Reflection and Refraction

01 / UNDERSTAND & APPLY

1. A ray diagram is a geometric model with a sign convention

Geometrical optics treats light as rays when relevant apertures and curvature scales are large compared with wavelength and diffraction can be neglected for the question. A ray indicates local propagation direction. A real image forms where rays physically converge; a virtual image is located where backward extensions appear to meet. A virtual image can be seen by an eye even though it cannot be caught directly on a screen at that apparent position.

Adopt one Cartesian convention for this lesson: light initially travels left to right, the pole or optical centre is the origin, rightward axial distances are positive and upward heights positive. Real objects on the incoming left side have u < 0. For lenses, a real image to the right has v > 0. For a mirror, a real reflected image in front on the left has v < 0 because the coordinate axis is still the original one.

A focal length is signed according to the focus location in this convention. A converging thin lens in air has f > 0 and a diverging lens f < 0. A concave mirror facing incoming light has f < 0 while a convex mirror has f > 0. These signs differ from some textbooks’ positive-real conventions; never combine equations from one convention with signs from another.

Paraxial formulas assume rays close to the principal axis and small angles so sinθ ≈ tanθ ≈ θ in radians. Spherical aberration means rays far from the axis may not share one image point. A clean schematic ray diagram illustrates the model, but it does not remove the approximation behind the compact imaging equation.

02 / UNDERSTAND & APPLY

2. Reflection and refraction depend on the interface normal

The law of reflection gives equal incident and reflected angles measured from the local normal. The incident ray, reflected ray and normal lie in one plane. For a plane mirror, the image is as far behind the mirror as the object is in front, has the same size and is virtual for an ordinary real object. Apparent left–right reversal is best understood as reversal of the coordinate perpendicular to the mirror rather than a mysterious exchange of every horizontal direction.

At a stationary interface between isotropic transparent media, Snell’s law is n₁ sin i = n₂ sin r. Angles are measured from the normal, not the surface. A ray entering higher refractive index bends toward the normal unless it is normally incident. Frequency remains fixed; speed and wavelength decrease when index increases.

For a plane-parallel slab surrounded by the same medium on both sides, the emergent ray is parallel to the incident ray but laterally displaced. For slab thickness t, incidence i and refraction r, displacement magnitude is t sin(i − r)/cos r. This result assumes flat parallel faces and the stated surrounding media. A prism generally changes the final direction because its faces are not parallel.

For near-normal viewing through a plane boundary from index n₁ toward a viewing medium n₂, an object at real depth d in n₁ appears at depth approximately d n₂/n₁. The approximation becomes angle-dependent away from the paraxial direction. Distinguish apparent depth from lateral displacement and from optical path length.

03 / UNDERSTAND & APPLY

3. Total internal reflection and prism deviation have different conditions

Total internal reflection requires light to travel from higher index n₁ toward lower index n₂ and incidence above the critical angle ic, where sin ic = n₂/n₁. At the critical angle, the refracted ray grazes the interface. Below it, some light transmits and some reflects; above it, the ideal lossless interface has no propagating refracted ray in the second medium.

Optical fibres guide light using an appropriate higher-index core and lower-index cladding. The ray model uses repeated total internal reflection, while a full waveguide description is needed for detailed modes. A prism can also use total internal reflection to redirect a beam efficiently. “Large incidence angle” alone is insufficient without the high-to-low index condition.

For a prism of apex angle A, internal refraction angles satisfy r₁ + r₂ = A, and deviation is δ = i + e − A. At minimum deviation in a symmetric path, i = e and r₁ = r₂ = A/2. For a prism in air, its index is n = sin[(A + δmin)/2]/sin(A/2). The thin-prism approximation δ ≈ (n − 1)A uses small angles in radians.

Dispersion arises because refractive index depends on wavelength. In common normal-dispersion glass, shorter visible wavelengths typically have greater index and deviation, but the detailed material dispersion matters. A prism changes the propagation directions of colours without changing their frequencies at stationary boundaries.

04 / UNDERSTAND & APPLY

4. Spherical mirrors and thin lenses use different signed equations

For a spherical mirror in the stated Cartesian convention, 1/v + 1/u = 1/f with f = R/2, and transverse magnification m = himage/hobject = −v/u. The radius R is signed by the centre of curvature’s location. A concave mirror can make real inverted or virtual upright images depending on object position; a convex mirror gives a reduced upright virtual image for an ordinary real object.

For a thin lens with the same surrounding medium on both sides, 1/v − 1/u = 1/f and m = v/u. A converging lens with an object beyond its focal distance forms a real inverted image. With the object between the lens and its focus, it forms a virtual upright magnified image. A diverging lens forms a reduced upright virtual image for a real object in the usual setup.

Three useful paraxial lens rays are: a ray parallel to the axis refracts through the far focus for a converging lens; a ray through the optical centre is approximately undeviated; and a ray directed through the near focus emerges parallel. For a diverging lens, use backward extensions of the appropriate refracted rays. At least two rays locate the image, and a third can check the construction.

For mirrors, a parallel incident ray reflects through the focus for a concave mirror; a ray through the centre of curvature retraces its path; and a ray through the focus reflects parallel. Keep actual ray segments and backward extensions visually distinct. A dashed construction behind a mirror or on the object side of a lens is not an actual light path.

05 / UNDERSTAND & APPLY

5. Curvature and refractive index set lens power

For refraction at a spherical interface from index n₁ to n₂ in the Cartesian convention, n₂/v − n₁/u = (n₂ − n₁)/R. The centre of curvature to the right gives R > 0. This paraxial relation can be applied successively to the two surfaces of a lens, tracking the first image as the second surface’s object.

For a thin lens of refractive index nlens in a surrounding medium nmedium on both sides, the lens-maker relation is 1/f = (nlens/nmedium − 1)(1/R₁ − 1/R₂). Surface radii must be signed with the same global axis. A symmetric biconvex lens in air has R₁ > 0 and R₂ < 0, giving positive power when nlens > nmedium.

A lens’s shape alone does not decide convergence independently of the surrounding medium. If its index equals the medium’s, its refractive power vanishes in the ideal model. If immersed in a medium with larger index, the sign can reverse. This is a useful limiting-case test of the relative-index factor.

Optical power is P = 1/f with f in metres, measured in dioptres. Thin lenses in contact have powers that add: Ptotal = P₁ + P₂, provided the ordinary thin-lens same-medium approximation applies. Separated lenses require propagation between them; simply adding powers then generally misses the separation term.

06 / UNDERSTAND & APPLY

6. A first image becomes the next element’s object

For separated optical elements, solve one at a time. Locate the first image in a common coordinate system, then measure its distance from the second element. If rays arriving at the second element are already converging toward a point beyond it, that point is a virtual object for the second element, with positive u in the stated left-to-right convention.

An intermediate image need not be physically formed before the second element intercepts the rays. The would-be convergence point still determines the incoming wavefront curvature and serves as the second element’s object. This is why negative or positive object distances cannot be chosen solely from the word “image” in the previous step.

Total transverse magnification is the product of the successive signed magnifications when image heights are consistently referenced. A negative product indicates overall inversion. Distances must be recomputed after every reflection if the convenient propagation-axis convention changes; either maintain a global Cartesian axis carefully or explicitly restart the convention at each stage.

07 / UNDERSTAND & APPLY

7. Optical instruments magnify angle, not just image height

The eye responds to the angular size of an object’s image on the retina. A simple magnifier lets an object be placed closer than the unaided near-point distance D while forming a virtual image the eye can focus on. For relaxed-eye viewing with the object near the magnifier’s focal plane, angular magnification is approximately D/f. With the final virtual image at the near point, it is approximately 1 + D/f under the usual close-lens geometry.

A compound microscope uses a short-focal-length objective to form a real enlarged intermediate image, then an eyepiece as a magnifier. For normal adjustment and a suitable long-tube paraxial approximation, magnification magnitude is approximately (L/fobjective)(D/feyepiece), where L is the relevant optical tube-length approximation. Exact problems should use the objective’s actual image/object distances rather than treating any physical tube measurement as interchangeable with L.

An astronomical refracting telescope for a distant object uses the objective to form an image near its focal plane and the eyepiece to make the final image at infinity in normal adjustment. Angular magnification is −fo/fe for two converging lenses, with the minus indicating inversion. Lens separation is approximately fo + fe. The objective aperture controls light collection and diffraction-limited resolution, not only magnification.

For basic vision correction, myopia requires a diverging lens to make a distant object appear at the eye’s far point, while hypermetropia/near-vision difficulty may require a converging lens to place a desired near object’s virtual image at the eye’s accessible near point. These are idealised optical exercises, not a prescription for an actual person; eye examination also considers other optical and health factors.

08 / UNDERSTAND & APPLY

7a. A microscope needs an actual intermediate-image calculation

For a specified microscope geometry, calculate the objective image distance before using any tube-length approximation. In normal adjustment the real intermediate image must lie one eyepiece focal length before the eyepiece, so the physical lens separation is v_objective + f_eyepiece. The objective magnification is v_objective/u_objective in the Cartesian convention. Multiplying it by D/f_eyepiece gives the signed angular magnification for a relaxed eye close to the eyepiece.

The familiar L/f_objective approximation replaces the exact objective image/object distance ratio using a long-tube regime. A given mechanical tube length may include distances that differ from the optical quantity used in that approximation. An explicit separation and exact thin-lens calculation prevent that ambiguity. The intermediate image is real and inverted; the eyepiece supplies angular enlargement rather than turning that image upright.

09 / UNDERSTAND & APPLY

7b. Reflection sends the real image back toward the object side

A concave mirror with an ordinary real object beyond the centre of curvature forms a smaller real image between the focus and centre of curvature. Both object and real image lie in front of the mirror, even though the incident and reflected rays travel in opposite directions. In the global left-to-right Cartesian convention used here, those object-side coordinates are negative.

The diagram uses an incident ray parallel to the principal axis, which reflects through the focus, and a ray aimed through the centre of curvature, which retraces its incoming line. Their reflected segments meet at the inverted real image. This is qualitatively different from transmission through a lens: a real mirror image does not have to lie to the right of the optical element. The paraxial construction represents a spherical mirror; wide-angle rays do not all meet at one exact point.

10 / UNDERSTAND & APPLY

8. Magnification, brightness and resolution are not synonyms

A large image is not necessarily a detailed image. Diffraction limits the smallest angular separation an aperture can resolve; geometric aberrations can blur it further. Increasing an eyepiece’s magnification beyond the available information merely enlarges blur. Lens diameter, optical quality and wavelength matter alongside focal length.

Spherical aberration arises when rays far from the principal axis focus differently from paraxial rays at a spherical surface. Chromatic aberration arises because refractive index and thus focal length vary with wavelength. Aperture stops can reduce some geometric aberrations at the cost of light collection, while compound lens designs can correct selected aberrations over a useful range.

Ray diagrams should therefore label which rays are actual and which are extensions, and whether the model is thin, paraxial and monochromatic. A mathematically divergent image distance at u = −f for the stated lens convention means emerging rays are parallel in the ideal model, not that energy or image brightness becomes infinite.

Use limiting cases to check signs: a distant real object should focus near the image-side focus of a converging lens; an index-matched lens should have zero power; a plane mirror should preserve size; and a diverging lens with a real object should not suddenly produce a real image in the same simple setup.

11 / UNDERSTAND & APPLY

9. Original practice: geometry before substitution

Use the Cartesian convention stated in section 1. Distances are measured from the relevant optical centre or mirror pole; all lenses are thin and rays paraxial unless stated otherwise.

• Angles of incidence/refraction are measured from normals.
• Total internal reflection needs high-to-low index and incidence above critical.
• Mirror and lens equations have different signs in the same Cartesian convention.
• An intermediate image can become a virtual object for the next element.
• Instrument magnification is angular and has viewing-condition assumptions.
• Ray optics does not remove diffraction or aberration limits.