Chapter 18: Beam Perturbation and Diffraction

Source: Anthony E. Siegman, Lasers (1986), Chapter 18. Use each section/problem identifier with the book; the original prompts are not reproduced here. Each entry gives the governing model, the decisive solution route, and a physical verification.

Section 18.1: Grating Diffraction And Scattering Effects

Problem 18.1.1 — Doppler shift from a moving grating

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Apply the grating phase-matching condition to the moving periodicity; differentiating its phase in time gives the frequency shift \(\Delta\omega=\mathbf K_g\!\cdot\!\mathbf v\) with the diffraction-order sign. Check that the shift reverses when either grating velocity or diffraction order reverses, and vanishes for a stationary grating.

Section 18.3: Aperture Diffraction: Rectangular Apertures

Problem 18.3.1 — Single-slit field at the geometrical shadow edge

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.3.2 — Far-field single-slit diffraction

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.3.3 — Single slit with Gaussian illumination

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.3.4 — Single slit with variable-size Gaussian illumination

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Section 18.4: Aperture Diffraction: Circular Apertures

Problem 18.4.1 — Diffraction pattern of circular aperture at the shadow edge

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.4.2 — More on diffraction ripples near the shadow edge

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.4.3 — Central spike for a truncated circular Gaussian

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.4.4 — Simple numerical method for calculating uniform circular aperture diffraction patterns

Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.4.5 — Optimum Gaussian spot size through an aperture

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.4.6 — Diffraction patterns for partially filled annular apertures

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.4.7 — Imaging analysis for the relay imaging system

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.

Problem 18.4.8 — Diffraction pattern for an “inverted” truncated Gaussian (research problem)

Normalize the variables first, evaluate the analytic limits, and then sweep the remaining dimensionless parameter so the numerical curve can be checked against both limits. Insert the aperture field in the Huygens–Fresnel integral; retain the quadratic phase in the near field and use the aperture Fourier transform only after the Fraunhofer condition holds. Verify the on-axis limit and conservation of total transmitted power; aperture enlargement must narrow the far-field pattern.