Chapter 21: Nonlinear Optics
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 21.
In-text exercises
Exercise 21.1-1 — Intensity needed for nonlinearity
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.
Figure 113 — Exercise 21.1-1: Intensity needed for nonlinearity. 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 exponential, logarithmic, and phasor identities and 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.
Since \(P_L=\epsilon_0(n^2-1)E\), \(P_2=2dE^2\), \(P_3=4\chi^{(3)}E^3\), and \(I=E^2/(\eta_0/n)\), setting the requested ratios to 0.01 gives
The enormous values explain why focused laser fields are normally required.
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 (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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Exercise 21.2-1 — Non-collinear type-II SHG
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.
Figure 114 — Exercise 21.2-1: Non-collinear type-II SHG. 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 exponential, logarithmic, and phasor identities, trigonometric and small-angle identities, and 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.
With \(n_e(\vartheta)^{-2}=\cos^2\vartheta/n_e^2+ \sin^2\vartheta/n_o^2\), solve
Sellmeier values for KDP at 1.06 and 0.53 micrometres inserted in these two equations give the complete one-parameter family; a root finder over \((\theta,\theta_1,\theta_2)\) reproduces it. The collinear endpoint is \(\boxed{\theta\simeq41^\circ,\ \theta_1=\theta_2=0}\), and continuation from that root gives the non-collinear branches.
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 (2) 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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Exercise 21.3-1 — DC-field-induced Kerr effect
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.
Figure 115 — Exercise 21.3-1: DC-field-induced Kerr effect. 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 exponential, logarithmic, and phasor identities and 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.
Expanding \(4\chi^{(3)}[E(0)+E(\omega)]^3\), the terms at \(\omega\) are \(12\chi^{(3)}E^2(0)E(\omega)\). Equating this to \(2n\epsilon_0\Delta nE(\omega)\) gives \(\Delta n=6\chi^{(3)}E^2(0)/(n\epsilon_0)=-s n^3E^2(0)/2\), hence \(\boxed{s=-12\chi^{(3)}/(\epsilon_0n^4)}\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (3) 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 21.3-2 — Optical Kerr lens
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.
Figure 116 — Exercise 21.3-2: Optical Kerr lens. 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 exponential, logarithmic, and phasor identities and 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.
Near the beam axis, \(n(I)d\simeq\mathrm{constant}-n_2I_0d(x^2+y^2)/W^2\). Matching its phase to \(\exp[jk_0(x^2+y^2)/(2f)]\) gives \(\boxed{f=W^2/(2n_2I_0d)}\); the sign follows the sign convention for propagation and \(n_2\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (4) 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 21.3-3 — Self- and cross-phase modulation
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.
Figure 117 — Exercise 21.3-3: Self- and cross-phase modulation. 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 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.
Collecting all cubic products at \(\omega_1\) gives a self term \(|E_1|^2E_1\) and two permutations for each other wave. Converting field strength to intensity yields \(\boxed{\Delta n_1=n_2(I_1+2I_2+2I_3)}\). Therefore wave 1 propagates at \(c_0/(n+\Delta n_1)\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (5) 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 21.4-1 — Degenerate three-wave mixing
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.
Figure 118 — Exercise 21.4-1: Degenerate three-wave mixing. 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 product, quotient, and chain rules, exponential, logarithmic, and phasor identities, and 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.
Put \(E=E_1e^{j\omega t}+E_3e^{j2\omega t}+\mathrm{c.c.}\) in \(P_{NL}=2dE^2\). Terms at \(\omega\) occur twice, whereas the \(2\omega\) product \(E_1E_1\) occurs once. Applying \(S=\mu_0\partial_t^2P_{NL}\) gives \(\boxed{S_1=2\mu_0\omega^2dE_3E_1^*,\ S_3=4\mu_0\omega^2dE_1^2}\), equivalent to Eqs. (21.4-14)–(21.4-16).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (6) 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 21.4-2 — Manley–Rowe relation
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.
Figure 119 — Exercise 21.4-2: Manley–Rowe relation. 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 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.
Multiply each coupled equation by its conjugate amplitude and add the complex conjugate. The common interaction term then gives \(\boxed{d|a_1|^2/dz=d|a_2|^2/dz=-d|a_3|^2/dz}\). These are photon-flux changes: one photon from each lower-frequency wave makes one sum-frequency photon.
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (7) 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 21.4-3 — Energy conservation
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.
Figure 120 — Exercise 21.4-3: Energy conservation. 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 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.
Weight the preceding derivatives by \(\hbar\omega_q\). Because \(\omega_1+\omega_2=\omega_3\), their sum vanishes: \(\boxed{d[\hbar\sum_q\omega_q|a_q|^2]/dz=0}\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (8) 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 21.4-4 — SHG envelope equations
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.
Figure 121 — Exercise 21.4-4: SHG envelope equations. 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 common differential-equation solutions, integration identities, and exponential, logarithmic, and phasor identities.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Substitute \(E_q=a_q(z)e^{-jk_qz}\) in the two Helmholtz equations and use \(|a_q''|\ll|k_qa_q'|\). Division by \(-2jk_q\) gives \(da_1/dz=-j2g a_1^*a_3e^{j\Delta kz}\) and \(da_3/dz=-jg a_1^2e^{-j\Delta kz}\), with the chapter normalization of \(g\); the factor two occurs only in the fundamental equation.
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (9) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiating the proposed solution and substituting it into the original differential equation verifies the functional form.
Exercise 21.4-5 — Infrared up-conversion
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.
Figure 122 — Exercise 21.4-5: Infrared up-conversion. 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 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.
Frequency addition gives \(\lambda_3^{-1}=\lambda_1^{-1}+\lambda_2^{-1}\), hence \(\boxed{\lambda_3=0.9636\ \mu\mathrm m}\). With the undepleted-pump formula and the given \(d^2/n^3\), area, power, and 1-cm length, \(\boxed{\eta_{\rm OFC}=5.31\times10^{-3}}\) (0.531%).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (10) 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 21.4-6 — KTP parametric amplifier
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.
Figure 123 — Exercise 21.4-6: KTP parametric amplifier. 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 power and decibel conversions, integration identities, and trigonometric and small-angle identities.
Step 3 — Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
Difference-frequency conservation gives \(\boxed{\lambda_i=1.852\ \mu\mathrm m}\). From Eq. (21.4-47), \(C=[2\omega_s\omega_i(\eta_0/n)^3d^2]^{1/2}= \boxed{8.99\times10^{-5}}\) in its stated SI normalization. A 3-dB power gain requires \(\cosh^2(C L\sqrt{P/A})=2\), so \(\boxed{P/A=2.40\times10^{11}\ \mathrm{W/m^2}}\); for example, a 1-W pump focused to \(4.16\times10^{-12}\ \mathrm{m^2}\) satisfies it.
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (11) 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.
Exercise 21.5-1 — Undepleted-pump THG
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.
Figure 124 — Exercise 21.5-1: Undepleted-pump THG. 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 integration identities, exponential, logarithmic, and phasor identities, and 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.
Retaining the cubic source terms at \(\omega\) and \(3\omega\), then applying SVEA, gives \(da_3/dz=-jg a_1^3e^{-j\Delta kz}\) when \(a_1\) is undepleted, with \(\boxed{g=3\chi^{(3)}\omega_3(\eta_1^3\eta_3)^{1/2}/2}\) under the chapter’s flux-amplitude normalization. Integration adds the familiar \(L\,\mathrm{sinc}(\Delta kL/2)\) phase-matching factor.
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (12) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Exercise 21.7-1 — Anharmonic oscillator polarization
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.
Figure 125 — Exercise 21.7-1: Anharmonic oscillator polarization. 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 common differential-equation solutions, exponential, logarithmic, and phasor identities, and 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.
Start with \(m\ddot x+m\gamma\dot x+Kx+K_2x^2=-eE\) and set \(P=-Nex\). Multiplication by \(-Ne/m\) gives Eq. (21.7-8), with \(\boxed{\omega_0^2=K/m,\ \chi_0=Ne^2/(\epsilon_0m\omega_0^2),\ b=K_2/(e^3N^2)}\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (13) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiating the proposed solution and substituting it into the original differential equation verifies the functional form.
Exercise 21.7-2 — Miller’s rule
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.
Figure 126 — Exercise 21.7-2: Miller’s rule. 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 common differential-equation solutions and 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.
The first iteration supplies \(P_1(\omega)=\epsilon_0\chi(\omega)E(\omega)\). Driving the linear oscillator at \(\omega_3=\omega_1+\omega_2\) with \(-bP_1^2\) supplies one susceptibility at each of the three frequencies: \(\boxed{d(\omega_3;\omega_1,\omega_2)=C_M \chi(\omega_3)\chi(\omega_1)\chi(\omega_2)}\), where the material constant \(C_M\) follows from \(b,\chi_0\), proving Miller’s rule.
End-of-chapter problems
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (14) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiating the proposed solution and substituting it into the original differential equation verifies the functional form.
Problem 21.2-2 — Up-conversion power exchange
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 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.
\(1/\lambda_p=1/0.5-1/1.3\) gives \(\boxed{\lambda_p=0.8125\ \mu\mathrm m}\). One lost 1.3-micrometre photon creates one 0.5-micrometre photon while consuming one pump photon. Thus a 1-mW signal loss gives \(\boxed{2.60\ \mathrm{mW}}\) at 0.5 micrometres and \(\boxed{1.60\ \mathrm{mW}}\) pump loss.
Numbered result. The principal result obtained in the working is
Check. Equation (15) 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 21.2-3 — Collinear type-II KDP matching
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 trigonometric and small-angle identities and 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.
For each trial \(\theta\), evaluate the extraordinary index and solve \(n_o(\omega)+n_e(\theta,\omega)=2n_o(2\omega)\) for o-e-o, or replace the right side by \(2n_e(\theta,2\omega)\) for o-e-e. Bracketing \(0<\theta<90^\circ\) with the Table 5.5-1 Sellmeier equations gives the requested cut angles; substitution back into the equation is the residual check (a configuration with no sign change has no physical cut angle).
Numbered result. The principal result obtained in the working is
Check. Equation (16) 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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 21.2-4 — Degenerate KDP down-conversion
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 trigonometric and small-angle identities and 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.
Try the allowed type-I condition \(n_e(\theta,0.6)=n_o(1.2)\). The given indexes bracket 1.490, and \(n_e(\theta)^{-2}=\cos^2\theta/1.468^2+sin^2\theta/1.509^2\) gives \(\boxed{\theta=47.2^\circ}\). The 0.6-micrometre pump is extraordinary; both collinear 1.2-micrometre daughter waves are ordinary.
Numbered result. The principal result obtained in the working is
Check. Equation (17) 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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 21.2-5 — Linear-dispersion matching obstruction
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 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.
With \(n(\lambda)=n_0-\beta\lambda\) and \(1/\lambda_3=1/\lambda_1+1/\lambda_2\), the \(n_0\) terms satisfy energy conservation but the three \(-\beta\) contributions leave a nonzero constant in \(k_1+k_2-k_3\); co-propagating exact matching is impossible for \(\beta\ne0\). Reversing one wave changes a wavevector sign and can supply a root, so counter-propagating matching is possible.
Numbered result. The principal result obtained in the working is
Check. Equation (18) 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 21.2-6 — Finite-volume phase mismatch
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 integration identities and 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.
In the radiation integral use \(|\mathbf r-\mathbf r'|\simeq r-\hat{\mathbf r}\cdot\mathbf r'\). For a uniform rectangular source the remaining integral factorizes into \(V\prod_i\mathrm{sinc}(\Delta k_iL_i/2)\). Thus intensity contains \(\prod_i\mathrm{sinc}^2(\Delta k_iL_i/2)\) and the longitudinal first zero is \(|\Delta k_z|=2\pi/L_z\), quantifying phase-mismatch tolerance.
Numbered result. The principal result obtained in the working is
Check. Equation (19) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 21.2-7 — Backward quasi-phase-matched SHG
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 vector-calculus identities and 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.
Momentum conservation including grating vector \(K=2\pi/\Lambda\) is \(-k_{2\omega}=2k_\omega-mK\), so \(\boxed{mK=2k_\omega+k_{2\omega}}\). If dispersion is neglected, \(k_{2\omega}=2k_\omega\); for \(m=7\), \(\boxed{\Lambda/\lambda_\omega=7/4}\) where \(\lambda_\omega\) is the fundamental wavelength in the crystal.
Numbered result. The principal result obtained in the working is
Check. Equation (20) 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 21.3-4 — Four-wave Manley–Rowe invariants
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 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.
One elementary event destroys photons 1 and 2 and creates photons 3 and 4: \(d\Phi_1=d\Phi_2=-d\Phi_3=-d\Phi_4\). Therefore \(\Phi_1-\Phi_2\), \(\Phi_3-\Phi_4\), and \(\Phi_1+\Phi_3\) are invariant; multiplying by photon energies and using \(\omega_1+\omega_2=\omega_3+\omega_4\) proves energy conservation.
Numbered result. The principal result obtained in the working is
Check. Equation (21) 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 21.3-5 — Spatial-soliton power
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 integration identities and 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.
For \(I(x)=I_0\,\mathrm{sech}^2(x/W_0)\), integration gives \(P'=\int I\,dx=2I_0W_0\). The soliton condition has \(I_0\propto W_0^{-2}\), so \(\boxed{P'\propto W_0^{-1}}\) (power per unit extent in the invariant transverse direction).
Numbered result. The principal result obtained in the working is
Check. Equation (22) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 21.3-6 — Light-controlled phase modulator
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 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.
\(n_2=3\eta_0\chi^{(3)}/(\epsilon_0n^2)=2.19\times10^{-18}\) \(\mathrm{m^2/W}\). Setting \(k_0n_2IL=\pi\) gives \(I_\pi=1.24\times10^{12}\ \mathrm{W/m^2}\) and, for a square \(0.1\)-mm beam, \(\boxed{P_\pi\simeq12.4\ \mathrm{kW}}\) (multiply by \(\pi/4\) instead for a circular diameter convention).
Numbered result. The principal result obtained in the working is
Check. Equation (23) 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 21.3-7 — DC-assisted SHG
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.
The cubic product \(4\chi^{(3)}[E_0+E_\omega]^3\) contains a \(2\omega\) term proportional to \(E_0E_\omega^2\). It acts like an effective quadratic coefficient \(d_{\rm eff}\propto\chi^{(3)}E_0\); conversion therefore scales as \(|\chi^{(3)}|^2E_0^2I_\omega L^2\) times the phase-matching sinc-squared factor.
Check. For a qualitative conclusion, test every absolute statement against the stated assumptions and at least one limiting case or counterexample.
Problem 21.4-7 — KDP amplifier gain
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 power and decibel conversions, trigonometric and small-angle identities, and 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 idler wavelength is \(0.8570\) micrometres. Equation (21.4-47) gives \(C=1.295\times10^{-4}\) and \(\gamma=2C\sqrt I=25.90\ \mathrm{m^{-1}}\) at \(10^6\ \mathrm{W/cm^2}\). Thus \(\boxed{G=\cosh^2(\gamma L/2)=1.293=1.12\ \mathrm{dB}}\).
Numbered result. The principal result obtained in the working is
Check. Equation (24) 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 21.4-8 — Degenerate down-converter
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 exponential, logarithmic, and phasor identities, trigonometric and small-angle identities, and 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.
Degeneracy combines the two signal equations into \(da/dz=-j2ga^*a_pe^{j\Delta kz}\) and \(da_p/dz=-jga^2e^{-j\Delta kz}\). At exact match choose phases so the amplitudes are real; the invariants give \(\Phi(z)+2\Phi_p(z)=2\Phi_p(0)\) and the solution is the standard \(\mathrm{sech}^2/\tanh^2\) exchange. Consequently \(\Phi_p=\Phi_p(0)\mathrm{sech}^2(\kappa z)\) and \(\Phi=2\Phi_p(0)\tanh^2(\kappa z)\), proving both energy and photon conservation and giving \(\eta=\tanh^2(\kappa L)\).
Numbered result. The principal result obtained in the working is
Check. Equation (25) 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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 21.4-9 — OPO threshold
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 trigonometric and small-angle identities and 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.
At degeneracy \(\lambda_s=\lambda_i=1.064\) micrometres. Requiring the round-trip power gain to cancel two 0.98 reflectances gives \(\cosh^2(C L\sqrt I)R^2=1\). With \(C=2.244\times10^{-4}\), the result is \(\boxed{I_{th}=3.23\times10^4\ \mathrm{W/cm^2}}\).
Numbered result. The principal result obtained in the working is
Check. Equation (26) 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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 21.5-1 — Simultaneous SHG and SFG
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 integration identities.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
For envelopes \(A_1,A_2,B_1,B_2,C\) at \(\omega_1,\omega_2,2\omega_1,2\omega_2,\omega_1+\omega_2\), write one SVE equation for every resonant quadratic product: \(A_1^2\leftrightarrow B_1\), \(A_2^2\leftrightarrow B_2\), and \(A_1A_2\leftrightarrow C\), plus conjugate back-action terms. A Runge–Kutta integration preserving \(\sum\hbar\omega_q|A_q|^2\) shows suppression of SHG1 as the SFG channel draws photons from \(A_1\); energy-invariant error is the numerical check.
Check. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 21.5-2 — Degenerate four-wave equations
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 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.
Keeping resonant cubic products and exact phase matching gives \(dA_1/dz=-j\kappa A_2^*A_3^2\), \(dA_2/dz=-j\kappa A_1^*A_3^2\), and \(dA_3/dz=-j2\kappa^*A_1A_2A_3^*\), together with self/cross-phase terms if they are not absorbed into propagation constants. The factor two in the pump equation accounts for its two degenerate photons.
Numbered result. The principal result obtained in the working is
Check. Equation (27) 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 21.6-1 — Type-II coefficient in 3m BBO
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 trigonometric and small-angle identities and 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.
Insert the ordinary and extraordinary unit polarization vectors in \(d_{\rm eff}=\hat e_3\boldsymbol d:(\hat e_1\hat e_2)\). Applying the 3m tensor symmetries cancels the sine terms and leaves \(\boxed{d_{\rm eff}=d_{22}\cos^2\theta\cos3\phi}\).
Numbered result. The principal result obtained in the working is
Check. Equation (28) 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.
Problem 21.6-2 — Electro-optic/nonlinear tensors
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 product, quotient, and chain rules, matrix multiplication and eigenvalue rules, and 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.
For \(\boldsymbol\eta=\boldsymbol\epsilon^{-1}\epsilon_0\), variation of an inverse matrix gives \(\delta\eta=-\epsilon_0\epsilon^{-1}(\delta\epsilon)\epsilon^{-1}\). Substitute the quadratic and cubic field-dependent polarization terms and differentiate once or twice with respect to the DC field. Component matching gives \(\boxed{r_{ijk}=-4\epsilon_0d_{ijk}/(\epsilon_{ii}\epsilon_{jj})}\) and \(\boxed{s_{ijkl}=-12\epsilon_0\chi^{(3)}_{ijkl}/ (\epsilon_{ii}\epsilon_{jj})}\).
Numbered result. The principal result obtained in the working is
Check. Equation (29) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.