BACKGROUND OF THE INVENTION

[0001]
Power splitters/combiners are essential components in optical systems, and especially integrated, planar waveguide systems. They are important where coherent signal splitting, signal combining, or both signal splitting and combining are required. Splitters are typically used to split a single optical signal into two signals. Combiners are typically used to combine two signals into a single signal. Splitter/combiner pairs are often used to control power flow through optical elements. Splitters and combiners can also be configured as multifunctional splitter/combiners, where the operation performed depends upon the direction of the input signals and the number of inputs provided.

[0002]
A frequently used splitter/combiner is composed of two waveguides that are sidecoupled to each other, forming a directional coupler. Light traveling in one waveguide couples to the other waveguide so that splitting or combining can occur. For an input signal directed into a first waveguide of the directional coupler, the power splitting ratio is defined as the ratio of the percentage of power at the output of the first waveguide to the percentage of power at the output of the second waveguide. The power splitting ratio, P, is related to the coupling value, φ, as

P=cos^{2 }(φ)/sin^{2}(φ) (1)

[0003]
where φ is the product of the coupling coefficient, μ, and the coupling length, L, of the device.

[0004]
To achieve a desired power splitting ratio, the coupling value for the coupler has to be exact. For example, if the coupling value of the coupler in a 50/50 splitter is smaller than the ideal value by 10%, the power splitting ratio becomes 57.8/42.2; and if the coupling value is smaller by 20%, the power splitting ratio becomes 65.5/34.5.

[0005]
The performance of directional couplers is inherently sensitive to fabrication errors. The coupling value is a function of physical parameters, such as waveguide crosssection, effective index, and waveguidetowaveguide separation. Fabrication errors can also lead to undesired dependencies between the coupling value and the frequency and polarizations of the input optical signals, on otherwise nominally spectrally flat, polarization insensitive designs.

[0006]
In order to improve performance, splitters/combiners have been proposed that include cascaded splitters and phase shifters. These systems are typically optimized for wavelength insensitivity, and sometimes robustness against uniform fabrication errors.
SUMMARY OF THE INVENTION

[0007]
Previously proposed power splitter/combiner systems with phase shifts are not adequately robust against errors for many fabrication processes. This is especially true for planar waveguide fabrication in high index contrast material systems. Lithographic patterning and material deposition processes used to control waveguide crosssections, effective indices, waveguide lengths, and waveguidetowaveguide separations, for example, do not always provide enough control to fabricate commercially relevant systems with adequate yields.

[0008]
The invention is directed to a splitter/combiner that is tolerant of parameter deviations. It is specifically applicable to integration into the transmission and/or reflection light paths of external systems. In this power splitter/combiner system, each splitter/combiner is comprised of two or more directional couplers serially connected to two or more phase shifts in an alternating order (e.g., directional coupler, phase shift, directional coupler, phase shift, directional coupler).

[0009]
The invention addresses the problem of parameter deviations in splitter/combiners by selecting specific coupling and phase values for the directional couplers and phase shifts to minimize changes in an output signal in response to changes in the coupling coefficients and the propagation constants, for example. Parameter deviations, such as coupling value deviations and phase value deviations from the nominal design due to fabrication errors, for example, typically give rise to frequency dependency, polarization dependency in the output signal and are specifically addressed in the invention by designing the system so that the sensitivity of the output signal to these changes is minimized.

[0010]
Additionally, the invention can be applied to address the specific problem of matched deviations in coupling values and phase values in a splitter/combiner system. The invention also addresses the problem of differential deviations in the phase values of the phase shifts by providing tunable phase shifts, employing controlled effects, such as electrooptic or thermooptic refractive index changes.

[0011]
In an exemplary embodiment of the invention, two splitter/combiners are coupled to an external system and engineered for desired reflection and transmission spectra. The splitter/combiner is designed for maximal flatness of the spectrum with respect to the deviations in the parameters in one example. Other methods, such as designing for equalripple of the spectrum for the desired ranges of the values for the parameters, can also be implemented. In another exemplary embodiment, two splitter/combiners are coupled to two external systems that have nominally identical reflection and transmission spectra.

[0012]
The splitter/combiner system of the invention can also be designed for a wide range of ratios for splitting and/or combining signals. The invention can be used for both monodirectional or bidirectional splitting and combining.

[0013]
In general, according to one aspect, the invention features a splitter/combiner system that comprises a serial connection of at least two couplers and at least two phase shifts. The coupling coefficients of the couplers and propagation constants of the phase shifts are selected to minimize changes in an output signal in response to changes in the coupling coefficients and the propagation constants.

[0014]
In general, according to another aspect, the invention can be characterized as a design process for a splitter/combiner system, in which a serial connection of at least two couplers and at least two phase shifts are provisioned to connect an input waveguide to an external system. Coupling coefficients of the couplers and propagation constants of the phase shifts are proscribed to minimize changes in an output signal in response to changes in the coupling coefficients and the propagation constants.

[0015]
The above and other features of the invention including various novel details of construction and combinations of parts, and other advantages, will now be more particularly described with reference to the accompanying drawings and pointed out in the claims. It will be understood that the particular method and device embodying the invention are shown by way of illustration and not as a limitation of the invention. The principles and features of this invention may be employed in various and numerous embodiments without departing from the scope of the invention.
BRIEF DESCRIPTION OF THE DRAWINGS

[0016]
In the accompanying drawings, reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale; emphasis has instead been placed upon illustrating the principles of the invention.

[0017]
[0017]FIG. 1 is a block diagram of a splitter/combiner coupled to an external system having a reflection and transmission spectrum, in accordance with the invention;

[0018]
[0018]FIG. 2 is a plan view of the generalized configuration of the directional couplers and phase shifts in accordance with the invention;

[0019]
[0019]FIG. 3 is a plan view of a splitter/combiner system coupled to a transmissive external optical system, according to a first embodiment of the invention;

[0020]
[0020]FIG. 4 is a plan view illustrating the layout of the waveguides in the first embodiment splitter/combiner system;

[0021]
[0021]FIG. 5 is a plot of the power of output signal C as a function of the coupling coefficient, showing a region in which changes in the coupling coefficient yields only small changes in the output signal;

[0022]
[0022]FIG. 6 is a plan view of a splitter/combiner coupled to a reflective external optical system in accordance with a second embodiment of the invention;

[0023]
[0023]FIG. 7 is a plan view of two exemplary splitter/combiners coupled an external system, in accordance with a third embodiment of the invention;

[0024]
[0024]FIG. 8 is a block diagram of a splitter/combiner system coupled to two external optical systems (subelements) having the same reflection and transmission spectra, in accordance with a fourth embodiment of the invention;

[0025]
[0025]FIGS. 9A and 9B are schematic block diagrams illustrating the design process for the fourth embodiment in view of the reflection and transmission characteristics of the external system;

[0026]
[0026]FIG. 10 is a plan view of a simplified splitter/combiner configuration for reflection/transmission spectra in accordance with the invention;

[0027]
[0027]FIG. 11 is a plan view of an alternate splitter/combiner configuration for reflection and transmission spectra in accordance with the invention; and

[0028]
[0028]FIG. 12 is a plan view illustrating the layout of the waveguides in still another embodiment of the splitter/combiner system.
DETAILED DESCRIPTION OF THE INVENTION

[0029]
This invention describes a splitter/combiner system that, when coupled to an external system having a reflection and/or transmission response, has a high level of tolerance to parameter deviations for the desired spectra of the output signal.

[0030]
[0030]FIG. 1 illustrates the typical application for the invention. FIG. 1 is a block diagram of one or more of splitter/combiners 100 are coupled to an external system 10 that has reflection (R(ω)) and transmission (T(ω)) spectra.

[0031]
The splitter/combiners 100 are monolithically fabricated from planar waveguides in a high index contrast material system in the current implementation. That is, the material system provides an index contrast between the refractive indices of the waveguides and the cladding that is greater than 1%, or preferably a higher contrast of greater than 2%. Presently, a silicon oxynitride system is used in which the refractive index of the waveguides is 1.60 and the refractive index of the cladding layers is about 1.44. Thus, Δn/n_{cladding }is greater than about 10%.

[0032]
The external system 10 is usually comprised of one or more devices, such as, but not limited to, waveguides, filters, and amplifiers. The splitter/combiner system can include one or more splitter/combiners 100 that are coupled to the external system 10. As a result of increasing the tolerance of the splitter/combiner systems 100, the performance and/or yield of the entire system is improved.

[0033]
[0033]FIG. 2 is a plan view of the generalized configuration of the directional couplers and phase shifts in accordance with the invention. FIG. 2 shows the general design of the splitter/combiners 100 of FIG. 1. Each splitter/combiner 100 is preferably comprised of two or more directional couplers 110 and two or more phase shifts 112.

[0034]
Arbitrarily combining directional couplers and phase shifts in a sequential manner will generally not improve tolerance, however. Instead, the coupling value, φ, of each directional coupler 110 and the phase value, θ, of each phase shift 112 must be chosen carefully so that the output signals of interest have a desired power splitting ratio and a high tolerance to a particular type, or types, of parameter deviation.

[0035]
Various mathematical criteria are used in designing a high parameter tolerance for the spectra of interest, such as designing for maximal flatness of the spectrum with respect to the changes in the parameters and designing for equalripple of the spectrum for the desired ranges of the values for the parameters. In designing for maximal flatness, the derivatives of the signal of interest with respect to the parameters of interest are engineered to be preferably negligible. In designing for equal ripple, the deviation of the spectrum of interest, from the ideal spectrum, is engineered to be within the desired tolerance for the desired parameter range. Combinations of spectra of interest, such as the reflection spectrum, the transmission spectrum, or simultaneously both the reflection and transmission spectra, can be designed for high parameter tolerance.

[0036]
[0036]FIG. 3 is a plan view of a splitter/combiner system coupled to a transmissive external optical system in accordance with a first exemplary embodiment of the invention. The splitter/combiner system includes one splitter/combiner 100 providing a high parameter tolerance transmission spectrum at a desired power splitting ratio.

[0037]
The exemplary splitter/combiner 100 includes three directional couplers 110 and two phase shifts 112, which are serially connected to each other in alternating order. The splitter/combiner system is coupled to an external system 10, which, in this embodiment, is comprised of two waveguides that have no reflection and unity transmission over the frequency range of interest, in this specific example.

[0038]
[0038]FIG. 4 is a plan view illustrating the layout of the waveguides in the first embodiment splitter/combiner system 100 of FIG. 3. Specifically, each of the directional couplers 110 comprises a region in which the waveguides AC and BD are in close physical proximity to each other. This enables the coherent combination and crosscoupling of energy between the waveguides. Each of the phase shifts 112 comprises segments of the waveguides AC and BD that have different lengths with respect to each other. This introduces a relative phase shift between signals propagating in the waveguides.

[0039]
Using a transfer matrix method, the spectra of the splitter/combiner system are expressed as a function of parameters, such as the coupling values of the directional couplers
110 and the phase values of the phase shifts
112. The output signals C and D for the splitter/combiner system are related to the input signals A and B as follows:
$\begin{array}{cc}\left[\begin{array}{c}C\\ D\end{array}\right]=\underset{\underset{\mathrm{waveguides}}{\uf613}}{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]}\ue89e\underset{\underset{\mathrm{third}\ue89e\text{\hspace{1em}}\ue89e\mathrm{directional}\ue89e\text{\hspace{1em}}\ue89e\mathrm{coupler}}{\uf613}}{[\text{\hspace{1em}}\ue89e\hspace{1em}\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{3}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{3}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{3}\right)& \mathrm{cos}\ue8a0\left({\phi}_{3}\right)\end{array}]}\ue89e\underset{\underset{\underset{\mathrm{shift}}{\mathrm{second}\ue89e\text{\hspace{1em}}\ue89e\mathrm{phase}}}{\uf613}}{[\text{\hspace{1em}}\ue89e\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{2}}& 0\\ 0& 1\end{array}]}\ue89e\text{\hspace{1em}}\ue89e\underset{\underset{\mathrm{second}\ue89e\text{\hspace{1em}}\ue89e\mathrm{directional}\ue89e\text{\hspace{1em}}\ue89e\mathrm{coupler}}{\uf613}}{[\text{\hspace{1em}}\ue89e\hspace{1em}\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{2}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{2}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{2}\right)& \mathrm{cos}\ue8a0\left({\phi}_{2}\right)\end{array}\ue89e\text{\hspace{1em}}]}\ue89e\underset{\underset{\mathrm{first}\ue89e\text{\hspace{1em}}\ue89e\mathrm{phase}\ue89e\text{\hspace{1em}}\ue89e\mathrm{shift}}{\uf613}}{\left[\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{1}}& 0\\ 0& 1\end{array}\right]}\ue89e\underset{\underset{\mathrm{first}\ue89e\text{\hspace{1em}}\ue89e\mathrm{directional}\ue89e\text{\hspace{1em}}\ue89e\mathrm{coupler}}{\uf613}}{\left[\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{1}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{1}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{1}\right)& \mathrm{cos}\ue8a0\left({\phi}_{1}\right)\end{array}\right]}\ue8a0\left[\begin{array}{c}A\\ B\end{array}\right]& \left(2\right)\end{array}$

[0040]
where θ_{1 }and θ_{2 }are the phase values of the phase shifts and φ_{1}, φ_{2}, and φ_{3 }are the coupling values of the directional couplers. In this embodiment, an input A is preferably equally split between outputs C and D. Therefore, for an input signal A with a normalized power of 1 and an input signal B with negligible power, the outputs C and D have normalized powers of ½. The output signal C is given by,

C=cos(φ_{1})cos(φ_{2})cos(φ_{3})e ^{iθ} ^{ 1 } ^{+iθ} ^{ 2 }−sin(φ_{1})sin(φ_{2})cos(φ_{3})e ^{iθ} ^{ 2 }−cos(φ_{1})sin(φ_{2})sin(φ_{3})e ^{iθ} ^{ 1 }−sin(φ_{1})cos(φ_{2})sin(φ_{3}) (3)

[0041]
The output signal D can be described by a similar equation.

[0042]
The splitter/combiner 100 in this embodiment has five designable parameters: 1) the coupling value of the first directional coupler, θ_{1}; 2) the phase value of the first phase shift, φ_{1}; 3) the coupling value of the second directional coupler, φ_{2}; 4) the phase value of the second phase shift, θ_{2}; and 5) the coupling value of the third directional coupler, φ_{3}. The coupling values of the directional couplers 110 are affected by the respective coupling coefficients and the coupling lengths of the directional couplers. The phase values of the phase shifts 112 are affected by the respective propagation constants and the propagation lengths of the phase shifts. Deviations in the coupling coefficients, the coupling lengths, the propagation constants, and the propagation lengths away from the nominal design will, therefore, affect the output signals C, D. Additionally, the deviations may be matched for similar devices. These effects are common to monolithic fabrication of planar waveguide systems due to the fabrication errors in layer thickness, doping to control refractive indices, and waveguide etching, for example.

[0043]
In this first embodiment, the coupling coefficients, of the directional couplers 110 are assumed to deviate by the same amount and the propagation constants p of both phase shifts 112 deviate by the same amount, which is typical for deviations resulting from common fabrication errors that can give rise to undesired frequency dependencies and polarization dependencies.

[0044]
According to the invention, the high tolerances for the output signals C, D are achieved by engineering the output signals C, D relative to the deviations in the coupling coefficients and the propagation constants, and specifically to matched deviations in the coupling coefficients and propagation constants, in this example.

[0045]
In this embodiment, the coupling coefficients of the directional couplers 110 are assumed to be susceptible to a matched deviation, δμ, and the propagation constants of the phase shifts 112 are assumed to be susceptible to a matched deviation, δβ. To achieve a desired insensitivity to the matched deviations in this embodiment, the derivatives of the output signal C, with respect to μ and β, are preferably negligible so that the spectrum is maximally flat. The output signal C has preferably a normalized power of ½.

[0046]
As a result of this design, the deviations of the parameter, μ, preferably do not change the normalized power of output C by greater than a desired amount, ε, over the desired range of the parameter, Δμ, as shown in FIG. 5. FIG. 5 is a plot of the power of output signal C as a function of the coupling coefficient, showing a region in which changes in the coupling coefficient yields only small changes in the output signal. The resulting equations are,
$\begin{array}{cc}{\uf603C\uf604}^{2}=\frac{1}{2}& \left(4\right)\\ \frac{\partial C}{\partial \mu}=0& \left(5\right)\\ \frac{\partial C}{\partial \beta}=0& \left(6\right)\end{array}$

[0047]
Solving these equations results in the following definitions for the coupling values and the phase values in this embodiment:
$\begin{array}{cc}{\phi}_{1}=+\frac{\pi}{8}& \left(7\right)\end{array}$
φ
_{2}=π (8)

φ_{3}=φ_{1} (9)

[0048]
[0048]
$\begin{array}{cc}{\theta}_{1}={\mathrm{cos}}^{1}\ue8a0\left(\frac{1}{4}\right)& \left(10\right)\end{array}$
θ
_{2}=−θ
_{1} (11)

[0049]
The method described above can be used to obtain other exemplary embodiments of the invention. For example, the number of directional couplers and phase shifts may be extended to any desired number or other splitting ratios can be obtained.

[0050]
[0050]FIG. 6 is a plan view of a splitter/combiner 100 coupled to a reflective external optical system 10 in accordance with a second embodiment of the invention. A high parameter tolerance reflection spectrum is obtained by engineering a splitter/combiner 100 that is coupled to an external system 10 with a desired reflection spectrum in the frequency range of interest. In this embodiment, a single splitter/combiner 100 preferably contains two directional couplers 110 and two phase shifts 112, which are serially connected to each other in alternating order.

[0051]
Using the transfer matrix method, reflected output signals A
_{R }and B
_{R }are related to input signals A
_{in }and B
_{in }by the following equation,
$\begin{array}{cc}\left[\begin{array}{c}{A}_{R}\\ {B}_{R}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{1}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{1}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{1}\right)& \mathrm{cos}\ue8a0\left({\phi}_{1}\right)\end{array}\right][\text{\hspace{1em}}\ue89e\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{1}}& 0\\ 0& 1\end{array}][\text{\hspace{1em}}\ue89e\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{2}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{2}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{2}\right)& \mathrm{cos}\ue8a0\left({\phi}_{2}\right)\end{array}][\text{\hspace{1em}}\ue89e\hspace{1em}\text{\hspace{1em}}\ue89e\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{2}}& 0\\ 0& 1\end{array}\ue89e\text{\hspace{1em}}]\ue89eR(\text{\hspace{1em}}\ue89e\hspace{1em}\omega )[\text{\hspace{1em}}\ue89e\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{2}}& 0\\ 0& 1\end{array}][\text{\hspace{1em}}\ue89e\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{2}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{2}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{2}\right)& \mathrm{cos}\ue8a0\left({\phi}_{2}\right)\end{array}][\text{\hspace{1em}}\ue89e\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{1}}& 0\\ 0& 1\end{array}][\text{\hspace{1em}}\ue89e\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{1}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{1}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{1}\right)& \mathrm{cos}\ue8a0\left({\phi}_{1}\right)\end{array}]\ue8a0\left[\begin{array}{c}{A}_{\mathrm{in}}\\ {B}_{\mathrm{in}}\end{array}\right]& \left(12\right)\end{array}$

[0052]
In this second embodiment, the reflection spectrum, R(ω), of the external system
10 is preferably
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\ue89er\ue8a0\left(\omega \right)$

[0053]
—the external system is preferably completely reflective, r(ω)=1, over the frequency range of interest. The input signals, A_{in }and B_{in}, are directed in ports A and B, and the reflected output signals, A_{R }and B_{R}, are also produced at ports A and B. In this embodiment, the reflected output signal A_{R }is preferably negligible so that the entire input signal A_{in }is directed to reflected output signal B_{R}. The input signal A_{in }preferably has a normalized power of 1 and the input signal B_{in }is preferably negligible. The reflected output signal A_{R }is given by,

A _{R} =e ^{2iθ} ^{ 2 } [e ^{iθ} ^{ 1 }cos(φ_{1})cos(φ_{2})−sin(φ_{1})sin(φ_{2})]^{2} −[e ^{iθ} ^{ 1 }cos(φ_{1})sin(φ_{2})+sin(φ_{1})cos(φ_{2})]^{2} (13)

[0054]
In this embodiment, the coupling coefficients of the directional couplers 110 are engineered for a matched deviation, δμ, and the propagation constants of the phase shifts 112 are engineered for a matched deviation, δβ. To achieve the desired insensitivity to the matched deviations in this embodiment, the reflected output signal A_{R }and the derivatives of A_{R}, with respect to μ and β, are preferably negligible. The resulting equations are:

A _{R} =[e ^{2iθ} ^{ 2 }cos(φ
_{1})cos(φ
_{2})−sin(φ
_{1})sin(φ
_{2})]
^{2} −[e ^{iθ} ^{ 1 }cos(φ
_{1})sin(φ
_{2})+sin(φ
_{1})cos(φ
_{2})]
^{2}=0 (14)
$\begin{array}{cc}\frac{\uf74c{A}_{R}}{\uf74c\mu}=0& \left(15\right)\\ \frac{\uf74c{A}_{R}}{\uf74c\beta}=0.& \left(16\right)\end{array}$

[0055]
These equations result in the following two sets of definitions for the coupling ratios and the phase shifts in this embodiment:

[0056]
Set #1—
$\begin{array}{cc}{\phi}_{1}=\left(\frac{2\ue89en+1}{4}\right)\ue89e\pi & \left(17\right)\\ {\phi}_{2}=\left(\frac{2\ue89em+1}{2}\right)\ue89e\pi & \left(18\right)\\ {\theta}_{1}=2\ue89eq\ue89e\text{\hspace{1em}}\ue89e\pi \pm {\mathrm{cos}}^{1}\ue8a0\left(\frac{2\ue89en+1}{4\ue89em+2}\right)& \left(19\right)\\ {\theta}_{2}={\theta}_{1}& \left(20\right)\end{array}$

[0057]
where n, m are nonnegative integers, 2n+1<4m+2 and q is any integer.

[0058]
Set #2—
$\begin{array}{cc}{\phi}_{1}=\left(\frac{2\ue89en+1}{4}\right)\ue89e\pi & \left(21\right)\\ {\phi}_{2}=m\ue89e\text{\hspace{1em}}\ue89e\pi & \left(22\right)\\ {\theta}_{1}=2\ue89eq\ue89e\text{\hspace{1em}}\ue89e\pi \pm {\mathrm{cos}}^{1}\ue8a0\left(\frac{2\ue89en+1}{4\ue89em}\right)& \left(23\right)\\ {\theta}_{2}={\theta}_{1}& \left(24\right)\end{array}$

[0059]
where n is a nonnegative integer and m is a positive integer, 2n+1<4m and q is

[0060]
Using Set # 1 in this exemplary embodiment, the first directional coupler is preferably a 50% coupler and the second directional coupler is preferably a 100% coupler. The first and second phase shifts are preferably both
$+\frac{2\ue89e\pi}{3}\ue89e\text{\hspace{1em}}\ue89e\mathrm{or}\ue89e\text{\hspace{1em}}\ue89e\mathrm{both}\ue89e\text{\hspace{1em}}\frac{2\ue89e\pi}{3}.$

[0061]
[0061]FIG. 7 is a plan view of two exemplary splitter/combiners 100 coupled an external system 10, in accordance with a third embodiment of the invention. The splitter/combiner system is designed for a high parameter tolerance transmission spectrum. In this embodiment, two splitter/combiners 100 are coupled to an external system 10 with a transmission spectrum. The splitter/combiners 100 in this embodiment preferably include two directional couplers 110 and two phase shifts 112 that are serially connected to each other in alternating order.

[0062]
Using transfer matrix method, the output signals C
_{T }and D
_{T }are related to the input signals A
_{in }and B
_{in }by the following equation,
$\begin{array}{cc}\left[\begin{array}{c}{C}_{T}\\ {D}_{T}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{4}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{4}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{4}\right)& \mathrm{cos}\ue8a0\left({\phi}_{4}\right)\end{array}\right]\ue8a0\left[\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{4}}& 0\\ 0& 1\end{array}\right]\ue8a0\left[\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{3}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{3}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{3}\right)& \mathrm{cos}\ue8a0\left({\phi}_{3}\right)\end{array}\right]\ue8a0\left[\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{3}}& 0\\ 0& 1\end{array}\right]\times T\ue8a0\left(\omega \right)\ue8a0\left[\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{2}}& 0\\ 0& 1\end{array}\right]\ue8a0\left[\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{2}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{2}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{2}\right)& \mathrm{cos}\ue8a0\left({\phi}_{2}\right)\end{array}\right]\ue8a0\left[\begin{array}{cc}{\uf74d}^{{\mathrm{\uf74e\theta}}_{1}}& 0\\ 0& 1\end{array}\right]\ue89e\hspace{1em}\left[\begin{array}{cc}\mathrm{cos}\ue8a0\left({\phi}_{1}\right)& i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{1}\right)\\ i\ue89e\text{\hspace{1em}}\ue89e\mathrm{sin}\ue8a0\left({\phi}_{1}\right)& \mathrm{cos}\ue8a0\left({\phi}_{1}\right)\end{array}\right]\ue8a0\left[\begin{array}{c}{A}_{\mathrm{in}}\\ {B}_{\mathrm{in}}\end{array}\right]& \left(25\right)\end{array}$

[0063]
In this embodiment, the external system
10 is comprised of waveguides coupling the first splitter/combiner to the second splitter/combiner. For this embodiment, the transmission spectrum of the waveguides, T(ω), is
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\ue89et\ue8a0\left(\omega \right)$

[0064]
—the waveguides are preferably completely transmissive at the frequency or frequencies of interest, so that t(ω) is preferably 1. The input is preferably directed in port A and the output is directed out ports C and D. Similar to the previous embodiment, the response of the output signal C_{T }is preferably negligible and the derivatives for signal C_{T }are preferably negligible. This embodiment can be used as a waveguide cross. In a fourth exemplary embodiment of the invention, a splitter/combiner system is designed for high parameter tolerance reflection and transmission spectra. Two splitter/combiners are coupled to an external system, such as a directionalcoupler assisted add/drop filter having reflection and transmission spectra, as described, for example, in U.S. Pat. Appl. No. Unknown, entitled “Directionalcoupler Assisted Add/Drop Filter With IQ Induced On/Off Switching and Modulation” filed Mar. 7, 2002 by common assignee, which is incorporated herein by reference in its entirety.

[0065]
[0065]FIG. 8 is a block diagram of a splitter/combiner system coupled to two external optical systems (subelements) having the same reflection and transmission spectra, in accordance with a fourth embodiment of the invention. The DCA filter resonatorsystem is reflective at particular frequencies and transmissive at others. The splitter/combiners 100 have 50/50 power splitting ratios and couple to the two DCA filter resonator subelements 101, 102.

[0066]
[0066]FIGS. 9A and 9B are schematic block diagrams illustrating the design process for the fourth embodiment in view of the reflection and transmission characteristics of the external system. Splitter/combiners 100 are preferably designed separately for the reflection and the transmission spectra. Specifically, the first splitter/combiner 1001 is designed using the parameter tolerance requirements for the reflection spectrum, similar to the second embodiment, as illustrated in FIG. 9A. Once the first splitter/combiner 1001 is designed based on the reflection spectrum, the second splitter/combiner 1002 is designed to satisfy the parameter tolerance requirements on the transmission spectrum as illustrated in FIG. 9B.

[0067]
In this embodiment, the desired values of output signal C_{T }and output signal A_{R }are simultaneously negligible, so that there is no crosstalk and no leakage. To achieve high tolerance to common deviations in the coupling coefficients and the propagation constants, the derivatives of signal C_{T }and signal A_{R }with respect to these parameters are preferably negligible. These requirements result in definitions for the coupling values and phase values.

[0068]
[0068]FIG. 10 is a plan view of a simplified splitter/combiner configuration for reflection/transmission spectra in accordance with the invention. The design is made for a splitter/combiner for reflection and then the design is mirrored for transmission. For example, the splitter/combiner described in the second embodiment, for reflection, can also be used for transmission, as shown in FIG. 10. In this embodiment, the output signal A_{R }and output signal C_{T }are preferably negligible. The input signal A_{in }preferably has a normalized power of 1. The input signal B_{in }preferably has negligible power. The reflection signal A_{R is: }

A _{R} ={e ^{2iθ} ^{ 2 } [e ^{iθ} ^{ 1 }cos(φ_{1})cos(φ_{2})−sin(φ_{1})sin(φ_{2})]^{2} −[e ^{iθ} ^{ 1 }cos(φ_{1})sin(φ_{2})+sin(φ_{1})cos(φ_{2})]^{2} }r(ω)=0 (26)

[0069]
and the transmission signal C_{T }is:

C _{T} ={e ^{2iθ} ^{ 2 } [e ^{iθ} ^{ 1 }cos(φ_{1})cos(φ_{2})−sin(φ_{1})sin(φ_{2})]^{2} −[e ^{iθ} ^{ 1 }cos(φ_{1})sin(φ_{2})+sin(φ_{1})cos(φ_{2})]^{2} }t(ω)=0 (27)

[0070]
Since the two signals have the same form, it is clear that a solution to one equation is also a solution to the other equation. A further alternate embodiment using this method is shown in FIG. 11. FIG. 11 is a plan view of an alternate splitter/combiner configuration for reflection and transmission spectra in accordance with the invention.

[0071]
In another exemplary embodiment of the invention, the phase values of the phase shifts in the splitter/combiner systems are controlled by an effect, such as electrooptic or thermooptic refractive index changes. Specifically, even with the foregoing parameter tolerant designs, in high index contrast planar waveguide systems, yields may not be high enough. This embodiment enables the phase values to vary by different amounts, addressing a differential error, for example. The differential error is corrected by tuning the phase values using a refractive effect, typically in a postfabrication tuning step. The tolerance of the phase value, and the splitter/combiner, is thereby increased.

[0072]
[0072]FIG. 12 shows one such implementation. FIG. 12 is a plan view illustrating the layout of the waveguides in still another embodiment of the splitter/combiner system. A post fabrication tuning mechanism is provided. Specifically, heaters 210 are integrated with the waveguides, and specifically on the phase shifts to enable post fabrication control over the refractive indices of waveguide A. This allows tuning of the propagation constants of the phase shifts 112.

[0073]
In the typical application, the operation of the splitter/combiner 100 is tested post fabrication. In the calibration sequence, the refractive indices of the waveguides are temperature tuned to optimize the splitter/combiner's performance. During normal operation, the heaters 210 maintain the waveguides at these calibrated temperatures in a steady state. Other techniques are used to finetune the operation of the splitter/combiner 100 in other implementations. For example, instead of temperature tuning the refractive indices, ultraviolet exposure is used to modify the waveguides in a post fabrication calibration operation.

[0074]
It should be appreciated that this post fabrication finetuning technique is applicable to any of the previously described embodiments to improve performance of the final system.

[0075]
Further, in all of the abovedescribed embodiments, the splitter/combiner systems can be used for signal propagation in either direction. As well, each port can be used as an input port, an output port, or both an input and an output port.

[0076]
The methods described above may be extended to any number of directional couplers and phase shifts. The methods may also be extended to any desired power splitting ratio and a wide range of parameter deviations. As well, the designs are not limited to any specific tolerance values or any specific set of parameters.

[0077]
Although the present invention has been shown and described with respect to several preferred embodiments thereof, various changes, omissions and additions to the form and detail thereof, may be made therein, without departing from the spirit and scope of the invention.