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
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.
For \(W_0=0.05\ \mathrm{mm}\) and \(\lambda=633\ \mathrm{nm}\), \(\theta_0=\lambda/(\pi W_0)=4.03\ \mathrm{mrad}\) and \(2z_0=2\pi W_0^2/\lambda=24.8\ \mathrm{mm}\). At the Moon the diameter is approximately \(2\theta_0z=\boxed{2.82\times10^3\ \mathrm{km}}\). \(R(0)=\infty\), \(R(z_0)=2z_0\), and \(R(2z_0)=2.5z_0\). 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
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.
For a Gaussian beam, transverse variation is \(\partial_xA\sim A/W\) and longitudinal variation is \(\partial_zA\sim A/z_0\). 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 \(W,R\)
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.
Use \(q^{-1}=R^{-1}-j\lambda/(\pi W^2)\) and write \(q=z+jz_0\). 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
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.
The given values give \(q_1=[1/R_1-j\lambda/(\pi W_1^2)]^{-1}=0.908000+j0.289025\) m. Since \(q_2=q_1+0.1\), 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
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.
Write \(R_i=z_i+z_0^2/z_i\) with \(z_2=z_1+d\). Eliminating \(z_0\) gives the printed \(z_1=-d(R_2-d)/(R_2-R_1-2d)\); back-substitution gives the stated \(z_0^2\). 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
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.
Apply \(q'=(Aq+B)/(Cq+D)\) to one lens-spacing cell and impose the same waist after the cell. 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
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.
With incident \(q=z+jz_0\), a lens gives \(q'=q/(1-q/f)\). Separating its real part (new waist position) and imaginary part reproduces Eq. (3.2-18). 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
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.
Apply the ABCD law to \(M=L(f_2)P(s)L(f_1)\). In the collimated limit \(s=f_1+f_2\), 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
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.
Multiplying the incident field by \(r(\rho)=e^{-\rho^2/W_m^2}\) adds reciprocal squared widths, 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
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.
The complete air–plate–air ABCD matrix is \(\begin{bmatrix}1&d/n\\0&1\end{bmatrix}\) for reduced angle. 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
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.
Adding the (1,0) and (0,1) intensities gives \(I\propto\rho^2e^{-2\rho^2/W_0^2}\). 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
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.
The stated full divergence is \(2\theta_0=1\) mrad, so \(W_0=\lambda/(\pi\theta_0)=\boxed{0.675\ \mathrm{mm}}\), \(2z_0=2\pi W_0^2/\lambda=\boxed{2.70\ \mathrm m}\), and \(I_{max}=2P/(\pi W_0^2)=\boxed{1.40\times10^6\ \mathrm{W,m^{-2}}}\). 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
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.
Solve \(W_i^2=W_0^2[1+(z_i/z_0)^2]\), \(z_2=z_1+0.1\) m, and \(z_0=\pi W_0^2/\lambda\). 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
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.
Each axis propagates independently: \(z_{0i}=\pi W_{0i}^2/\lambda\), \(\theta_i=\lambda/(\pi W_{0i})\), and \(R_i=z[1+(z_{0i}/z)^2]\). If \(W_{0x}=2W_{0y}\), the waist is twice as wide in \(x\), 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
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.
\(z_0=\pi W_0^2/\lambda=1.609\) m. Requiring \(W'_0/W_0=0.1\) in \(z'_0/z_0=f^2/[(z-f)^2+z_0^2]\) gives the existence condition \(f\geq z_0/10\). Therefore the shortest lens is \(\boxed{f=160.9\ \mathrm{mm}}\), 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
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.
Plot \(W'_0=W_0f/[ (z-f)^2+z_0^2]^{1/2}\) with \(W_0=\sqrt{\lambda z_0/\pi}\). Direct expansion gives the distant-beam and geometric-focus limits of Eqs. (3.2-10), (3.2-12), while \(z\ll z_0\) gives \(W'_0/W_0=f/\sqrt{f^2+z_0^2}\), 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
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.
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 \(n\). 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
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
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.
At radius \(sW\), the fractions are \(F_{00}=1-e^{-2s^2}\), \(F_{10}=F_{01}=1-e^{-2s^2}(1+2s^2)\), and \(F_{11}=1-e^{-2s^2}(1+2s^2+2s^4)\). Thus at \(s=1\) they are \(\boxed{0.8647,0.5940,0.5940,0.3233}\). 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)
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.
Equal complex coefficients give field \(U\propto(x+y)e^{-\rho^2/W^2}\). 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
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.