The geometric phase : interferometric observations with white light P . HARIHARAN

Observations on white-light interference fringes show that the effects due to the introduction of a variable geometric phase, which is independent of the wavelength, differ significantly from those due to a change in the optical path difference . 1 . Introduction Berry [1] has shown that the wavefunction of a quantum system may acquire an additional phase factor (the geometric phase) when the system is taken around a circuit in parameter space . In the case of photons, such a phase shift can be produced by a cyclic change in their state of polarization (the Pancharatnam phase) [2-5] . Several experimental measurements of the Pancharatnam phase have been made [6-11] with monochromatic light ; some preliminary observations have also been made using a simple white-light interferometer [12] . This letter presents quantitative measurements on the interference fringes obtained with white light in an interferometer using an achromatic phase shifter operating on the Pancharatnam phase . 2 . Optical system The apparatus used in these experiments is shown schematically in figure 1 . Light from a tungsten lamp, linearly polarized at 45° to the plane of the figure by a polarizer P 1i is divided at a polarizing beam splitter into two orthogonally polarized beams that traverse the same closed triangular path in a Sagnac interferometer in opposite directions . A second polarizer P 2 , with its axis at 45° to the plane of the figure, brings the two beams leaving the interferometer into a condition to interface . The two beams in this interferometer always emerge parallel to one another and can be made to coincide by adjusting the beam splitter and the mirrors . This adjustment can be made conveniently with a laser as the light source . Interference fringes can then be obtained quite easily with the white-light source . Since both the beams traverse the same optical circuit, the optical path difference is always equal to zero at the centre of the field of view . When a small lateral shear was introduced between the beams, an interference pattern consisting of equally spaced, straight fringes was formed on the charge-coupled device (CCD) array placed in the focal plane of the imaging lens L 2 . This interference pattern could be viewed conveniently on the television monitor ; the intensity distribution in the pattern could also be recorded and stored for further analysis on the computer . 0950-0340/94 $10. 00 © 1994 Taylor & Francis Ltd . 664 P. Hariharan et al . I ∎r Polarizing beam splitter Figure 1 . Schematic diagram of the optical system of the interferometer . The phase difference between the beams was varied by a system that operated on the geometric phase, consisting of a half-wave retarder (HWR), which could be rotated by known amounts, located between two circular polarizers CP 1 and CP2 . Since the retardation produced by a single birefringent plate is inversely proportional to the wavelength, we used for CP I and CP2 a combination of a half-wave plate and a quarter-wave plate which yields an achromatic circular polarizer [13], and for the HWR a sandwich consisting of two quarter-wave plates and a half-wave plate, which is a good approximation to an achromatic HWR [14] . 3. Theory The phase change introduced by moving a polarization state around a circuit on the Poincare sphere [15, 16] is equal to half the solid angle subtended by the circuit at the centre of the sphere. In the present arrangement the linearly (s-)polarized beam reflected by the polarizing beam splitter passes through the circular polarizer CP 1 , the HWR and the circular polarizer CP2, in that order and, as shown in figure 2, the polarization state of this beam traces out the closed circuit A 1 SA2 NA1 on the Poincare sphere. If the HWR is set with its principal plane at an angle +0 to the principal planes of CP 1 and CP 2 , the transmitted beam acquires a phase shift of 20 . The transmitted (p-polarized) beam traverses the interferometer in the opposite direction, and its polarization state traces out the path B 1 SB2NB 1 on the Poincare sphere. Since the circuit B 1 SB2NB1 is identical with the circuit A 1 SA2NA1 but is traversed in the opposite sense, this beam experiences a phase shift of -20 . These operations on the geometrical phase lead to an additional phase difference Acp0 =40 between the two beams in the interferometer, without introducing any change in the optical paths . Because of the topological nature of the geometric phase, the phase difference that can be introduced in this manner is unbounded and its sign can be reversed [10] . The geometric phase 665