Light & Ray Optics – Comprehensive Study Notes

Learning Outcomes

  • By the end of this topic students should be able to:

    • Explain and apply the principles of reflection, refraction, and total internal reflection (TIR).

    • Analyse image formation by plane, concave and convex mirrors; distinguish between real and virtual images.

    • Apply the mirror equation and magnification relationship to solve quantitative problems.

    • Describe thin-lens behavior for converging (convex) and diverging (concave) lenses; derive image characteristics with the thin-lens equation.

    • Combine multiple lenses to predict final image position, orientation and magnification.

    • Recognise and explain the operation of common optical instruments (camera, human eye, magnifier, microscope, telescope, prism binoculars, periscope, optical fibres).

Reflection Fundamentals

  • Reflection types

    • Specular (smooth surface): Incident and reflected rays lie in the same plane ⟂ to surface.

    • Diffuse (rough surface): Local law of reflection obeyed at each microfacet; macroscopic scattering in many directions.

  • Law of Reflection (plane mirror)

    • \thetai = \thetar (angle measured from the normal).

    • Incident ray, reflected ray, and normal are coplanar.

  • Plane mirror image properties

    • Image distance v equals object distance u: u = v.

    • Image is virtual, upright, same size, laterally reversed; appears behind mirror.

    • Rays diverge from the virtual image point P′; eye back-traces them as straight lines.

    • No convergence of light; cannot project on a screen.

  • Ray-tracing essentials (plane mirror)

    • Draw at least two incident rays from tip of object; reflect using equal angles; extend backward to locate image.

Spherical Mirrors

  • Terminology

    • Concave mirror: reflecting surface on interior of sphere; converging mirror.

    • Convex mirror: reflecting surface on exterior; diverging mirror.

    • Centre of curvature C, vertex V, principal axis, focal point F, radius of curvature R.

  • Focal length (paraxial approximation)

    • f = \frac{R}{2} for both concave and convex mirrors (sign conventions apply).

Image Characteristics by Concave Mirror

  • Object at \infty: image at F, real, inverted, highly diminished.

  • Object beyond C (u>2f): image between F and C, real, inverted, diminished.

  • Object at C (u=2f): image at C, real, inverted, same size.

  • Object between F and C (f<u<2f): image beyond C, real, inverted, magnified.

  • Object at F: image at \infty.

  • Object between F and mirror (u<f): image behind mirror, virtual, upright, magnified.

Image Characteristics by Convex Mirror

  • Object anywhere in front: image behind mirror, virtual, upright, diminished; 0<v<f.

Mirror Equation & Magnification

  • Spherical mirror equation (Gaussian form)

    • \frac{1}{f}=\frac{1}{u}+\frac{1}{v}.

    • Sign convention (front of mirror = positive):

    • u>0 for real objects.

    • v>0 for real images (in front); v<0 for virtual (behind).

    • f>0 concave; f<0 convex.

  • Linear magnification

    • M = \frac{hi}{ho}= -\frac{v}{u}.

    • M>0 → upright; M<0 → inverted.

Refraction Basics

  • When light crosses an interface two phenomena occur: partial reflection & refraction (transmission with direction change).

  • Index of refraction: n = \frac{c}{v_{medium}}; vacuum n=1 (exact).

  • Representative n values: air ~1.00, water 1.33, glass 1.50, diamond 2.42, silicon (IR) 3.50.

Snell’s Law

  • n1\sin\theta1 = n2\sin\theta2.

  • For n2>n1 light bends toward the normal; for n21 it bends away.

Total Internal Reflection (TIR)

  • Occurs only when n1>n2 (light from denser → rarer medium).

  • Critical angle \thetac: \thetac = \sin^{-1}\left(\frac{n2}{n1}\right) when \theta_2 = 90^{\circ}.

  • Criteria: \thetai > \thetac → 100 % reflection, 0 % transmission.

  • Applications

    • Prism binoculars & periscopes (multiple TIR surfaces for path folding).

    • Optical fibres: core n1, cladding n2; repeated TIR confines light.

Thin Lenses

  • Lens types

    • Converging (convex, thicker centre) vs diverging (concave, thinner centre).

    • Forms: biconvex, plano-convex, meniscus; analogous concave shapes.

  • Double refraction: light bends twice (air→glass, glass→air) but treated with single effective surface for thin-lens model.

Ray-Tracing Rules (convex lens)

  1. Ray parallel to axis → refracts through far focal point F_2.

  2. Ray through near focal point F_1 → emerges parallel to axis.

  3. Ray through optical centre passes undeviated.

  • Diverging lens: focal points are virtual; rays appear to originate from F on incoming side.

Thin-Lens Equation & Magnification

  • Lens equation (Gaussian form)

    • \frac{1}{f}=\frac{1}{u}+\frac{1}{v}.

    • Sign conventions (optical axis direction to right):

    • Object: u>0 if object is on incident-light side (real); u<0 for virtual object.

    • Image: v>0 if on opposite side from object (real); v<0 if on same side (virtual).

    • Focal length: f>0 converging; f<0 diverging.

  • Magnification

    • M = \frac{hi}{ho}=\frac{v}{u} (positive ⇒ upright, negative ⇒ inverted).

Image Characteristics (Convex Lens)

  • u>2f: real, inverted, diminished, f<v<2f.

  • u=2f: real, inverted, same size, v=2f.

  • f2f.

  • u=f: image at infinity (used in telescope eyepiece for relaxed eye).

  • u<f: virtual, upright, magnified (magnifying glass).

Image Characteristics (Concave Lens)

  • Object anywhere: virtual, upright, diminished; image forms between lens and near focal point.

Multiple-Lens Systems

  • Procedure

    1. Treat first lens with given object distance u1, find v1.

    2. Image I1 becomes object for second lens; determine new object distance u2 (with proper sign, measured from lens 2).

    3. Continue for additional lenses.

  • Overall magnification: M = M1\times M2 \times \dots \times M_n.

Optical Instruments & Applications

  • Camera

    • Converging lens focuses real inverted image onto a sensor/film (fixed focal length; variable lens-to-film distance for focus).

    • Resolution set by pixel size (e.g., 4600\times3500 px, 12 cm sensor width).

  • Human Eye

    • Cornea + lens act as variable-focal length system; retina is fixed image plane.

    • Defects: myopia (near-sighted, corrected with diverging lens), hyperopia (far-sighted, converging lens), presbyopia (bifocals).

  • Magnifying Glass (simple loupe)

    • Virtual upright image at least 25 cm from eye (near-point) or at infinity for relaxed vision.

  • Compound Microscope

    • Objective (short f_o) forms real, inverted, magnified image at tube distance L.

    • Eyepiece (lens of f_e) treats this image as object, producing highly magnified virtual image at near-point or infinity.

  • Astronomical Refracting Telescope

    • Objective (large f_o) forms real image near its focal plane.

    • Eyepiece set so final image at infinity, angular magnification M=\frac{fo}{fe}.

  • Prism Binoculars / Periscope

    • Employ TIR in right-angle prisms to invert image and shorten optical path.

  • Optical Fibres

    • Light launched into high-n core undergoes continuous TIR at core-cladding interface; allows low-loss data transmission.

Consolidated Formula List

  • Law of Reflection: \thetai = \thetar.

  • Mirror/Lens Equation: \frac{1}{f}=\frac{1}{u}+\frac{1}{v}.

  • Magnification: M = -\frac{v}{u} (mirrors), M = \frac{v}{u} (lenses, sign conventions as above).

  • Radius–Focal relationship (spherical mirror): f = \frac{R}{2}.

  • Snell’s Law: n1\sin\theta1 = n2\sin\theta2.

  • Critical angle (TIR): \thetac = \sin^{-1}\left(\frac{n2}{n1}\right), \; n1>n_2.

  • Multiple lenses: M{total}=\prod{i=1}^{n} M_i.

These bullet-point notes capture all major and minor concepts, equations, examples, and applications discussed in the transcript, providing a stand-alone study resource for Light & Ray (Geometrical) Optics.