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Take a beam of light, change its polarisation step by step, and bring it back to exactly the polarisation it started with. You might expect it to be the same light. S. Pancharatnam showed in 1956 that it can carry an extra phase shift, and that the shift depends only on the shape of the path the polarisation took. Decades later this became one of the best-known examples of a geometric phase in physics. This post covers who he was, the idea, a worked example and what students can learn from his work.
Who was S. Pancharatnam?
A review of his Collected Works in Physics Bulletin (1976) says he moved from C. V. Raman's crystal optics to work with G. W. Series at Oxford on optical pumping and resonance, and that he died at the age of 35 in 1969. By then he had published 22 papers, and he left three partly finished manuscripts that Series edited and published after his death. A 2013 article in Resonance by Nityananda, Ramaseshan and Madhusudana gives his dates as 1934 to 1969.
His most famous paper, Generalized theory of interference, and its applications, appeared in the Proceedings of the Indian Academy of Sciences. It was received on 30 October 1956, numbered as a memoir of the Raman Research Institute, and communicated by Sir C. V. Raman. A year earlier he had published on achromatic combinations of birefringent plates, stacks of crystal plates that change polarisation in the same way across many colours.
What is the Poincare sphere?
Every state of polarisation can be drawn as a point on a sphere. Linear polarisations, horizontal, vertical and the diagonals, lie around the equator. Right and left circular polarisation sit at the two poles. Elliptical states fill the rest. Changing the polarisation, for example with a wave plate, moves the point across the sphere.
Pancharatnam used this sphere to treat a general question: what happens when two coherent beams in different elliptical polarisation states are superposed? He expressed the intensity and polarisation of the result in terms of the sides of spherical triangles drawn on the sphere.
What is the Pancharatnam phase?
To compare the phases of two beams in different polarisation states, Pancharatnam needed a rule for when they count as "in phase". The rule he chose has a surprising consequence. If you go from state A to B, B to C, and C back to A, the beam need not come back in phase with how it started. Berry describes this as the "non-transitivity" of Pancharatnam's connection between states.
Michael Berry's paper The adiabatic phase and Pancharatnam's phase for polarized light (1987) puts it this way: Pancharatnam showed that a cyclic change in the state of polarisation is accompanied by a phase shift "determined by the geometry of the cycle as represented on the Poincare sphere". Berry had found a similar geometric phase for slowly changed quantum systems in Quantal phase factors accompanying adiabatic changes (1984), and the 1987 paper set out the precise relation between the two. That is why it is often called the Pancharatnam-Berry phase.
In 1988 Chyba, Simon, Wang and Mandel measured it with a Michelson interferometer. They took a beam's polarisation round a closed circuit on the Poincare sphere and confirmed that "the phase shift equals half of the solid angle subtended by the closed circuit at the center of the sphere".
Worked example: half the solid angle
Take a loop that visits three points on the sphere: horizontal linear polarisation, +45° linear polarisation, and right circular polarisation, joined by arcs of great circles. Each of the three points is 90° from the other two on the sphere, so the loop marks out one eighth of the sphere.
The whole sphere subtends a solid angle of 4π steradians at its centre.
One eighth of it: Ω = 4π / 8 = π/2 steradians.
Geometric phase = Ω/2 = π/4 radians = 45°.
As an optical path difference for red laser light (λ = 633 nm): a phase of 2π matches one wavelength, so π/4 matches λ/8 = 633 / 8 ≈ 79 nm.
So, without any extra glass in the path, the beam behaves as if it had travelled about 79 nm further. That is enough to shift interference fringes visibly.
Common mistakes: using the full solid angle instead of half, measuring angles on the Poincare sphere as if they were the angles of the polarisation itself (a 45° turn of linear polarisation is a 90° arc on the sphere), and forgetting that 4π, not 2π, is the solid angle of a full sphere. If polarisation is new to you, start with Etienne-Louis Malus and polarisation by reflection. The Poincare sphere and the geometric phase are university-level topics, so treat this as reading beyond your syllabus.
What can a student take from how Pancharatnam worked?
Use a picture to think. The sphere turned messy algebra about ellipses into geometry of triangles.
Take definitions seriously. His careful rule for "in phase" is exactly what revealed the new effect.
Good work can wait to be understood. Its wider meaning was recognised almost thirty years later.
Polarisation states are much easier to follow when you can rotate a plate and watch the result. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
Next step: work out the geometric phase for a loop that encloses a quarter of the sphere, and check that you get π/2.
