Optics · Chapter 17

Geometric Optics

When the wavelength of light is much smaller than the optical elements — lenses, mirrors, apertures — we can treat light as rays traveling in straight lines. This is geometric optics.

PrerequisitesWaveproperties(Ch.8)forthephysicalpicture⋅TrigonometryforSnell′slawcalculatioWave properties (Ch. 8) for the physical picture \cdot Trigonometry for Snell's law calculations
Learning Goals
  • Apply the law of reflection and the mirror equation to find image properties.
  • Use Snell's law to calculate refraction angles at a boundary between two media.
  • Determine the critical angle for total internal reflection and explain its applications.
  • Solve the thin lens equation for image distance and magnification.
  • Trace principal rays through lens and mirror systems to locate images.

17.1 Reflection

When light strikes a smooth surface, it reflects. The law of reflectionstates that the angle of incidence equals the angle of reflection, both measured from the normal to the surface:

θi=θr\theta_i = \theta_r(17.1)

For a flat mirror, the image appears as far behind the mirror as the object is in front — virtual (no real light passes through it), upright, and the same size as the object. For a curved mirror, we use the mirror equation:

1/do+1/di=1/f=2/R1/d_{o} + 1/d_{i} = 1/f = 2/R(17.2)

where d_o is the object distance, d_i is the image distance, f is the focal length, and R is the radius of curvature. A concave mirror has f > 0; convex has f < 0. The magnification m = −d_i/d_o: negative means inverted.

17.2 Refraction and Snell's Law

Light bends when it passes from one medium to another because its speed changes. The ratio of light's speed in vacuum to its speed in the medium is the index of refractionn = c/v. The bending is governed by Snell's law:

n1sinθ1=n2sinθ2n_{1} sin \theta_{1} = n_{2} sin \theta_{2}(17.3)

Light bends toward the normal when entering a denser medium (larger n) and away from the normal when entering a less dense medium. Common indices: air ≈ 1.00, water ≈ 1.33, glass ≈ 1.5, diamond ≈ 2.42.

Definition 17.1 — Total Internal Reflection
Whenlighttravelsfromadensemedium(n1)toalessdensemedium(n2When light travels from a dense medium (n_{1}) to a less dense medium (n_{2} < n1),thereexistsacriticalangleθcabovewhichalllightisreflectedandnonetransmin_{1}), there exists a critical angle \theta_c above which all light is reflected and none transmited:θc=arcsin(n2/n1\theta_c = arcsin(n_{2}/n_{1}Forglass−air(n1=1.5,n2=1.0):θc=arcsin(1/1.5)=41.8°.ThisistheprinciplebehindFor glass-air (n_{1}=1.5, n_{2}=1.0): \theta_c = arcsin(1/1.5) = 41.8°. This is the principle behind optical fiber communication: light is trapped inside the fiber by total internal reflection around every bend.
Figure 17.1. Rayopticssimulation.Refractiontab:adjusttheincidentangleandindexn2toseebendiRay optics simulation. Refraction tab: adjust the incident angle and index n_{2} to see bending and total internal reflection. Thin lens tab: principal rays show how a converging lens forms a real image. Mirror tab: concave mirror with mirror equation.
Example 17.1 — Snell's Law at a Glass Surface

Alightrayinairstrikesaglasssurface(n=1.52)atθ1=45°.FindtherefractedanglA light ray in air strikes a glass surface (n = 1.52) at \theta_{1} = 45°. Find the refracted angle. What is the critical angle for this glass?

Snell's law:n1sinθ1=n2sinθ2→1.00×sin45°=1.52×sinθ2n_{1} sin \theta_{1} = n_{2} sin \theta_{2} \to 1.00 \times sin 45° = 1.52 \times sin \theta_{2}
Refracted angle:sinθ2=sin45°/1.52=0.707/1.52=0.465→θ2=27.7°sin \theta_{2} = sin 45° / 1.52 = 0.707 / 1.52 = 0.465 \to \theta_{2} = 27.7°
Critical angle:θc=arcsin(n2/n1)=arcsin(1/1.52)=arcsin(0.658)=41.1°\theta_c = arcsin(n_{2}/n_{1}) = arcsin(1/1.52) = arcsin(0.658) = 41.1°
Interpretation:Any ray inside the glass hitting the surface at > 41.1° will be totally internally reflected.

17.3 Thin Lenses

A thin lens refracts light at two surfaces. For a lens much thinner than its focal length, both refractions are treated as occurring at the lens plane. The thin lens equation is the same form as the mirror equation:

1/do+1/di=1/f1/d_{o} + 1/d_{i} = 1/f(17.4)

A converging (convex) lens has f > 0. A diverging (concave) lens has f < 0. The focal length is related to the lens geometry by the lensmaker's equation:

1/f=(n−1)×(1/R1−1/R2)1/f = (n-1) \times (1/R_{1} - 1/R_{2})(17.5)

where R₁ and R₂ are the radii of curvature of the two surfaces (positive if center of curvature is to the right). The power of a lens is P = 1/f measured in diopters (D = m⁻¹).

Theorem 17.1 — Image Properties for a Converging Lens
Object beyond 2f: real, inverted, reduced, on far side of lensObject at 2f: real, inverted, same size, at 2fObject between f and 2f: real, inverted, enlarged (projector)Object inside f: virtual, upright, enlarged (magnifying glass)
Example 17.2 — Image Location from a Thin Lens

Aconverginglenshasf=20cm.Anobjectisplaced60cmfromthelens.FindtheimagedA converging lens has f = 20 cm. An object is placed 60 cm from the lens. Find the image distance and magnification.

Thin lens equation:1/di=1/f−1/do=1/20−1/60=3/60−1/60=2/601/d_{i} = 1/f - 1/d_{o} = 1/20 - 1/60 = 3/60 - 1/60 = 2/60
Image distance:di=60/2=30cm(positive:real,onfarsided_{i} = 60/2 = 30 cm (positive: real, on far side
Magnification:m=−di/do=−30/60=−0.5(inverted,halfthesizem = -d_{i}/d_{o} = -30/60 = -0.5 (inverted, half the size
Example 17.3 — Combination of Two Lenses

Twoconverginglenses,f1=30cmandf2=10cm,are20cmapart.Objectis45cmtotheTwo converging lenses, f_{1} = 30 cm and f_{2} = 10 cm, are 20 cm apart. Object is 45 cm to the left of lens 1. Find the final image.

Lens 1:1/di1=1/30−1/45=3/90−2/90=1/90→di1=90cm1/di_{1} = 1/30 - 1/45 = 3/90 - 2/90 = 1/90 \to di_{1} = 90 cm
Object for lens 2:do2=20−90=−70cm(virtualobject,behindlens2do_{2} = 20 - 90 = -70 cm (virtual object, behind lens 2
Lens 2:1/di2=1/10−1/(−70)=7/70+1/70=8/70→di2=8.75cm1/di_{2} = 1/10 - 1/(-70) = 7/70 + 1/70 = 8/70 \to di_{2} = 8.75 cm
Final image:8.75 cm to the right of lens 2. Real and inverted.
Definition 17.2 — Common Traps
  • Angles are measured from the normal: not from the surface.
  • Virtual images have sign meaning: track the sign convention before interpreting image distance.
  • Total internal reflection needs high-to-low index: there is no critical angle going from air into glass.
  • Thin-lens systems are sequential: the first image becomes the object for the second lens.
Exercises — 17.1–17.3 Geometric Optics
1.
Arayinairstrikesglass(n=1.52)atanangleofincidenceof45°.FindtherefractedA ray in air strikes glass (n = 1.52) at an angle of incidence of 45°. Find the refracted angle.
°
Straightforward
2.
Aconverginglens(f=20cm)hasanobject60cmaway.FindtheimagedistanceA converging lens (f = 20 cm) has an object 60 cm away. Find the image distance.
cm
Straightforward
3.
Whatisthecriticalanglefortotalinternalreflectionataglass(n=1.5)toairinterWhat is the critical angle for total internal reflection at a glass (n = 1.5) to air interface?
°
Intermediate
4.An object is 15 cm from a converging lens of focal length 20 cm. Find the image location and magnification. What type of image is it?
Intermediate
5.Derive the angular magnification of a simple two-lens telescope with objective focal length f_andeyepiecefocallengthfeye.Iffobj=100cmandfeye=5cm,whatisthemagnificaand eyepiece focal length f_{eye}. If f_{obj} = 100 cm and f_{eye} = 5 cm, what is the magnification and tube length?
Challenging
Key Takeaways
  • Lawofreflection:θi=θr—angleinequalsangleout,measuredfromnormalLaw of reflection: \theta_i = \theta_r — angle in equals angle out, measured from normal.
  • Snell′slaw:n1sinθ1=n2sinθ2—lightbendstowardnormalenteringadensermediumSnell's law: n_{1} sin \theta_{1} = n_{2} sin \theta_{2} — light bends toward normal entering a denser medium.
  • Totalinternalreflectionatθ>θc=arcsin(n2/n1)—basisofopticalfiberTotal internal reflection at \theta > \theta_c = arcsin(n_{2}/n_{1}) — basis of optical fiber.
  • Thinlensequation:1/do+1/di=1/f,sameformasmirrorequationThin lens equation: 1/d_{o} + 1/d_{i} = 1/f, same form as mirror equation.
  • Magnificationm=−di/do—negativemeansinvertedMagnification m = -d_{i}/d_{o} — negative means inverted.
  • Object inside focal length gives a virtual, upright, magnified image (magnifying glass).