Chapter 6: More on Geometrical Optics

Source: Eugene Hecht, Optics, fifth Global Edition, Chapter 6. Prompts are paraphrased by topic rather than reproduced. An asterisk in the heading preserves the book’s marker for a problem omitted from its selected solutions; the derivation below is supplied independently.

End-of-chapter problems

Problem 6.1* — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.2 — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. The book’s selected-answer check begins From Eq. (6.8), 1/ƒ = 1/ƒ′ + 1/ƒ′ d/ƒ′ƒ′ = 2/ƒ′ - 2/3ƒ′, ƒ = 3ƒ′/4 From Eq. (6.9), H11H1 = (3ƒ′/4)(2ƒ′/3)/ƒ′ = ƒ′/2. From Eq. (6.10), H22H2 = -(3ƒ′/4)(2ƒ′/3)/ƒ′ = -ƒ′/2. Z03_HECH6933_05_GE_SOL.indd 694 08/09/16 9:14 pm Solutions to Selected Problems 695 7.8 E = E1 = E2 = E01{sin[vt k(x + ∆x) + sin(vt kx)}. Since sinb +. Substitute the result back into the governing relation to verify its units and sign.

Problem 6.3 — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 6.4* — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.5 — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 6.6* — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.7 — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 6.8* — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.9* — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.10* — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.11 — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 6.12* — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.13 — thick lenses and cardinal points: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Multiply the two refractions and internal translation; obtain power and principal-plane offsets from the ABCD elements. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 6.14* — analytical ray tracing: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.15* — analytical ray tracing: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.16* — analytical ray tracing: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.17* — analytical ray tracing: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.18* — analytical ray tracing: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.19* — analytical ray tracing: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.20* — analytical ray tracing: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.21* — analytical ray tracing: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.22 — analytical ray tracing: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 6.23 — analytical ray tracing: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 6.24 — analytical ray tracing: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Propagate height and reduced angle surface by surface and impose the requested imaging or afocal matrix condition. The book’s selected-answer check begins 𝒟1 = (1.5 - 1)/0.06 = 8.33 m-1 ; 𝒟2 = (1.5 - 1)/ - (-0.25) = 2 m-1 . 𝒜 = c 1 - (2)0.03/1.5 -8.33 - 2 + 8.33(2)0.03/1.5 0.03/1.5 1 - (8.33)0.03/1.5 d and 𝒜 = c 0.96 - 10 0.02 0.833 d. The determinant of this matrix (rounded to three decimal places) is exactly 1.000, which confirms our calculations.. Substitute the result back into the governing relation to verify its units and sign.

Problem 6.25* — aberrations and image quality: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.26* — aberrations and image quality: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.27* — aberrations and image quality: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.28* — aberrations and image quality: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.29* — aberrations and image quality: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.30 — aberrations and image quality: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. The book’s selected-answer check begins See E. Slayter, Optical Methods in Biology. PC/CA = (n1/n2)R/R = n1/n2, while CA/P′C = n1/n2. Therefore triangles ACP and ACP′ are similar; using the sine law sin∡PAC PC = sin∡APC CA or n2 sin∡PAC = n1 sin∡APC but ui = ∡PAC, thus ut = ∡APC = ∡P′AC, and the refracted ray appears to come from P9.. Substitute the result back into the governing relation to verify its units and sign.

Problem 6.31 — aberrations and image quality: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. The book’s selected-answer check begins From Eq. (5.6), let cosw = 1 w2 /2; then /o = [R2 + (so + R)2 - 2R(so + R) + R(so + R)w2 ]1/2 /o -1 = [so 2 + R(so + R)w2 ]-1/2 /i -1 = [si 2 - R(si - R)w2 ]-1/2 where the first two terms of the binomial series are used, /o -1 ≈ so -1 - (so + R)h2 /2so 3 R where w ≈ h/R, /i -1 ≈ si -1 + (si - R)h2 /2si 3 R Substituting. Substitute the result back into the governing relation to verify its units and sign.

Problem 6.32 — aberrations and image quality: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. Substitution back into the starting relation supplies the final sign and dimensional check.

Problem 6.33* — aberrations and image quality: derivation

Start from the governing relation rather than the desired result; rearrange until the requested form follows, so the argument is not circular. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.

Problem 6.34* — aberrations and image quality: calculation

List the supplied quantities with units, substitute only after the symbolic relation is isolated, and retain guard digits until the final result. Expand optical path or transverse error to the required order, identify the aberration coefficient, and test its symmetry and scaling. Finish by checking the governing equation, the dimensions, and the zero/large-parameter limit; these checks replace reliance on an unavailable answer-key entry.