Refraction, lenses and optical instruments

Refraction, lenses and optical instruments

What changes at an optical boundary

Light crossing between transparent materials can change speed and wavelength. Frequency remains fixed at a stationary boundary because oscillations must connect continuously. Except at normal incidence, the speed change usually changes ray direction. Entry into a slower medium bends toward the normal; entry into a faster one bends away. Draw the normal perpendicular to the surface before labelling incidence and refraction angles. They are measured from that normal, not the surface. Optical density refers to the relevant propagation behaviour rather than an automatic comparison of mass density. A ray can change speed without bending when it crosses normally.

Index and Snell's law

Absolute index compares vacuum speed with material speed. For ordinary transparent materials in the visible range it exceeds one. Snell's law n₁sin uses both normal-measured angles. Air can often be approximated by. A parallel-sided slab in matching surrounding media gives an emergent ray parallel to the incident ray but laterally shifted. That shift depends on slab thickness and angles and is not a frequency change. At a minimum-deviation passage through a suitable prism, relative to air under the standard assumptions. Identify prism angle A and deviation D before substitution.

Critical angle and internal reflection

Total internal reflection requires travel from greater refractive index toward lower index and incidence above critical angle. At the critical angle the refracted ray lies along the boundary, so for n₁ >n₂. Below it there can be both reflected and transmitted light. Optical fibres use a higher-index core with lower-index cladding to guide suitable rays, although real transmission has losses and design constraints. Reflecting prisms can redirect beams within instruments. Do not apply the critical-angle equation with the larger index in the numerator and accept an impossible sine above one; such a result signals that the chosen travel direction cannot produce the proposed critical condition.

Apparent depth, dispersion and mirages

For near-normal viewing through a flat water-to-air boundary, real depth/apparent depth approximates the index relative to air. The underwater object appears shallower because backward extensions of emergent rays meet above it. This is not a general formula at every viewing angle. A prism disperses white light because index varies with wavelength; ordinary glass generally deviates violet more than red. A mirage arises from light travelling through a varying-index air structure associated with temperature gradients, rather than an actual pool of reflecting water. Refraction and internal-reflection ray models help describe it without treating air as a uniformly indexed slab everywhere.

Constructing lens ray diagrams

A thin converging lens brings initially parallel paraxial rays toward a far principal focus; a diverging lens spreads them as though from a near focus. Draw axis, optical centre and focal points on both sides. A centre ray proceeds approximately undeviated in the thin-lens model. A parallel ray through a converging lens goes through the far focus, while one aimed through the near focus emerges parallel. Two rays can locate an image; dashed backward extensions show a virtual meeting distinct from actual convergence. The diagram must correspond to the object position and lens type rather than a memorised picture reused for every question.

An explicit lens convention

Here the real-is-positive convention uses positive real-object distance u and positive focal length f for a converging. Positive v gives a real image on the far side and negative v a virtual image on the object side. Size-magnification magnitude is|v/u|. Beyond2f the image is diminished, real and inverted; at2f equal size; between f and2f magnified, real and inverted; inside f magnified, upright and virtual. At f emergent rays are parallel. Newton's lens form uses distances a and b from the appropriate focal points for the matching real-object/real-image geometry, not ordinary lens-centre distances.

Power and close-lens combinations

Power is1/f with f in metres and unit dioptre. A converging lens has positive and diverging lens negative power under the usual convention. A25 cm converging lens is0.25 m and has+4 D power. Thin close lenses approximately add powers, so+5 D and−2 D give+3 D net. Their focal lengths do not directly add. The approximation assumes negligible lens separation; separated optical systems need a fuller model. A larger power magnitude means shorter focal-length magnitude, not a physically thicker image. Retain the sign when calculating diverging correction or a combination that may have a negative net power.

Eye, camera and projector

The eye focuses a real image on the retina and adjusts mainly through lens-shape changes. A camera forms a real image on film or sensor and uses focusing and aperture controls; a projector forms an enlarged real image on a screen from a suitable object. Short-sightedness involves distant rays focusing before the retina in the simple model and is corrected by suitable diverging lenses. Long-sightedness and near-focus difficulties use appropriate converging correction in the corresponding simple treatment. Near and far points describe limits of clear viewing for a given state of the eye. These principles do not replace a professional personal prescription.

Magnifiers and compound instruments

A magnifying glass is a converging lens with its object within the focal length, giving an upright virtual image of larger angular size. A compound microscope's objective gives a magnified real intermediate image and its eyepiece magnifies that. An astronomical refracting telescope in normal adjustment has objective and eyepiece arranged around the intermediate focal image, with angular-magnification magnitude fo/fe in the simple model. Image inversion is often acceptable for astronomy. Angular magnification compares viewing angles and differs from the image-height ratio of one lens. Identify which lens forms the first image and where the next lens uses it as its object.

Self-check and experimental care

For f =10 cm and, so v =15 cm. The image is real, inverted and half size. For glass-to-air n =1.5, critical angle is sin⁻¹(1/1.5), about41.8°. In a focal-length experiment align object, lens and screen, locate a sharp image and measure from the lens centre under the thin-lens approximation. Repeat with different object distances and compare results. A ray sketch can expose an impossible sign or image location before lengthy arithmetic. Never look directly at the Sun through an unfiltered lens or optical instrument; concentrated sunlight can injure the eye.