Spectral drill: a geometrical phase shifter within a Fabry-P&#x000E9;rot cavity

Seigo Ohno

doi:10.1364/OSAC.1.000136

1. Introduction

Owing to their regular frequency intervals and sharp linewidths, the axis modes of a Fabry-Pérot (FP) cavity provide ruler markings along the spectral region of a spectrometer. For instance, a tandem-type FP spectrometer has resolved the fine spectral information in spectroscopic measurements over a long term [1]. In a laser system based on an FP cavity, the lasing frequency can be controlled by tuning the axis modes to achieve a particular effective cavity length [2]. Among the many techniques used to change the effective cavity length are sweeping of the mirror position [3], controlling the refractive index of the in-cavity medium using the electro-optic (EO) effect [4], and adjusting the gas pressure [5]. As these methods change the physical conditions of the cavity, the continuous-sweep range of the axis modes is limited by the parameters of the cavity system, such as the mirror spacing, the maximum voltage of the EO crystal, and the maximum pressure of the gas medium. Furthermore, in such cases, scanning is accompanied by a slight but exact expansion of the mode intervals. This article presents a novel method that sweeps the axis modes continuously and unidirectionally, returning to the original condition of the cavity without expanding the mode intervals. The method is applicable to novel tunable continuous wave (CW) lasers and precise spectroscopy to resolve narrower than the free spectral range (FSR) of the cavity.

Our concept is simply based on the geometrical phase of circularly polarized light (CPL). The FP interference of the electronic wave-function in two dimensional electron gas (2DEG) has been already investigated to sweep the cavity modes by the magnetic flux through Aharonov-Bohm phase, the geometrical phase experienced by an electron in the vector potential [6, 7]. Here, we consider the re-import of the geometric phase in a FP cavity from 2DEG system to optics, analogously. The geometric phase in optics is so-called Pancharatnam-Berry phase, which has been studied in the context of Berry curvature on a Poincaré sphere [8]. The geometrical phase has been exploited in an achromatic phase shifter that precisely controls the phase in optical-coherence tomography [9]. These principles of CPL-phase control have also been applied to a design strategy for artificial structures [10,11]. Structural designs have benefited from recent advances in metasurfaces. Examples of such metasurfaces include those for splitting the CPL according to its handedness [12] and for generating optical vortices [12] and vortex arrays [13].

This article discusses a GPS consisting of fixed phase plates and a rotational phase plate inserted within an FP cavity, in which the light makes round trips with multiple reflections, as conceptually shown in Fig. 1. The additional phase delay caused by the GPS induces a shift in the axis modes. The phase value can be continuously changed by altering the rotation angle of a phase plate; as this plate is rotated, the cavity-axis modes shift in one direction, like the threads that appear to move unidirectionally on a rotating drill. Analogously to the equal-interval pitch of the drill thread, the FSR remains constant (does not expand) with rotation of the phase plate. Indeed, the some patents have been already claimed the similar configurations where a GPS is placed in a pulse laser cavity or an optical comb with remembering the time domain behavior of them [14, 15]. These techniques will be usable to improve the performance of a frequency comb for a particular frequency standard [16, 17] and for a modern high-level spectroscopic technique [18]. In this article, we focused on the CW response of a cavity including a GPS. The behaviors of two GPSs with different configurations are discussed. The GPSs are investigated by both traditional analytical and numerical calculations and experimental demonstrations.

Fig. 1 Concept of a geometrical phase shifter (GPS) consisting of fixed quarter-wave plates (QWPs) and a rotating half-wave plate (HWP) in a Fabry-Pérot cavity confined by two mirrors. During multi-reflection in the cavity, a light experiences a geometric phase of 2θ with every pass through the GPS. The spectral behavior of the separated axis modes in the free spectral range (FSR) resembles that of the spiral thread on a rotating drill.

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2. Type I configuration

In the Type-I configuration (see Fig. 2), a GPS comprising two quarter-wave plates (QWP1 and QWP2) and a half-wave plate (HWP) are inserted into a cavity in which light is confined using a front mirror, M1, and an end mirror, M2. The fast axes of QWP1 and QWP2 are aligned along the x- and y-axes, respectively. The HWP is rotated by angle θ between its fast axis and the x-axis. This GPS configuration was previously investigated in Yang et al. [9]. The light-path length in the cavity (excluding the phase retardation caused by the phase plates) is denoted by d; when a linearly polarized light propagates in a cavity and passes through the GPS, the complex amplitude of the electric field changes from $E_{0} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 1 \end{matrix})$ to

E_{0}^{'} = Q_{-} [R^{- 1} (θ) H R (θ)] Q_{+} E_{0} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ - 1 \end{matrix}) \exp 2 i θ,

where Q and H are the Jones matrices of the quarter-wave and half-wave plates, respectively, and R(θ) denotes the rotation of the coordinates through θ. The subscript + (−) on Q represents the direction of the fast axis toward the x-(y-) direction. For the calculation

Q_{\pm} = (\begin{matrix} 1 & 0 \\ 0 & \pm i \end{matrix})

,

H = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})

, and

R (θ) = (\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix})

were used [19]. We further assumed that the phase plates are anti-reflection-coated, such that the reflection loss on the phase-plate surfaces and the complexity of cavity-mode analysis are negligible.

Fig. 2 Type-I configuration of a geometric phase shifter (GPS) in a cavity formed by two mirrors (Ml and M2). The GPS consists of two quarter-wave plates (QWP1, QWP2) and a half-wave plate (HWP). The fast axes of QWP1 and QWP2 are oriented along the x and y axes, respectively. The HWP rotates through an angle θ from the x-axis. An incident light is linearly polarized in the direction of (x, y) = (1, 1). d is the optical-path length between the mirrors (excluding the phase retardation imposed by the phase plates)

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The complex amplitude means that the GPS rotates the polarization of light through 90 degrees. The phase retardation depends on the HWP-rotation angle θ. After passing through the GPS, the light confined in the cavity reflects from the end mirror M2, is again phase-retarded by the GPS, and then reflects from the front mirror M1, thus returning to its initial position. Consequently, the complex amplitude, E₁, at the initial position after a single round trip is given as

\begin{array}{l} E_{1} = M Q_{+} [R^{- 1} (- θ) H R (- θ)] Q_{-} M E_{0}^{'} \\ r^{2} \exp 4 i θ E_{0}, \end{array}

where the reflection matrix M is given by

(\begin{matrix} - r & 0 \\ 0 & r \end{matrix})

and r is the amplitude reflectivity of the mirrors. Note that, in this calculation, the matrices of the phase plates are invariant during the reflection, whereas the coordinate system’s rotation changes from θ to −θ. Equation (2) implies that the GPS affects only the 4θ phase within a typical FP cavity. Hence, considering multiple reflections within a cavity, the amplitude of the transmitted light is given by

\begin{array}{l} E_{T} = (1 - R) e^{i (k d + 2 θ)} \sum_{n = 0}^{\infty} {[R e^{2 i (k d + 2 θ)}]}^{n} \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ - 1 \end{matrix}) \\ = \frac{(1 - R) \exp i (k d + 2 θ)}{1 - R \exp i (2 k d + 4 θ)} \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ - 1 \end{matrix}), \end{array}

where k is the wavenumber of light and R = |r|²is the reflectivity of the mirrors. The mirror transmittance is assumed to be 1 − R. The transmittance of this system is therefore given as

T = {| E_{T} |}^{2} = {[1 + \frac{4 R}{{(1 - R)}^{2}} \sin^{2} \frac{δ}{2}]}^{- 1},

where, δ = 2kd + 4θ is the total phase retardation of a single round trip. This formulation is the same as a typical FP-cavity formulation [19], except for the inclusion of an additional term (4θ) in δ. The resonant condition of the cavity is kd = nπ − 2θ, where n is an integer-valued mode number. Therefore, the resonance frequency of the cavity can be tuned to an arbitrary frequency by rotating the HWP. For each revolution of the HWP, i.e., for each 2π change of θ, the axis modes shift by four times the width of the FSR given by ck/δ ∼ c/2d.

3. Type-II configuration

We next introduce the Type-II configuration. Two QWPs are arranged in the cavity, as shown in Fig. 3. In this case, QWP1 is fixed with its fast axis aligned along the x axis, and QWP2 is rotated through an angle θ. The complex amplitude after passing through a Type-II GPS is given by

E_{0}^{'} = [R^{- 1} (θ) Q_{+} R (θ)] Q_{+} E_{0} = (\begin{matrix} \cos θ - \frac{π}{4} \\ \sin θ - \frac{π}{4} \end{matrix}) \exp i θ .

Fig. 3 Type-II configuration of a GPS in a cavity. The GPS consists of two quarter-wave plates labeled QWP1 and QWP2. The fast axis of QWP1 is oriented along the x axis, while QWP2 rotates through angle θ from the x-axis. The incident light is polarized identically to its polarization in the Type-I configuration illustrated in Fig. 2. The polarization of the transmitted light depends on the rotation angle θ, as depicted in Eq. (7)

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The passing light reflects from the end mirror, M2, again passes through the GPS, and reflects from the front mirror, M1. After returning to its initial position, the light’s complex amplitude is derived as

E_{1} = M Q_{+} [R^{- 1} (θ) Q_{+} R (- θ)] M E_{0}^{'} = r^{2} \exp 2 i θ E_{0} .

Note that, in the Type-II configuration, the phase is half that in the Type-I configuration [cf. Eqs. (2) and (6)]. This result can be explained by the maximum retardation imposed by the GPS. The maximum retardation for a single pass is π/2 + π + π/2 = 2π in the Type-I configuration, and π/2 + π/2 = π in the Type-II configuration. Similarly to Eqs. (3) and (4), the complex amplitude of the light transmitted in the cavity is given by

\begin{array}{l} E_{T} = (1 - R) e^{i (k d + θ)} \sum_{n = 0}^{\infty} {[R e^{2 i (k d + θ)}]}^{n} (\begin{matrix} \cos θ - \frac{π}{4} \\ \sin θ - \frac{π}{4} \end{matrix}) \\ = \frac{(1 - R) \exp i (k d + θ)}{1 - R \exp i (2 k d + 2 θ)} (\begin{matrix} \cos θ - \frac{π}{4} \\ \sin θ - \frac{π}{4} \end{matrix}) \end{array}

and the transmittance is derived as the same formulation with Eq. (4) with differing δ = 2kd + 2θ. The Type-II configuration has a simpler design than the Type-I, but as the polarization of the transmitted light depends upon the rotation angle of the phase plate, the tunable range of the axis modes per rotation in the type II configuration is double that in the FSR. Hence, the rotation-angle sensitivity of the Type I configuration is twice that of Type II.

4. Experiments and discussion

We experimentally demonstrated the type-I and -II configurations. A single-mode laser (Anritsu, MG9541A) of wavelength 1.55 µm was passed through the GPS-containing cavity confined with 99 % reflective dielectric multi-layer mirrors. The cavity length was set to 155 mm, and a frontal mirror was attached to a piezo-actuator scanning along the optical axis to fine-tune the cavity length. The scanning direction of the mirror was set such that the cavity length shortened with increasing the voltage applied to the piezo-actuator. In both configurations, the phase plate to be rotated was mounted on an auto-rotational stage (Sigma, OSMS-40YAW), and an InGaAs PIN photodiode was used to monitor the transmitted intensity. The transmitted intensities in both configurations were two-dimensionally investigated while the cavity length was scanned three times longer than the FSR by the piezo-actuator and the phase plate was rotated through 360 degrees.

The results for types I and II are shown in the upper panels of Fig. 4(a) and (b), respectively. The horizontal axis denotes the rotation angle of the phase plate and the vertical axis denotes piezo-voltage, while the lower voltage corresponds to the longer cavity length. Four peaks observed in the vertical direction form a negatively slanted-stripe pattern as the rotation angle increases. The slant angles for the different types agree well with the (dashed) iso-phase lines, which are determined using the formulas 2kd + 4θ = const. and 2kd + 2θ = const. for types I and II, respectively, as well as the mode intervals of 7.1 V along the piezo-scanning. The slight deviation of the axis modes from the lines seems to be caused by frequency drift of the single-mode laser during the measurement.

Fig. 4 Two-dimensional interferograms obtained by rotating the phase plate and scanning the piezo-actuator for (a, upper panel) type I and (b, upper panel) type II. The dashed lines depict the expected isophase lines. Typical interferograms for the phase-plate rotation are extracted on the lower panels.

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The significant interferograms measured at 25 and 18 V for types I and II, respectively, are extracted in the lower panels. Four and two peaks observed are consistent with θ dependence of δ in both cases. The linewidth of the peaks in type I were around half of the width of type II; this means that the finesse of the cavity independent of the configurations of both types. Though it is not essential to the discussion of the tunablity of the axis modes, the reason why the finesse is too low for high-reflectivity cavity mirrors seems to be due to imperfection of the beam collimation and inescapable reflection loss at each surface of the phase plates in spite of AR coating on them. The effective reflectivity in our experiments was estimated to be around 70%.

Additionally, We investigate the effect of the phase-retardation deviated from λ/2 or λ/4 in the phase plates. Since infinite possible deviation can exist, here, we restrict our discussion only to the deviation of the rotating phase plate and assume that other phases are appropriate. The phase deviations, Δϕ, for the HWP in type I and the QWP in type II are introduced as $H = (\begin{matrix} 1 & 0 \\ 0 & e^{i (π + Δ ϕ)} \end{matrix})$ and $Q_{+} = (\begin{matrix} 1 & 0 \\ 0 & e^{i (π / 2 + Δ ϕ)} \end{matrix})$ , respectively. When Δϕ = 0, the HWP and QWP work properly. Considering a similar situation to our experiment, the transmittance is numerically calculated by two-dimensionally changing the cavity length and the rotation angle of a phase plate under various conditions of the phase deviation. The reflectivities of the mirrors were set to 70 % with reference to the effective reflectivity in our experiment. The wavelengths of the incident beam and the cavity length were 1.55 µm and 155 mm, respectively. From each interferogram obtained by scanning the cavity length, we extracted the peak positions and plotted them on the two-dimensional maps shown in Fig. 5(a) and 5(b) for types I and II, respectively. In both (a) and (b), the upper and lower panels show the ranges of 0 ≤ Δϕ ≤ π and π ≤ Δϕ ≤ 2π, respectively. In the figure, lighter colors for curves indicate lower relative peak heights.

Fig. 5 Two-dimensional mapping of the axis modes with scanning of the cavity length and a rotation angle that depends on the phase retardation of the rotating phase plate in the type-I (a) and -II (b) configurations. Lighter-color curves indicate relatively low peak intensities of the modes.

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In the type-I configuration, the negative-slope-stripe pattern of the axis modes is observed at Δϕ = 0. If Δϕ deviates from 0, the axis modes will split every 90 degrees and an axis mode with positive slope will be allowed with a tiny intensity. Both negative and positive slopes become shallow with increasing Δϕ. When Δϕ becomes just π, the axis modes degenerate and become independent of the rotation angle, because the Jones matrix of the deviated HWP becomes a unit matrix. Further increasing Δϕ modulates the axis modes and gradually approaches the original stripe pattern until Δϕ = 2π. Note that, in our type-I configuration, a perfect positive slope cannot to be allowed. The positive-slope modes can be allowed for the polarized incident beam along (x, y) = (1, −1).

In type II, the splitting behavior every 180 degrees under a slight deviation from Δϕ = 0 is similar to that for type I. The axis modes have degeneracy at Δϕ = π 2, where the Jones matrix of the deviated QWP corresponds to an appropriate HWP. Because the/ Jones matrix for an HWP in front of a mirror is generally the same as with a single mirror, namely HMH = M, it does not change with rotation. By further increasing Δϕ axis modes with positive slope become allowed and perfect positive-stripe patterns appear at Δϕ = π. In the range of π ≤ Δϕ ≤ 2π, the symmetrical modes with the range 0 ≤ Δϕ ≤ π are allowed and, finally, the axis modes return to the negative-slope-stripe pattern. These common behaviors to type I and II show that the geometrical phase resolves the degeneracy on the independent resonance modes and can seamlessly interconnect each other.

5. Conclusion

The resonant frequency of a GPS-based FP cavity can be tuned by controlling the rotation angle of the GPS phase plate. The sensitivity of the frequency shift of the axis modes to changes in the rotation angle depends upon the GPS configuration.

The next scope of the presenting work is to control the lasing frequency of a FP laser. The seamless tuning of the resonant frequencies will be a novel technique for the precise laser spectroscopy. Moreover, the characteristic angle-to-frequency conversion enables precise angle determination from frequency measurements by combining a GPS-based FP cavity and modern RF-technology to measure the frequency.

Funding

Konica Minolta Science and Technology Foundation (Konica Minolta Imaging Science Encouragement Award); Grant-in-Aid for Scientific Research(C) Grant Number 18K04967.

Acknowledgments

The author would like to thank Prof. Katsuhiko Miyamoto in Chiba university for fruitful discussions and experimental help. The author also thanks Mr. Makoto Saito in Tohoku university for his help in figure preparation.

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Spectral drill: a geometrical phase shifter within a Fabry-Pérot cavity

Abstract

1. Introduction

2. Type I configuration

3. Type-II configuration

4. Experiments and discussion

5. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (5)

Equations (7)

OSA Continuum