Chapter 23: Optical Interconnects and Switches =============================================== Source: Saleh and Teich, *Fundamentals of Photonics*, second edition, Chapter 23. Section 23.1 Formulation — Interconnection Matrix ------------------------------------------------- In optical communication and photonic integrated circuits, multi-port optical couplers and routers connect :math:`M` input channels to :math:`N` output channels. In *Fundamentals of Photonics* (Saleh & Teich, 2nd ed., Section 23.1), the interconnection matrix is introduced via three baseline equations: 1. **Coherent field superposition** (Eq. 23.1-1): .. math:: U_\ell^{(o)} = \sum_{m=1}^M T_{\ell m} U_m^{(i)} 2. **Ideal 3-dB coupler example** (Eq. 23.1-2): .. math:: \mathbf{T} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & j \\ j & 1 \end{bmatrix} 3. **Incoherent power distribution** (Eq. 23.1-3): .. math:: P_\ell^{(o)} = \sum_{m=1}^M |T_{\ell m}|^2 P_m^{(i)} While these three relations provide the starting point, practical photonic design requires: * The explicit **coherent output power** including cross-channel interferometric beating terms. * The physical constraints imposed by **energy conservation**, which dictate that lossless transmission matrices :math:`\mathbf{T}` are unitary and the corresponding power transfer matrices :math:`\mathbf{S}` are orthostochastic. * A unified formulation using the **mutual coherence matrix** (or cross-spectral density matrix) that smoothly bridges fully coherent, partially coherent, and fully incoherent regimes. 1. Coherent Multiport Fields and Output Power ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 1.1 Matrix Formulation of Linear Superposition ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Let the monochromatic (or quasi-monochromatic) complex field amplitudes at the :math:`M` input ports and :math:`N` output ports be represented by column vectors: .. math:: \mathbf{U}^{(i)} = \begin{bmatrix} U_1^{(i)} \\ U_2^{(i)} \\ \vdots \\ U_M^{(i)} \end{bmatrix} \in \mathbb{C}^M, \qquad \mathbf{U}^{(o)} = \begin{bmatrix} U_1^{(o)} \\ U_2^{(o)} \\ \vdots \\ U_N^{(o)} \end{bmatrix} \in \mathbb{C}^N. For any linear, time-invariant optical interconnect, the output field vector is related to the input field vector by an :math:`N \times M` complex interconnection matrix :math:`\mathbf{T}`: .. math:: :label: eq-coherent-field-matrix \mathbf{U}^{(o)} = \mathbf{T} \mathbf{U}^{(i)}, \qquad U_\ell^{(o)} = \sum_{m=1}^M T_{\ell m} U_m^{(i)}. Each matrix element :math:`T_{\ell m}` is a complex transmission coefficient characterizing the path from input port :math:`m` to output port :math:`\ell`: .. math:: T_{\ell m} = |T_{\ell m}| e^{j \theta_{\ell m}}, where :math:`|T_{\ell m}|` represents the field amplitude coupling ratio and :math:`\theta_{\ell m} = \arg(T_{\ell m})` is the optical phase delay through the device between port :math:`m` and port :math:`\ell`. 1.2 Expansion of Coherent Output Power ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Each input field phasor can be expressed in terms of its optical power :math:`P_m^{(i)} = |U_m^{(i)}|^2` and initial phase :math:`\phi_m`: .. math:: U_m^{(i)} = \sqrt{P_m^{(i)}} \, e^{j \phi_m}. The optical power emerging at output port :math:`\ell` is the squared modulus of the total superposed field: .. math:: P_\ell^{(o)} = |U_\ell^{(o)}|^2 = \left( \sum_{m=1}^M T_{\ell m} U_m^{(i)} \right) \left( \sum_{k=1}^M T_{\ell k}^* U_k^{(i)*} \right). Expanding the double summation into self-power terms (:math:`m = k`) and mutual cross terms (:math:`m \neq k`): .. math:: P_\ell^{(o)} = \sum_{m=1}^M |T_{\ell m}|^2 |U_m^{(i)}|^2 + \sum_{m=1}^M \sum_{k \neq m}^M T_{\ell m} T_{\ell k}^* U_m^{(i)} U_k^{(i)*}. Pairing complex conjugate indices (:math:`m < k`) gives the complete coherent power relation: .. math:: :label: eq-coherent-power-complete \boxed{P_\ell^{(o)} = \underbrace{\sum_{m=1}^M |T_{\ell m}|^2 P_m^{(i)}}_{\text{Direct power contribution}} + \underbrace{2 \sum_{m < k}^M |T_{\ell m}| |T_{\ell k}| \sqrt{P_m^{(i)} P_k^{(i)}} \cos\Big( (\phi_m - \phi_k) + (\theta_{\ell m} - \theta_{\ell k}) \Big)}_{\text{Interferometric cross-terms}}.} .. note:: * The first summation matches the incoherent power equation (Eq. 23.1-3). * The second summation represents **constructive and destructive interference**. It depends directly on the relative input phases :math:`\Delta \phi_{mk} = \phi_m - \phi_k` as well as the internal device differential phase delays :math:`\Delta \theta_{\ell, mk} = \theta_{\ell m} - \theta_{\ell k}`. 1.3 Example: Symmetrical 2 × 2 3-dB Coupler ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For an ideal symmetric 3-dB directional coupler (Saleh & Teich Eq. 23.1-2): .. math:: \mathbf{T} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & j \\ j & 1 \end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & e^{j\pi/2} \\ e^{j\pi/2} & 1 \end{bmatrix}. The output fields are: .. math:: U_1^{(o)} = \frac{1}{\sqrt{2}}\left( U_1^{(i)} + j U_2^{(i)} \right), \qquad U_2^{(o)} = \frac{1}{\sqrt{2}}\left( j U_1^{(i)} + U_2^{(i)} \right). Applying Eq. :eq:`eq-coherent-power-complete`: .. math:: P_1^{(o)} &= \frac{1}{2} P_1^{(i)} + \frac{1}{2} P_2^{(i)} + \sqrt{P_1^{(i)} P_2^{(i)}} \cos\left( (\phi_1 - \phi_2) - \frac{\pi}{2} \right) \\ &= \frac{1}{2}\left( P_1^{(i)} + P_2^{(i)} \right) + \sqrt{P_1^{(i)} P_2^{(i)}} \sin(\phi_1 - \phi_2), .. math:: P_2^{(o)} &= \frac{1}{2} P_1^{(i)} + \frac{1}{2} P_2^{(i)} + \sqrt{P_1^{(i)} P_2^{(i)}} \cos\left( (\phi_1 - \phi_2) + \frac{\pi}{2} \right) \\ &= \frac{1}{2}\left( P_1^{(i)} + P_2^{(i)} \right) - \sqrt{P_1^{(i)} P_2^{(i)}} \sin(\phi_1 - \phi_2). If the two input powers are equal (:math:`P_1^{(i)} = P_2^{(i)} = P_0`) with relative phase :math:`\Delta \phi = \phi_1 - \phi_2`: .. math:: P_1^{(o)} = P_0 \big(1 + \sin \Delta \phi\big), \qquad P_2^{(o)} = P_0 \big(1 - \sin \Delta \phi\big). * When :math:`\Delta \phi = +\pi/2`: :math:`P_1^{(o)} = 2 P_0` (all light exits port 1) and :math:`P_2^{(o)} = 0`. * When :math:`\Delta \phi = -\pi/2`: :math:`P_1^{(o)} = 0` and :math:`P_2^{(o)} = 2 P_0` (all light exits port 2). * Total power is conserved at every phase: :math:`P_1^{(o)} + P_2^{(o)} = 2 P_0 = P_1^{(i)} + P_2^{(i)}`. 2. Incoherent Power Formulation ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 2.1 Mutual Incoherence and Time Averaging ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ When the input beams originate from independent, uncorrelated sources, or when their frequency difference exceeds the detector integration bandwidth, the relative optical phases fluctuate randomly over time: .. math:: \left\langle e^{j(\phi_m(t) - \phi_k(t))} \right\rangle = 0 \quad \text{for } m \neq k, where :math:`\langle \cdot \rangle` denotes statistical / time averaging. Under this condition, the cross-interference terms in Eq. :eq:`eq-coherent-power-complete` average strictly to zero: .. math:: :label: eq-incoherent-power P_\ell^{(o)} = \left\langle |U_\ell^{(o)}(t)|^2 \right\rangle = \sum_{m=1}^M |T_{\ell m}|^2 P_m^{(i)}. This is Equation (23.1-3) of Saleh & Teich. 2.2 Power (Intensity) Transfer Matrix ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Defining the input power vector :math:`\mathbf{P}^{(i)} = [P_1^{(i)}, \dots, P_M^{(i)}]^T` and output power vector :math:`\mathbf{P}^{(o)} = [P_1^{(o)}, \dots, P_N^{(o)}]^T`, the incoherent mapping is: .. math:: :label: eq-power-transfer-matrix \mathbf{P}^{(o)} = \mathbf{S} \mathbf{P}^{(i)}, where :math:`\mathbf{S}` is the real, non-negative :math:`N \times M` **power transfer matrix**: .. math:: S_{\ell m} = |T_{\ell m}|^2 \ge 0. For an ideal 2 × 2 3-dB coupler: .. math:: \mathbf{S} = \begin{bmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{bmatrix} \implies \begin{bmatrix} P_1^{(o)} \\ P_2^{(o)} \end{bmatrix} = \begin{bmatrix} \frac{1}{2}(P_1^{(i)} + P_2^{(i)}) \\ \frac{1}{2}(P_1^{(i)} + P_2^{(i)}) \end{bmatrix}. Regardless of the relative phases or internal phase delays, mutually incoherent inputs always split evenly across the output ports. 3. Unified Formulation: Mutual Coherence Matrix ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ To describe fields with arbitrary spatial and temporal coherence, we use the second-order **mutual coherence matrix** (or spatial cross-spectral density matrix). 3.1 Coherence Matrix Definition ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The mutual spatial coherence among the :math:`M` input ports is captured by the matrix :math:`\mathbf{J}^{(i)} \in \mathbb{C}^{M \times M}`: .. math:: \mathbf{J}^{(i)} \equiv \left\langle \mathbf{U}^{(i)} \mathbf{U}^{(i)\dagger} \right\rangle, \qquad J_{mn}^{(i)} = \left\langle U_m^{(i)} U_n^{(i)*} \right\rangle, where :math:`\dagger` denotes the Hermitian conjugate (transpose conjugate). * The diagonal entries are the optical powers: :math:`J_{mm}^{(i)} = \langle |U_m^{(i)}|^2 \rangle = P_m^{(i)}`. * The off-diagonal entries represent mutual correlation: .. math:: J_{mn}^{(i)} = \sqrt{P_m^{(i)} P_n^{(i)}} \, \gamma_{mn}^{(i)}, where :math:`\gamma_{mn}^{(i)}` is the complex degree of spatial coherence between ports :math:`m` and :math:`n` (:math:`0 \le |\gamma_{mn}^{(i)}| \le 1`). 3.2 Transformation through the Linear Interconnect ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Applying the linear field transformation :math:`\mathbf{U}^{(o)} = \mathbf{T} \mathbf{U}^{(i)}`: .. math:: \mathbf{J}^{(o)} = \left\langle \mathbf{U}^{(o)} \mathbf{U}^{(o)\dagger} \right\rangle = \left\langle (\mathbf{T} \mathbf{U}^{(i)}) (\mathbf{T} \mathbf{U}^{(i)})^\dagger \right\rangle = \mathbf{T} \left\langle \mathbf{U}^{(i)} \mathbf{U}^{(i)\dagger} \right\rangle \mathbf{T}^\dagger. .. math:: :label: eq-coherence-transformation \boxed{\mathbf{J}^{(o)} = \mathbf{T} \mathbf{J}^{(i)} \mathbf{T}^\dagger.} The output power at port :math:`\ell` is simply the :math:`\ell`-th diagonal element of :math:`\mathbf{J}^{(o)}`: .. math:: P_\ell^{(o)} = J_{\ell \ell}^{(o)} = \left[ \mathbf{T} \mathbf{J}^{(i)} \mathbf{T}^\dagger \right]_{\ell \ell} = \sum_{m=1}^M \sum_{n=1}^M T_{\ell m} T_{\ell n}^* J_{mn}^{(i)}. Expanding this in terms of :math:`\gamma_{mn}^{(i)}`: .. math:: :label: eq-general-power-master \boxed{P_\ell^{(o)} = \sum_{m=1}^M |T_{\ell m}|^2 P_m^{(i)} + 2 \sum_{m < n}^M |T_{\ell m}| |T_{\ell n}| \sqrt{P_m^{(i)} P_n^{(i)}} \, \mathrm{Re}\Big( \gamma_{mn}^{(i)} e^{j(\theta_{\ell m} - \theta_{\ell n})} \Big)} 3.3 Limiting Cases of the Master Equation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1. **Fully Coherent Inputs** (:math:`|\gamma_{mn}^{(i)}| = 1`): Here :math:`\gamma_{mn}^{(i)} = e^{j(\phi_m - \phi_n)}`, and the matrix :math:`\mathbf{J}^{(i)}` has rank 1: .. math:: \mathbf{J}^{(i)} = \mathbf{U}^{(i)} \mathbf{U}^{(i)\dagger}. Equation :eq:`eq-general-power-master` exactly reproduces the coherent interference expansion in Eq. :eq:`eq-coherent-power-complete`. 2. **Mutually Incoherent Inputs** (:math:`\gamma_{mn}^{(i)} = \delta_{mn}`): The off-diagonal terms vanish, and :math:`\mathbf{J}^{(i)}` becomes purely diagonal: .. math:: \mathbf{J}^{(i)} = \operatorname{diag}\left( P_1^{(i)}, P_2^{(i)}, \dots, P_M^{(i)} \right). Equation :eq:`eq-general-power-master` collapses to Eq. (23.1-3): .. math:: P_\ell^{(o)} = \sum_{m=1}^M |T_{\ell m}|^2 P_m^{(i)}. 3. **Partially Coherent Inputs** (:math:`0 < |\gamma_{mn}^{(i)}| < 1`): Interference fringes persist, but their visibility (contrast) is scaled by :math:`|\gamma_{mn}^{(i)}|`. 4. Energy Conservation and Mathematical Constraints ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 4.1 Lossless Interconnects: Unitarity of Matrix T ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For an :math:`N \times N` coupler without dissipation, scattering, or radiation loss, the total output power must equal the total input power for **any** arbitrary input field state :math:`\mathbf{U}^{(i)}`: .. math:: P_{\mathrm{total}}^{(o)} = \mathbf{U}^{(o)\dagger} \mathbf{U}^{(o)} = \mathbf{U}^{(i)\dagger} (\mathbf{T}^\dagger \mathbf{T}) \mathbf{U}^{(i)} = \mathbf{U}^{(i)\dagger} \mathbf{U}^{(i)} = P_{\mathrm{total}}^{(i)}. This condition holds for all :math:`\mathbf{U}^{(i)}` if and only if :math:`\mathbf{T}` is a **unitary matrix**: .. math:: :label: eq-unitarity \mathbf{T}^\dagger \mathbf{T} = \mathbf{I} \quad \text{and} \quad \mathbf{T} \mathbf{T}^\dagger = \mathbf{I}. In terms of elements: * **Column orthonormality** (conservation from input ports): .. math:: \sum_{\ell=1}^N T_{\ell m} T_{\ell k}^* = \delta_{mk}. * **Row orthonormality** (conservation to output ports): .. math:: \sum_{m=1}^N T_{\ell m} T_{k m}^* = \delta_{\ell k}. 4.2 Orthostochasticity of Power Transfer Matrix S ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Evaluating the column and row conditions for :math:`m = k` and :math:`\ell = k`: .. math:: \sum_{\ell=1}^N |T_{\ell m}|^2 = \sum_{\ell=1}^N S_{\ell m} = 1 \quad (\forall m), \qquad \sum_{m=1}^N |T_{\ell m}|^2 = \sum_{m=1}^N S_{\ell m} = 1 \quad (\forall \ell). Therefore: * Every column sums to 1 (all power injected into input :math:`m` is distributed among outputs). * Every row sums to 1. A non-negative matrix whose rows and columns both sum to 1 is called **doubly stochastic** (or bistochastic). A matrix whose elements are the squared moduli of a unitary matrix (:math:`S_{\ell m} = |T_{\ell m}|^2`) is specifically known as an **orthostochastic matrix**. 4.3 Lossy Couplers and Singular Value Limits ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Real couplers exhibit excess loss and finite port isolation. For passive lossy systems: * The eigenvalues of :math:`\mathbf{T}^\dagger \mathbf{T}` are bounded: :math:`0 \le \lambda_j(\mathbf{T}^\dagger \mathbf{T}) \le 1`. * The singular values of :math:`\mathbf{T}` satisfy :math:`\sigma_j(\mathbf{T}) \le 1`. * The column sums of :math:`\mathbf{S}` satisfy :math:`\sum_{\ell=1}^N S_{\ell m} \le 1`. 5. Coupler Performance Specifications ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ In optical engineering, practical multi-port couplers and switches are specified by standard logarithmic power ratios: * **Insertion Loss (IL)** between input port :math:`m` and output port :math:`\ell`: .. math:: \mathrm{IL}_{\ell m} = -10 \log_{10}\left( \frac{P_\ell^{(o)}}{P_m^{(i)}} \right) = -10 \log_{10}|T_{\ell m}|^2 = -10 \log_{10} S_{\ell m} \quad [\mathrm{dB}]. * **Splitting Ratio** for port :math:`\ell`: .. math:: \mathrm{SR}_\ell = \frac{P_\ell^{(o)}}{\sum_{k=1}^N P_k^{(o)}} = \frac{S_{\ell m}}{\sum_{k=1}^N S_{k m}}. * **Excess Loss (EL)** of the device when driven by port :math:`m`: .. math:: \mathrm{EL}_m = -10 \log_{10}\left( \frac{\sum_{\ell=1}^N P_\ell^{(o)}}{P_m^{(i)}} \right) = -10 \log_{10}\left( \sum_{\ell=1}^N S_{\ell m} \right) \quad [\mathrm{dB}]. * **Crosstalk / Isolation** between unintended output port :math:`\ell` and intended output port :math:`k`: .. math:: \mathrm{XT}_{\ell k} = 10 \log_{10}\left( \frac{P_\ell^{(o)}}{P_k^{(o)}} \right) \quad [\mathrm{dB}]. 6. Summary Comparison Table ^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. list-table:: Comparison of Coherent, Incoherent, and Partially Coherent Regimes :widths: 22 26 26 26 :header-rows: 1 * - Characteristic - Coherent Regime (Eq. 23.1-1) - Incoherent Regime (Eq. 23.1-3) - General / Partially Coherent * - **Primary Variable** - Field vector :math:`\mathbf{U} \in \mathbb{C}^N` - Power vector :math:`\mathbf{P} \in \mathbb{R}^N` - Coherence matrix :math:`\mathbf{J} \in \mathbb{C}^{N \times N}` * - **Governing Relation** - :math:`\mathbf{U}^{(o)} = \mathbf{T} \mathbf{U}^{(i)}` - :math:`\mathbf{P}^{(o)} = \mathbf{S} \mathbf{P}^{(i)}` - :math:`\mathbf{J}^{(o)} = \mathbf{T} \mathbf{J}^{(i)} \mathbf{T}^\dagger` * - **Matrix Nature** - Complex :math:`\mathbf{T}` (unitary if lossless) - Real :math:`S_{\ell m} = |T_{\ell m}|^2` (orthostochastic) - Bilinear transformation with :math:`\mathbf{T}, \mathbf{T}^\dagger` * - **Output Power** - :math:`|\sum_m T_{\ell m} U_m^{(i)}|^2` - :math:`\sum_m |T_{\ell m}|^2 P_m^{(i)}` - :math:`[\mathbf{T} \mathbf{J}^{(i)} \mathbf{T}^\dagger]_{\ell\ell}` * - **Phase Dependence** - Fully sensitive to relative source & device phases - Insensitive to optical phases - Scaled by degree of coherence :math:`|\gamma_{mn}|` * - **Interference Terms** - Present (:math:`\propto 2 \sqrt{P_m P_k} \cos \Delta \Phi`) - Strictly zero - Scaled (:math:`\propto 2 \sqrt{P_m P_k} |\gamma| \cos \Delta \Phi`) In-text exercises ----------------- Exercise 23.1-1 — Interconnection capacity ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Step 1 — Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. .. _fop-exercise-23-1-1-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_23_01_01.svg :alt: Illustrated calculation map for Exercise 23.1-1, Interconnection capacity :align: center :width: 95% **Figure 129 — Exercise 23.1-1: Interconnection capacity.** The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself. **Step 2 — Mathematical formulas used.** The working uses :ref:`stationary-value condition ` and :ref:`algebraic rearrangement and dimensional checks `. **Step 3 — Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. An aperture of width :math:`a` contains :math:`Ba` independent grating samples in either transverse coordinate, hence :math:`(Ba)^2` independent space-frequency cells. Assigning :math:`M` directions to each of :math:`L` inputs consumes :math:`ML` cells, proving :math:`\boxed{ML\le(Ba)^2}`. If every input is connected to every output, the maximum density is therefore :math:`\boxed{B^2=10^6\ \mathrm{interconnections/mm^2}}` for 1000 lines/mm. **Step 4 — State the numbered result.** The principal result obtained in the working is .. math:: :label: fop-exercise-23-1-1-result \boxed{B^2=10^6\ \mathrm{interconnections/mm^2}} **Step 5 — Check.** Equation :eq:`fop-exercise-23-1-1-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Exercise 23.1-2 — Separable logarithmic map ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Step 1 — Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. .. _fop-exercise-23-1-2-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_23_01_02.svg :alt: Illustrated calculation map for Exercise 23.1-2, Separable logarithmic map :align: center :width: 95% **Figure 130 — Exercise 23.1-2: Separable logarithmic map.** The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself. **Step 2 — Mathematical formulas used.** The working uses :ref:`product, quotient, and chain rules `, :ref:`exponential, logarithmic, and phasor identities `, and :ref:`algebraic rearrangement and dimensional checks `. **Step 3 — Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Differentiate the proposed phase: .. math:: :label: fop-exercise-23-1-2-eq-1 {\partial\phi\over\partial x}={2\pi\over\lambda d}(\ln x-x),\qquad {\partial\phi\over\partial y}={2\pi\over\lambda d}(\ln y-y). Equation (23.1-7) then gives :math:`x'=x+(\lambda d/2\pi)\phi_x=\ln x` and likewise :math:`y'=\ln y`, which proves the map. **Step 4 — Interpret the result.** The final relation or conclusion in Step 3 is the requested result. Read its sign, scale, or physical classification using the conventions fixed in Step 1. **Step 5 — Check.** Equation :eq:`fop-exercise-23-1-2-eq-1` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Exercise 23.4-1 — Bistable nonlinearities ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Step 1 — Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. .. _fop-exercise-23-4-1-illustration: .. figure:: /_static/knowledge_base/worked_exercises/fundamentals_of_photonics/exercise_illustrations/exercise_23_04_01.svg :alt: Illustrated calculation map for Exercise 23.4-1, Bistable nonlinearities :align: center :width: 95% **Figure 131 — Exercise 23.4-1: Bistable nonlinearities.** The diagram identifies the input quantities, physical operation, requested result, variable meanings, and an independent verification route. Every symbol in the variable strip is labeled on the model itself. **Step 2 — Mathematical formulas used.** The working uses :ref:`stationary-value condition `, :ref:`product, quotient, and chain rules `, and :ref:`algebraic rearrangement and dimensional checks `. **Step 3 — Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. For each candidate plot :math:`y(x)=x/\eta(x)` and locate folds from :math:`dy/dx=0`; two folds delimit a three-valued output interval. Working choices are (a) :math:`a=0.2`, (b) :math:`a=5,\theta=0`, (c) :math:`\theta=0`, (d) :math:`a=0.5`, and (e) :math:`a=10`. For example, case (e) has :math:`y=x(x+a)^2/(x+1)^2` and its stationary numerator is :math:`x^2+(3-a)x+a`; at :math:`a=10` the folds are exactly :math:`\boxed{x=2,5}`. The same derivative test, rather than visual guesswork, verifies the other four plots. End-of-chapter problems ----------------------- **Step 4 — State the numbered result.** The principal result obtained in the working is .. math:: :label: fop-exercise-23-4-1-result \boxed{x=2,5} **Step 5 — Check.** Equation :eq:`fop-exercise-23-4-1-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Problem 23.1-3 — Conformal-map hologram ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`product, quotient, and chain rules `, :ref:`vector-calculus identities `, and :ref:`exponential, logarithmic, and phasor identities `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. For a single continuous phase mask, Eq. (23.1-7) would require :math:`\phi_x\propto\ln r-x` and :math:`\phi_y\propto\tan^{-1}(y/x)-y`. But .. math:: :label: fop-problem-23-1-3-eq-1 \partial_y(\ln r-x)={y\over r^2},\qquad \partial_x[\tan^{-1}(y/x)-y]=-{y\over r^2}. The mixed derivatives disagree, so :math:`\boxed{\text{no scalar phase function exists for this map in one thin hologram}}`. It requires at least a two-element coordinate transformer (or a segmented/nonlocal implementation); the curl test is the essential design result. **Check.** Equation :eq:`fop-problem-23-1-3-eq-1` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Problem 23.2-1 — Four-channel cascaded MZIs ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Near :math:`\lambda_0`, :math:`\Delta\nu=c\Delta\lambda/\lambda_0^2= 24.96` GHz. Adjacent channels must swap ports in the first MZI, so :math:`\Delta d=c/(2n\Delta\nu)=\boxed{2.612\ \mathrm{mm}}`. Each second- stage MZI separates channels spaced by :math:`2\Delta\nu`, so both use :math:`\boxed{1.306\ \mathrm{mm}}`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-23-2-1-result \boxed{1.306\ \mathrm{mm}} **Check.** Equation :eq:`fop-problem-23-2-1-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Problem 23.2-2 — WGR wavelength increment ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`optical path and Fermat's principle ` and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Adjacent WGR outputs require an optical path increment equal to the channel spacing: :math:`n\Delta d_b=\Delta\lambda`. Hence :math:`\boxed{\Delta d_b=0.2\ \mathrm{nm}/2.3=0.08696\ \mathrm{nm}}` in the star-coupler material. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-23-2-2-result \boxed{\Delta d_b=0.2\ \mathrm{nm}/2.3=0.08696\ \mathrm{nm}} **Check.** Equation :eq:`fop-problem-23-2-2-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Problem 23.2-3 — Two-by-two wavelength transpose ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. An :math:`l\to m` path transmits wavelength :math:`\lambda` when :math:`n\Delta d_{lm}=q_{lm}\lambda` for an integer order, while the rejected wavelength is not an integer divisor. Choose :math:`\Delta d_{11}` resonant for :math:`\lambda_1`, :math:`\Delta d_{12}` for :math:`\lambda_2`, :math:`\Delta d_{21}` for :math:`\lambda_3`, and :math:`\Delta d_{22}` for :math:`\lambda_4`; explicitly :math:`\boxed{n\Delta d_{11}=q_1\lambda_1, n\Delta d_{12}=q_2\lambda_2,n\Delta d_{21}=q_3\lambda_3, n\Delta d_{22}=q_4\lambda_4}`. Selecting integers that make every unwanted ratio nonintegral completes the router. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-23-2-3-result \boxed{n\Delta d_{11}=q_1\lambda_1, n\Delta d_{12}=q_2\lambda_2,n\Delta d_{21}=q_3\lambda_3, n\Delta d_{22}=q_4\lambda_4} **Check.** Equation :eq:`fop-problem-23-2-3-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Problem 23.3-1 — Cascaded-switch loss and crosstalk ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`power and decibel conversions ` and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. The worst route through the five-element 4-by-4 network traverses three 2-by-2 switches, so loss is :math:`\boxed{3(0.5)=1.5\ \mathrm{dB}}`. Adding three independent :math:`10^{-3}` leakage powers gives crosstalk :math:`10\log_{10}(3\times10^{-3})=\boxed{-25.2\ \mathrm{dB}}`; a deliberately conservative coherent phase alignment would instead bound it at -20.5 dB. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-23-3-1-result 10\log_{10}(3\times10^{-3})=\boxed{-25.2\ \mathrm{dB}} **Check.** Equation :eq:`fop-problem-23-3-1-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Converting the final decibel value back to a linear power ratio checks the logarithm, sign, and accumulated loss budget. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Problem 23.3-2 — MZI voltage-error crosstalk ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`power and decibel conversions `, :ref:`trigonometric and small-angle identities `, and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. The cross state needs :math:`\boxed{V=V_\pi}`. A 1% error leaves fractional leakage ratio :math:`\tan^2(0.01\pi/2)=2.468\times10^{-4}`, hence :math:`\boxed{XT=-36.08\ \mathrm{dB}}`. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-23-3-2-result \boxed{XT=-36.08\ \mathrm{dB}} **Check.** Equation :eq:`fop-problem-23-3-2-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Converting the final decibel value back to a linear power ratio checks the logarithm, sign, and accumulated loss budget. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension. Problem 23.3-3 — TSI with programmable delays ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Demultiplex the incoming frame into :math:`N` spatial lanes. Program lane :math:`i` with delay :math:`d_i=(\pi(i)-i)\bmod N` slots for the requested permutation :math:`\pi`; a second bank adds a common frame delay so all values are causal. Remultiplex lanes in their fixed order. This :math:`\boxed{\text{DEMUX}\to\text{programmable delays}\to\text{MUX}}` construction absorbs the original fixed-delay/space-switch/fixed-delay stages into the addressable delay settings. **Numbered result.** The principal result obtained in the working is .. math:: :label: fop-problem-23-3-3-result \boxed{\text{DEMUX}\to\text{programmable delays}\to\text{MUX}} **Check.** Equation :eq:`fop-problem-23-3-3-result` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale. Problem 23.4-2 — Threshold optical logic ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses the physical definitions stated in the item; no separate calculus or algebraic identity is required for this qualitative comparison. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. Sum equal optical inputs and choose a threshold between levels: between one and two units gives AND, while between zero and one gives OR. Complement the threshold device's output (or exchange bright/dark ports) for NAND and NOR. One scalar threshold cannot implement XOR because its truth set is not linearly separable; use an OR followed by suppression of the two-input level, or two threshold stages. The same sum with threshold :math:`0.5` implements OR for any :math:`N`. **Check.** For a qualitative conclusion, test every absolute statement against the stated assumptions and at least one limiting case or counterexample. Problem 23.4-3 — Kerr-feedback interferometer ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ **Definitions and setup.** Symbols are local to this item and follow the chapter convention. Each physical quantity and supplied numerical value is introduced at its first use below; angles are in radians unless a degree symbol is shown, and units are retained through numerical substitution. **Mathematical formulas used.** The working uses :ref:`stationary-value condition `, :ref:`trigonometric and small-angle identities `, and :ref:`algebraic rearrangement and dimensional checks `. **Worked derivation.** The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity. The Kerr arm phase is :math:`\Delta\phi=\pi I_o/I_\pi+\phi`, so MZI interference gives :math:`\boxed{I_o/I_i=[1+\cos(\pi I_o/I_\pi+\phi)]/2}`. For :math:`\phi=0`, write :math:`x=I_o/I_\pi` and :math:`y=I_i/I_\pi=2x/[1+\cos(\pi x)]`. Then .. math:: :label: fop-problem-23-4-3-eq-1 {dI_o\over dI_i}= \left\{{2\over1+\cos\pi x}+{2\pi x\sin\pi x\over(1+\cos\pi x)^2}\right\}^{-1}. The ideal differential gain diverges at fold points where the denominator vanishes; physical loss and finite response time cap that formal maximum. **Check.** Equation :eq:`fop-problem-23-4-3-eq-1` can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model.