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Optically heterodyned laser interferometry, as applied to measuring linear displacements, requires different optical frequencies to be encoded onto unique polarization states. To eliminate non-linear contributions to the interferometer signal, the frequency difference must be introduced after beam splitting and the interfering beams must be recombined via spatially separated paths. The polarization jitter of the frequency-shifted beams still originates a noise in the beat-signal phase. A formula is given expressing the noise amplitude in terms of the illuminating beam’s extinction ratio.
© 2013 Optical Society of America
1. Introduction
Laser interferometry and optical heterodyning, in which photons having two different frequencies are used, allow displacement measurements to be made with the greatest resolution [1–4]. In order to deliver photons through the reference and measurement arms according to their lower or higher frequency, it is usual to encode frequency into orthogonal polarization states, so that photons having different frequencies can be carried by a single beam and separated by polarization-dependent optics.
Geometric phase limits the accuracy of displacement measurements [5]; applications of heterodyne laser-interferometry are impaired by this problem. In a seminal paper, Lawall and collaborators described an interferometer which is free of periodic nonlinearity [6, 7]; an additional concept can be found in [8]. In these interferometers, two separate beams with different frequencies – produced by two acousto-optic modulators – are delivered to the interferometer via spatially separated paths. Hence, polarization encoding of the photon frequency is no longer necessary and the geometric contribution to the fringe phase is eliminated. As shown in Fig. 1, the frequency shifted beams are split in A and B, delivered through the interferometer arms, and combined in C and D to originate reference and measurement beat-signals. The phase difference between these signals is proportional to the difference between the optical length of the measurement and reference paths of the photons thorough the interferometer. From a topological viewpoint, we can equivalently describe the interferometer as a single beam that is split in C, delivered through the reference and measurement arms, and recombined in D.
Fig. 1 Topology of a laser interferometer applying optical heterodyne. Low- and high-frequency photons (red and blue) enter the interferometer in A and B, where they are split and delivered via spatially separated paths to the output ports C and D. We can equivalently say that a beam is split in C, delivered through the reference ( ) and measurement ( ) arms, and recombined in D. The phase difference between the beat signal detected in C and D is proportional to the difference between the optical lengths of the measurement and reference arms.
We realized a variant of the Lawall’s interferometer for the use in the measurement of the molar Planck constant by γ-ray diffractometry [9]. In our interferometer, though the two low- and high-frequency beams are independently generated and kept spatially separated up to the beat-signal detectors, polarization division was still used to avoid back-reflections and light recycling and to minimize losses of optical power [10]. Consequently, the differential retardation of the reference and measurement photons is recovered after the projection of the photon states in the same polarization. To our disappointment, an unexpected phase noise was observed, which was traced back to the polarization noise of the photons entering the interferometer. Variations of the beat-signal phase induced by changes of the polarization states at the interferometer input ports were observed and reported also by Weichert [11]. The necessity of understanding and reducing this noise prompted this investigation.
2. Interferometer concept
γ-ray diffractometry requires high resolution angle measurements, which are traced back to laser-interferometry measurements of the displacement of two retro-reflectors pairs mounted at the ends of two beams embedded in the diffractometer spindles. A detailed description of the diffractometer and angle interferometer can be found in [9]. In our Laue-Laue setup, depicted in Fig. 2, the relevant measurement quantity is the angle between the two diffractometer crystals. Two acousto-optic modulators generate two beams, linearly polarized and shifted in frequency, from a stabilized single frequency laser source. These beams are delivered to the interferometer by two single-mode and polarization-preserving fibers (not shown in Fig. 2). They enter the interferometer linearly polarized along a direction forming a 45° angle with the interferometer reflection-plane, are delivered to the diffractometer retro-reflectors according to horizontal and vertical polarizations, are back-reflected by two fixed reference mirrors, and are combined coming out of the interferometer. Since the measurand is the angle between the two crystals, both the interferometer arms can serve as the reference or the measurement purpose. Auxiliary monolithic arms, ending in the RRP reference roof-prism, are added for redundancy and testing.
Fig. 2 Interferometer layout. Retro reflectors are mounted at the ends of beams embedded in the spindles that rotate the Si crystals A and B; γ rays propagate from right to left. The low- and high-frequency beams enter the “low” and “high” ports and interfere in the detectors D1 and D2. The optical elements (grey) have plane-parallel surfaces with coatings: mirror (solid line), polarization splitting (dotted line), non-polarization splitting (dashed line), anti-reflection (no line). Retarder plates are shown by thick lines (orange, λ/4) and double thin lines (blue, λ/2). LMM and UMM are the low-middle and up-middle mirrors; CP are compensating plates. The reference roof prism RRP raises the back-reflected beams of the auxiliary monolithic arms to a parallel plane; the λ/2 plates in front of it intercept only the incoming beams in the lowest plane. The detectors of the auxiliary monolithic arms, on the top of D1 and D2, are not indicated.
The reference and measurement beat-signals are detected after projections on linear polarizations forming again 45° angles with the reflection plane. Figure 3 shows the Lissajous curve of the beat-signals originated by the two auxiliary monolithic arms. Since no variation of the optical paths in expected, the two signals would have jointly gone along the same perfect ellipse again and again. Contrary, an unexpected noise is observed that was traced back to perturbations of the optical fibers delivering the low- and high-frequency beams to the interferometer.
Fig. 3 Observed Lissajous curve of the measurement beat signal vs. the reference one. For an ideal system it would have been a perfect ellipse.
3. Interferometer model
3.1. Jones’ formalism
The analysis, both experimental and theoretical, of the observed noise by means of the interferometer shown in Fig. 2 was hampered by its complexity. Therefore, we cut the interferometer design drastically by turning to the set-up shown in Fig. 4. The low- and high-frequency beams (linearly polarized at 45° angles with respect to the reflection plane) enter the polarizing beam-splitter and two (reference and measurement) beams, obtained by recombining photons having vertical and horizontal linear polarizations, exit from the polarizing beam-splitter. This skeletal model proved capable to reproduce the noise observed and, being any internal disturbance eliminated, confirmed its origin in the fiber delivering of the low- and high-frequency beams. To put it simply, a polarization jitter implies variations of the phase difference between the linear (vertical and horizontal) polarization components of the input beams and, consequently, between the interfering photons, because they stem from different components of each beam. Presently, we develop the relevant mathematical model and calculate the extinction ratio required to minimize the differential phase-noise of the beat signals; a detailed description of the formalism needed to describe polarization encoding can be found in [5, 12].
The beams entering the interferometer are
where [1, δi exp(iαi)]T are two-components spinor describing the polarization of the fields emerging from the fibers, ϕ is the retardation difference at the interferometer entrance, Ω is the heterodyne frequency, and the matrix rotates the polarizations by π/4 rad. The [1, δi exp(iαi)]T spinors are represented by using orthogonal (horizontal and vertical) linear polarization states as basis. The beams entering the interferometer are quasi linearly polarized; hence δ2 ≪ 1, where δ2 is the extinction ratio. The phases α1,2 are arbitrary, they vary randomly between zero and 2π, e.g., they depend on the thermal and mechanical stress of the fibers.Fig. 4 Model of the heterodyne interferometer. PBS is the polarizing beam-splitter, D are the detectors of the beat signals, P are linear polarizers oriented at 45° with respect to the reflection plane, νlow and νhigh are the low- and high-frequency beam entering the interferometer.
The input fields are split according to the horizontal and vertical polarizations and recombined in two detectors via separate (reference and measure) paths. Let us assume that the reference path is traversed by the first and second beam’s linear horizontal and vertical components, respectively. Hence, the reference field is
where are projectors on the π/4, horizontal, and vertical linear polarization states and the optical-path difference between the two reference arms has been assumed zero. Since only the first order terms will be retained, small aberrations of P∠, P||, and P⊥ contribute only through terms independent of the extinction ratios. Therefore, fixing Eqs. (4a–c) at the specified values, do not jeopardize the model generality.The measurement path is traversed by the first and second beam’s linear vertical and horizontal components, respectively. Hence, the measurement field is
where x is the retardation between the two interferometer arms. The interfering beams are where |∠〉 is a linear polarization at 45° with respect to the reflection plane and only the leading terms have been retained. Hence, the normalized beat-signals are Some Lissajous curves of the measurement beat-signal vs. the reference one – Eqs. (7a) and (7b), respectively – are shown in Fig. 5. If the field entering the interferometer is linearly polarized, that is, if δ1 = δ2 = 0, the beat signals are as expected. It is worth noting that the differential phase noise ϕ(t) in beam delivering shifts the heterodyne frequency from Ω to Ω + ∂tϕ. The phase difference between the beat signals (7a) and (7b) isThe same result would be obtained if the beams had been differently recombined, e.g.,
No extra phase arises if the beams are recombined by amplitude division, that is,The relation (9) is central in our analysis. Since the α1 and α2 phases walk randomly in the [0, 2π] interval, by assuming δ1 ≈ δ2 ≈ δ and by observing that Var[sin(α1)] = Var[sin(α2)] = 1/2, the root-mean-square noise of the phase-difference measurement is
Therefore, the extinction ratio of the polarized beams entering the interferometer must be better than , where is the targeted noise power of the phase-difference measurement. To give a numerical example, if the targeted standard-deviation is σx = 2π10−4 rad, the extinction ratio of the beams illuminating the interferometer must be at least δ2 ≈ 10−7. It is hard to have polarizers with such a high extinction ratio. This problem can be relieved by a careful alignment of the (linear) polarization of the delivered beams to the slow axis of the polarization preserving fibers to make to make the extinction ratio at the fiber output as small as possible and by using multiple polarizers. Whenever it is possible, polarization division must be avoided.Fig. 5 Lissajous curves of Ir(t) vs. Im(t) with α1 varying from zero to 2π. Parameter values are α2 = 0, ϕ = 0, δ1 = δ2 = 0.1, and x = 0 (left), π/4 (middle), and π/2 right.
3.2. Berry’s geometrical-phase
A complementary model considers the transport of polarizations on the Poincaré sphere and the associated Berry’s phase. Photon separation and recombination occur also on the Poincaré sphere – the space of the polarization states. Since polarizations are made to follow a closed path joining the input and output states (see Fig. 6), the photons acquire a cyclic phase – named after Berry – equal to half the solid angle subtended at the origin of the sphere by the contour obtained by closing the polarization path through geodesic arcs [13, 14]. As a consequence, if frequency tagging and photon separation and recombination are not ideal, the Berry’s phase gives rise to periodic errors [5, 15].
Fig. 6 Circuits of the polarization states on the Poincaré sphere. Left: ideal beam splitting and recombination. Right: aberrated beam splitting and recombination. |u1〉 and |u1〉 are the polarization states entering the interferometer. The split states – the north and south poles |||〉 and |⊥〉 – are horizontal and vertical linear polarizations. The detection state – the intersection of the zero meridian with the equator, |∠〉 – is a linear polarization at 45° with respect to the reflection plane. The beat signals acquires a geometric phase-shift equal to half the solid angle subtended by the polarization circuit at the origin of the sphere.
Figure 6 (left) shows the topology of the polarization transport when the input states – |u1〉 and |u2〉 – are split and recombined according to (4a–c). The |||〉 and |⊥〉 split states are the north and south poles; the intersection of the zero meridian with the equator, |∠〉, is the detection state of the beat signals. To calculate the geometric phase – the angle subtended at the origin by the path of the polarization states – let us observe that the longitudes of the input states are 2δ1 sin(α1) and 2δ2 sin(α2). Hence, with a unit radius and up the first order in δ1 and δ2, the area of the spherical wedge shown in Fig. 6 is 4[δ1 sin(α1) + δ2 sin(α2)]. This shows that the phase noise in Eq. (9) originates from the transport of the polarization states and it is caused by the jitter of the |u1〉 and |u2〉 location on the sphere.
The influence of aberrations (e.g., misalignments of the input beams and optical components and imperfect beam splitting and recombination) can be understood by looking at the geometrical representation of the interferometer on the Poincaré sphere. A quantitative aberration-analysis requires the use of the relevant expressions of the P||, P⊥, and P∠ projectors in Eqs. (3) and (5). Figure 6 (right) shows the circuit of the polarization states when beam splitting and recombination is slightly perturbed. It is worth noting that, with a first order calculation, the circuit area is still 4[δ1 sin(α1) + δ2 sin(α2)]. This explains why Eq. (9) does not lose generality when the Eqs. (4a–c) are fixed at the specified values.
4. Conclusions
The delivery of the photon polarization through a heterodyne interferometer, where the input beams are split and recombined by polarization-dependent optics, has been investigated by using the Jones’s formalism and a two-component spinor representation of the polarization states. Polarization jitter implies different phases of the components, vertically and horizontally polarized, of the laser beams at the interferometer input ports and, consequently, it originates variations of the beat-signal phase. A complementary model of the interferometer operation traces back both the non-linearity and phase noise of the beat signal to the phase holonomy of the transport of the polarization states around a closed loop on the Poincaré sphere. Under the assumption of quasi-linear polarization states at the interferometer input ports, a formula has been obtained expressing the root-mean-square noise in terms of the extinction ratio of the input beams.
Acknowledgment
This work was funded by the Italian ministry of education, university, and research; awarded projects 2011.
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