Physics · Ch 9 — Optics

Defects of lenses (aberrations of optical images)

9.10

Defects of lenses (aberrations of optical images)

Just as spherical mirrors of finite aperture suffer aberrations, so do lenses of finite aperture and finite thickness -- but lenses additionally suffer a defect mirrors never do, precisely because lens action depends on REFRACTION (which varies by refractive index, and hence by colour), whereas mirror action depends only on REFLECTION (which involves no colour-dependent dispersion at all).

CHROMATIC ABERRATION: a convex lens can be usefully thought of as built from two thin prisms joined base-to-base, and a concave lens as two thin prisms joined vertex-to-vertex; since a prism disperses different colours by different amounts, a lens likewise brings each colour of a polychromatic (e.g. white) beam to a slightly DIFFERENT focus -- violet, being deviated the most, focuses CLOSEST to the lens, and red, deviated least, focuses FARTHEST. Longitudinal chromatic aberration (the axial spread between the different colours' foci), transverse chromatic aberration, and the circle of least confusion are all defined for lenses in exactly the same way as for spherical aberration in mirrors.

Chromatic aberration cannot be eliminated for ALL colours simultaneously, but it CAN be corrected for one chosen pair of extreme colours (which incidentally reduces it substantially for the colours in between too), using an ACHROMATIC combination -- either a convex and a concave lens held in contact, or two convex lenses with a carefully chosen separation. For two thin lenses of dispersive powers $\omega_1$ and $\omega_2$ held in CONTACT, requiring the combination's resultant focal length to be identical for violet and red light ( $f_V=f_R$ ), and applying the lens maker's equation to each colour in turn, leads (after simplification) to the ACHROMATISM CONDITION $\dfrac{K_1}{K_2}=-\dfrac{\omega_1(n_{2Y}-1)}{\omega_2(n_{1Y}-1)}$ , where $K=\frac{1}{R_1}-\frac{1}{R_2}$ for each lens. Since dispersive power $\omega$ is always a positive quantity, this condition FORCES one of the two lenses to be convex and the other concave. For the overall combination to be CONVERGING, the FLINT-glass lens (higher refractive index, around 1.655, and higher dispersion) must be the DIVERGING (concave) member, and the CROWN-glass lens (lower refractive index, around 1.517, lower dispersion) must be the CONVERGING (convex) member.

Worked illustration: cataract-correction concavo-convex spectacles, radii of curvature 10 cm and 50 cm, made of crown glass with $n_R=1.51$ and $n_V=1.53$ , convex face receiving the incident rays. Using the lens maker's equation $\dfrac{1}{f}=(n-1)\left(\dfrac{1}{10}-\dfrac{1}{50}\right)=(n-1)(0.08)$ for each colour gives $f_R$ and $f_V$ separately, and their difference -- the longitudinal chromatic aberration -- comes out to about 1.93 cm, described as 'quite appreciable' for a spectacle lens. …

Figure 9.23Figs 9.23(a) and (b): A lens modelled as two prisms joined together

What this figure shows. Part (a): a convex (converging) lens shown in cross-section, with two triangular wedge shapes (representing thin prisms) overlaid on its upper and lower halves, the two wedges drawn joined BASE-to-BASE at the lens's central thick region -- illustrating that a convex lens can be thought of as a stack of prism-like wedges of increasing angle away from the centre, each behaving like a small prism that would, on its own, deviate light toward its base, i.e. toward the lens's central axis (hence a convex lens converges light). Part (b): a concave (diverging) lens shown in cross-section, with two triangular wedge shapes overlaid on its upper and lower halves, this time drawn joined VERTEX-to-VERTEX at the lens's thin central region -- illustrating that a concave lens likewise behaves like prism-wedges, but here each wedge's base points AWAY from the axis (toward the thick …

Figure 9.24Figs 9.24(a) and (b): Chromatic aberration for convex and concave lenses

What this figure shows. Part (a): a convex lens with a beam of parallel WHITE light incident along (or parallel to) the principal axis; after passing through the lens, the beam is shown converging to NOT one but several closely-spaced focal points along the axis, one per colour -- violet (V), having the highest refractive index and hence deviated most, focuses CLOSEST to the lens (shortest focal length), red (R) focuses FARTHEST from the lens (longest focal length), with the other spectral colours' foci falling in between along a short segment of the axis; a small circle (the circle of least confusion) is marked at the position along the axis where the overall image blur is smallest. Part (b): the mirror-image situation for a concave lens, where the same parallel white beam diverges after the lens, and the several colours' virtual foci (traced backward as dashed lines to points on the axis in FRONT of the lens, i.e. on the same side as the incident light) are …

Misc Ex.9.11Longitudinal chromatic aberration of cataract-correction spectacles

Worked out. Cataract-correction spectacles are concavo-convex, with radii of curvature 10 cm and 50 cm, made of crown glass with refractive indices 1.51 for red and 1.53 for violet, and the convex face receiving the incident rays (so both radii are taken as positive, R1=10 cm, R2=50 cm, per the diagram's sign convention for this shape). Applying the lens maker's equation 1/f=(n-1)(1/R1-1/R2)=(n-1)(1/10-1/50)=(n-1)(0.08) separately for each colour gives fR and fV, and the longitudinal chromatic aberration (the difference fV minus fR, i.e. the axial separation between the violet and red foci) comes out to about 1.93 cm -- described in the book as 'quite appreciable' for a spectacle lens, and the same result is noted to hold even if t …

Figure 9.25Figs 9.25(a) and (b): Spherical aberration for convex and concave lenses

What this figure shows. Part (a): a convex lens with a WIDE beam of monochromatic parallel rays incident along the principal axis -- the PARAXIAL rays (close to the axis) are shown converging to one focal point Fp on the axis, while the MARGINAL rays (farther from the axis, near the lens's outer edge) are shown converging to a DIFFERENT point Fm closer to the lens, so the two foci Fp and Fm are axially separated; a small circle (the circle of least confusion) is marked at the position between them where the beam's cross-section is narrowest, and the axial distance between Fm and Fp is marked as the longitudinal spherical aberration. Part (b): the analogous diagram for a concave lens, where the wide beam diverges after the lens and the paraxial and marginal rays' virtual foci (traced backward, on the same side as the incident light) are similarly shown at two different axial posi …

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