Chapter 3: Beam Optics
Source: Saleh and Teich, Fundamentals of Photonics, second edition, Chapter 3. Beam radius \(W\) is the field’s \(1/e\) radius.
In-text exercises
Exercise 3.1-1 — He–Ne Gaussian beam
Brief solution
1. Method. The working uses integration identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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 29 — Exercise 3.1-1: He–Ne Gaussian beam. 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 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. For \(W_0=0.05\ \mathrm{mm}\) and \(\lambda=633\ \mathrm{nm}\), \(\theta_0=\lambda/(\pi W_0)=4.03\ \mathrm{mrad}\) and \(2z_0=2\pi W_0^2/\lambda=24.8\ \mathrm{mm}\).
Detailed step 2. At the Moon the diameter is approximately \(2\theta_0z=\boxed{2.82\times10^3\ \mathrm{km}}\). \(R(0)=\infty\), \(R(z_0)=2z_0\),
Detailed step 3. and \(R(2z_0)=2.5z_0\).
Detailed step 4. For \(P=1\ \mathrm{mW}\), \(I(0,0)=2P/(\pi W_0^2)=25.5\) \(\mathrm{W,cm^{-2}}\) and \(I(0,z_0)=12.7\) \(\mathrm{W,cm^{-2}}\); a 100-W isotropic source at \(z_0\) gives only \(5.14\ \mathrm{W,cm^{-2}}\).
Step 4 — State the numbered result. The principal result obtained in the working is
Step 5 — Check. Equation (1) 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 3.1-2 — Paraxial validity
Brief solution
1. Method. The working uses product, quotient, and chain rules, integration identities, and trigonometric and small-angle identities.
Show 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 30 — Exercise 3.1-2: Paraxial validity. 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, 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.
Detailed step 1. For a Gaussian beam,
Detailed step 2. transverse variation is \(\partial_xA\sim A/W\) and longitudinal variation is \(\partial_zA\sim A/z_0\).
Detailed step 3. Their ratio in the neglected Helmholtz term is \(1/(kz_0)\sim\theta_0^2/2\); therefore \(\theta_0\ll1\) implies \(|\partial_z^2A|\ll|2k\partial_zA|\).
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. The zero-angle or paraxial limit supplies an independent sign and magnitude check whenever that limit is part of the model.
Exercise 3.1-3 — Recovering a beam from \(W,R\)
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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 31 — Exercise 3.1-3: Recovering a beam from W,R. 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. Use \(q^{-1}=R^{-1}-j\lambda/(\pi W^2)\) and write \(q=z+jz_0\).
Detailed step 2. Taking real and imaginary parts gives \(z=R/[1+(\lambda R/\pi W^2)^2]\) and \(W_0=W/[1+(\pi W^2/\lambda R)^2]^{1/2}\) (with the propagation-side sign set by \(R\)).
Step 4 — State the numbered result. The principal result obtained in the working is
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 3.1-4 — Propagating known width and curvature
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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 32 — Exercise 3.1-4: Propagating known width and curvature. 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. The given values give \(q_1=[1/R_1-j\lambda/(\pi W_1^2)]^{-1}=0.908000+j0.289025\) m.
Detailed step 2. Since \(q_2=q_1+0.1\),
Detailed step 3. conversion back yields \(\boxed{W_2=1.10046\ \mathrm{mm}}\) and \(\boxed{R_2=1.09087\ \mathrm m}\).
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.
Exercise 3.1-5 — Two measured curvatures
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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 33 — Exercise 3.1-5: Two measured curvatures. 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. Write \(R_i=z_i+z_0^2/z_i\) with \(z_2=z_1+d\).
Detailed step 2. Eliminating \(z_0\) gives the printed \(z_1=-d(R_2-d)/(R_2-R_1-2d)\); back-substitution gives the stated \(z_0^2\).
Detailed step 3. The physical root requires \(z_0^2>0\).
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.
Exercise 3.2-1 — Periodic beam relay
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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 34 — Exercise 3.2-1: Periodic beam relay. 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. Apply \(q'=(Aq+B)/(Cq+D)\) to one lens-spacing cell and impose the same waist after the cell.
Detailed step 2. The resulting real \(z_0\) contains \(\sqrt{d(4f-d)}\); hence a physical self-reproducing beam exists only for \(\boxed{0\leq d\leq4f}\).
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 3.2-2 — Lens and collimation
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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 35 — Exercise 3.2-2: Lens and collimation. 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. With incident \(q=z+jz_0\),
Detailed step 2. a lens gives \(q'=q/(1-q/f)\).
Detailed step 3. Separating its real part (new waist position) and imaginary part reproduces Eq. (3.2-18).
Detailed step 4. Setting the outgoing waist far away requires the incident wavefront curvature at the lens to satisfy \(R(z)=f\); real solutions exist only when \(f\geq2z_0\).
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.
Exercise 3.2-3 — Two-lens beam expander
Brief solution
1. Method. The working uses Fourier-transform and convolution identities, matrix multiplication and eigenvalue rules, and vector-calculus identities.
2. Reasoning and answer.
Show 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 36 — Exercise 3.2-3: Two-lens beam expander. 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 Fourier-transform and convolution identities, matrix multiplication and eigenvalue rules, and vector-calculus identities.
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. Apply the ABCD law to \(M=L(f_2)P(s)L(f_1)\).
Detailed step 2. In the collimated limit \(s=f_1+f_2\),
Detailed step 3. the waist and divergence transform as \(\boxed{W_{0,out}/W_{0,in}=f_2/f_1}\) and \(\theta_{out}/\theta_{in}=f_1/f_2\); the product \(W_0\theta=\lambda/\pi\) is unchanged.
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.
Exercise 3.2-4 — Gaussian-reflectance mirror
Brief solution
1. Method. The working uses integration identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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 37 — Exercise 3.2-4: Gaussian-reflectance mirror. 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.
Detailed step 1. Multiplying the incident field by \(r(\rho)=e^{-\rho^2/W_m^2}\) adds reciprocal squared widths,
Detailed step 2. while the spherical phase changes the curvature as reflection does: \(\boxed{W_2^{-2}=W_1^{-2}+W_m^{-2}}\) and \(\boxed{R_2^{-1}=2R^{-1}-R_1^{-1}}\) in the reflected propagation coordinate.
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. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Exercise 3.2-5 — Plane-parallel plate
Brief solution
1. Method. The working uses matrix multiplication and eigenvalue rules, vector-calculus identities, and trigonometric and small-angle identities.
2. Reasoning and answer.
Show 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 38 — Exercise 3.2-5: Plane-parallel plate. 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, vector-calculus 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.
Detailed step 1. The complete air–plate–air ABCD matrix is \(\begin{bmatrix}1&d/n\\0&1\end{bmatrix}\) for reduced angle.
Detailed step 2. Thus \(\boxed{q_{out}=q_{in}+d/n}\): the beam emerging in air has the original waist and divergence but is advanced relative to free propagation by \(d(1-1/n)\).
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. Multiply the matrices independently in the stated input-to-output order and verify that every product has compatible dimensions.
Exercise 3.3-1 — Incoherent donut beam
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
Show 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 39 — Exercise 3.3-1: Incoherent donut beam. 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.
Detailed step 1. Adding the (1,0) and (0,1) intensities gives \(I\propto\rho^2e^{-2\rho^2/W_0^2}\).
Detailed step 2. Its peak is at \(\boxed{\rho=W_0/\sqrt2=0.7071\ \mathrm{mm}}\); solving \(I/I_{max}=e^{-2}\) gives the two radii \(\boxed{0.1620\ \mathrm{mm}}\) and \(\boxed{1.5009\ \mathrm{mm}}\) for \(W_0=1\) mm.
End-of-chapter problems
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.
Problem 3.1-6 — Nd:YAG beam parameters
Brief solution
1. Method. The working uses vector-calculus identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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 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.
Detailed step 1. The stated full divergence is \(2\theta_0=1\) mrad,
Detailed step 2. so \(W_0=\lambda/(\pi\theta_0)=\boxed{0.675\ \mathrm{mm}}\), \(2z_0=2\pi W_0^2/\lambda=\boxed{2.70\ \mathrm m}\),
Detailed step 3. and \(I_{max}=2P/(\pi W_0^2)=\boxed{1.40\times10^6\ \mathrm{W,m^{-2}}}\).
Detailed step 4. At \(z=1\) m multiply by \([1+(z/z_0)^2]^{-1}\) to obtain \(\boxed{9.05\times10^5\ \mathrm{W,m^{-2}}}\).
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. 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 3.1-7 — Beam from two widths
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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. Solve \(W_i^2=W_0^2[1+(z_i/z_0)^2]\), \(z_2=z_1+0.1\) m,
Detailed step 2. and \(z_0=\pi W_0^2/\lambda\).
Detailed step 3. The solution is \(\boxed{W_0=0.2000\ \mathrm{mm}}\); the waist lies \(\boxed{100.0\ \mathrm{mm}}\) before the first measurement plane.
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 3.1-8 — Elliptic Gaussian beam
Brief solution
1. Method. The working uses integration identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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. Each axis propagates independently: \(z_{0i}=\pi W_{0i}^2/\lambda\), \(\theta_i=\lambda/(\pi W_{0i})\),
Detailed step 2. and \(R_i=z[1+(z_{0i}/z)^2]\).
Detailed step 3. If \(W_{0x}=2W_{0y}\),
Detailed step 4. the waist is twice as wide in \(x\),
Detailed step 5. but the far field is twice as wide in \(y\); the ellipse rotates its major-axis orientation by \(90^\circ\).
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 3.2-6 — Smallest focusing lens
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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. \(z_0=\pi W_0^2/\lambda=1.609\) m.
Detailed step 2. Requiring \(W'_0/W_0=0.1\) in \(z'_0/z_0=f^2/[(z-f)^2+z_0^2]\) gives the existence condition \(f\geq z_0/10\).
Detailed step 3. Therefore the shortest lens is \(\boxed{f=160.9\ \mathrm{mm}}\),
Detailed step 4. placed \(z=f\) from the original waist; it produces the requested \(100\ \mathrm{\mu m}\) diameter.
Numbered result. The principal result obtained in the working is
Check. Equation (14) 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 3.2-7 — Focused-waist plot
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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. Plot \(W'_0=W_0f/[ (z-f)^2+z_0^2]^{1/2}\) with \(W_0=\sqrt{\lambda z_0/\pi}\).
Detailed step 2. Direct expansion gives the distant-beam and geometric-focus limits of Eqs. (3.2-10), (3.2-12),
Detailed step 3. while \(z\ll z_0\) gives \(W'_0/W_0=f/\sqrt{f^2+z_0^2}\),
Detailed step 4. Eq. (3.2-13).
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 3.2-8 — Refraction of a waist
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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. The transverse waist is continuous at the plane boundary and the wavelength falls to \(\lambda_0/n\); consequently \(\boxed{\theta_t=\theta_i/n=0.667\ \mathrm{mrad}}\) and the Rayleigh range grows by \(n\).
Detailed step 2. The sketch is the same waist followed by a more slowly expanding cone.
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. 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 3.2-9 — Gaussian beam in a GRIN slab
Brief solution
1. Method. The working uses matrix multiplication and eigenvalue rules, integration identities, and trigonometric and small-angle identities.
2. Key step.
With \(q_0=jz_0\) and the stated matrix, \(q(d)=[jq_0\cos(ad)+\sin(ad)/a]/[\cos(ad)-ja z_0\sin(ad)]\). Use \(W^2=-\lambda/[\pi\operatorname{Im}(1/q)]\); simplification gives
3. Answer.
The width breathes periodically; it is constant only for \(az_0=1\).
Show 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 matrix multiplication and eigenvalue rules, integration identities, and trigonometric and small-angle identities.
Worked derivation. The calculation is kept in symbolic form until the governing relation has been rearranged for the requested quantity.
With \(q_0=jz_0\) and the stated matrix, \(q(d)=[jq_0\cos(ad)+\sin(ad)/a]/[\cos(ad)-ja z_0\sin(ad)]\). Use \(W^2=-\lambda/[\pi\operatorname{Im}(1/q)]\); simplification gives
The width breathes periodically; it is constant only for \(az_0=1\).
Check. Equation (17) 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 3.3-2 — Enclosed Hermite–Gaussian power
Brief solution
1. Method. The working uses integration identities, exponential, logarithmic, and phasor identities, and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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, exponential, logarithmic, and phasor 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 radius \(sW\),
Detailed step 2. the fractions are \(F_{00}=1-e^{-2s^2}\), \(F_{10}=F_{01}=1-e^{-2s^2}(1+2s^2)\),
Detailed step 3. and \(F_{11}=1-e^{-2s^2}(1+2s^2+2s^4)\).
Detailed step 4. Thus at \(s=1\) they are \(\boxed{0.8647,0.5940,0.5940,0.3233}\).
Detailed step 5. At \(s=\sqrt2\), \(F_{00}=\boxed{0.9817}\) and \(F_{11}=\boxed{0.7619}\).
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. Differentiation of an antiderivative, or normalization of a definite integral, checks the integration step.
Problem 3.3-3 — Coherent (1,0)+(0,1)
Brief solution
1. Method. The working uses exponential, logarithmic, and phasor identities and algebraic rearrangement and dimensional checks.
2. Reasoning and answer.
Show 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 exponential, logarithmic, and phasor 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. Equal complex coefficients give field \(U\propto(x+y)e^{-\rho^2/W^2}\).
Detailed step 2. Its intensity has a dark diagonal \(y=-x\) and two lobes along \(y=x\); rotating coordinates by \(45^\circ\) identifies it as a first-order HG mode.
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. 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 3.3-4 — Axial phase modes
Brief solution
1. Method. The working uses algebraic rearrangement and dimensional checks.
2. Key step.
Between \(-z_0\) and \(z_0\), the phase is \(2kz_0-(l+m+1)\pi/2\). Setting it to \(N\pi\) gives
3. Answer.
For \(z_0=0.30\) m, adjacent \(N\) values are 250 MHz apart; retain the integers whose values fall in \(10^{14}\pm2\) GHz.
Show 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.
Between \(-z_0\) and \(z_0\), the phase is \(2kz_0-(l+m+1)\pi/2\). Setting it to \(N\pi\) gives
For \(z_0=0.30\) m, adjacent \(N\) values are 250 MHz apart; retain the integers whose values fall in \(10^{14}\pm2\) GHz.
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. Repeat the substitution with unrounded intermediate values and retain the displayed units; the final unit must have the requested dimension.