feature of polynomial (nonlinear) permittivities with positive terms.
One of the most known peculiarities of nonlinear guided waves is power-dependent PCs; this is clearly shown in Figures 2, 4, 6, 8, 10, and 12(b). It is interesting that varying the value , one strongly affects the corresponding PC and therefore the eigenmode. The fact that the power-dependent PCs can have some potential for optical signal processing is pointed out in many papers (see, e.g., [28–34]). For additional description of applications of nonlinear guided waves and Kerr effect, see [32, 34], where many useful references are also given.
The result given in the paper clearly shows that nonlinear problems can have solutions that cannot be considered as perturbations of solutions of corresponding linear problems. Thus, it is necessary to be careful when one linearizes a nonlinear problem and considers the linearized problem without proving that there are no other solutions.
Of course, it is impossible to expect the existence of purely nonlinear waves for infinitely many PCs. However, it is possible that purely nonlinear waves can be observed in an experiment for some first purely nonlinear PCs. Indeed, as is seen from Figures 5, 7, and 9, max values of few first purely nonlinear eigenmodes are not too big (in comparison with the linear eigenmode). This probably gives an opportunity to observe such waves in an experiment. For the rest of the PCs, the value is so high that the Kerr law is no longer valid.
As a matter of fact, different types of nonlinearities admit nonlinear solutions that become linear ones in the linear limit. For example, for a wide range of saturated nonlinearities, there exist only a finite number of PCs [40, 41]. Thus, there is a qualitative difference between saturated and unbounded (Kerr, qubic-quintic-septic, and, more generally, power and polynomial) nonlinearities. It seems that this difference can be used in order to understand what kinds of nonlinear permittivities are closer to real situations.
If the purely nonlinear guided modes are confirmed by experiment, the theory of nonlinear guided wave propagation will definitely advance. If they are not observed in experiments, then well-known and widespread formulas for nonlinear permittivities must be changed so that mathematical analysis of these models can give results that better satisfy reality.
In spite of the fact that in some papers the nonlinear eigenmodes are searched in an explicit form (see, e.g., [7, 16, 17, 20–22, 34]), this way is not an appropriate one. Indeed, on one hand, explicit solutions to nonlinear equations are often complicated special functions (if it is possible to find them at all) and, therefore, it is almost impossible to study such solutions. On the other hand, many properties/characteristics of eigenwaves can be calculated from original differential equations and boundary conditions, like it is done in this paper. In order to check calculations, one solves numerically the DE (58) with respect to in a prescribed segment; then, for each found PC, one solves the Cauchy problem for (7) and (8) with initial data at and transmission conditions (11) at . In this case, the second condition (10) is fulfilled indispensably (obviously, in numerical calculations, cannot be exactly zero). Thus, Theorems 1, 2, and 3 together with computations allow one to find all nonlinear eigenwaves for which the PCs are calculated.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study is supported by the Russian Foundation for Basic Research (Grant no. 15-01-00206), the Russian Federation President Grant (Project no. MK-4684.2016.1), and the Ministry of Education and Science of the Russian Federation (Agreement no. 1.894.2017/4.6).