Chapter 19: Acousto-Optics

Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 19.

In-text exercises

Exercise 19.2-1 — Modulator parameters

Brief solution

1. Method. The working uses trigonometric and small-angle identities and algebraic rearrangement and dimensional checks.

2. Key step.

Use the internal optical wavelength \(\lambda=\lambda _0/n\), \(\sin\theta_B=\lambda f/(2v_s)\), and \(B=v_s\delta\theta/\lambda\) (or \(B=v_s/D\)). The two designs give

3. Answer.

\[\boxed{\theta_{B1}=0.1035^\circ,\quad B_1=13.84\ {\rm MHz}}, \qquad \boxed{\theta_{B2}=2.877^\circ,\quad B_2=2.20\ {\rm MHz}}.\]

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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 107 — Exercise 19.2-1: Modulator parameters. 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 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.

Use the internal optical wavelength \(\lambda=\lambda _0/n\), \(\sin\theta_B=\lambda f/(2v_s)\), and \(B=v_s\delta\theta/\lambda\) (or \(B=v_s/D\)). The two designs give

(1)\[\boxed{\theta_{B1}=0.1035^\circ,\quad B_1=13.84\ {\rm MHz}}, \qquad \boxed{\theta_{B2}=2.877^\circ,\quad B_2=2.20\ {\rm MHz}}.\]

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. 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 19.2-2 — Scanner parameters

Brief solution

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

2. Reasoning and answer.

\[\boxed{1.445\ \mathrm{mrad}=0.0828^\circ}\]

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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 108 — Exercise 19.2-2: Scanner parameters. 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.

Detailed step 1. Here \(N=BT=D B/v_s\) and \(\Delta\theta=(\lambda_0/n)B/v_s\).

Detailed step 2. With \(B=20\) MHz,

Detailed step 3. fused quartz therefore needs \(\boxed{D=30.0\ \mathrm{mm}}\) and scans \(\boxed{1.445\ \mathrm{mrad}=0.0828^\circ}\).

Detailed step 4. Flint glass needs only \(15.5\) mm for the same 100 spots and scans \(2.797\) mrad; slower sound gives both a smaller aperture and a larger scan.

Step 4 — State the numbered result. The principal result obtained in the working is

(2)\[\boxed{1.445\ \mathrm{mrad}=0.0828^\circ}\]

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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.

Exercise 19.2-3 — Filter resolving power

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.

\[\boxed{\lambda/\Delta\lambda=fT}\]

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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 109 — Exercise 19.2-3: Filter resolving power. 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, 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.

Detailed step 1. Differentiate \(\sin\theta=\lambda f/(2v_s)\) at fixed angle: \(|\Delta\lambda|/\lambda=|\Delta f|/f\).

Detailed step 2. A finite interaction time \(T\) resolves acoustic frequencies no closer than \(1/T\); hence \(\boxed{\lambda/\Delta\lambda=fT}\).

Step 4 — State the numbered result. The principal result obtained in the working is

(3)\[\boxed{\lambda/\Delta\lambda=fT}\]

Step 5 — Check. Equation (3) 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.

Exercise 19.3-1 — Transverse strain in a cubic crystal

Brief solution

1. Method. The working uses matrix multiplication and eigenvalue rules and algebraic rearrangement and dimensional checks.

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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 110 — Exercise 19.3-1: Transverse strain in a cubic crystal. 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 matrix multiplication and eigenvalue rules 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.

Detailed step 1. The shear wave has only \(s_{13}=s_{31}=S\).

Detailed step 2. The impermeability block in the \(x\)–\(z\) plane is \(\bigl[\begin{smallmatrix}n^{-2}&p_{44}S\\p_{44}S&n^{-2}\end{smallmatrix}\bigr]\).

Detailed step 3. Its eigenvectors are at \(\pm45^\circ\),

Detailed step 4. with eigenvalues \(n^{-2}\pm p_{44}S\); the unchanged \(y\) eigenvalue is \(n^{-2}\).

Detailed step 5. Thus the crystal is biaxial and,

Detailed step 6. to first order, \(\boxed{n_\pm\simeq n\mp n^3p_{44}S/2,\ n_y=n}\).

End-of-chapter problems

Step 4 — State the numbered result. The principal result obtained in the working is

(4)\[\boxed{n_\pm\simeq n\mp n^3p_{44}S/2,\ n_y=n}\]

Step 5 — Check. Equation (4) 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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.

Problem 19.1-1 — Four periodic structures

Brief solution

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

2. Reasoning and answer.

\[\omega_r=\omega\pm\Omega\]

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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. A traveling acoustic grating supplies \((\mathbf q,\Omega)\),

Detailed step 2. so the two Bragg choices have \(\mathbf k_r=\mathbf k\pm\mathbf q\) and optical frequency \(\omega_r=\omega\pm\Omega\).

Detailed step 3. A standing acoustic wave is the sum of both traveling gratings and produces both shifts.

Detailed step 4. A static sinusoidal index grating supplies \(\pm\mathbf q\) but zero frequency,

Detailed step 5. while a static layered lattice supplies reciprocal vectors \(m\mathbf q\) and elastic diffraction orders with no optical frequency shift.

Numbered result. The principal result obtained in the working is

(5)\[\omega_r=\omega\pm\Omega\]

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.

Problem 19.1-2 — Bragg scattering integral

Brief solution

1. Method. The working uses integration identities, trigonometric and small-angle identities, and algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{R\simeq(\kappa D)^2}\]

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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, 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. At exact phase matching the source phase cancels the Green-function phase,

Detailed step 2. so the far-field amplitude is proportional to source volume \(V=AD\).

Detailed step 3. Dividing the scattered flux by incident flux gives \(R=\sin^2(\kappa D)\) in the coupled-wave solution and \(\boxed{R\simeq(\kappa D)^2}\) in the first-Born limit.

Detailed step 4. With the chapter’s photoelastic perturbation, \(\kappa=\pi\Delta n/\lambda\); this is the small-signal expansion of Eq. (19.1-22).

Numbered result. The principal result obtained in the working is

(6)\[\boxed{R\simeq(\kappa D)^2}\]

Check. Equation (6) 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 19.1-3 — Raman–Nath width limit

Brief solution

1. Method. The working uses trigonometric and small-angle identities and algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{D_s\lesssim\Lambda^2/(2\pi\lambda)}\]

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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.

Detailed step 1. The thin-grating condition is that diffraction separation accumulated across the sound width remain below one acoustic period.

Detailed step 2. Using diffraction angle \(\lambda/\Lambda\) gives the Klein–Cook parameter \(Q=2\pi\lambda D_s/\Lambda^2\);

Detailed step 3. Raman–Nath operation requires \(Q\lesssim1\),

Detailed step 4. or \(\boxed{D_s\lesssim\Lambda^2/(2\pi\lambda)}\) (order-one conventions move the numerical boundary but not the scaling).

Numbered result. The principal result obtained in the working is

(7)\[\boxed{D_s\lesssim\Lambda^2/(2\pi\lambda)}\]

Check. Equation (7) 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 19.1-4 — Combined lithium-niobate modulation

Brief solution

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

2. Reasoning and answer.

\[\boxed{\theta_B=3.198^\circ}\]

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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. \(\Lambda=v_s/f=2.467\ \mu\mathrm m\) and \(\lambda=633/2.3=275.2\) nm,

Detailed step 2. so \(\boxed{\theta_B=3.198^\circ}\).

Detailed step 3. Electro-optic phase modulation produces carrier-centered sidebands \(\omega+m\Omega\); reflection translates every one by \(\pm\Omega\).

Detailed step 4. For a short microwave pulse the electro-optic sidebands appear immediately,

Detailed step 5. whereas the delayed acoustic packet contributes only after its sound transit time through the illuminated region.

Numbered result. The principal result obtained in the working is

(8)\[\boxed{\theta_B=3.198^\circ}\]

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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.

Problem 19.2-4 — Producing sinusoidal amplitude modulation

Brief solution

1. Method. The working uses exponential, logarithmic, and phasor identities, trigonometric and small-angle identities, and algebraic rearrangement and dimensional checks.

2. Reasoning and answer.

\[\boxed{U_o=A\cos(\Omega t)e^{j\omega t}}\]

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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.

Detailed step 1. Split the input equally and send the branches to oppositely oriented Bragg cells.

Detailed step 2. Their first orders are \((A/2)e^{j(\omega+\Omega)t}\) and \((A/2)e^{j(\omega-\Omega)t}\).

Detailed step 3. Recombine them in phase: \(\boxed{U_o=A\cos(\Omega t)e^{j\omega t}}\).

Numbered result. The principal result obtained in the working is

(9)\[\boxed{U_o=A\cos(\Omega t)e^{j\omega t}}\]

Check. Equation (9) 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 19.2-5 — Deflection without frequency translation

Brief solution

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

2. Reasoning and answer.

\[(+\Omega)+(-\Omega)=0\]

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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. Cascade two equal Bragg cells with acoustic wavevectors chosen so both deflections add,

Detailed step 2. but drive one on its upshift order and the other on its downshift order.

Detailed step 3. The net wavevector changes by the desired two grating momenta while \((+\Omega)+(-\Omega)=0\); spatially filter the selected first-order path after each cell.

Numbered result. The principal result obtained in the working is

(10)\[(+\Omega)+(-\Omega)=0\]

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.

Problem 19.3-2 — Front Bragg diffraction

Brief solution

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

2. Reasoning and answer.

\[\boxed{\Lambda=141.1\ \mathrm{nm}}\]

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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. The incident extraordinary and reflected ordinary waves obey \(q=k_o+k_e\); hence \(\boxed{\Lambda=\lambda_0/(n_o+n_e)}\) and the reflected wave is polarized ordinary,

Detailed step 2. perpendicular to the optic-axis plane.

Detailed step 3. With the stated indexes and \(\lambda_0=633\) nm, \(\boxed{\Lambda=141.1\ \mathrm{nm}}\).

Numbered result. The principal result obtained in the working is

(11)\[\boxed{\Lambda=141.1\ \mathrm{nm}}\]

Check. Equation (11) 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.