Method for calibrating a free-space-coupled fiber-optic transmission system

[0002] The invention relates to optical switching and free-space coupling of fiber optic waveguides in a single-mode fiber-optic transmission system, and specifically to calibrating the model terms of such systems. The invention finds application to micro-electromechanical systems (MEMS), but it is not so limited.

[0003] The technique of using open-loop oscillatory mirror commands (dithers) may be used in optical switching for: (1) detection of alignment errors through synchronous detection such as is described in a patent application of the present inventor entitled METHOD AND APPARATUS FOR OPTICAL BEAM ALIGNMENT DETECTION AND CONTROL (TTC Docket 20974-1.00), and for (2) intentional optical power attenuation through induced constant-power coupling loss such as is described in a patent application of the present inventor entitled METHOD AND APPARATUS FOR OPTICAL BEAM POWER ATTENUATION (TTC Docket 20974-8.00). In both examples, performance is limited by errors in the kinematics models that relate actuation commands to coupling loss. In the case of synchronous detection, modeling errors may create a cross-coupled closed-loop response that would necessitate reducing the servo bandwidths to achieve stable response. A A reduced bandwidth translates directly to an increased time required to nullify the alignment errors detected after the open-loop portion of a switch, which increases the overall switch time. In both of the detection and attenuation examples, modeling errors can cause undesirable signal content in the coupled power.

[0004] The kinematical relationship between mirror actuation commands and coupling loss is a strong function of the optical design (nominal path lengths, focal length, wavelength, and fiber properties) as well as unavoidable small errors in the component prescriptions and in the time-varying and temperature-dependent alignment of the optical components. The calibration values determined during an initial off-line calibration might provide acceptable performance for only a relatively short time after the calibration, after which the system might violate one or more performance specifications. If the system is required to operate longer than the time period of validity of the initial calibration, then the approach of using an initial off-line calibration is not acceptable. Taking a connection off-line for the purpose of calibration or injecting calibration-specific signals into the system may also be unacceptable options.

[0005] What is needed is a non-interference calibration technique designed to continuously detect and tune the parameters of a kinematical model used as part of a control system for a fiber-optic switch or similar optical system. This non-interference calibration technique must not require that any calibration-specific signals be injected into the system, or that the operation of primary functions such as alignment control and/or power attenuation be halted for the purpose of calibration.

[0006] According to the invention, a method is provided for calibrating a free-space-coupled fiber-optic transmission system. The method uses signal content in the measured coupled power at frequencies equal to the sum, difference, and first harmonics of the frequencies of control-related dither signals to infer and correct errors in the kinematical model used to apply the dithers. The technique can operate independently or simultaneously with dither-based functions such as alignment control and power attenuation to provide continuous tracking of model terms. The calibration method does not require that any calibration-specific signals be injected into the system, or that the operation of primary functions be halted for the purpose of calibration.

[0007] The invention is explained with respect to a structure with two beam-steering elements having four degrees of freedom. The invention is readily generalized to a single beam-steering element or to a larger number of beam-steering elements.

[0008] The invention will be better understood by reference to the following detailed description in connection with the accompanying embodiments.

[0009] FIG. 1 is a block diagram of a four-port MEMS mirror array fiber optic switch in which the present invention may be implemented.

[0010] FIG. 2 is a diagram to illustrate beam alignment errors at the output lens as used in the present invention.

[0011] FIG. 3 is a three dimensional graph to illustrate constant-power dithering.

[0012] FIG. 4 is a schematic diagram to illustrate the integration of the calibration technique into a control system according to the invention.

[0013] FIG. 5 is a schematic diagram to illustrate the function of the calibration law.

[0014] Free-Space Coupled Fiber Optic Switch Using MEMS

[0015] Referring to FIG. 1 there is shown an example of a four-port MEMS mirror array fiber-optic switch 10 in which the present invention may be implemented. The function of the fiber-optic switch 10 is to produce desired free-space couplings between the input fibers (in a first fiber array 12) and output fibers (in a second fiber array 14) via two-axis steering mirrors on a first mirror array 16 and a second mirror array 18. In the embodiment illustrating the invention, the optical path is unidirectional between input fibers and output fibers, although the invention is not so limited. Expanding laser beams emanating from the input fiber array 12 are substantially collimated using a first lens array 20 confronting the first fiber array 12. Mirrors on the first or input mirror array 16 steer the collimated beams 23-26 from the first lens array 20 toward the appropriate mirrors on the second or output mirror array 18. The mirrors on the output mirror array 18 steer their incident beams 23′-26′ into the corresponding lenses on a second or output lens array 30. The output lenses of the second lens array 30 produce converging beams necessary for coupling power into the output fibers 23″-26″ of the second fiber array 14.

[0016] Output Beam Alignment Geometry

[0017] Referring to FIG. 2, there is shown a diagram defining the alignment of an output beam relative to its output lens/fiber port in accordance with the invention. This technique may be applied in a variety of geometries, only one of which is that of FIG. 1. The axes “x”, “y”, and “z” define an “output” frame. The axis z is defined as the optical axis of the particular lens/fiber port (which may vary across the array of lens/fiber ports), and the perpendicular axes x and y are in the plane perpendicular to z. The beam intersects the output x-y plane at the linear displacements P_xand Py in x and y, respectively. The two angles θ_xand θ_ydefine the orientation of the beam in the output x-y-z frame. With θ_xand θ_yequal to zero, the beam is parallel to the z axis. θ_xand θ_yare the rotations of the beam in the x and y directions, respectively, where a small-angle approximation (valid for the purpose of analyzing optical coupling) eliminates the need to define the order of the rotations. To clarify the sense of the rotations, the small-angle approximation for the unit vector in the direction of the beam expressed in the output frame is given by:

[0018] u_beam^out=[θ_y−θ_x1].

[0019] Gaussian Power Coupling

[0020] Assuming that longitudinal misalignments are zero and that the beam is matched to the mode field radius of the output fiber, the total coupled power P_out(a scalar quantity measured at the output fiber) can be approximated in a Gaussian form in terms of an input power P_inand four normalized beam alignment errors:

[0021] P_out=P_ine⁻(α²+β²+ρ²σ²),  (1)

[0022] where P_inis the optical power before loss due to alignment errors, and the four normalized errors α, β, ρ, and σ are given by: $\alpha =\frac{f}{{\omega }_0}{\theta }_x,\beta =\frac{f}{{\omega }_0}{\theta }_y,\rho =\frac{n_{\mathrm{gap}}\pi {\omega }_0}{\lambda f}\left(P_x-f{\theta }_y\right),\mathrm{and}\sigma =\frac{n_{\mathrm{gap}}\pi {\omega }_0}{\lambda f}\left(P_y+f{\theta }_x\right),$

[0023] where:

[0024] f is the lens focal length,

[0025] w₀is the beam radius at 1/e²power density,

[0026] λ is the laser wavelength, and

[0027] n_gapis the index of refraction of the medium in the lens/fiber gap.

[0028] Constant-Power Dithering

[0029] In the case of a MEMS fiber-optic switch for which the coupled power is Gaussian in the four normalized alignment errors, quadrature (sine and cosine signals at a given frequency) dithering of the beam alignments produces a constant coupled power when the alignment errors (ignoring the dither component) are zero and the model parameters of the system are properly tuned. FIG. 3 illustrates for two of the four axes the concept of using quadrature dithering to provide a constant output power when the coupled power is Gaussian in the two alignment errors shown. When the alignment errors (ignoring the dither component) are zero as in FIG. 3, the sine and cosine dithers at the given frequency, whether above or below the natural resonance of the device, produce a circular trajectory in the two alignment errors. Since the sum of the squares of these two alignment errors is constant (due to the sine and cosine dithering), the result, as dictated by Equation 1, is a constant attenuated coupled power even though the individual errors are changing. If the other two dithers are also in quadrature form, these dithers will contribute a second constant loss term, such that the overall attenuation level due to the four dithers is constant.

[0030] Kinematics Modeling

[0031] Dithering in the coupling space requires a model of the kinematical mapping between the mirror angles and the four coupling errors so that the mirrors can work in concert to produce the desired orthogonal beam motions at the output.

[0032] Specifically, the kinematical mapping between the four angular displacements of the two mirrors in an optical path and the four displacements in the coupling space is expressed as:

[0033] p=MQθ,

[0034] where p is defined as a vector of the four normalized beam alignment components:

[0035] p=[ασβρ]^T,

[0036] and θ as a vector of the four mirror angles:

[0037] θ=[x_inx_outy_iny_out]^T,

[0038] where x_in, x_out, y_in, and y_outare the mirror rotations in the directions of x₁, x₂, y₁, and y₂, respectively, as shown in FIG. 1. The 4×4 axis-misalignment matrix Q expresses any misalignments between the four mirror angles with respect to the effect on motion of the beam at the output lens. The matrix Q maps the mirror angles θ to a set of coordinates u:

[0039] u=Qθ,

[0040] where u is given by:

[0041] u=[v₁v₂w₁w₂]^T.

[0042] By definition, the coordinates v₁, and v₂are combinations of the mirror angles that map uniquely to variations in the two coupling coordinates α and σ, which produce motion of the output beam in a plane defined as P_v. The coordinates w₁and w₂map uniquely to variations in the two coupling coordinates β and ρ, which produce motion of the output beam in a plane P_wthat is perpendicular to P_v. (Both P_vand P_ware parallel to the output optical axis.)

[0043] The misalignment matrix Q is assumed to have the form: $Q=\left[\begin{array}{cc}{I}_2& Q_A\\ Q_B& I_2\end{array}\right],$

[0044] where I₂is an identity matrix of dimension two, and the 2×2 coupling matrices Q_Aand Q_Bare ideally zero The matrix Q_Adefines the non-ideal coupling of y_inand y_outinto v₁and v₂, and the matrix Q_Bdefines the non-ideal coupling of x_inand x_outinto w₁and w₂.

[0045] The 4×4 kinematics matrix M that maps the decoupled coordinates u to the beam alignments p as in:

[0046] p=Mu

[0047] is given by: $M=\left[\begin{array}{cc}{M}_v& 0_{2\times 2}\\ 0_{2\times 2}& M_w\end{array}\right],M_v=\left[\begin{array}{cc}{M}_{\mathrm{v11}}& M_{\mathrm{v12}}\\ M_{\mathrm{v21}}& M_{\mathrm{v22}}\end{array}\right],M_w=\left[\begin{array}{cc}{M}_{\mathrm{w11}}& M_{\mathrm{w12}}\\ M_{\mathrm{w21}}& M_{\mathrm{w22}}\end{array}\right],$

[0048] where the scalar terms M_vijand M_wijare predicted analytically as:

[0049] M_v11=2A cos(φ);

[0050] M_v21=−2B cos(φ)(h₂+h₃)/f−1);

[0051] M_v12=2A cos(φ);

[0052] M_v22=−2B cos(φ)(h₃/f−1);

[0053] M_w11=−2A;

[0054] M_w21=−2B(h₂+h₃)/f−1);

[0055] M_w12=2A;

[0056] M_w22=2B(h₃/f−1);

[0057] where

[0058] A=f/w₀;

[0059] B=n_gapπw₀/λ;

[0060] where

[0061] f=lens focal length;

[0062] λ=laser wavelength;

[0063] w₀=beam radius at 1/e²power density;

[0064] n_gap=index of refraction of the medium in the gap between the lens and fiber;

[0065] φ=y-angle rotation of mirror arrays from perpendicular to lens/fiber optical axes;

[0066] h₂=path distance between mirrors in a given connection; and

[0067] h₃=path distance between the output mirror and output lens in a given connection.

[0068] The matrices M and Q together define the overall mapping from mirror angles θ to the coupling variables p, as repeated here:

[0069] p=MQθ.

[0070] The inverse reflection kinematics matrix N is defined as the inverse of M:

[0071] N=m⁻¹,

[0072] such that the mirror joint angles θ can be computed from the vector of reference output variables p_refand model terms Q_modand N_mod:

[0073] θ=Q⁻¹_modN_modp_ref,

[0074] where the terms p_ref, Q⁻¹_mod, and N_modare given by:

[0075] $p_{\mathrm{ref}}={\left[{\alpha }_{\mathrm{ref}}{\sigma }_{\mathrm{ref}}{\beta }_{\mathrm{ref}}{\rho }_{\mathrm{ref}}\right]}^T,Q_{\mathrm{mod}}=\left[\begin{array}{cc}{I}_2& Q_{\mathrm{Amod}}\\ Q_{\mathrm{Bmod}}& I_2\end{array}\right],\mathrm{and}{N}_{\mathrm{mod}}=M_{\mathrm{mod}}^{-1}=\left[\begin{array}{cc}{N}_{\mathrm{vm}}& 0_{2\times 2}\\ 0_{2\times 2}& N_{\mathrm{wm}}\end{array}\right],$

[0076] where N_vmand N_wmare composed of eight scalar quantities N_vmijand N_wmij(i,j=1:2): $N_{\mathrm{vm}}=\left[\begin{array}{cc}{N}_{\mathrm{vm11}}& N_{\mathrm{vm12}}\\ N_{\mathrm{vm21}}& N_{\mathrm{vm22}}\end{array}\right],\mathrm{and}{N}_{\mathrm{wm}}=\left[\begin{array}{cc}{N}_{\mathrm{wm11}}& N_{\mathrm{wm12}}\\ N_{\mathrm{wm21}}& N_{\mathrm{wm22}}\end{array}\right].$

[0077] Coupled Power as a Function of Actual and Model Terms

[0078] The model of the kinematics matrix M_modis given by: $M_{\mathrm{mod}}=\left[\begin{array}{cc}{M}_{\mathrm{vmod}}& 0_{2\times 2}\\ 0_{2\times 2}& M_{\mathrm{wmod}}\end{array}\right],$

[0079] where the sub-matrices M_vmodand M_wmodare defined in terms of eight scalar quantities: $M_{\mathrm{vmod}}=\left[\begin{array}{cc}{M}_{\mathrm{vm11}}& M_{\mathrm{vm12}}\\ M_{\mathrm{vm21}}& M_{\mathrm{vm22}}\end{array}\right]M_{\mathrm{wmod}}=\left[\begin{array}{cc}{M}_{\mathrm{wm11}}& M_{\mathrm{wm12}}\\ M_{\mathrm{wm21}}& M_{\mathrm{wm22}}\end{array}\right],$

[0080] and the misalignment model Q_mod, following the form of Q, is defined as: $Q_{\mathrm{mod}}=\left[\begin{array}{cc}{I}_2& Q_{\mathrm{Amod}}\\ Q_{\mathrm{Bmod}}& I_2\end{array}\right].$

[0081] The terms Q_Amodand Q_Bmodare defined in terms of four axis-misalignment parameters δ_mod1, δ_mod2, δ_mod3, and δ_mod4as: $Q_{\mathrm{Amod}}=\left[\begin{array}{cc}{\delta }_{\mathrm{mod1}}& 0\\ 0& {\delta }_{\mathrm{mod2}}\end{array}\right],\mathrm{and}{Q}_{\mathrm{Amod}}=\left[\begin{array}{cc}0& {\delta }_{\mathrm{mod3}}\\ {\delta }_{\mathrm{mod4}}& 0\end{array}\right].$

[0082] Other definitions of Q_modusing alternate placements of δ_modk(k=1,2,3,4) are possible but would require further analysis to verify functionality.

[0083] Combining the forward kinematics equation p=MQθ and the inverse kinematics equation for θ produces a relationship between the actual and reference dither states:

[0084] p=MQQ⁻¹_modN_modp_ref

[0085] Since the coupled output power (ignoring alignment errors) is given by:

[0086] P_out=P_ine^−|p|,

[0087] the log term −log(P_out) is equal to:

[0088] −log(P_out)=|p|²−log(P_in),

[0089] where |p|², which can be formed as p^Tp, is given by:

[0090] |p|²=p^Tp=p^T_refN^T_modQ^−T_modQ^TM^TMQQ⁻¹_modN_modp_ref.

[0091] The equation for |p|²can be used to derive the sensitivity of −log(P_out) to modeling errors in Q_modand M_mod, providing methods to detect and reduce the modeling errors, which will be discussed next.

[0092] Detection of Axis Misalignments and Coupling Matrix Errors

[0093] According to the invention, the approach presented here for the detection of axis misalignments and coupling-matrix errors is based on linearizing the equation for |p|²with respect to errors in model parameters. The relevant error matrices Q_errand M_errare defined as the differences between the model and actual terms:

[0094] Q_err=Q_mod−Q

[0095] and

[0096] M_err=M_mod−M.

[0097] The linearized form of the equation for |p|²applicable to the general case of one or more steering elements can be expressed as a function of model and error terms:

[0098] |p|²_lin=p^T_refp_ref+−p^T_ref(M_modQ_errN_mod+N^T_modQ_errM^T_mod)p_ref+−p^T_ref(M_errN_mod+N^T_modM^T_err)p_ref.

[0099] For the case of an optical switch or similar beam-steering mechanism employing two beam-steering elements, the terms Q_errand M_errare 4×4 matrices having the block forms consistent with the assumed forms of Q and M: $Q_{\mathrm{err}}=\left[\begin{array}{cc}{0}_{2\times 2}& Q_{\mathrm{Aerr}}\\ Q_{\mathrm{Berr}}& 0_{2\times 2}\end{array}\right],\mathrm{and}{M}_{\mathrm{err}}=\left[\begin{array}{cc}{M}_{\mathrm{verr}}& 0_{2\times 2}\\ 0_{2\times 2}& M_{\mathrm{werr}}\end{array}\right].$

[0100] Defining p_vrefand p_wrefas sub-vectors of p_ref:

[0101] p_vref=[α_refσ_ref]^T,

[0102] and

[0103] p_wref=[β_refρ_ref]^T,

[0104] the linearized form of |p|²can be expressed in terms of the 2×2 model and error matrices:

[0105] |p|²_lin=p^T_vrefp_vref+p^T_wrefp_wref−2p^T_vref(M_vmodQ_AerrN_wmod+N^T_vmodQ^T_BerrM^T_wmod)p_wref−2p^T_vrefM_verrN_vmodp_vref−2p^T_wrefM_werrN_wmodp_wref.

[0106] Defining the four errors in the misalignment model as:

[0107] ε_k=δ_modk−δ_k(k=1,2,3,4),

[0108] the error matrices Q_Aerrand Q_Berr, consistent with the assumed forms of Q_Amodand Q_Bmod, are given by: $Q_{\mathrm{Aerr}}=\left[\begin{array}{cc}{ϵ}_1& 0\\ 0& {ϵ}_2\end{array}\right],\mathrm{and}{Q}_{\mathrm{Berr}}=\left[\begin{array}{cc}0& {ϵ}_3\\ {ϵ}_4& 0\end{array}\right].$

[0109] The 2×2 coupling-matrix errors are expressed as: $M_{\mathrm{verr}}=\left[\begin{array}{cc}{ϵ}_5& {ϵ}_7\\ {ϵ}_6& {ϵ}_8\end{array}\right]\mathrm{and}{M}_{\mathrm{werr}}=\left[\begin{array}{cc}{ϵ}_9& {ϵ}_{11}\\ {ϵ}_{10}& {ϵ}_{12}\end{array}\right].$

[0110] Assuming constant-power dithering (using sine and cosine signals at two frequencies for the four reference signals α_ref, β_ref, ρ_ref, and ρ_ref), and using the above definitions of the error terms, the coupled-power term |p|²_lincan be written as a function of the dither references p_vrefand p_wrefand a linear combination of twelve unbiased mode functions G_k(k=1,2, . . . , 12) and twelve bias terms B_k(k=1,2, . . . , 12) scaled by the corresponding error terms: ${\left|p\right|}_{\mathrm{lin}}^2=p_{\mathrm{vref}}^T{p}_{\mathrm{vref}}+p_{\mathrm{wref}}^T{p}_{\mathrm{wref}}+\sum _{i=1}^{12}{G}_i{ϵ}_i+\sum _{i=1}^{12}{B}_i{ϵ}_i,$

[0111] where the twelve unbiased mode functions G_k(k=1,2, . . . , 12) are given by:

[0112] G₁=−2(α_refM_vm11+σ_refM_vm21)(β_refN_wm11+ρ_refN_wm12)

[0113] G₂=−2(α_refM_vm12+σ_refM_vn22)(β_refN_wm21+ρ_refN_wm22)

[0114] G₃=−2(α_refN_vm11+σ_refN_vm12)(β_refM_wm12+ρ_refM_wm22)

[0115] G₄=−2(α_refN_vm21+σ_refN_vm22)(β_refM_wm11+ρ_refM_wm21)

[0116] G₅=−2(α²_refN_vm11+α_refσ_refN_vm12)+N_vm11

[0117] G₆=−2(α_refσ_refN_vm11+σ²_refN_vm12)+N_vm12

[0118] G₇=−2(α²_refN_vm21+α_refσ_refN_vm22)+N_vm21

[0119] G₈=−2(α_refσ_refN_vm21+σ²_refN_vm22)+N_vm22

[0120] G₉=−2(β²_refN_wm11+β_refρ_refN_wm12)+N_wm11

[0121] G₁₀=−2(β_refρ_refN_wm11+ρ²_refN_wm12)+N_wm12

[0122] G₁₁=−2(β²_refN_wm21+β_refρ_refN_wm22)+N_wm21

[0123] G₁₂=−2(β_refρ_refN_wm21+ρ²_refN_wm22)+N_wm22

[0124] and the twelve bias functions B_k(k=1,2, . . . , 12) are given by:

[0125] B₁=0

[0126] B₂=0

[0127] B₃=0

[0128] B₄=0

[0129] B₅=−N_vm11

[0130] B₆=−N_vm12

[0131] B₇=−N_vm21

[0132] B₈=−N_vm22

[0133] B₉=−N_wm11

[0134] B₁₀=−N_wm12

[0135] B₁₁=−N_wm21

[0136] B₁₂=−N_wm22

[0137] The mode functions G_k(k=1,2, . . . , 12) and bias functions B_k(k=1,2, . . . , 12) represent the AC and DC sensitivities, respectively, of |p|²_linto the twelve errors ε_k(k=1,2, . . . , 12). However, while the first four mode functions G₁-G₄(associated with axis misalignments) are orthogonal, the structure of G₅-G₁₂(associated with terms of M_verr) results in two of the four mode functions G₅-G₈being linear combinations of the other two, and two of the four mode functions G₉-G₁₂(associated with terms of M_werr) being linear combinations of the other two. Thus, only eight linear combinations of the twelve errors ε_k(k=1,2, . . . , 12) are observable in |p|²_lin. As a result, a choice must be made as to which two combinations of ε₅-ε₈and which two combinations of ε₉-ε₁₂are to be considered observable in |p|²_lin.

[0138] For the purpose of simplifying this discussion, ε₅and ε₆will be considered the two observable combinations of ε₅-ε₈, and ε₉and ε₁₀will be considered the two observable combinations of ε₉-ε₁₂. Other choices for observable quantities using combinations of all four respective ε_kare possible, and some choices might be advantageous or convenient, depending on the system under consideration. The specific choice does not alter the general approach for detecting the observable combinations of ε_k, which will now be discussed.

[0139] The eight error terms ε_k(k=1−6,9−10) represent the correlation between |p|²_linand the respective mode function G_k(k=1−6,9−10). As such, the errors ε_k(k=1−6,9−10) can be detected as the correlation between |p|²_linin and the respective mode functions G_k(k=1−6, 9−10).

[0140] According to the invention, the correlation approach presented here for finding ε_k(k=1−6,9−10) is to use a moving average filter operating on the product of the mode functions G_kand the negative of the logarithm of the power:

[0141] ε_k=MoveAvgFilt(−log(P_out)G_k)(k=1−6,9−10).

[0142] The moving average filter averages −log(P_out)G_kover a time period T_avg, where T_avgis chosen to be equal to the period over which the mode functions G_k(k=1−6,9−10) complete one cycle of relative phasing. Such a filter preserves the low-frequency band of its input signal, while squelching signals at the averaging frequency f_avg=1/T_avgand its harmonics. Filtering −log(P_out)G_kwith this filter thus extracts the detected error ε_kwhile squelching residual signal content at f_avgand its harmonics.

[0143] As an example, the two dither frequencies f₁and f₂are given by:

[0144] f₁=2f_avg,

[0145] and

[0146] f₂=3f_avg.

[0147] and the four dither references are given by sine and cosine signals at the two frequencies:

[0148] α_ref=sin(2πf₁t),

[0149] σ_ref=cos(2πf_{t1i t}),

[0150] β_ref=sin(2πf₂t),

[0151] and

[0152] ρ_ref=cos(2πf₂t).

[0153] Observation of G_k(k=1−12) indicates that the mode functions G_k(k=1−4) contain only cross-frequency terms resulting in signal content at f₂−f₁and f₂+f₁, while functions G_k(k=5−12) contains signal content at 2f₁and 2f₂. The averaging period T_avgover which the mode functions repeat one cycle of relative phasing is given by 1/f_avg, in which time the signals at the six frequencies f₂−f₁, f₁, f₂, 2f₁, f₂+f₁, and 2f₂repeat 1, 2, 3, 4, 5, and 6 times, respectively.

[0154] As a note, the DC-dependence terms B_kprovide no information for detecting ε_k, since individual DC contributions to log(P_out) from multiple ε_kvalues cannot be differentiated. Also, the DC contributions have no detrimental effect on the correlation detection between −log(P_out) and G_k, since correlation with G_kignores the DC component in −log(P_out) due to the unbiased nature of G_k.

[0155] According to the invention, the eight detected error terms ε_k(k=1−6, 9−10) are used to update the four model terms δ_modk(k=1, 2, 3, 4) and the four observable components of M_vmodand M_wmodas follows. The detected errors ε_kare a measure of the modeling errors in the system at the time of the detection, so the eight elements ε_k(k=1−6, 9−10) diminish to zero when the model terms are at the correct static values. Numerically integrating the eight instantaneous estimates ε_k(k=1−6, 9−10) yields smoothed correction terms λ_k:

[0156] λ_k=∫Kε_kdt

[0157] The correction terms λ_k(k=1−6, 9−10) can be subtracted from the values of the eight initial estimates to form the eight model terms. The convergence rate of the eight model terms is set by the integral gain K used to incrementally add ε_k(k=1−6, 9−10) to the existing correction terms λ_k(k=1−6, 9−10). An excessive gain value K will yield unstable response. Assuming the use of a reasonable gain, the model terms converge to steady-state values as the use of the model terms successfully minimizes the signal content in −log(P_out) correlated to the eight independent mode functions G_k(k=1−6, 9−10).

[0158] Since only two combinations of the four model errors between M_vand M_vmodand two combinations of the four model errors between M_wand M_wmodare observable, the steady-state values of M_vmodand M_wmodwill not in general be equal to M_vand M_w, respectively. As a result, the model matrices M_vmodand M_wmodwill not represent the intended mapping between the mirror angles θ=[x_inx_outy_iny_out]^T(assuming no misalignments) and the output coordinates p=[ασβρ]^T. However, the steady-state values of M_vmodand M_wmodwill provide a mapping between the mirror angles and orthogonal combinations of α, σ, β, and, ρ, which is sufficient for use with controllers governing alignment or attenuation. After an initial calibration phase, rows of M_vmodand M_wmodcan be scaled up or down to normalize the amplitude of the AC power drop resulting from the application of individual dither references p_ref.

[0159] System with Calibration

[0160] FIG. 4 is a schematic diagram of a control system 100 with calibration according to the invention illustrated in interaction with elements of a fiber optic switch of FIG. 1. The control system 100 supplies the actuation voltages for a pair of MEMS mirrors on mirror arrays 16 and 18 of an electro-opto-mechanical plant. The MEMS mirrors produce the four mirror angles as a function of the actuation voltages. Reflection kinematics 109 specific to the optical design define the mapping of the four mirror angles to the four beam alignment errors at the output lens of lens array 30 (as shown in FIG. 2). As defined in Equation 1, the coupling of power into the output fiber of output fiber array 14 is Gaussian in the four beam alignment errors. FIG. 4 schematically illustrates the transformation shown in FIG. 1 in which beam 23 is steered by mirrors 16 and 18 to yield beam 23′ incident at a lens in the output lens array 30. FIG. 4 also illustrates coupling of incident beam 23′ through the lens into the associated fiber 23″ of the output fiber array 14. A power tap 108 at the optical output supplies the feedback signal, which is converted to an electrical signal by a photodetector 110.

[0161] Within the control system 100, the feedback power signal is supplied to a calibration law unit 300 and to a control law unit 102. Similarly, the dither references 120 are provided to the calibration law unit 300 and the control law unit 102. The control law unit 102 produces a set of feedback control signals p_com, which is a vector of commanded values of the output-space components α, β, ρ, and σ. The inverse kinematics and actuation model 107 converts the output-space commands into actuator drive voltages, which, upon application to the plant, are intended to produce output-space errors p that track the commands p_com. According to the invention, the calibration law unit 300 modifies parameters of the inverse kinematics/actuation model 107.

[0162] Referring to FIG. 5, there is shown a detail of the calibration law unit 300 referenced in FIG. 4. According to the invention, the calibration law unit 300 is provided with the dither references 120 and the measured power, and produces the parameters used in the inverse kinematics/actuation model 107 referenced in FIG. 4. The mode function generator 301 forms the eight mode functions G_k(k=1−6, 9−10) (the analytical sensitivities of the log of the coupled power to modeling error ε_k) using products of the dither references scaled by a set of model parameters, which can be updated (optionally) using the modified model parameters. The mode functions G_kare multiplied at the multiplier 305 by the negative of the log of the measured power supplied by log function 303. The moving average filter 307 filters the output of the multiplier to pass the low-frequency components while squelching signal content at frequencies equal to the dither frequencies, the sum and difference of the dither frequencies, and harmonics of the dither frequencies. The signals ε_k(k=1−6, 9−10) passed by the filter 307 are the instantaneous estimates of the errors in the model parameters. The integrator block 309 integrates these error estimates to produce smoothed corrections to the model parameters, λ_k(k=1−6, 9−10), which are subtracted from the initial parameter estimates at summer junction 311 to produce the parameters used in the inverse kinematics/actuation model 107 as shown in FIG. 4.

[0163] Novel Features

[0164] The calibration technique detects modeling errors through the observation of undesirable signal content in the coupled power at frequencies equal to the sum, difference, and harmonics of the dithers injected for the purpose of primary functions such as alignment detection and power attenuation. Thus, the technique does not require that any calibration-specific signals be injected into the system, or that the operation of primary functions be halted for the purpose of calibration. The calibration function can track model terms during operation to ensure continuous optimal performance. The technique can run independently as well, requiring only the presence of the dithers.

[0165] Alternate Implementations

[0166] The invention has been explained with reference to specific embodiments. Other embodiments will be evident to those of ordinary skill in the art. For instance, the estimation of model terms can be implemented in batch mode rather than continuously as described. Also, whereas the coupling model and associated derivatives were derived for the case of an analytical model, the model and its derivatives may also be formed using a numerical approach using experimental or simulated data. The approach used to estimate modeling errors was presented for the case of constant-power dithering using sine and cosine excitation at two frequencies, and for a system with a pair of beam-steering elements having four degrees of freedom, but the approach is not limited to this case. The equations may be readily generalized for systems using other forms of dithering and employing one or more beam-steering elements. It is therefore not intended that this invention be limited, except as indicated by the appended claims.