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.

It is well known that material absorption and dispersion are connected by the Kramers-Kronig relations. This fact can be viewed as a result of microscopic causality; the medium polarization cannot precede the electric force [1–3]. Alternatively, the Kramers-Kronig relations can be viewed as results of relativistic causality; no signal can propagate faster than the vacuum velocity of light [4]. The Kramers-Kronig relations have long constituted fundamental tools in the investigation of linear optical properties of condensed matter, gasses, molecules, and liquids, as well as finding applications within high-energy physics, acoustics, statistical physics, and signal processing [5].

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, 7], 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]. 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 (Ω₁), and one range close to material resonances, where completeness is not guaranteed to hold in general (Ω₂). Each of the intervals Ω₁ and Ω₂ do not have to be connected. As an example, let Ω₁ = [0, ω ₁) ⋃ (ω ₂,∞) and Ω₂ = [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 Ω₁, 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 ∈ Ω₁, 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 ∈ Ω₂, 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], 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 ∈ Ω₁.

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 ∈ Ω₁ 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.

The inverse Fourier transform of Eq. (6) is

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]. 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 ∈ Ω₁. 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 Ω₁ 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 - ${\omega }_p^2$/(2ω ²) in this limit, where ω_p is the plasma frequency of the medium [9]. This ensures that the high frequency region belongs to Ω₁, 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 : ${\int }_0^{\propto }$|〈ψ ₀(ψ ₀)|ψ_kt(ω)〉|² exp{i[n_kt(ω) - 1]ωL/c}dk_t , where n_kt = (1 - c ² $k_t^2$ /ω ²)^1/2 [8]. If |〈ψ ₀(ω ₀)|ψ_kt(ω)〉|² → 0 sufficiently fast to ensure convergence of ${\int }_0^{\infty }$ |〈ψ ₀(ω ₀)|ψ_kt(ω)〉|²dk_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 ∈ Ω₁, 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 Ω₁.) We define $n_j^r$ (ω) ≡ Re n_j (ω), $n_j^i$ (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 Ω₂, allowing us to drop the principal value in the integral over Ω₂. Defining

we can rewrite Eq. (15) as

where a common factor ω has been canceled. Eq. (20) is not so useful in this form in general, due to the term δ. By taking the derivative of this equation with respect to ω, evaluated at ω = ω ₀, and remembering that c_j (ω ₀, ω ₀) = 0 for all j except for j = 0 where c ₀(ω ₀, ω ₀) = 1, we obtain

In obtaining Eq. (21), we have made use of the fact that c_j (ω ₀, ω′) has a zero of more than first order at ω′ = ω ₀ for the contributing modes, meaning that the point ω′ = ω ₀ does not lie in the region of integration. Eq. (21) expresses that the variation in the mode index has two contributions: One term related to the evanescent modes in Ω₁ (waveguide dispersion term), and one term related to the material absorption in Ω₂.

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 ∉ Ω₂, 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 Ω₁ = [0,ω ₁) ⋃ (ω ₂, ∞) and Ω₂ = [ω ₁, ω ₂], where 0 < ω ₁ < ω ₂ < ∞. If the maximum mode index for the contributing modes in Ω₁ is, say, 10, we must choose L ~ c/(10ω ₁) so that the expansion in Eq. (14) is valid. Assuming ω≪ω ₁ we obtain |∂δ/∂ω| ≲ 8ω/${\omega }_1^2$.

Thus, assuming negligible material dispersion and loss in the frequency range of interest, we have

This equation is a more useful form of the Kramers-Kronig relation for waveguides, since the term δ (or ∂δ/∂ω) is not present. Eq. (23) shows that the change in mode index with frequency for a given, bound mode at a given frequency is determined by the properties of the evanescent modes, through the overlap c_j .

If the material absorption is not negligible in the frequency range of interest, there are still cases where the waveguide modes are known to form a complete set [6,7]. In such cases, we can set Ω₁ = [0, ∞) and ignore Ω₂. Setting ω = ω ₀ in Eq. (15) we obtain

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 Ω₁. 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]. 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

Eqs. (15)–(16) then reduce to

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

The fact that Kramers-Kronig relations exist for the modes of a hollow metallic waveguide might appear a little surprising. However, below cutoff, the mode index is imaginary, and the imaginary values in this region are related to the real values of the mode index above cutoff. Thus the evanescent damping below cutoff can be determined from a dispersion measurement above cutoff, and vice versa. In Fig. 1 we plot the real and imaginary values of n_j (ω), as well as d$n_j^r$ dω, and demonstrate Eqs. (30), (31), and (32) numerically.

Use of Eqs. (30) and (31) is however not limited to hollow metallic waveguides. As an example, we consider a dielectric-filled metallic waveguide. In this case the mode fields are exactly the same as for the hollow metallic waveguide, so we can use Eqs. (30)–(31) directly. The relative permittivity ε_r of the fill material is assumed to be described by a one-resonance Lorentz model

where ω_p = 2ω _j,c, ω _res = 3ω _j,c, and γ= ω _j,c. Solving the wave equation, the real and imaginary part of the mode indices can be found exactly in this case. Figure 2 shows the result when determining the real part of the mode index from the imaginary part, and vice versa, using Eqs. (30) and (31). The results from the Kramers-Kronig calculations are seen to be in excellent agreement with the results obtained from the exact solution of the wave equation.

 

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.

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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.

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2.2. An index-guiding waveguide

Consider a planar, dielectric, index-guiding waveguide. For simplicity, the refractive index of the core is taken to be 1.01, and the loss is assumed negligible in the frequency range of interest. The cladding is air (with refractive index equal to unity). The constant refractive index 1.01 refers to the observation frequencies; on the other hand material dispersion and loss are necessarily present for larger frequencies. Here we will just assume that the refractive index has a physical behavior for large frequencies; the detailed form is not relevant, as shown above. The core thickness is denoted d. Only TE modes are considered. The mode ψ ₀ in Eq. (11) is first taken to be the fundamental mode. An example of the field of the fundamental mode is shown in Fig. 3(a) for ωd/c = 40.

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 [8,11]. 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).

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For evanescent modes, we have $n_j^i$ = c|β_j |/ω. This means that

where we have used Eq. (35) to express $n_j^i$ 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) d$n_0^r$/dω is shown, based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.

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Figure 4 shows the result of using this formula to calculate d$n_0^r$/dω. The exact mode index, found by solving the wave equation, is shown in Fig. 4(a). The derivative of the exact mode index with respect to frequency is shown in Fig. 4(b), together with results found using Eq. (39). The results agree, with a maximum discrepancy of 6%. The discrepancy is expected to be due to the free space approximation for the evanescent modes as well as the numerical resolution in the calculations. In a similar manner, Eq. (39) is used to calculate the derivative of the mode index with respect to frequency for the second order symmetric mode in the planar waveguide. The results are shown in Fig. 5.

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) d$n_2^r$/dω is shown, based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.

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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]. 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].

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]. K is the Bloch wavenumber, which is real outside the band gaps, while it can be written

in the band gaps. The imaginary part, Kⁱ , of K results in an evanescent cladding field in the bandgaps. In the core, the field can be written

where $k_1=\sqrt{{\left(\frac{\omega }{c}\right)}^2-{\beta }^2}$. 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 ${\mathit{\text{dn}}}_0^r$/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).

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Fig. 7. The mode index $n_0^r$ 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.

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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.

3. Discussion

In the examples above, the waveguide dispersion of an effectively lossless guided mode is obviously not caused by its loss. Instead, for the dielectric waveguides, the imaginary mode indices associated with the evanescent modes enter into the Kramers-Kronig relation Eq. (23). In the limit of small index contrast (weakly guiding waveguides), the evanescent modes are independent of the waveguide structure. As a result, the dispersion of a certain guided mode is determined solely by its mode field as a function of frequency. This might find practical applications. Note that the first order derivative of the mode index with respect to frequency is only dependent on the mode field at a single frequency.

We say that a mode propagates causally if it can be excited (and detected) separately and causally. By this we mean that the mode can be excited separately at z = 0 by a field which is zero for all points in the xy-plane for t < 0. This is manifestly satisfied in Eqs. (4)–(5). Assuming that a mode with a frequency-dependent mode profile propagates causally, we could choose a frequency-dependent ψ _det = ψ _src that matches the mode in question for all frequencies. This would lead to a transfer function G(ω) on the form Eq. (25), implying that Eqs. (30)–(31) would be valid for this mode. This shows that if a mode propagates causally, its mode index must satisfy the usual Kramers-Kronig relations Eqs. (30)–(31). On the other hand, if a mode does not satisfy Eqs. (30)–(31), it cannot be excited separately and causally at z = 0. This means that a band of radiation modes, as well as bound modes, are excited as well. The interference of all these modes at all frequencies gives the causal signal at z = L, delayed by L/c compared to the excitation. For example, for the index-guiding waveguide in Section 2.2, one finds that the fundamental mode does not propagate causally: Using Eq. (30) to determinate the real part of the mode index from the imaginary part, which is zero at observation frequencies, leads erroneously to a constant $n_0^r$ at observation frequencies.

We observe from Eq. (23) that a frequency-dependent mode profile is necessary to obtain a nontrivial dispersion for modes which are effectively lossless at observation frequencies. This shows that it is impossible to obtain dispersion tailoring over a frequency band without sacrificing coupling efficiency.

4. Conclusion

We have derived Kramers-Kronig relations for the effective index of modes in optical waveguides. When material dispersion and absorption can be neglected in the frequency range of interest, the evanescent modes enter as an effective loss term in the Kramers-Kronig relations, meaning that Kramers-Kronig relations exist even if material absorption is negligible in the frequency range of interest. The theory is successfully applied to metallic waveguides, index-guiding waveguides, and bandgap-guiding waveguides. The theory may give a deeper understanding of the origin of waveguide dispersion, and its connection to causality.