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

Brief solution

1. Method. The working uses algebraic rearrangement and dimensional checks.

2. Key step.

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,

3. Answer.

\[\boxed{a=z/z_0=-51.02},\qquad \boxed{T=T_0\sqrt{1+a^2}=510.3\ \mathrm{ps}}.\]

Returning to the original width requires zero net GVD: \(20(100)-100d_2=0\), so \(\boxed{d_2=20.0\ \mathrm{km}}\).

Show detailed stepsHide detailed steps

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,

(1)\[\boxed{a=z/z_0=-51.02},\qquad \boxed{T=T_0\sqrt{1+a^2}=510.3\ \mathrm{ps}}.\]

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

Brief solution

1. Method. The working uses algebraic rearrangement and dimensional checks.

2. Key step.

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

3. Answer.

\[\boxed{\zeta=-{2a\over T^2} =-{2d/z_0\over T_0^2[1+(d/z_0)^2]}}.\]

The next length \(2d\) brings the pulse to the same width and opposite pre-modulator chirp, proving periodicity.

Show detailed stepsHide detailed steps

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

(2)\[\boxed{\zeta=-{2a\over T^2} =-{2d/z_0\over T_0^2[1+(d/z_0)^2]}}.\]

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

Brief solution

1. Method. The working uses product, quotient, and chain rules, Fourier-transform and convolution identities, and exponential, logarithmic, and phasor identities.

2. Key step.

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

\[\widetilde A(f)\propto e^{-\pi^2T^2f^2} +(1-ja)^{-1/2}e^{-\pi^2T^2f^2/(1-ja)};\]

3. Answer.

its squared magnitude and argument are the requested spectral intensity and phase, and differentiating that argument twice gives the spectral chirp.

Show detailed stepsHide detailed steps

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

(3)\[\widetilde A(f)\propto e^{-\pi^2T^2f^2} +(1-ja)^{-1/2}e^{-\pi^2T^2f^2/(1-ja)};\]

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

Brief solution

1. Method. The working uses Fourier-transform and convolution identities, integration identities, and algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{\Delta f=0.179/T}\]

Show detailed stepsHide detailed steps

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.

Detailed step 1. Half intensity satisfies \(\mathrm{sech}^2(t/T)=1/2\),

Detailed step 2. so \(T_{FWHM}=2T\operatorname{arcosh}\sqrt2=\boxed{1.763T}\).

Detailed step 3. 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 \(\boxed{\Delta f=0.179/T}\).

Detailed step 4. Thus \(T_{FWHM}\Delta f=0.315\),

Detailed step 5. compared with 0.441 for a Gaussian.

Numbered result. The principal result obtained in the working is

(4)\[\boxed{\Delta f=0.179/T}\]

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

Brief solution

1. Method. The working uses product, quotient, and chain rules, trigonometric and small-angle identities, and algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{4/\alpha^2}\]

Show detailed stepsHide detailed steps

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.

Detailed step 1. Differentiate Snell’s law at both faces while imposing Brewster incidence and the symmetric-ray condition.

Detailed step 2. The two angular derivatives are equal and add,

Detailed step 3. giving \(d\theta_d/dn=-2\).

Detailed step 4. Substitution in the angular-dispersion chirp formula gives \(\boxed{b=-4(n-N)^2R_0\lambda_0/(\pi c^2)}\).

Detailed step 5. The thin-prism result has the same numerator multiplied by apex-angle squared,

Detailed step 6. so the ratio is \(\boxed{4/\alpha^2}\).

Numbered result. The principal result obtained in the working is

(5)\[\boxed{4/\alpha^2}\]

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

Brief solution

1. Method. The working uses product, quotient, and chain rules, Fourier-transform and convolution identities, and exponential, logarithmic, and phasor identities.

2. Reasoning and answer.

\[\boxed{\Lambda_{max}=1001.67/(2n_{eff})\ \mathrm{nm}}\]

Show detailed stepsHide detailed steps

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.

Detailed step 1. The 0.44-ps transform-limited pulse has \(\Delta\nu=1.00\) THz and \(\Delta\lambda\simeq3.33\) nm about 1 micrometre.

Detailed step 2. The group-delay sweep of \(H(f)=e^{-jb\pi^2f^2}\) is \(\Delta\tau=\pi b\Delta\nu=12.57\) ps.

Detailed step 3. A reflection grating therefore needs \(\boxed{L=c\Delta\tau/(2n_g)=1.884/n_g\ \mathrm{mm}}\),

Detailed step 4. and its local pitch must cover \(\boxed{\Lambda_{min}=998.33/(2n_{eff})\ \mathrm{nm}}\) through \(\boxed{\Lambda_{max}=1001.67/(2n_{eff})\ \mathrm{nm}}\).

Detailed step 5. Reverse the pitch gradient to reverse the chirp sign.

Numbered result. The principal result obtained in the working is

(6)\[\boxed{\Lambda_{max}=1001.67/(2n_{eff})\ \mathrm{nm}}\]

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

Brief solution

1. Method. The working uses stationary-value condition and algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{2\pi|b|/T=2|D_\nu|z/T}\]

Show detailed stepsHide detailed steps

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.

Detailed step 1. The input spectrum is \(T\,\mathrm{sinc}(fT)\).

Detailed step 2. For \(|b|\gg T^2\),

Detailed step 3. stationary phase maps frequency to time by \(f=t/(\pi b)\),

Detailed step 4. so the output envelope is proportional to \(\boxed{\mathrm{sinc}[tT/(\pi b)]}\): the stated sinc-shaped pulse.

Detailed step 5. Its first zeros are at \(t=\pm\pi|b|/T\),

Detailed step 6. 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

(7)\[\boxed{2\pi|b|/T=2|D_\nu|z/T}\]

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

Brief solution

1. Method. The working uses algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{|M|=T_2/T_1=d_2/d_1}\]

Show detailed stepsHide detailed steps

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.

Detailed step 1. In the far-dispersion limit the first fiber maps input time to frequency,

Detailed step 2. the quadratic modulator supplies a time-lens phase,

Detailed step 3. and the second maps frequency back to time.

Detailed step 4. Cancellation of the residual quadratic phase requires \(\boxed{1/d_1+1/d_2=1/f}\),

Detailed step 5. with \(f=-\pi/(\zeta D_\nu)\).

Detailed step 6. The remaining kernel is a scaled delta function,

Detailed step 7. 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

(8)\[\boxed{|M|=T_2/T_1=d_2/d_1}\]

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

Brief solution

1. Method. The working uses algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{a_3/T_3^2=a_1/T_1^2+a_2/T_2^2}\]

Show detailed stepsHide detailed steps

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.

Detailed step 1. Instantaneous energy conservation requires \(\omega_{3i}(t)=\omega_{1i}(t)+\omega_{2i}(t)\).

Detailed step 2. 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.

Detailed step 3. Therefore,

Detailed step 4. for example, \(a_1/T_1^2=a_3/T_3^2-a_2/T_2^2\),

Detailed step 5. which can exceed the pump rate when the idler chirp has the opposite sign.

Detailed step 6. This enables chirp magnification before dispersive pulse compression or time-to-frequency conversion.

Numbered result. The principal result obtained in the working is

(9)\[\boxed{a_3/T_3^2=a_1/T_1^2+a_2/T_2^2}\]

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

Brief solution

1. Method. The working uses product, quotient, and chain rules, Fourier-transform and convolution identities, and exponential, logarithmic, and phasor identities.

2. Key step.

Expand \(k_q(\omega_q+\Omega)\) through \(\Omega^2\), apply SVEA, and inverse transform. Each envelope satisfies

\[\left(\partial_z+v_q^{-1}\partial_t +{j\beta_{2q}\over2}\partial_t^2\right)A_q =-j\kappa_q A_r^{(*)}A_s^{(*)}e^{\pm j\Delta kz},\]

3. Answer.

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).

Show detailed stepsHide detailed steps

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

(10)\[\left(\partial_z+v_q^{-1}\partial_t +{j\beta_{2q}\over2}\partial_t^2\right)A_q =-j\kappa_q A_r^{(*)}A_s^{(*)}e^{\pm j\Delta kz},\]

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

Brief solution

1. Method. The working uses integration identities and algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{2,\ 1/2,\ 1/\sqrt2,\ \sqrt2,\ 2}\]

Show detailed stepsHide detailed steps

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.

Detailed step 1. For a fundamental soliton, \(I_0T_0^2\propto|D|\) and energy \(E\propto I_0T_0\).

Detailed step 2. Holding energy fixed gives \(T_0\propto|D|\), \(I_0\propto1/|D|\),

Detailed step 3. field peak \(|A_0|\propto|D|^{-1/2}\),

Detailed step 4. amplitude area \(\int|A|dt\propto|D|^{1/2}\),

Detailed step 5. and soliton distance \(z_0\propto|D|\).

Detailed step 6. Thus the 20-versus-10 case has ratios \(\boxed{2,\ 1/2,\ 1/\sqrt2,\ \sqrt2,\ 2}\),

Detailed step 7. respectively.

Numbered result. The principal result obtained in the working is

(11)\[\boxed{2,\ 1/2,\ 1/\sqrt2,\ \sqrt2,\ 2}\]

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

Brief solution

1. Method. The working uses algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{I_0=1.29\times10^8\ \mathrm{W/m^2}=12.9\ \mathrm{kW/cm^2}}\]

Show detailed stepsHide detailed steps

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.

Detailed step 1. Combining the fundamental-soliton condition with \(z_0=\pi T_0^2/D_\nu\) eliminates pulse width and gives \(\boxed{I_0|z_0|=\lambda_0/(4\pi n_2)}\).

Detailed step 2. At 1.55 micrometres, \(n_2=3.19\times10^{-20}\ \mathrm{m^2/W}\),

Detailed step 3. 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

(12)\[\boxed{I_0=1.29\times10^8\ \mathrm{W/m^2}=12.9\ \mathrm{kW/cm^2}}\]

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

Brief solution

1. Method. The working uses integration identities and algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{255\ \mathrm{fs}}\]

Show detailed stepsHide detailed steps

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.

Detailed step 1. Convolving two equal Gaussian intensities broadens FWHM by \(\sqrt2\),

Detailed step 2. so the measured trace is Gaussian with \(\boxed{70.7\ \mathrm{fs}}\) FWHM.

Detailed step 3. For one pulse to broaden fivefold,

Detailed step 4. the 800-nm dispersion gives \(z=z_0\sqrt{5^2-1}=\boxed{0.118\ \mathrm m}\).

Detailed step 5. Correlating 50-fs and 250-fs Gaussians gives \(\boxed{255\ \mathrm{fs}}\).

Detailed step 6. The unequal-arm interferometric trace can reveal spectral phase sensitivity,

Detailed step 7. but reduced overlap,

Detailed step 8. fiber loss/nonlinearity,

Detailed step 9. and unknown fiber dispersion make inversion less robust.

Numbered result. The principal result obtained in the working is

(13)\[\boxed{255\ \mathrm{fs}}\]

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

Brief solution

1. Method. The working uses integration identities.

Show detailed stepsHide detailed steps

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.

Detailed step 1. Write the recombined field as \(E_1+E_2\).

Detailed step 2. A two-photon detector measures \(\int|E_1+E_2|^4dt\); expansion contains the two intensity-autocorrelation terms,

Detailed step 3. phase-sensitive \(E_1^2E_2^{*2}+\mathrm{c.c.}\),

Detailed step 4. and oscillatory cross terms.

Detailed step 5. SHG followed by a square-law detector produces the same fourth- order structure after its generated field is integrated,

Detailed step 6. apart from the SHG phase-matching/filter response.

Detailed step 7. 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.