Concept 01 / 07Wave Optics
Huygens' Principle and Wavefronts: Deriving Reflection and Refraction from a Wave Picture
A wavefront is a surface of constant phase; treating every point on it as a source of secondary wavelets reproduces the laws of reflection and refraction and explains why wavelength, not frequency, changes in a medium.
A wavefront is the locus of all points of a wave that are vibrating in the same phase at a given instant. A source emits continuously, so there is a whole family of such surfaces, one for every value of the phase; we usually draw the ones passing through the crests, separated by one wavelength .
The perpendicular to the wavefront at a point gives the direction in which energy travels there, and that perpendicular is exactly what geometrical optics calls a ray. Rays and wavefronts are therefore two dialects of the same physics. Rays are convenient when is negligible compared with the apertures and obstacles; wavefronts become indispensable the moment it is not, which is the whole of wave optics.
Shapes and amplitudes. A point source in a homogeneous isotropic medium sends out spherical wavefronts; a long narrow slit source sends out cylindrical wavefronts; a source at infinity, or a point source at the focus of a converging lens, gives plane wavefronts. Energy conservation fixes how the amplitude behaves. The same power crosses every wavefront, so for a sphere , giving and amplitude . For a cylinder the area goes as , so and . For a plane wavefront the area does not change at all and the amplitude stays constant, which is why a well collimated laser beam does not fade with distance.
Huygens' principle has three clauses. First, every point of a wavefront acts as an independent source of secondary spherical wavelets. Second, these wavelets travel forward with the speed of the wave in that medium, so in time each has radius . Third, the new position of the wavefront after time is the forward envelope, that is, the common tangent surface, of all these secondary wavelets.
Law of reflection. Let a plane wavefront strike a plane mirror, with touching the mirror first and still a distance away. Let reach the mirror at the point after a time , so . During that same time the secondary wavelet born at has grown into a hemisphere of radius on the same side of the mirror. The reflected wavefront is the tangent drawn from to that hemisphere.
Now compare the two right triangles and . They share the hypotenuse , the angle at and the angle at are both right angles, and . So the triangles are congruent, which forces the angle to equal the angle . These two angles are precisely the angle of incidence and the angle of reflection , because the angle between a wavefront and the surface always equals the angle between the corresponding ray and the normal. Hence . Equivalently and , and since we get . The incident ray, the normal and the reflected ray also lie in one plane because the whole construction does.
Law of refraction. Repeat the construction at a boundary between two media with speeds and , as in the diagram above. In the time taken by to travel to in medium 1, the wavelet from spreads into medium 2 with the smaller speed , so its radius is only while . The refracted wavefront is the tangent .
From the two right triangles sharing the hypotenuse , and . Dividing, , a constant for the given pair of media. Writing for the refractive index, , so . Snell's law drops out of the geometry with no extra assumptions, and it comes with a bonus that the ray picture could never supply: light must travel slower in the optically denser medium. Newton's corpuscular theory predicted the opposite, and Foucault's 1850 measurement in water settled the argument in favour of waves.
What changes at a boundary and what does not. The boundary is driven by the incident wave and re-radiates at exactly the frequency that shakes it, so frequency is the same in both media. Since and drops by the factor , the wavelength must drop by the same factor, . This single fact drives a large part of wave optics: it is why a slab of thickness and index contains waves instead of , and hence why it adds an extra optical path compared with the same thickness of vacuum. Optical path is always geometric path, and phase difference is always (optical path difference), with measured in vacuum.
Total internal reflection also follows. Going from denser to rarer, with . Beyond the critical angle the required exceeds , so no tangent plane to the secondary wavelets exists in the second medium: no refracted wavefront can be constructed, and all the energy goes into the reflected wavefront.
Gotchas that JEE Advanced likes. First, Huygens' construction as originally stated also predicts a backward wave, because a spherical wavelet spreads in all directions. The cure is Fresnel's obliquity factor , which multiplies the amplitude of a wavelet emitted at an angle to the forward direction; it equals straight ahead and straight back, killing the backward wave. Second, Huygens' principle is a construction for where the wavefront goes, not a dynamical law: by itself it gives directions, not intensities. The moment you let the secondary wavelets interfere with each other rather than merely taking their envelope, you get the Huygens-Fresnel principle, and out of it come interference and diffraction. Third, shapes of wavefronts after optical elements are a favourite: a plane wavefront becomes converging spherical after a converging lens, diverging spherical after a diverging lens, and stays plane but tilted after a prism, because the part of the wavefront that traverses the thicker part of the glass is delayed most.
Key Takeaways and Formulae:
A wavefront is a surface of constant phase; rays are its normals. Intensity .
Spherical wave: , . Cylindrical wave: , . Plane wave: constant.
Huygens: new wavefront = forward envelope of secondary wavelets of radius .
Reflection: , from . Refraction: .
In a medium: unchanged, , .
Optical path geometric path; a slab of thickness adds ; phase difference .
Common mistake: writing inside a medium as the vacuum value. Always convert, or work with optical paths and keep as the vacuum wavelength.
Common mistake: thinking the secondary wavelets are real new sources that can be detected individually. They are a bookkeeping device that reproduces the correct envelope.
Brain Teaser 1: A plane wavefront in air strikes a still water surface at an angle of incidence of . For water . Find the acute angle between the reflected wavefront and the refracted wavefront.
Solution: First get the refraction angle. , so .
Now work with rays, because the angle between two planes equals the angle between their normals, and the rays are exactly the normals to the wavefronts. The reflected ray leaves at on the other side of the normal, in medium 1; the refracted ray goes on at from the same normal, into medium 2. Going round from the reflected ray to the refracted ray through the interface, the three angles at the point of incidence add to : angle between them .
Since each wavefront is perpendicular to its own ray, the two wavefront planes meet at the same pair of angles, and its supplement . The acute angle asked for is therefore about .
A useful check on the method: if the second medium had , then and the angle between the rays would become , which is right, since reflected and refracted rays then differ only by the sign of the normal component.
Brain Teaser 2: Monochromatic light of vacuum wavelength nm passes through a glass slab of thickness m and refractive index . How many complete waves does the light contain inside the slab, how many would the same m of vacuum contain, and what is the resulting extra optical path? Use the result to explain why frequency, and not wavelength, is the quantity conserved at a boundary.
Solution: Inside the glass the wavelength is nm. Number of waves in the slab .
In the same geometric thickness of vacuum, the number of waves is .
The slab therefore packs extra waves into the same space, corresponding to an extra optical path of m. Check against the formula: m. They agree.
Why frequency is conserved: wavefronts cannot be created or destroyed at the boundary, and they cannot pile up there, because the boundary has no room to store them. So the number of wavefronts arriving per second in medium 1 must equal the number leaving per second in medium 2, which is the statement that is the same on both sides. Physically, the electrons at the surface are driven by the incident field and, in a linear medium, re-radiate at precisely the driving frequency. Since changes and cannot, is forced to change, exactly as the count above shows.
Brain Teaser 3: A plane wavefront falls normally on a thin converging lens of focal length made of glass of index . Using only the idea that light is slowed inside glass, show that the emergent wavefront is spherical with its centre at the focus, and prove that every part of the beam arrives at the focus in phase.
Solution: The lens is thickest on the axis and thins out towards the rim. Light through the centre spends the longest time in the slow medium, so the central part of the wavefront is delayed most and lags behind the edges. A plane wavefront with its middle pulled back is a wavefront curved so as to converge, so the emergent wavefront is spherical and closing in on a point on the axis.
To prove that point is a genuine common focus, compare optical paths from the emergent wavefront to the axial point at distance . Take a ray at height from the axis. Its remaining geometric path to is , which is longer than the axial ray's path by .
That excess must be cancelled by the extra glass the axial ray had to cross. For a thin lens of surface radii and , the thickness at height is , so the extra optical path collected by the central ray relative to the ray at height is , using the lensmaker's formula .
The two excesses are equal and opposite, so the total optical path from the incoming plane wavefront to is the same for every . Equal optical path means equal phase, so all the wavelets arrive at in step and interfere constructively. This is the wave-optics meaning of a focus, and it shows that the lensmaker's formula is exactly the condition for the lens to convert a plane wavefront into a spherical one.
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