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Abstract
Young’s dual-pinhole interference experiment with arbitrary fully correlated and polarized vector light fields leads to a Pancharatnam–Berry geometric phase that is related to the associated dynamical phase. We demonstrate theoretically and experimentally how the dynamical phase across the interference pattern can be deciphered from the total phase, thereby leaving only the geometric phase. Our results constitute the first genuine interferometric phase measurements that yield the Pancharatnam–Berry geometric phase in Young’s two-beam interference setup.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
The Pancharatnam–Berry geometric phase describes an additional phase contribution that a beam of light acquires when its polarization state undergoes a cyclic in-phase evolution [1,2]. The phase originates from the curvature of the Poincaré sphere. Its magnitude equals half the solid angle enclosed by the polarization-state trajectory on the sphere. The geometric phase in optics was introduced by Pancharatnam in 1956 [3] and its connection to the adiabatic phase [4] in slowly cycled quantum systems was pointed out by Berry in 1987 [5]. Since then the geometric phases appearing in various physical contexts have been extensively studied [6–8]. In optics, the geometric phase has been demonstrated through several experiments [9–11]. It has been shown to exist when the propagation direction of a light beam is slowly changed (spin-redirection phase) [12] with single photons [13], surface plasmon polaritons [14], metasurfaces [15–17], and even in the temporal interference (beating) of electromagnetic beams [18]. The geometric phase is central in novel concepts such as breaking time-reversal symmetry and unidirectional propagation of light fields in photonic crystals [19]. Recently, the phenomenon has found commercial applications in augmented reality (AR) devices in the form of geometric phase lenses that utilize the engineered birefringence of optical materials [20–22].
The existence of the Pancharatnam–Berry geometric phase in Young’s two-pinhole interference pattern was reported recently [23]. The phase appears in the periodic intensity and polarization-state profile on a screen behind the pinholes. In [23], two different formulas for the geometric phase were established in terms of the field intensities in the openings together with either the pinhole-field polarization states or the observation-screen intensity fringe information. The consistency of these two formulations was also demonstrated both numerically and experimentally. The previously introduced methods are, however, indirect as the geometric phase is not measured using interference experiments. In this work we show theoretically and experimentally how the dynamical phase accumulates over a closed polarization-state path in the interference pattern, enabling us to separate the dynamical and geometric phases. Our results represent the first true interferometric phase measurement that yields the unambiguous Pancharatnam–Berry geometric phase in Young’s two-beam interference, to the best of our knowledge. Further, we identify a fundamental connection between the geometric and dynamical phases present in Young’s two-beam interference. Both phases depend on the intensities and polarization states of the two beams, and a change in one unavoidably alters the other.
2. THEORY
A. Young’s Two-Beam Interference
The (modified) Young’s double-pinhole interference experiment is depicted in Fig. 1. A monochromatic (fully coherent and polarized) light beam of angular frequency $\omega$ is directed onto plane $A$ with two identical pinholes at fixed positions. A $2f$ lens system transforms the emitted spherical waves into plane waves whose interference pattern is observed on plane $B$. Two movable pinholes are placed on $B$ at coordinates ${x_n}$ and ${x_{n + 1}}$ with $n \in \{1, \ldots ,N - 1\}$, where $N$ is a large integer, generating another interference pattern on plane $C$.
Fig. 1. Schematic representation of the modified Young’s interference experiment used in the measurements. The setup consists of plane $A$ with two fixed pinholes, plane $B$ with movable pinholes at ${x_n}$ and ${x_{n + 1}}$, observation plane $C$, and a lens of focal length $f$. The pinhole-field intensities and Poincaré vectors are ${S_{0i}}$ and ${{\textbf{P}}_i}$, $i \in \{1,2\}$. The propagation angle of the plane waves is $\theta$ and $\Lambda$ denotes the period of the interference pattern on $B$ with end points ${x_1}$ and ${x_N} = {x_1} + \Lambda$.
The Stokes parameters [24,25] describing the intensity, ${S_0}(x)$, and the polarization state, ${S_m}(x)$, $m \in \{1,2,3\}$, of the electric field on $B$ are given by the electromagnetic interference law [26,27]
B. Pancharatnam–Berry Phase
Provided the polarization states at the pinholes are different, Eq. (1) implies a spatially periodic state of polarization on plane $B$ with a period of $\Lambda = \pi /k\sin \theta$. Between the end points of a single period, the polarization state experiences a closed-path evolution of incremental in-phase changes on the Poincaré sphere, inducing a Pancharatnam–Berry geometric phase difference to the electric fields separated by $\Lambda$. The magnitude of this phase is [23]
Let us next consider the detection of ${\Phi _{{\rm{PB}}}}$ based on genuine phase measurements. According to Pancharatnam, the phase difference of the electric field vectors at points $x$ and ${x^\prime}$ on $B$ is $\Phi (x,{x^\prime}) = \arg [{{\textbf{E}}^ *}(x) \cdot {\textbf{E}}({x^\prime})]$. By analyzing the interference of the plane waves on $B$, the magnitude of the phase difference for points separated by $\Lambda$ is found to be $|{\Phi _{{\rm{tot}}}}| = |\Phi ({x_1},{x_1} + \Lambda)| = \pi$. This phase difference consists of two contributions: a dynamical phase, ${\Phi _{{\rm{dyn}}}}$, and the Pancharatnam–Berry geometric phase, ${\Phi _{{\rm{PB}}}}$. In order to extract ${\Phi _{{\rm{PB}}}}$, we must determine how the dynamical phase accumulates on moving from some ${x_1}$ to ${x_1} + \Lambda$ and subtract it from the total phase. We do this by dividing the period $\Lambda$ on screen $B$ into $N$ equal intervals and calculate how the phase of the electric field is acquired from the incremental parts. As a result, the geometric phase over one period can be expressed as
The incremental phase difference obeys $\Phi ({x_n},{x_{n + 1}}) = \arg [{\mu _0}({x_n},{x_{n + 1}})]$, where ${\mu _0}({x_n},{x_{n + 1}})$ is defined in analogy to Eq. (2a) but with the two pinholes at ${x_n}$ and ${x_{n + 1}}$ on $B$, resulting in the intensity fringes on screen $C$ (see Fig. 1). The $C$-plane counterpart of Eq. (1), with $n = 0$, implies that
C. Implications
We emphasize a fundamental, distinct feature of the geometric phase in Young’s interference pattern. According to Eq. (3), the phase depends on the intensities and polarization states at the pinholes. However, the total phase difference over one period is $\pi$ regardless of the pinhole fields. Therefore, the geometric and the dynamical phases are linked and a change in one phase necessarily alters the other. Consequently, not just the geometric phase but also the dynamical phase depends on the intensities and polarization states of the pinhole fields. This curious feature has not been recognized before in the context of Young’s experiment. The unique measurement scheme presented here enables the direct measurement of the formation of the dynamical phase at plane $B$. With this advancement, we can separately evaluate, by true phase measurements, the contributions of the dynamical phase and the geometric phase to the overall phase. We also remark that the total phase value $\pi$ is due to the plane-wave interference at $B$. If the lens is removed, the total phase over one period depends on the location of the period [23].
3. ARRANGEMENT FOR PHASE MEASUREMENT
In order to determine the Pancharatnam–Berry phase through phase measurements we make use of Eq. (4) and a novel experimental system depicted in Fig. 2(a). The device consists of two separate pinhole pairs on two planes, $A$ and $B$. The first pair on plane $A$ is used to generate a desired Pancharatnam–Berry phase to the interference pattern on plane $B$, while the second, movable pair on this plane enables the determination of the incremental phase differences $\Phi ({x_n},{x_{n + 1}})$ from the interference intensity fringes on screen $C$.
Fig. 2. Experimental system. (a) Layout of the setup is as follows: HeNe laser source, beam splitters BS1-2, tilted glass sheet GS, mirrors M1-2, circular polarizers CPL1-2, linear polarizers LPL1-2, pinholes PH1-2 (plane $A$), lenses L1-3, digital micromirror device array DMD (plane $B$), and detectors D1-3 (plane $C$ at D1). (b) Close-up of the DMD and double slits. (c) Focused image of the $A$-plane pinholes. (d) Interference pattern on the DMD plane and the two slits (small black dots marked by the red arrows).
A single-mode linearly polarized helium-neon (HeNe, wavelength 633 nm) laser beam is divided by a nonpolarizing beam splitter BS1 into two arms. The beam intensities in the arms are adjusted independently using circular polarizers CPL1 and CPL2, while their polarization states are set with linear polarizers LPL1 and LPL2. The beams are guided with mirrors M1 and M2; their alignment and position is controlled with a tilted glass sheet GS. The arms include the first pair of pinholes PH1 and PH2 of diameters of 150 µm whose lateral positions can be adjusted with two micrometer screws (plane $A$ in Fig. 1). The beams are recombined with beam splitter BS2 and the output is collimated with a 300 mm focal length lens L2 (lens in Fig. 1). To observe the pinhole positions, their focused image [see Fig. 2(c)] is captured with camera detector D3 and objective lens L1 of a focal length of 55 mm. The spacing between the pinholes is 354 µm.
The movable pair of pinholes (plane $B$ in Fig. 1) is implemented by using a digital micromirror device array (DMD, Texas Instruments DLP3000) [31]. The DMD consists of $608 \times 684$ mirrors of 10.8 µm size. In order to increase the power of diffracted light we use slits instead of pinholes in the experiments. One slit in plane $B$ consists of five micro mirrors in a row, as depicted in Fig. 2(b). Camera detector D2 is focused on the DMD plane with an objective lens L3 to image the interference pattern [see Fig. 2(d)]. The chosen spacing between the first pinholes together with the Fourier-transforming lens results in the fringe period of 536 µm on the DMD plane. The two small black dots in Fig. 2(d) indicate the $B$-plane slits. For the photograph, the number of mirrors is larger than in the actual phase measurements to make the slits more visible. Light reflected from the mirror slits forms an interference pattern on the camera detector D1 (plane $C$ in Fig. 1). The intensity fringes at D1 provide the required information on ${\mu _0}({x_n},{x_{n + 1}})$ via Eq. (5) to determine ${\Phi _{{\rm{PB}}}}$. We remark that the DMD system could also be employed to measure the other normalized coherence Stokes parameters ${\mu _m}({x_n},{x_{n + 1}})$, $m \in \{1,2,3\}$, as described in [32].
4. MEASUREMENTS RESULTS
Figure 3 displays the magnitudes of the Pancharatnam–Berry phases ${\Phi _{{\rm{PB}}}}$ obtained for three different pinhole-field polarizations in plane $A$ as a function of the intensity ratio ${S_{01}}/{S_{02}}$ at the apertures. We consider only the phase magnitude, which is the quantity implied by Eq. (3). To generate the polarization states at the pinholes a $y$-polarized field in PH1 is kept fixed, while the linear polarization state in PH2 is rotated using the polarizer LPL2. The intensity ratio of the pinhole fields is adjusted using the circular polarizer CPL2. The blue (uppermost) objects in Fig. 3 correspond to orthogonal $y$- and $x$-polarized pinhole fields with the angle difference of 90°, while the red (middle) and green (lowest) objects denote the situations at 45° and 20° angles, respectively. The lines correspond to the theoretical results obtained with Eq. (3) in terms of the intensity and polarization information at the apertures PH1 and PH2, while the dots mark the averaged results evaluated from Eq. (4) using the measured intensity information on screen $C$. The error bars represent the related standard deviations. The theoretical values were obtained by assuming ideal polarization elements before the pinholes PH1 and PH2.
Fig. 3. Magnitude of the Pancharatnam–Berry phase as the ratio of the $A$-plane pinhole intensities is varied. Blue (uppermost), red (middle), and green (lowest) objects correspond to linear polarization states at the pinholes with the angle difference of 90°, 45°, and 20°, respectively. The three cases under closer examination in Fig. 4 are numbered and marked with circles, with the inset showing the corresponding loops on the Poincaré sphere.
In order to ascertain the Pancharatnam–Berry phase by means of Eq. (4) (dots in Fig. 3) we measured the total dynamical phase, ${\Phi _{{\rm{dyn}}}}$, by scanning the DMD pinhole pairs over a single $B$-plane period $\Lambda$, whose length corresponds to $N = 24$. This procedure yields the values $\Phi ({x_n},{x_n} + \Delta x)$, $n \in (1, \ldots ,24)$, where the fixed pinhole separation is $\Delta x = 2 \times 10.8\;{{\unicode{x00B5}{\rm m}}}$ (two mirror rows). The total phase over the period was evaluated as ${\Phi _{{\rm{tot}}}} = \Phi ({x_1},{x_1} + 49 \times 10.8\;{{\unicode{x00B5}{\rm m}}})$. The measurements were done for several (interlaced) periods to evaluate the average values and assess the error limits. The different periods are $\Lambda$-length portions between ${-}648 \;{{\unicode{x00B5}{\rm m}}}$ and 648 µm (${\pm}60$ mirror rows), corresponding to 72 samples for ${\Phi _{{\rm{dyn}}}}$ and ${\Phi _{{\rm{tot}}}}$. Overall, the theoretical and experimental values in Fig. 3 match well for all intensity ratios and pinhole-field polarizations.
Fig. 4. Incremental dynamical phases (left column) and their accumulation (right column) in the cases one (upper row), two (middle row), and three (lowest row) marked with circles in Fig. 3. On left, the measured and numerical values are indicated with dots and curves, respectively, while vertical lines exemplify one period with length $\Lambda$. On right, the theoretical and experimental results are indicated with the dashed blue line and the solid red line, respectively.
To examine how the dynamical phase is accumulated over the interference period of length $\Lambda$, we select three individual measurements for closer inspection. These are numbered and marked with circles in Fig. 3. For the cases one, two, and three, the polarization angle differences and pinhole intensity ratios are (20º, 0.386), (90º, 1.01), and (45º, 1.72), respectively. The inset in Fig. 3 depicts the corresponding numerically calculated polarization paths traced by the Poincaré vector on the Poincaré sphere on traversing one period in the interference pattern on plane $B$. Figures 4(a), 4(c), and 4(e) show the measured phases $\Phi ({x_n},{x_n} + \Delta x)$ after the correction of a systematic phase error (see Supplement 1 for details). The dots correspond to the measured quantities while the lines give the theoretical values. The vertical black lines exemplify one period $\Lambda$ over which the incremental phases $\Phi ({x_n},{x_n} + \Delta x)$ are summed to obtain ${\Phi _{{\rm{dyn}}}}$ and ${\Phi _{{\rm{tot}}}}$. Figures 4(b), 4(d), and 4(f) display how the dynamical phases are acquired over the periods marked in 4(a), (c), and (e), respectively. The solid red lines depict the measured values while the dashed blue lines show the theoretical phases. The corresponding phases ${\Phi _{{\rm{tot}}}}$ and ${\Phi _{{\rm{dyn}}}}$ are marked with separate points at the end of the curves. The magnitude of the Pancharatnam–Berry phase, ${\Phi _{{\rm{PB}}}} = {\Phi _{{\rm{tot}}}} - {\Phi _{{\rm{dyn}}}}$, is also given. We see from Fig. 4 that the agreement between the measured and theoretical results is good in all three cases. The main difference is the tilt of the experimental curves in (a), (c), and (e) caused by a slight residual spherical phase present in the measurements. A similar tilt is also found for the other cases plotted in Fig. 3. It should be noted that the spherical phase front, beam tilt, and other distortions affect ${\Phi _{{\rm{dyn}}}}$ and ${\Phi _{{\rm{tot}}}}$ equally, and hence do not influence the value of ${\Phi _{{\rm{PB}}}}$. The observed small discrepancy in the theoretical and measured values of $|{\Phi _{{\rm{PB}}}}|$ originates mainly from the optical path-length changes due to air flow and thermal expansions, as well as the limited resolution of the camera detector.
5. CONCLUSIONS
We examined the Pancharatnam–Berry geometric phase in the interference pattern of Young’s double-pinhole experiment. We demonstrated the foundational inseparable character of the geometric and dynamical phases in such an arrangement, from which it follows that not only the geometric phase but also the dynamical phase depends on the intensities and polarization states of the electric fields at the pinholes. Further, we introduced theoretically and presented experimentally a direct phase measurement technique for the incremental determination of the dynamical phase over one polarization cycle on the Poincaré sphere, thereby rendering a purely interferometric detection of the geometric phase. The validity of the method was shown with different pinhole-field intensities and polarizations. The results corroborate a deep connection between two foundational concepts of physics, the Pancharatnam–Berry geometric phase and Young’s two-beam interference.
Funding
Academy of Finland (308393, 310511, 320166, 322002); Emil Aaltosen Säätiö.
Disclosures
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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