Chapter 22: Ultrafast Optics
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 22.
In-text exercises
Exercise 22.3-1 — Two-fiber dispersion compensation
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 127 — Exercise 22.3-1: Two-fiber dispersion compensation. 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.
Convert \(D_\lambda\) with \(D_\nu=-(\lambda_0^2/c)D_\lambda=-1.603\times10^{-25}\) \(\mathrm{s^2/m}\). Thus \(z_0=\pi T_0^2/D_\nu=-1.960\) km and, after 100 km,
Returning to the original width requires zero net GVD: \(20(100)-100d_2=0\), so \(\boxed{d_2=20.0\ \mathrm{km}}\).
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 22.3-2 — Periodic phase compensation
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 128 — Exercise 22.3-2: Periodic phase compensation. 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.
Propagation from a pulse waist through distance \(d\) gives \(a=d/z_0\) and \(T^2=T_0^2[1+(d/z_0)^2]\). A quadratic phase changes the chirp by \(\Delta a=\zeta T^2\). Reflection symmetry about each modulator demands \(a\mapsto-a\), hence
The next length \(2d\) brings the pulse to the same width and opposite pre-modulator chirp, proving periodicity.
End-of-chapter problems
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. Check that dimensions agree term by term, then test the simplest symmetry or limiting case for the expected sign and scale.
Problem 22.1-1 — Sum of unchirped and chirped Gaussians
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, Fourier-transform and convolution identities, and 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.
Let \(x=t^2/T^2\). The sum envelope is \(A=A_0e^{-x}(1+e^{jax})=2A_0e^{-x}e^{jax/2}\cos(ax/2)\). Therefore \(I=4|A_0|^2e^{-2x}\cos^2(ax/2)\), phase is \(ax/2\) between cosine zeros, and the local chirp coefficient is \(a/2\) (with pi phase jumps at zeros). Fourier transforming each term gives
its squared magnitude and argument are the requested spectral intensity and phase, and differentiating that argument twice gives the spectral chirp.
Check. Equation (3) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors.
Problem 22.1-2 — Hyperbolic-secant pulse
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 Fourier-transform and convolution identities, 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.
Half intensity satisfies \(\mathrm{sech}^2(t/T)=1/2\), so \(T_{FWHM}=2T\operatorname{arcosh}\sqrt2=\boxed{1.763T}\). The transform pair \(\mathrm{sech}(t/T)\leftrightarrow\pi T\, \mathrm{sech}(\pi^2Tf)\) gives spectral intensity proportional to \(\mathrm{sech}^2(\pi^2Tf)\) and \(\boxed{\Delta f=0.179/T}\). Thus \(T_{FWHM}\Delta f=0.315\), compared with 0.441 for a Gaussian.
Numbered result. The principal result obtained in the working is
Check. Equation (4) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 22.2-1 — Symmetric Brewster prism
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, 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.
Differentiate Snell’s law at both faces while imposing Brewster incidence and the symmetric-ray condition. The two angular derivatives are equal and add, giving \(d\theta_d/dn=-2\). Substitution in the angular-dispersion chirp formula gives \(\boxed{b=-4(n-N)^2R_0\lambda_0/(\pi c^2)}\). The thin-prism result has the same numerator multiplied by apex-angle squared, so the ratio is \(\boxed{4/\alpha^2}\).
Numbered result. The principal result obtained in the working is
Check. Equation (5) 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 22.2-2 — Chirped Bragg grating
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, Fourier-transform and convolution identities, and 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.
The 0.44-ps transform-limited pulse has \(\Delta\nu=1.00\) THz and \(\Delta\lambda\simeq3.33\) nm about 1 micrometre. The group-delay sweep of \(H(f)=e^{-jb\pi^2f^2}\) is \(\Delta\tau=\pi b\Delta\nu=12.57\) ps. A reflection grating therefore needs \(\boxed{L=c\Delta\tau/(2n_g)=1.884/n_g\ \mathrm{mm}}\), and its local pitch must cover \(\boxed{\Lambda_{min}=998.33/(2n_{eff})\ \mathrm{nm}}\) through \(\boxed{\Lambda_{max}=1001.67/(2n_{eff})\ \mathrm{nm}}\). Reverse the pitch gradient to reverse the chirp sign.
Numbered result. The principal result obtained in the working is
Check. Equation (6) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 22.3-3 — Dispersed rectangular pulse
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 stationary-value condition 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 input spectrum is \(T\,\mathrm{sinc}(fT)\). For \(|b|\gg T^2\), stationary phase maps frequency to time by \(f=t/(\pi b)\), so the output envelope is proportional to \(\boxed{\mathrm{sinc}[tT/(\pi b)]}\): the stated sinc-shaped pulse. Its first zeros are at \(t=\pm\pi|b|/T\), giving zero-to-zero width \(\boxed{2\pi|b|/T=2|D_\nu|z/T}\).
Numbered result. The principal result obtained in the working is
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Problem 22.3-4 — Temporal imaging
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.
In the far-dispersion limit the first fiber maps input time to frequency, the quadratic modulator supplies a time-lens phase, and the second maps frequency back to time. Cancellation of the residual quadratic phase requires \(\boxed{1/d_1+1/d_2=1/f}\), with \(f=-\pi/(\zeta D_\nu)\). The remaining kernel is a scaled delta function, so \(A_o(t)\propto A_i(-t/M)\) with \(\boxed{|M|=T_2/T_1=d_2/d_1}\).
Numbered result. The principal result obtained in the working is
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.
Problem 22.5-1 — Chirp matching and amplification
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.
Instantaneous energy conservation requires \(\omega_{3i}(t)=\omega_{1i}(t)+\omega_{2i}(t)\). For coincident pulses this means their chirp rates obey \(\boxed{a_3/T_3^2=a_1/T_1^2+a_2/T_2^2}\); group-delay matching keeps the three time coordinates overlapped. Therefore, for example, \(a_1/T_1^2=a_3/T_3^2-a_2/T_2^2\), which can exceed the pump rate when the idler chirp has the opposite sign. This enables chirp magnification before dispersive pulse compression or time-to-frequency conversion.
Numbered result. The principal result obtained in the working is
Check. Equation (9) 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 22.5-2 — Pulsed mixing with GVD
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, Fourier-transform and convolution identities, and 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.
Expand \(k_q(\omega_q+\Omega)\) through \(\Omega^2\), apply SVEA, and inverse transform. Each envelope satisfies
where conjugates are chosen for sum- or difference-frequency generation. The first derivative is group delay, the second is GVD, and the right side is exactly the resonant component of \(2dE^2\); these are Eqs. (22.5-3).
Check. Equation (10) can be checked by substituting it back into the preceding governing relation and reversing the algebraic steps. Transform dimensions and the expected even/odd or conjugate symmetry provide an independent check on signs and scale factors.
Problem 22.5-3 — Equal-energy solitons at two dispersions
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 a fundamental soliton, \(I_0T_0^2\propto|D|\) and energy \(E\propto I_0T_0\). Holding energy fixed gives \(T_0\propto|D|\), \(I_0\propto1/|D|\), field peak \(|A_0|\propto|D|^{-1/2}\), amplitude area \(\int|A|dt\propto|D|^{1/2}\), and soliton distance \(z_0\propto|D|\). Thus the 20-versus-10 case has ratios \(\boxed{2,\ 1/2,\ 1/\sqrt2,\ \sqrt2,\ 2}\), respectively.
Numbered result. The principal result obtained in the working is
Check. Equation (11) 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.
Problem 22.5-4 — Fiber-soliton intensity
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.
Combining the fundamental-soliton condition with \(z_0=\pi T_0^2/D_\nu\) eliminates pulse width and gives \(\boxed{I_0|z_0|=\lambda_0/(4\pi n_2)}\). At 1.55 micrometres, \(n_2=3.19\times10^{-20}\ \mathrm{m^2/W}\), and 30 km, \(\boxed{I_0=1.29\times10^8\ \mathrm{W/m^2}=12.9\ \mathrm{kW/cm^2}}\).
Numbered result. The principal result obtained in the working is
Check. Equation (12) 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 22.6-1 — Gaussian-pulse autocorrelation
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.
Convolving two equal Gaussian intensities broadens FWHM by \(\sqrt2\), so the measured trace is Gaussian with \(\boxed{70.7\ \mathrm{fs}}\) FWHM. For one pulse to broaden fivefold, the 800-nm dispersion gives \(z=z_0\sqrt{5^2-1}=\boxed{0.118\ \mathrm m}\). Correlating 50-fs and 250-fs Gaussians gives \(\boxed{255\ \mathrm{fs}}\). The unequal-arm interferometric trace can reveal spectral phase sensitivity, but reduced overlap, fiber loss/nonlinearity, and unknown fiber dispersion make inversion less robust.
Numbered result. The principal result obtained in the working is
Check. Equation (13) 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 22.6-2 — Two-photon versus SHG interferometry
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.
Write the recombined field as \(E_1+E_2\). A two-photon detector measures \(\int|E_1+E_2|^4dt\); expansion contains the two intensity-autocorrelation terms, phase-sensitive \(E_1^2E_2^{*2}+\mathrm{c.c.}\), and oscillatory cross terms. SHG followed by a square-law detector produces the same fourth- order structure after its generated field is integrated, apart from the SHG phase-matching/filter response. Hence an ideal instantaneous two-photon absorber is the broadband analogue; a real absorber replaces the SHG transfer function by its own two-photon spectral response.
Check. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.