Refraction at Plane Surfaces & TIR Refraction at Plane Surfaces & Total Internal Reflection When light goes from air into water or glass, it bends because its speed changes. That bending at the boundary is called refraction. Everyday sights like a pencil seeming bent in a glass of water, a coin looking shallower than it is, or the sparkling of a diamond all come from refraction and its extreme case: total internal reflection (TIR). At a simple flat interface, refraction is governed by Snell’s law, which relates the sines of the angles of incidence and refraction to the refractive indices of the media. If you can read the problem and translate it into an angle diagram with a normal at the surface, then most questions boil down to: choose the right formula, apply a sign or small-angle approximation carefully, and check the boundary conditions. The same simple ideas also give useful results like apparent depth (how far below the surface things look), lateral shift through a glass slab (how a parallel-sided block displaces a ray sideways), and the critical angle (beyond which the light does not emerge, but reflects completely inside the denser medium). Optical fibres use many successive TIRs to guide light over long distances with low loss. Heat shimmering on roads and mirages in deserts are also refraction stories, produced by continuous variation of refractive index with height. This chapter focuses on flat interfaces and TIR: definitions, equations, edge cases, and exam traps, with lots of worked examples to make the rules feel natural. remember Analogy: Think of a marching band entering mud at an angle. The side that hits mud slows first and the direction bends toward the normal. Going from mud to dry ground at an angle, the band turns away from the normal. That’s refraction. Bending of a light ray at the interface of two media of different optical density due to change in speed. Refraction Ratio of speed of light in vacuum to speed in the medium; larger n means light travels slower in that medium. Refractive index (n or ) Snell’s law For a ray crossing a plane interface: n 1 i = n 2 r , where i is the angle with the normal in medium 1 and r in medium 2. The shallower depth at which an underwater object appears when viewed from a rarer medium, under near-normal viewing. Apparent depth Sideways displacement of a ray emerging parallel to the incident ray after passing through a parallel-sided slab. Lateral shift Critical angle Angle of incidence in the denser medium for which the refracted ray just grazes along the interface ( r = 90 ). Complete reflection that occurs when light tries to pass from a denser to a rarer medium with incidence angle greater than the critical angle. Total internal reflection (TIR) A flexible, transparent waveguide (core with higher n and cladding with lower n ) that uses repeated TIR to confine light. Optical fibre Snell’s Law Angles are measured from the normal to the interface. Use this law to determine the angle of refraction or the index of the second medium when light crosses a boundary surface. Snell’s law links geometry to material property: if n 2>n 1 (rarer to denser), r<i (bends toward normal). If n 2<n 1 (denser to rarer), r>i (bends away). For small angles (paraxial rays), in radians, which simplifies many plane-surface results such as apparent depth. Keep a consistent sign convention: angles with the normal are positive and between 0° and 90°. The normal is your anchor for every drawing and calculation. tip Snell’s law applies at a sharp interface between transparent, isotropic media for the same monochromatic ray. It fails if the surface is rough (diffuse scattering), if absorption is strong, or if the medium is anisotropic (birefringence needs a different treatment). Planar interface; object in denser medium; observer in rarer medium (e.g., air) Small angles (paraxial approximation): Monochromatic light; isotropic media Apparent depth for observation from a rarer medium d app = d n Valid for near-normal viewing from air; object in a medium of refractive index . Apparent depth (air above) Apparent depth equals real depth divided by refractive index when viewed from air at small angles. d app A coin is at the bottom of a tank of water ( n=4/3 ). Its real depth is 4.0 cm. What is its apparent depth when viewed from above, near-normal? Use d app =d real /n for near-normal viewing from air. cm n = 4/3 d real = 4.0 ,cm NCERT Exemplar style: Apparent depth in water easy A parallel-sided glass slab does not change the direction of a ray overall; the emergent ray is parallel to the incident ray. But it does shift the ray sideways. This lateral shift depends on the slab thickness, refractive index, and the angle the ray makes at entry. The exact expression uses the refraction angle r inside the slab, which you get from Snell’s law. Plane-parallel slab of thickness t and refractive index n Surrounding medium is air ( n 1 ) Geometrical optics (no diffraction); planar faces Lateral shift through a parallel-sided slab = t , (i-r) r , r= -1 ! ( i n ) With r= -1 [( i)/n] . For small i , t ,i ,(1-1/n) (radians). Lateral shift medium cm A glass slab ( n=1.50 ) of thickness t=6.0 cm is struck by a ray at i=30 . Find the lateral shift. n = 1.50 t = 6.0 ,cm i = 30 Qualitative i–r relation from Snell’s law; slope depends on index ratio. For a fixed n2/n1, r increases sub-linearly with i; for n2>n1 the curve lies below y=x, for n2<n1 it lies above until reaching 90° at the critical angle. Angle of refraction r (degrees) deg n2/n1 control dependent custom deg Angle of incidence i (degrees) No bending at normal incidence Critical condition (denser to rarer) 90 i=C Wavelength matters: refractive index is slightly larger for violet than red in most glasses. As a result, the same incident angle gives a smaller refraction angle for violet than for red. That’s why different colours separate in prisms and also why the critical angle differs by colour. In TIR questions, a smaller critical angle for violet means it undergoes TIR more readily than red in the same geometry. Light travels from denser medium ( n ) to rarer medium (air, 1 ) Interface is plane; monochromatic light Critical angle relation for TIR C = 1 n ( for denser-to-air ) Critical angle C satisfies C=1/n (for denser-to-air). TIR occurs only for i>C in the denser medium. Total internal reflection requires two things: the ray is in the denser medium trying to enter a rarer medium, and the incidence angle exceeds the critical angle. At i=C , the refracted ray skims along the boundary; for any larger incidence, refraction is not possible and the ray reflects fully. Right-angled prisms exploit this to turn beams by 90° or 180° with almost no loss. Optical fibres use a high-index core and lower-index cladding to confine light by repeated TIR at the core–cladding boundary. Applications of TIR: optical fibres (telecom, endoscopes), diamond brilliance (small critical angle), periscopes with prisms, and retroreflectors. remember n 1 = 1.50 n 2 = 1.46 n 0 = 1.00 (air) hard C, NA , a An optical fibre has core index n 1=1.50 and cladding index n 2=1.46 . The fibre is in air. Find (a) the critical angle at the core–cladding boundary, (b) the numerical aperture (NA), and (c) the acceptance half-angle in air. At the core–cladding boundary, C = n 2/n 1 . For external coupling from air, NA = n 1 2 - n 2 2 and a = -1 ( NA ) (with surrounding air). A hot road or desert often shows a watery look (inferior mirage). Near the ground, hot air is less dense, so its refractive index is smaller. As light from the sky enters layers of decreasing index, it continuously bends away from the normal. For grazing rays, the bend is so large that they turn upward before reaching the ground, mimicking TIR in a graded index. Your eyes trace these rays straight, as if coming from a reflected sky on the ground, creating the illusion of water. neet-alert TIR does not occur at the critical angle but for incidence strictly greater than C . At i=C , the refracted ray is along the interface; a small increase in i triggers TIR. TIR needs light in the denser medium trying to enter a rarer medium. From air to glass, the ray is going from rarer to denser; TIR cannot occur. Total internal reflection happens when light goes from air into glass at a large angle. It is valid for near-normal viewing (paraxial rays). At large oblique angles, the apparent depth depends on the viewing angle via exact Snell refraction. Apparent depth formula d app =d/n holds for any viewing angle. For remembering TIR condition Dense to Rare, angle beware: if i grows past C , nothing emerges there. Refraction outcomes at a plane interface Rarer ( n 1<n 2 ) to Denser r<i Toward normal No TIR possible Denser ( n 1>n 2 ) to Rarer r>i Away from normal TIR possible if i>C Normal incidence i=0 No bending Only speed changes At critical angle i=C Refracted along interface Transition to TIR Case Relation Bending Notes At the polarizing (Brewster) angle i p , reflected and refracted rays are perpendicular and = i p . Transparent dielectrics; unpolarized light At Brewster’s angle, reflected and refracted rays are perpendicular Brewster’s law = i p (context: polarization at reflection) = i p Brewster’s angle concerns polarization by reflection, not TIR. Do not confuse it with the critical angle. At Brewster’s angle, reflection is minimized for light polarized in the plane of incidence, and the reflected light is fully plane-polarized perpendicular to that plane. Critical angle is about refraction ceasing altogether; Brewster’s angle is about the polarization state of the reflected ray. In YDSE, the angular fringe width is = /d . It depends on wavelength and inverse slit separation, not on screen distance. Two coherent slits separated by d ; screen far away Small-angle approximation Angular fringe width in YDSE = d For Fraunhofer single-slit diffraction, minima occur at a = n (n=1,2, ) . Condition for minima in single-slit diffraction Fraunhofer regime; slit width a ; monochromatic coherent light a = n Do not mix up Brewster’s angle and critical angle. Brewster’s sets polarization of reflected light ( = i p ); critical sets the onset of TIR ( C=1/n ). They are different phenomena and angles. neet-alert A straw in water appears bent toward the normal at the surface. A swimming pool looks shallower than its actual depth. Diamonds sparkle because their small critical angle traps rays for multiple internal reflections. Optical fibres carry internet data by TIR in a high-n core. Shimmering roads on hot days are mirage-like refraction effects. Real-life cues to check your intuition Draw the normal and mark i and r clearly. Apply Snell’s law to find r or i as needed. For slabs, use geometry to relate shift/deviation to i and r . For near-normal viewing, use d app =d/n ; else use exact trigonometry. For TIR, verify both conditions: denser-to-rarer and i>C . Steps to solve plane-surface refraction numericals Sometimes you need the more general small-angle relation for apparent depth across any two media: if the object is in medium 1 with index n 1 and the observer is in medium 2 with index n 2 , then near-normal viewing gives d app = d ,(n 2/n 1) . The popular d/n form is a special case with n 2=1 (air) and n 1=n . For stacked plane layers, apply this sequentially from the object’s layer to the observer’s side. General apparent depth (small angles) Object in n 1 , observer in n 2 ; reduces to d/n for n 2=1 . Equivalent statement: n = h/h' , i.e., refractive index equals real depth divided by apparent depth (viewing from air, small angles). Angles in optics are always measured from the normal, not from the surface. A common error is to plug the complement by mistake into Snell’s law. tip Sign conventions are simpler for plane surfaces: distances along the incident ray are positive if measured in the direction of the ray, but for most NEET-level plane refraction problems, you can work purely with geometry and sines of angles with the normal. Keep units consistent, and remember to convert degrees to radians when using small-angle approximations with trigonometric functions in calculators expecting radians. When solving multi-interface problems, keep track of where each refraction occurs and whether the emergent ray is parallel to the incident one (as in a parallel slab). Apparent position shifts but direction may not. If you must combine layers, move step by step using the general small-angle apparent depth relation between consecutive media or use exact Snell refraction at each surface if angles are not small. Normal shift of the inner surface as seen from air A plate glass window of thickness 8.0 mm ( n=1.50 ) covers an aquarium. A small mark on the inner surface of the glass is observed from air, near-normal. By how much does the mark’s apparent position shift from the actual front surface? mm For a point inside the slab at depth t , apparent depth from the observer in air is t/n . The shift is = t - t/n = t ,(1-1/n) . t = 8.0 ,mm n = 1.50 medium Common refractive indices at room temperature: air ≈ 1.0003, water ≈ 1.33, crown glass ≈ 1.52, flint glass ≈ 1.62, diamond ≈ 2.42. The larger the index, the smaller the critical angle into air: for diamond, C -1 (1/2.42) 24.4 , which is why light tends to stay inside and reflect multiple times, producing brilliance when the cut is optimized. Water 1.33 48.8° Crown Glass 1.52 41.1° Flint Glass 1.62 38.3° Diamond 2.42 24.4° Material Refractive Index (approx.) Critical Angle to Air (approx.) A right-angle prism (45°–45°–90°) can turn a beam by 90° using TIR: the hypotenuse acts as the reflecting face. If the prism glass has n> 2 1.414 , then the internal incidence at the hypotenuse (45°) is above the critical angle, guaranteeing TIR. Most optical glasses ( n 1.5 ) satisfy this, so such prisms make low-loss mirrors without silvering. How to use the visual tool: Set n and i . Read the internal refraction angle r from Snell’s law. Increase t to see the lateral shift grow while the emergent ray stays parallel. Observe that as i increases, shift increases nonlinearly; as n increases, r decreases and shift changes accordingly. Try n<1.2 vs n>1.8 to see the sensitivity. neet-alert Trap: In a slab, the emergent ray is parallel to the incident ray only if the faces are parallel. If the faces are not parallel (e.g., a prism), the emergent ray is deviated in direction as well. Energy considerations: At a real interface, reflection and refraction both occur for i<C , and the amplitudes depend on polarization and angle (Fresnel coefficients). At i>C , the transmitted wave in the rarer medium becomes evanescent (no net energy flow away), while all the incident power is reflected back—this is TIR. NCERT problems at this level typically ignore partial reflection for i<C unless polarization is being discussed. Frequency is invariant across media: the source sets it. When light slows down entering a denser medium, its wavelength shortens so that v= is satisfied. This is crucial in multi-colour problems: different wavelengths (colours) refract by different amounts because the refractive index depends on wavelength (dispersion). When solving acceptance angle problems for fibres in a medium other than air, remember to use the relative refractive indices: NA = n 1 2 - n 2 2 ,/ ,n 0 , where n 0 is the refractive index of the outside medium. Then a = NA if NA 1 . If NA >1 (which can happen only with n 0>1 in a limiting sense), it means all rays within a hemisphere are accepted from that external medium. Angle chasing tip: For multi-bounce TIR questions in prisms or fibres drawn in 2D, keep careful track of the angle with the normal at each hit. Internal reflection preserves angle of incidence equals angle of reflection. Translate those internal angles back to the external face when checking for escape or TIR. Apparent shift vs normal shift: Apparent depth d app is the perceived distance from the surface to the point. Normal shift is = d real - d app . In a slab, it is often easier to compute = t(1-1/n) for a mark on the back face viewed from air, rather than finding d app and subtracting from t . Oblique viewing caveat: If the line of sight is at a large angle to the normal, the image of an underwater point shifts not only in depth but also laterally; exact ray tracing with Snell’s law is then required. For NEET, unless stated otherwise, assume near-normal viewing for depth-type questions. Speed vs direction: The refraction law is essentially a consequence of Fermat’s principle (light takes the path of least time). At a flat boundary with different speeds on either side, that time-optimal path corresponds to Snell’s relation. You are not expected to derive this variational principle in NEET, but remembering the physical reason helps avoid rote confusion. Worked angle arithmetic practice: For n=1.52 glass to air, C= -1 (1/1.52) 41.1 . Any internal incidence larger than this yields TIR. For a 45°-45°-90° prism, the angle at the hypotenuse is 45°, so TIR is assured. Replace glass with water ( n=1.33 ) and 45° is now greater than water’s C 48.8 ? No—45° is less than 48.8°, so TIR would not occur for water at that angle. Always compute and compare. Precision note: Unless a problem specifies more, quote numerical answers to 2 significant figures for NEET. Use degrees unless radians are explicitly required; for small-angle approximations, ensure your calculator is in radians if you are substituting angle values directly into . Bending of light at a boundary due to speed change Refraction Material property n=c/v Refractive index Snell’s law n 1 i = n 2 r Apparent depth d app =d/n (air, small angles) =t , (i-r)/ r Lateral shift Critical angle C=1/n (to air) Complete reflection for i>C in denser medium Total internal reflection Numerical aperture NA = n 1 2-n 2 2 (in air) NA i p Brewster’s angle = i p for polarization by reflection a =n Diffraction minima Quick recap of key terms