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Abstract
An optical configuration for Fabry–Pérot cavity scanning using a geometric phase shifter, known as the “spectral drill,” is improved to acquire a spectrum in real-time. Previously, the resonance condition of the spectral drill is swept by the mechanical rotation of a phase plate comprising a geometric phase shifter, and the acquisition time is limited. In this work, using a q-plate and a camera instead of phase plate rotation and a photo detector, we remove all the spinning mechanics and increase the acquisition rate by a factor 720. This technique will be applied to locking laser frequency.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Regular separation of the resonance modes of a Fabry–Pérot (FP) cavity is used as a ruler in the spectral region and has been applied as a frequency marker for fine spectroscopy [1–3]. It also serves as the essential principle for frequency combs [4]. The resonance frequencies and their separations can be scanned by changing the cavity length [5,6]. The maximum scanning-frequency range in the FP cavity is restricted by the physical limitations on the mirror position within the cavity, and by the tunable range of the physical properties defining the effective light-path length [5,6]. For example, supposing that one of the cavity mirrors is mounted on a piezo manipulator for scanning along the optical axis, the scanning range is limited only to the micron order [7,8]. This is sufficient for visible light, but short for infrared light, which has a wavelength greater than one micron. Another problem with the piezo manipulator is its hysteresis. For a precise measurement, one must take care that the frequency axis is reproducible, and this cannot be assumed due to the hysteresis [9].
On the other hand, the geometric phase experienced by a light whose polarization state moves on a Poincaré sphere is understood to be analogous to the Berry phase experienced by the wavefunction of the electron propagation in a vector field. The geometric phase of light was investigated by Puncharatnum in the early days of the geometric phase research [10]. Spatially distributed subwavelength grating structures for controlling the geometric phase have been investigated for spatial control of optical phase [11] and the concept was applied to generate a beam with topological charge [12]. Recently, such approaches have attracted attention because the metasurface technology can improve the performance through the degree of freedom in the subwavelength structure design [13,14]. As an application of the geometric phase of light, a geometric phase shifter (GPS) consisting of a simple series of phase plates has also been proposed by some groups for precise control of the optical [15] and THz-wave phases [16]. We have previously reported on the “spectral drill” optical configuration (so called because the motion of the resonance modes moving in the spectral region resembles the apparent motion of the grooves on a drill). A GPS is placed into a FP cavity herein. In the spectral drill, the resonance frequency of the FP cavity can be swept continuously in a seamless manner by rotating a phase plate comprising the GPS [17]. In our previous system, the phase plate was mounted on an auto-rotational stage; thus, the scanning speed of the resonance modes was restricted by the stage’s rotational speed. In this paper, we propose an idea of breaking this limitation by replacing the phase plate and photo-receiver with a q-plate and camera, respectively and show the experimental proof of this concept. Since the presenting configuration excludes any mechanics in the interferometer, namely all optics are fixed, it has potential to be free from the mechanical fluctuations. We believe this method will enable new spectroscopic applications for the spectral drill.
2. Spectral distribution
A GPS consists of one quarter-wave plate (QWP), one half-wave plate (HWP), and another QWP in series. The fast axes of the two QWPs are respectively tilted by ±45° from the polarization direction of the incident light. In a typical GPS setup, a HWP can be rotated by mounting on a rotational stage [15,16,17]. When the rotational angle of the HWP is α, the light experiences a geometric phase delay of 2α through the GPS. Consequently, a GPS within a FP cavity causes the confined light to carry a phase delay of 4α in every round trip. In this case, the transmission intensity of the cavity is given as [17]
In this situation, one must scan the rotation angle of the HWP by at most 90° to sense the angle corresponding to the resonance frequency. If an auto-rotational stage is used, several seconds are spent on a single scan. In the present work, we propose a method for achieving faster measurements. Our concept is to use a q-plate instead of a rotational stage. A q-plate is made of anisotropic molecules such as liquid crystal or its polymerized materials. While it partially works as a HWP, the fast axis of the HWP gradually rotates m/2 times around the optical center, where m is an integer. For the case where m = 1, the orientation of the fast axis around the center of the q-plate is depicted using solid curves in the inset of Fig. 1. The q-plate is usually used to generate an optical vortex beam with a topological charge m [18,19].
Fig. 1. Concept of a zero-spindle spectral drill—the rotational stage for the half-wave plate and photo detector are respectively replaced with a q-plate and imaging camera [17]; the orientation of the fast axis around the center of the q-plate (m = 1) is depicted in the inset. The fast axis is tilting by α at the position (r, θ).
We consider replacing the HWP in the GPS with a q-plate, as shown in Fig. 1. When a GPS is introduced into the FP cavity, the transmission intensity distribution in the polar coordinates (r, θ), with an origin set at the center of the q-plate, can be derived by modifying Eq. (1) as
3. Experimental results and discussion
3.1 Setup
In the experiment, we actually constructed the ZSSD system and evaluated its spectral response by incident of a monochromatic laser with changing its frequency beyond the free spectral range (FSR) of the ZSSD cavity. The experimental setup is depicted in Fig. 2. A single-mode external cavity diode laser (ECDL, Newfocus, velocity) with a wavelength of 1.55 μm and an oscillation frequency that can be fine-tuned using an externally induced voltage was used as a light source to be measured for spectral information. The beam from the laser system was incident on the ZSSD with being parallel to its optical axis. In front of the ZSSD cavity, the beam was gradually diverged using a convex lens of short focusing length, f = 5 cm. A GPS consisting of two QWPs and a commercial q-plate (Thorlabs, WPV10L-1550, m = 1) was allocated within the cavity. The phase plates and the q-plate were placed around 3 cm apart from each other within the cavity. The cavity length d was set to 12 cm. Behind 8 cm from the cavity, the transmitted-beam image was measured using an InGaAs camera (Goodrich Corp., SU320KTS-1.7RT) that was sensitive to IR light. 320 × 256 pixel images were recorded in a personal computer with a frame rate of 60 fps. The spectral information can be acquired without any mechanical rotation of the HWP, whereas, in our prior works, a rotational stage was required for the HWP to resolve the spectrum [17].
Fig. 2. Experimental setup of a zero-spindle spectral drill—ECDL: external-cavity-diode laser; QWP: quarter-wave plate; M: high-reflectivity mirror.
3.2 Results
We swept the laser frequency beyond FSR of the ZSSD cavity at a rate of 0.624 GHz/s with recording a spectral image into a movie. The spectral images were extracted every 20 frames and shown in Figs. 3(a)–(g) with a corresponding frequency change of Δf against Fig. 3(a). In all the images, a couple of bright arms were observed, and their trajectory traced a spiral pattern from the center to the outer part of the image. These bright curves respectively correspond to the resonance of the cavity at that part. The two bright arms in these figures corresponded to our expected number of 2m, since m was unity in our q-plate. The bright arms were gradually rotated clockwise by sweeping the laser frequency. When Δf became about 1 GHz, as shown in Fig. 3(f), the arms return to a similar position to Fig. 3(a), owing to Δf reaching to the free spectral range of the FP cavity. Note that the spiral shape of the fringes is attributed to the slightly longer cavity length at the outer side from the central part, considering a gradually diverging beam within the cavity. Moreover, the rotational direction of the spiral patterns reverses when the fast axis of the QWP2 in Fig. 2 is replaced by the slow axis by rotating the QWP2 by 90°.
Fig. 3. (a) Acquired image under zero-bias induced to the fine-tunning port of the ECDL, (b–g) Images acquired when the laser frequency was gradually detuned from (a).
3.3 Analysis and discussions
We estimated the frequency sensitivity of the rotating spiral pattern under changes in the laser frequency. To quantitatively estimate the rotation angle of the bright arms, we defined a donut-shaped region of interest (ROI) on the spectral image, which is the green-shaded area between the inner and outer circles in Fig. 4. To investigate the argument angle of the bright arms and their motion, we define the averaged brightness in the radial direction as $I(\phi ) = \int_{{r_{\textrm{in}}}}^{{r_{\textrm{out}}}} {i(r,\phi )dr} $, where rin and rout are, respectively, the radii of the inner and outer circles of the ROI and i(r, ϕ) is the pixel value at position (r, ϕ). The central position of the ROI and radius values of rin = 41 pixels and rout = 90 pixels covering the most inner part of the arms are manually set such that the center corresponds to that of the outer fringes in the spectral images and the motion of the brightness signal can be clearly shown. We obtained I(ϕ) for each frame in the recorded movie and ordered them by laser frequency detuning.
The frequency dependence of I(ϕ) is depicted in Fig. 5. The upward-sloping striped pattern was observed with changing laser frequency. Two bright stripes are observed in the vertical direction owing to the two-arm spiral in the spectral image. In the horizontal direction, the bright part periodically appeared at an interval of about 1 GHz, corresponding to FSR of the cavity. The slope tilt agrees with the resonance condition $\phi = {\textstyle{\pi \over m}}\left( {{\textstyle{{\Delta f} \over {{f_{\textrm{FSR}}}}}} + n} \right)$, where fFSR is the FSR defined in the cavity and Δf denotes detuning of the laser frequency. The given FSR of 1 GHz was slightly lower than the expected FSR of 1.25 GHz for a cavity length d = 12 cm. This value is reasonable, considering the extension of the effective light-path length due to the insertion of the optics within the cavity and diverging beam, which obliquely propagates in the cavity. Therefore, the frequency change against the arm rotation is 1/2π GHz/rad. Since the reading accuracy of the argument is roughly estimated to be 0.2π rad., this yields a system frequency sensitivity of 0.5 GHz. This sensitivity can be increased by configuring a longer cavity length. Note that the fluctuation in the intensity of the stripe pattern with changing the frequency was attributed to the fringes disturbed by the light scattered from the optics other than the cavity mirrors.
In the present system, the spectral-image-acquisition rate was determined by the frame rate of the InGaAs camera (60 Hz). In our prior spectral drill [17], we used an auto-rotational stage to rotate an HWP. Compared with the rotational speed of the stage of 30°/s, a 720-times-faster acquisition rate was achieved. We consider this real-time measurement of a spectrum to be suitable for feedback control of laser frequency, for example, we expect that the ZSSD will be applicable to frequency locking of a single-mode laser for precise spectroscopy if its frequency sensitivity can be improved. To improve this sensitivity, a high q-value of the cavity or a long cavity length are necessary.
4. Conclusion
By improving the configuration of a spectral drill using a q-plate and a camera, we achieved real-time measurement of a spectrum in a fixed FP cavity. This acquisition time was 720 times faster than that of the prior spectral drill. In this demonstration, the accuracy of the frequency determination was about 0.5 GHz. Since the ZSSD includes no mechanics, it has a potential to be developed as an accurate frequency measurement system with high stability if the q-value of the cavity and some performances will be improved in the future work. This will allow a new methodology to control the laser frequency and its applications.
Funding
Japan Society for the Promotion of Science (JP18H01908, JP18K04967, JP19K05299); Tohoku University-NICT matching fund (2019).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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