1. Introduction
The advances in fabrication of sophisticated waveguides, such as microstructured fibers, have enabled a large degree of dispersion tailoring. The waveguide dispersion is crucial for a range of applications: dispersionless transmission, dispersion compensation, and enhancement or suppression of nonlinear effects. In order to exploit the potential of these waveguides, it is important to understand the origin of waveguide dispersion.
The main motivation of this paper is to obtain a deeper understanding of the origin of waveguide dispersion by investigating the connection between waveguide dispersion and causality. It is shown that Kramers-Kronig relations exist for waveguides, even when material dispersion and material loss is negligible in the frequency range of interest. To our knowledge, this result has not appeared previously. The theory is applied to hollow waveguides with perfectly conductive walls, index-guiding dielectric waveguides, and bandgap-guiding waveguides. We will demonstrate that for hollow waveguides with perfectly conductive walls, each mode propagates causally, and the associated mode index obeys the usual Kramers-Kronig relations. That is, the real part of the mode index is determined by the imaginary part, and vice versa. For dielectric waveguides, it turns out that the (real) mode index of a guided mode is related to the (imaginary) mode indices of the evanescent modes. For weakly guiding waveguides, we show that the derivative of the mode index with respect to frequency for a certain mode is given solely by the associated mode field profile. Thus, the derivative of the mode index can be calculated from a measurement of the mode profile at a single frequency.
The remaining of the paper is structured as follows: In Section 2 we use relativistic causality to derive Kramers-Kronig relations for waveguides. The theory is applied to hollow waveguides with perfectly conductive walls (Subsection 2.1), index-guiding dielectric waveguides (Subsection 2.2), and bandgap-guiding waveguides (Subsection 2.3). Finally, we discuss and interpret the result that modes with frequency-dependent profiles may propagate non-causally.
2. Causality and Kramers-Kronig relations
Relativistic causality means that no signal can propagate faster than the light velocity in vacuum,
c. Here we will examine the implications of this condition on the effective index of modes in passive waveguides. Only true modes are considered (leaky modes are not treated here). For notational simplicity, we consider planar waveguides, where the refractive index varies in the
y-direction and is constant in the
x- and
z-direction. However, the derivation can straightforwardly be extended to 2-d waveguides, yielding identical results. The signal is assumed to propagate in the positive
z-direction and the electric field is taken to be polarized in the
x-direction. To obtain the field at arbitrary
z, one could expand the field at
z = 0 into a complete set of modes and propagate each mode separately. Completeness is guaranteed for frequencies where the waveguide medium is lossless. However, since we will be concerned by all frequencies, from dc to infinity, we cannot assume that the waveguide medium is lossless. Indeed, it is the material resonances (and loss) that give rise to a refractive index different from unity at the frequencies of interest. While completeness is known to hold in some special cases for absorbing waveguides [
6
6
.
V. V.
Shevchenko
, “
Spectral expansions in eigenfunctions and associated functions of a non-self-adjoint Sturm-Liouville problem on the whole real line
,”
Differ. Equations.
15
,
1431
–
1443
(
1979
).
,
7
7
.
V. V.
Shevchenko
, “
On the completeness of spectral expansion of the electromagnetic field in the set of dielectric circular rod waveguide eigen waves
,”
Radio Science
17
,
229
–
231
(
1982
). [CrossRef]
], it is not guaranteed to hold in general. (For weakly absorbing media, one can
treat material absorption perturbatively, and let the mode fields be approximated by those of a lossless waveguide [
8
8
.
A. W.
Snyder
and
J. D.
Love
,
Optical Waveguide Theory
(
Chapman and Hall, New York
,
1983
).
]. This approximation cannot be expected to be accurate for frequencies near material resonances, where absorption is large.) We therefore divide the entire frequency range [0, ∞) in two separate ranges; one range where material absorption is negligible and completeness holds (Ω
_{1}), and one range close to material resonances, where completeness is not guaranteed to hold in general (Ω
_{2}). Each of the intervals Ω
_{1} and Ω
_{2} do not have to be connected. As an example, let Ω
_{1} = [0,
ω
_{1}) ⋃ (
ω
_{2},∞) and Ω
_{2} = [
ω
_{1},
ω
_{2}], where 0 <
ω
_{1} ≤
ω
_{2} < ∞. In this case, the interval [0,
ω
_{1}) could be the “frequency range of interest”, that is a range where we are interested in the waveguide dispersion, and where absorption is negligible. As we will discuss shortly, absorption is also negligible for sufficiently large frequencies (i.e., for
ω >
ω
_{2}). The time-dependent field at
z can now be expressed as
The notation ∫_{Ω1⋃(-Ω1)} means that we integrate over the positive frequencies Ω_{1}, in addition to the corresponding negative frequencies. However, the physical field E(z,t) is real, and consequently the contribution from the negative frequencies in the integral is the complex conjugate of the contribution from the positive frequencies. When ω ∈ Ω_{1}, the field is expanded into a complete set of modes, and the mode fields, propagation constants, and mode weights are denoted ψ_{j}
, β_{j}
, and A_{j}
, respectively. The y-dependence of the mode fields is omitted for simplicity. The implicit meaning of the ∑ symbol is a sum over bound modes and an integral over radiation modes. When ω ∈ Ω_{2}, the field profile at z is denoted χ(z, ω) exp[iωz/c], with weight B. The factor exp[iωz/c] is factorized out for later convenience.
Since
E(
z,
t) is real, the propagation constants
β_{j}
(
ω) satisfy
β_{j}
(-
ω) = -
β*
_{j}(
ω). The mode profiles satisfy an orthogonality relation [
8
8
.
A. W.
Snyder
and
J. D.
Love
,
Optical Waveguide Theory
(
Chapman and Hall, New York
,
1983
).
], which in our case can be written
where δ_{ij}
means a Dirac delta function for two radiation modes, and a Kronecker delta otherwise. For a normalized field profile ψ, the completeness relation implies that
for any ω ∈ Ω_{1}.
We imagine a time-dependent source at z = 0. The source is used to excite modes of the waveguide. We further imagine that the light is detected after propagating a distance L by means of a filter with a detector behind. The field at z = 0, as produced by the source, is written
Similarly, the field at z = L, as seen by the detector, is
Here
ψ
_{src} and
ψ
_{det} are arbitrary, normalizable, frequency-independent functions of
y that characterize the source and detector filter, respectively. The frequency independence of
ψ
_{src} and
ψ
_{det} ensures that the field is excited and detected in a causal manner. For example, by choosing a causal
A
_{in} (
ω) in Eq. (
4), one ensures that
E(0,
t) is zero for all points in the
xy-plane for
t < 0. Similar reasoning also applies to detection. By using the expansion Eq. (
1) together with the orthogonality condition Eq. (
2), and setting
χ(0,
ω) =
ψ
_{src} we obtain
where
The physical interpretation of this expression when ω ∈ Ω_{1} is that the source excites all modes overlapping with the source field. Each mode propagates independently through the waveguide. The detected signal results from an interference between all the excited modes overlapping with the filter profile at z = L.
in an obvious notation. Relativistic causality implies that g(τ) = 0 for τ < L/c. This means that
which gives
Eq. (
10) implies that
Ĝ(
ω) is analytic in the upper half of the complex
ω-plane [
3
3
.
H. M.
Nussenzveig
,
Causality and Dispersion Relations
(
Academic Press, New York
,
1972
).
]. We choose
where ψ
_{0}(ω
_{0}) is the field distribution of a guided mode (e.g. the fundamental mode) of the waveguide at a frequency ω
_{0} ∈ Ω_{1}. We then obtain
where the mode index n_{j}
is defined as
More interesting than a relation between the real and imaginary part of
Ĝ(
ω) is a relation between the real and imaginary part of the mode indices. Such a relation can be obtained by Taylor expanding the exponentials in Eq. (
12) to first order. To ensure the validity of this expansion, we must check that [
n_{j}
(
ω) - 1]
ω is bounded in Ω
_{1} for the contributing modes. Since the refractive index of the medium is bounded everywhere, [
n_{j}
(
ω) - 1]
ω is clearly bounded for guided modes and bounded
ω. Next, we consider what happens when
ω → ∞. For any dielectric medium, the refractive index can be written
n
_{diel} ≈ 1 -
ωp2/(2
ω
^{2}) in this limit, where
ω_{p}
is the plasma frequency of the medium [
9
9
.
J. D.
Jackson
,
Classical Electrodynamics
(
Wiley, New York
,
1999
).
]. This ensures that the high frequency region belongs to Ω
_{1}, and that (
n_{j}
- 1)
ω ∝ 1/
ω for all guided modes. Finally, consider the contribution to
G(
ω) from the radiation modes. In an obvious notation, this contribution can be written as an integral over transverse wavenumber
k_{t}
:
∫0∝|〈
ψ
_{0}(
ψ
_{0})|
ψ_{kt}(
ω)〉|
^{2} exp{
i[
n_{kt}(
ω) - 1]
ωL/
c}d
k_{t}
, where
n_{kt} = (1 -
c
^{2}
kt2
/
ω
^{2})
^{1/2} [
8
8
.
A. W.
Snyder
and
J. D.
Love
,
Optical Waveguide Theory
(
Chapman and Hall, New York
,
1983
).
]. If |〈
ψ
_{0}(
ω
_{0})|
ψ_{kt}(
ω)〉|
^{2} → 0 sufficiently fast to ensure convergence of
∫0∞ |〈
ψ
_{0}(
ω
_{0})|
ψ_{kt}(
ω)〉|
^{2}d
k_{t}
, the error by truncating the integral at a cutoff wavenumber
k_{t}
=
k_{c}
can be made arbitrarily small by choosing a large enough
k_{c}
. Using Parseval’s relation it can be shown that this condition is satisfied when
ψ
_{0} is normalizable. Thus, since
k_{t}
is finite for the contributing radiation modes, we find that |
n_{kt} - 1|
ω is finite for all
ω ∈ Ω
_{1}, and that (
n_{kt} - 1)
ω ∝ 1/
ω when
ω → ∞.
Since L can be chosen freely, it is therefore always possible to chose a small enough L such that the exponentials in Ĝ(ω) can be Taylor expanded to first order with negligible error. Defining
we observe that
F(
ω) is square integrable along the real axis. Thus, Eq. (
10) implies through Titchmarsh’s theorem that
where
P means principal value, and we have used the symmetry
n_{j}
(-
ω) =
n*
_{j}(
ω). Eqs.
(15)–(16) are the Kramers-Kronig relations for waveguides.
Eq. (
15) states that a weighted sum of the real part of all mode indices is given by a Kramers-Kronig integral involving a weighted sum of the imaginary part of all mode indices. These imaginary parts, in turn, may not only be a result of material absorption, but can also be due to evanescent modes. Since the material is assumed lossless in the frequency range of interest, only Eq. (
15) will be considered here. (Eq. (
16) is not relevant for guided modes that are lossless in Ω
_{1}.) We define
njr
(
ω) ≡ Re
n_{j}
(
ω),
nji
(
ω) ≡ Im
n_{j}
(
ω), and
The right-hand side of Eq. (
15) can now be written
where ∑′ means a sum over evanescent modes. The observation frequency ω is assumed to be outside Ω_{2}, allowing us to drop the principal value in the integral over Ω_{2}. Defining
we can rewrite Eq. (
15) as
The last term on the right-hand side of (21) can be neglected when material dispersion is negligible around
ω
_{0}: Take the derivative of Eq. (
19) with respect to
ω. Since
ω ∉ Ω
_{2}, we are allowed to take the derivative inside the integral. This gives
where we have used the Cauchy-Schwarz inequality in the last step, together with the fact that
ψ
_{0}(
ω
_{0}) and
χ(0,
ω) are normalized. (Note that since the waveguide is assumed passive, 〈
χ(
L,
ω′)|
χ(
L,
ω′)〉 ≤ 〈
χ(0,
ω′)|
χ(0,
ω′)〉 = 1.) It is now clear that |
∂δ/
∂ω| is negligible provided the material resonances (loss) are located far enough from
ω, in other words, when the material dispersion is small around
ω. For example, take Ω
_{1} = [0,
ω
_{1}) ⋃ (
ω
_{2}, ∞) and Ω
_{2} = [
ω
_{1},
ω
_{2}], where 0 <
ω
_{1} <
ω
_{2} < ∞. If the maximum mode index for the contributing modes in Ω
_{1} is, say, 10, we must choose
L ~
c/(10
ω
_{1}) so that the expansion in Eq. (
14) is valid. Assuming
ω≪
ω
_{1} we obtain |
∂δ/
∂ω| ≲ 8
ω/
ω12.
Thus, assuming negligible material dispersion and loss in the frequency range of interest, we have
In other words, in these cases the real part of the mode index is given by the associated imaginary part in addition to the imaginary parts of all other mode indices. The contributions from the different modes are again weighted by the overlaps
c_{j}
. Although Eq. (
24) remains valid for waveguides that are lossless in the frequency region of interest, it is fundamentally different from Eq. (
23) in that the loss far away from
ω cannot be neglected in the integral.
We will now provide examples of the application of Kramers-Kronig relations for waveguides.
2.1. A hollow waveguide with perfectly conductive walls
As a first example, we consider a hollow waveguide with perfectly conductive walls (hereby denoted a hollow metallic waveguide). This waveguide is particularly simple since all modes ψ_{j}
are independent of frequency and form a complete set for all fields satisfying the boundary condition of zero field in the walls. All frequencies therefore belong to Ω_{1}. By choosing ψ
_{src} = ψ
_{det} = ψ_{j}
, we obtain
The propagation constant of a hollow metallic waveguide mode is
where
ω
_{j,c} is the cutoff frequency of the mode [
10
10
.
D. K.
Cheng
,
Field and Wave Electromagnetics
(
Addison-Wesley, Reading, Massachusetts
,
1989
).
]. Note that the propagation constant is real for
ω ≥
ω
_{j,c}, while it is imaginary for 0 ≤
ω <
ω
_{j,c}. Using Eq. (
13) to find
n_{j}
from Eq. (
26), we obtain
It is clear that |
n_{j}
(
ω) - 1|
ω is bounded for all
ω, and that
n_{j}
(
ω) has the desired asymptotic behavior when
ω → ∞. Thus the exponential in Eq. (
12) can be Taylor expanded to first order, yielding
We observe that in this case, the real part of the mode index is determined directly from the imaginary part of the mode index, and vice versa.
Similarly, for
ω >
ω
_{j,c}, Eq. (
23) reduces to
Fig. 1. The real and imaginary part of the mode index for a hollow waveguide with perfectly conducting walls is shown in (a). The derivative of the real part of the mode index with respect to frequency is shown in (b). Also shown are results obtained using the Kramers-Kronig relations, where the real part of the mode index is determined from the imaginary part and vice versa. In all cases the results from the Kramers-Kronig relations are in excellent agreement with the exact results. We find that any discrepancy is only dependent on the numerical resolution in the calculations.
Fig. 2. Real and imaginary part of the mode index for a dielectric-filled waveguide with perfectly conducting walls. Exact results and results obtained using the Kramers-Kronig relations for waveguides are shown.
2.2. An index-guiding waveguide
Only symmetric evanescent modes are needed to be taken into account in Eq. (
23), since the overlap between a symmetric and an anti-symmetric mode is zero. In order to find
c_{j}
(
ω,
ω′) for the evanescent modes, one must solve the wave equation to obtain the mode field of the fundamental mode at the frequency
ω, and the mode field of the symmetric evanescent modes at the frequency
ω′. However, since the index contrast between the core and the cladding is small, we approximate the evanescent modes of the weakly guiding waveguide by the evanescent modes for free space [
8
8
.
A. W.
Snyder
and
J. D.
Love
,
Optical Waveguide Theory
(
Chapman and Hall, New York
,
1983
).
,
11
11
.
Z. H.
Wang
, “
Free Space Mode Approximation of Radiation Modes for Weakly Guiding Planar Optical Waveguides
,”
IEEE J. Quantum Electron.
34
,
680
–
685
(
1998
). [CrossRef]
]. The field of such modes can be written
where
and β_{j}
= i|β_{j}
|. Consequently, for evanescent modes we have
which shows that c(k_{t}
) is proportional to the absolute square of the Fourier transform of ψ
_{0} in the free space approximation. The mode profile ψ
_{0} is assumed to be known.
Fig. 3. The fundamental mode in a planar index-guiding waveguide for
ωd/
c = 40 (a), and the resulting integral Eq. (
38) over evanescent modes (b).
For evanescent modes, we have nji
= c|β_{j}
|/ω. This means that
where we have used Eq. (
35) to express
nji
in terms of
k_{t}
and
ω′. A plot of this integral is shown in
Fig. 3(b), where the field
ψ
_{0} shown in
Fig. 3(a) is used. Inserting Eq. (
38) into Eq. (
23) gives
Since the inner integral in (39) decays rapidly with ω, the upper limit in the outer integral has been set to infinity as an approximation.
Fig. 4. Mode index for the fundamental TE mode in a dielectric waveguide (a). In (b) dn0r/dω is shown, based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.
Note that the principal value in Eq. (
39) induces no numerical problems since, in practice, the integral is cut off at a frequency less than
ω. This is justified since numerical evaluation of Eq. (
38) in the examples above shows that the main contribution to the integral comes from frequencies much less than
ω.
It is apparent that even if a guided mode in an index-guiding waveguide is lossless in the frequency range of interest, its change in mode index with frequency can be determined using a Kramers-Kronig relation. The crucial point is that when exciting the waveguide using a signal with a frequency-independent field distribution that matches the guided mode at one frequency, one does not get perfect match at other frequencies, and may therefore excite evanescent modes as well. The evanescent modes have an imaginary propagation constant and enter the Kramers-Kronig relations for waveguides as an effective loss term. It is this effective loss that determines the change of mode index with frequency for the guided mode.
Fig. 5. Mode index for the second order symmetric TE mode in a dielectric waveguide (a). In (b) dn2r/dω is shown, based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.
2.3. A Bragg reflection waveguide
As a third example, we consider a symmetric waveguide with an air core of thickness
d = 8Λ, and a periodic cladding consisting of an infinite number of alternating high and low index layers, see
Fig. 6(a). Light is guided in the core by means of Bragg reflection from the periodic cladding [
12
12
.
P.
Yeh
,
A.
Yariv
, and
C.-S.
Hong
, “
Electromagnetic propagation in periodic stratified media. I. General theory
,”
J. Opt. Soc. Am.
67
,
423
–
438
(
1977
). [CrossRef]
]. The periodic cladding consists of high-index layers of thickness
b = 0.5Λ and refractive index 1.01 at observation frequencies, and low-index layers with thickness
b and refractive index 1. The low index contrast between the high and low index layers is chosen in order to use the free space approximation for the evanescent modes. This type of waveguide is a planar version of the low index contrast photonic bandgap fibers recently reported [
13–15
13
.
F.
Brechet
,
P.
Leproux
,
P.
Roy
,
J.
Marcou
, and
D.
Pagnoux
, “
Analysis of bandpass filtering behaviour of single-mode depressed-core-index photonic-bandgap fibre
,”
Electron. Lett.
36
,
870
–
872
(
2000
). [CrossRef]
].
We consider only symmetric modes. The field in the cladding can be written on the Bloch form
where
E_{K}
(
y) is periodic with period Λ, and
E_{K}
(
y) and
K can be found using the transfer-matrix method [
12
12
.
P.
Yeh
,
A.
Yariv
, and
C.-S.
Hong
, “
Electromagnetic propagation in periodic stratified media. I. General theory
,”
J. Opt. Soc. Am.
67
,
423
–
438
(
1977
). [CrossRef]
].
K is the Bloch wavenumber, which is real outside the band gaps, while it can be written
in the band gaps. The imaginary part, K^{i}
, of K results in an evanescent cladding field in the bandgaps. In the core, the field can be written
where
k1=(ωc)2−β2. By matching the cladding and core fields and their derivatives at the core-cladding boundary, one obtains the dispersion relation for symmetric guided modes. A necessary condition for obtaining a guided mode in the air core is that the corresponding
β and
ω results in a complex
K. An example of the mode field for the fundamental guided mode at a frequency
ωΛ/
c = 35 is shown in
Fig. 6(b). The mode index for the fundamental guided mode as a function of frequency is shown in
Fig. 7(a), together with the band edges of the first bandgap. By proceeding in the same manner as for the index-guiding waveguide, we use the Kramers-Kronig relation Eq. (
39) to calculate
dn0r/
dω for the fundamental mode in the Bragg reflection waveguide. The result is shown in
Fig. 7(b). There is good agreement between results found by solving the wave equation and results found using the Kramers-Kronig approach.
Fig. 6. The refractive index profile of a symmetric Bragg reflection waveguide is shown in (a). The fundamental guided mode for ωΛ/c = 35 is shown in (b).
Fig. 7. The mode index n0r of the fundamental guided mode as a function of frequency is shown in (a) together with the band edges of the first bandgap. dn_{r}
_{0}/dω is shown in (b), based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.
It is to be noted that in general for Bragg reflection waveguides, there exists a discrete set of localized evanescent modes. These modes are localized in the core region due to the bandgap, and have a pure imaginary propagation constant. They must in general be taken into account in the sum over evanescent modes in Eq. (
23), but do not contribute for the parameters chosen in the example above.