1. Introduction
The paper is structured as follows: In Section 2 we describe the system model and introduce some definitions required for the proper description of the output field’s quality. In Section 3 we outline the numerical method employed. Section 4 is dedicated to the discussions of the numerical results of slab and fiber waveguide. Various input coupling situations and the effect of an input pupil positioned just in front of the waveguide’s input facet are investigated. In Section 5 we compare the results obtained for the slab waveguide with those for the fiber waveguide, while Section 6 demonstrates the excellent agreement between numerical results and analytical solutions for some special cases. Finally, in Section 7 we present a summary and conclusions.
2. Waveguide modeling and definitions of characteristics
Fig. 1. System model. P_{in} .. input power, 2w .. beam width (defined by first zeros in the focal plane), 2a .. core width, D_{A} .. input pupil width, D_{B} .. output pupil width, D_{C} .. near-core power width, p(z) .. near-core power, E_{W} .. waveguide field just before output facet, E_{f} .. field of fundamental mode. The constructive and destructive superposition of field E_{w} with the fundamental mode E_{f} leading to output power P_{+ }and P- serves only to illustrate the definition of mode purity MP. The right part shows the cross section in case of a fiber waveguide.
Table 1. Waveguide parameters used for the numerical calculations (Δ .. relative index difference, V .. normalized frequency, VC .. cut-off frequency). Also given is the input pupil width DA (if not explicitly set to infinity) and the output pupil width DB. |
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An ideal lens focuses a truncated plane wave to a spot in the focal plane. This plane may be offset from the input plane by a distance Δz (“defocus”). (A positive Δz means that the focus is within the waveguide). In the two-dimensional case, i.e. the slab waveguide, the input field distribution is proportional to
i.e., a sinc distribution with the first zeros at |x’|=w. Thus the spot width of the input beam is 2w. A tilt represented by the angle ε≠0 in the field distribution is expressed by using a slightly rotated coordinate system with x’ instead of x. Since differences in the field distributions for the two possible polarization states TE (electric field polarized orthogonal to the plane of incidence) and TM (magnetic field polarized orthogonal to the plane of incidence) are rather small, we will, in the following, only consider the case of TE polarization, i.e. the input field is polarized along the y-direction.
In the three-dimensional case, i.e. the fiber waveguide, the incident radiation is again assumed to be focused by a lens. However, here we will only consider the case of normal incidence (ε=0). The input distribution is thus proportional to
i.e., an Airy distribution whose first zero is at r=w. In the fiber case, the quantity 2w denotes the spot diameter of the input beam. J_{1} represents the Bessel function of the first kind, order one.
For the characterization of the usefulness of a single-mode waveguide as a modal filter we introduce the following three quantities:
2.1. Near-core power
The near-core power p(z) represents the fraction P_{DC} of the total power in the waveguide that is found within a width D_{C} centered around the core, normalized to the input power P_{in}, i.e.
D_{C} should be large enough to contain essentially all the power of the steady-state fundamental mode. Our choice of D_{C}=8a=18 λ_{0} provides more than 99.99% of the mode’s power in PDC for z→∞ for an operation at λ=λ_{0}. Even for a wavelength λ=1.5 λ_{0}, more than 99.3% of the power of the fundamental mode is included. Diagrams presenting the function p(z) thus illustrate how fast the field in the waveguide approaches the fundamental mode.
In case of the fiber waveguide the calculation of power deserves special attention: Even if the plane wave incident on the lens is linearly polarized, say along the y-direction, as assumed for the numerical calculations, the field in the focal plane will also contain components - though small - in z and x direction, otherwise Maxwell’s equations would not be satisfied. These components (E
_{z}, E
_{x}) are not Airy-distributed. The above defined quantities P
_{DC}, P
_{in} - and hence also p(z) - contain all three field components. Along this line, the exact solution for the fundamental mode of a step index fiber does always contain all three field components [
88. A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman & Hall, 1983), p. 259 ff.
], [
99. G. Grau and W. Freude, Optische Nachrichtentechnik (Springer, 1991), pp. 51.
]. For the fiber parameters given in
Table 1, the maximal E
_{z} component is by a factor of 25 smaller than that of the dominating E
_{y} component, and the maximum amplitude of E
_{x} is smaller by another factor of 25. The numerical method we apply for calculating the field in the focal plane and in the fiber (see Section 3) does automatically yield the field components E
_{z} and E
_{x}.
2 2. Coupling efficiency
We define the coupling efficiency η as the ratio of the power carried by the waveguide’s fundamental mode to the input power P_{in}. However, as the mode extends to infinity in transverse direction, the numerical determination requires a slightly modified definition. We maintain sufficient accuracy when taking - instead of the total power of the fundamental mode - the power P_{DC} for D_{C}=18 λ_{0} in its limit for z→∞, designated P_{DC, z→∞}. The coupling efficiency is thus given as
2.3. Mode purity
So far we introduced quantities which characterize only the amount of light coupled into the fundamental mode. In contrast, the following definition of mode purity MP(L), with L as the waveguide length, offers a measure of how well the actual amplitude and phase distribution E
_{w}(x)=A
_{w}(x)exp(jφ
_{w}(x)) found just before the output pupil resemble the ideal distribution of the fundamental mode E
_{f}(x)=A
_{f}(x)exp(jφ
_{f}). In case of the slab, the quantities E
_{w}, A
_{w}, φ
_{w} and E
_{f}, A
_{f}, are functions of x, while for the fiber they depend on the coordinates r and φ. We define mode purity by means of output powers P+ and P-, as indicated in the right part of
Fig. 1. The first one, P
_{+}, is the power resulting from an in-phase superposition of the field E
_{w} with the ideal field E
_{f}, where for E
_{f} we will take the numerical solution of the steady-state fundamental mode. The second one is due to an out-of-phase superposition. For determining P
_{+} and P- we set - at the waveguide axis - the field amplitude Af equal to Aw. In the same way, the phase relationship for calculating constructive and destructive interference, i.e. φ
_{w} - φ
_{f}=0° and 180°, is defined at the waveguide axis. It turned out that the function MP(L) is almost independent on whether or not one includes the components E
_{z} and E
_{x} in the calculation of the three-dimensional case. To avoid unnecessary complexity, we based the calculation of mode purity MP of the fiber only on the amplitude and phase of the dominating y-components.
The powers P_{+} and P- depend on the pupil width D_{B} and are related to the corresponding field intensities I_{+} and I- by
for the slab and the fiber, respectively. For the slab the intensities I+ and I- are in turn given by the electric fields as
and correspondingly for the fiber. The mode purity MP then follows as
The quantity MP tends to infinity in the case where the field distribution E_{w} approaches that of the fundamental mode, E_{f}. In general, a smaller value of output pupil diameter D_{B} will result in a higher value of MP, however at the cost of reduced overall optical throughput.
3. Numerical method
which defines the m^{th} (eigen)mode for this specific segment. Mathematically, S_{m}(x,y) and b_{m} represent the eigenfunction and eigenvalue of the solution. The simple harmonic zdependence of the mode is the key to find solutions for long structures as efficiently as for short structures. The total field distribution in each segment is described via the complete basic set of modes which, in case of a waveguide structure, consists in general of a few guided modes and an infinite number of radiation modes. Fortunately, most of the higher-order radiation modes die off quickly with increasing propagation distance z and can therefore be neglected without sacrificing accuracy. For easy numerical handling, each mode is furthermore represented mathematically by a Fourier series. The x- and y-dependent material parameters are described by Fourier series as well. Each segment represents thus an autonomous eigenvalue problem whose solutions are the modal fields that are specific to the segment’s geometry and material parameters. The complete set of modes in segment l is described by
where A^{1}
_{m} and B^{1}
_{m} stand for the mode amplitudes of forward and backward propagating waves, and K=2π/Λ is the fundamental spatial frequency with Λ being the structure’s period in transverse direction. The 2M+1 eigenvectors summarized in S_{pqm} and the eigenvalues b_{m} represent now the complete set of solutions to the eigenvalue problem discussed above.
Although the use of Fourier series makes the system periodic in transverse direction, we are still able to analyze single aperiodic elements such as waveguides by introducing absorbing, perfectly matched layers along the rim of the calculation area parallel to the z-direction. The absorption and thickness of these layers must be chosen such that field contamination from neighboring waveguide structures due to the inherent periodicity is kept to a minimum. The condition of perfect matching with respect to the material parameters is needed for suppressing reflections from these additional layers [
1010. E. Silbersteinet al., “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]
].
4. Field behavior for various waveguide geometries and input coupling situations
In this section we present numerical results both for the slab and fiber single-mode waveguide:
First, two-dimensional color coded intensity distributions for various input coupling situations give a physical insight into how the incident free-space beam is either converted into the waveguide’s fundamental mode or distributed across the cladding in form of radiation modes. In accordance with
Fig. 1, the input radiation enters the waveguide from the left, passes through a transient region whose length mainly depends on the coupling situation, and eventually approaches the fundamental mode. To visualize the field behavior within a large dynamic range, the intensity distributions are normalized with respect to their maximum values and shown on a logarithmic scale from 0 to -40 dB. The waveguide’s core region appears therefore as a red horizontal bar, as this is the location of highest intensity. In these figures we have indicated the waveguide input facet and the core-cladding boundaries by thin black vertical and horizontal lines.
Second, more quantitative information about coupling efficiency and power variation within a cross section of width D_{C}=18 λ_{0} in core vicinity can be read off from graphs showing the near-core power p(z) along the z-direction.
Finally, the waveguide’s filter quality is estimated with the help of graphs showing the mode purity MP as a function of waveguide length L.
4.1 The slab waveguide
Fig. 2. Slab waveguide operated at λ=λ_{0}: Intensity distributions without input pupil for a beam width of 2w=11 λ_{0} and perfect alignment (Δz=0, ε=0, Δx=0). (a) extension in x-direction is 500 λ_{0}, (b) extension in x-direction is 100 λ_{0}.
When comparing
Figs. 2(b) and
3(a) it is evident that a pupil with a properly chosen width is capable of suppressing the radiation modes to some extent. We conclude that for waveguides without input pupil the number of radiation modes excited does not only increase with increasing mismatch between the sinc distribution’s central lobe and the waveguide’s fundamental mode but is also caused by the sinc distribution’s sidelobes, half of which are out-of-phase with the main lobe
Figure 4 shows how the near-core power p(z) gradually approaches the steady state for the five cases discussed above. The values on the right represent the coupling efficiencies η. In case of perfect alignment, these near-core power graphs confirm the positive effect of an input pupil in a twofold way: First, coupling efficiency is increased by some 2%, and second, the field approaches the steady state within a shorter distance.
Fig. 3. Slab waveguide operated at λ=λ_{0}: Intensity distributions with input pupil (D_{A}=11 λ_{0}) for a beam width of 2w=11 λ_{0}. (a) perfect alignment (Δz=0, ε=0, Δx=0) (b) defocus Δz=50 λ_{0} (c) angular misalignment ε=1° (d) lateral misalignment Δx=1 λ_{0}.
Fig. 4. Slab waveguide: Near-core power p(z) as a function of distance z from the input facet for the cases of
Fig. 2 and
Figs. 3(a) to
3(d). The numbers at the right give the coupling efficiencies η.
Fig. 5. Slab waveguide operated at λ=λ_{0}: Mode purity MP as a function of waveguide length L for a beam width of 2w=11 λ_{0} and for input pupil diameters of D_{A}=11 λ_{0} and D_{A}=100 λ_{0}. The latter case is practically identical to the case without input pupil (D_{A}=∞).
Fig. 6. Slab waveguide: Mode purity MP as a function of waveguide length L for the four cases of alignment specified in
Fig. 3. (a) for operation at λ=λ
_{0}, (b) for operation at λ=1.5 λ
_{0}.
4.2. Field behavior of the slab waveguide with core inhomogeneities
Next we investigated the effect of small variations of the core diameter and core refractive index on mode purity and near-core power p(z). To this end, we divided the waveguide into three large sections. The first and third section consist of a homogeneous waveguide region equal to the one discussed in Section 4.1. The middle section is subdivided into 400 small waveguide segments whose lengths vary randomly between 3 λ
_{0} and 7 λ
_{0}. Furthermore, each segment contains a core whose refractive index varies randomly between 1.503 and 1.50375 and whose width varies randomly between 4 λ
_{0} and 4.5 λ
_{0} (see
Fig. 7). The total length of the inhomogeneous section is approximately 2000 λ
_{0}.
Fig. 7. Schematic of inhomogeneous waveguide.
The mode purity as shown in
Fig. 8 was evaluated with respect to the fundamental mode in the homogeneous (first or third) region as a function of waveguide length L. Shortly after the onset of the inhomogeneous region the mode purity collapses to values below 10
^{4} and stays there as long as inhomogeneities are present. Recovery to higher values is only possible after the field enters again the homogeneous region.
Fig. 8. Slab waveguide operated at λ=λ_{0}: Mode purity as a function of L for a waveguide with (blue) and without (red) an inhomogeneous core with a length of about 2000 λ_{0}, starting at z=2000 λ_{0}. The input beam parameters are 2w=11 λ_{0} and perfect alignment.
While the effect of inhomogeneities on the mode purity is quite dramatic, the loss in near-core power remains small (of the order of 1%), as shown in
Fig. 9. Clearly, reflections from the inhomogeneous section (2000 λ
_{0}<z<4000 λ
_{0}) affect the field in the first homogeneous region (0<z<2000 λ
_{0}). The seemingly long period of some 200 λ
_{0} of the excited standing wave in this region is a numerical artifact caused by undersampling (one sample per 2 λ
_{0}) of the function p(z) in direction of increasing distance z and cannot be directly related to the waveguide geometry or material parameters.
Fig. 9. Slab waveguide operated at λ=λ_{0}: Near-core power p(z) as a function of distance z for a waveguide with (blue) and without (red) inhomogeneities. The input beam parameters are 2w=11 λ_{0} and perfect alignment. The values at the right of the diagram represent the coupling efficiencies η to the fundamental mode in the third section.
4.3. Fiber waveguide
Fig. 10. Fiber operated at λ=λ_{0}: Intensity distributions without input pupil, input beam diameter 2w=8.8 λ_{0}. (a) perfect alignment (b) defocus Δz=50 λ_{0},.
Fig. 11. Fiber operated at λ=λ_{0}: Intensity distributions with input pupil, input beam diameter 2w=8.8 λ_{0}. (a) perfect alignment (b) defocus Δz=50 λ_{0},.
Fig. 12. Fiber operated at λ=λ_{0}: Near-core power p(z) as a function of distance z for 2w=8.8 λ_{0}. The numbers at the right give the coupling efficiencies η. Dotted red line: perfect alignment without input pupil (D_{A}=∞) Solid red line: perfect alignment with input pupil (D_{A}=8.8 λ_{0}) Blue line:defocus Δz=50 λ_{0},, with input pupil (D_{A}=8.8 λ_{0}).
Fig. 13. Fiber operated at λ=λ_{0} with input beam diameter 2w=8.8 λ_{0}: Mode purity MP as a function of waveguide length L for input pupil diameters of D_{A}=8.8 λ_{0} and D_{A}=100 λ_{0}. The latter case is practically identical to the case without input pupil (D_{A}=∞).
Fig. 14. Fiber: Mode purity MP as a function of waveguide length L for perfect alignment (red line) and for a defocus of Δz=50 λ_{0}, (blue line) when operated at λ=λ_{0}. The dotted lines give MP for operation at λ=1.5 λ_{0} (green line: perfect alignment, magenta line: defocus of Δz=50 λ_{0},).
5. Comparison of fields in slab and fiber waveguide
Fig. 15. Comparison of near-core power p(z) for slab (dotted lines) and fiber (solid lines) with input pupils, operated at λ=λ_{0}. The red lines are for perfect alignment, the blue lines for a defocus of Δz=50 λ_{0}.
Fig. 16. Comparison of mode purity MP for slab (dotted lines) and fiber (solid lines) with input pupils, operated at λ=λ_{0}. The red lines are for perfect alignment, the blue lines for a defocus of Δz=50 λ_{0}.
6. Comparison of numerical results and analytical solutions
Table 2: . Comparison of results (perfect alignment). |
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1) including truncation by output aperture of width D_{B}=18 λ_{0}
2) here truncation is due to the definition of “near-core power”, i.e. by the width D_{C}
7. Summary and conclusion
When the field at the input facet of a single-mode waveguide does not perfectly match the one of the waveguide’s fundamental mode, the actual formation of the fundamental mode is a process which may take place over a length of few hundreds to a few thousands of wavelengths. Using the Fourier Modal Method we have calculated the transient fields after the input facets of both slab and fiber step-index waveguides. While color-coded diagrams showing the intensity distribution in the meridional plane give a good qualitative picture of how the unguided power is radiated off, the near-core power is a quantitative measure of the development of the fundamental mode along the waveguide axis. Lastly, our definition of mode purity allows us to determine to a very high degree the identity of the actual waveguide field with the ideal mode. In this context, the definition of an exit pupil turned out to be essential.
For the numerically evaluated examples we have assumed parameters leading to a fundamental mode with a normalized frequency a few percent below the next mode’s cut-off. We showed that the establishment of the fundamental mode strongly depends on both the distribution of the input field and any misalignment of the input wave. Mode purity is very sensitive to even small misalignments of input field and to any variations in waveguide geometry. For the typical input field distribution containing side lobes, the placement of a pupil just in front of the waveguide’s facet is advantageous when striving for single-mode purity in short waveguide lengths. Such a pupil could be realized by evaporating a metal film onto the input facet of the slab waveguide or fiber. The pupil helps to remove unwanted field components from the input field that may interfere with the fundamental mode. It also somewhat relaxes the dependence of mode purity on misalignments. However, it lowers the coupling efficiency for larger wavelengths in a broadband application.
The treatment of the three-dimensional case (i.e. the fiber) is computationally much more extensive than that of the two-dimensional case (i.e. the slab). However, we found that fiber waveguides do not exhibit significantly different behavior in the characteristics of near-core power and mode purity when compared to their two-dimensional counterparts. Thus an insight into the performance of the fiber can indeed be obtained by first studying the equivalent slab problem.