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Abstract
Residual amplitude modulation is one of the major sources of instability in many precision measurements using frequency modulation techniques. Although a transverse and inhomogeneous distribution of residual amplitude modulation has long been observed, the underlying mechanism is not well understood. We perform measurement and analysis of this spatial inhomogeneity using several electro-optic crystals of different types. Two distinct components are identified in the spatial distributions, and their detailed properties, some of which are previously unnoticed, are mapped out and analyzed, showing that the spatial inhomogeneity can be explained by acousto-optic interaction inside the crystal. Moreover, this spatial inhomogeneity can be further suppressed, improving the 1000-s stability of residual amplitude modulation to 3×10−7 (8×10−8) at modulation frequency of 11 MHz (120 kHz), corresponding to a frequency instability of 1×10−17 (3×10−18), estimated for a cavity-stabilized laser using a Pound-Drever-Hall discrimination slope of 1×10−4 V/Hz.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical frequency modulation (FM) is widely used in a broad range of research fields such as modulation transfer spectroscopy (MTS) [1], ultra-stable lasers [2,3], and gravitational wave detection [4]. In FM detection schemes, electro-optic modulators (EOMs) are frequently used to obtain phase modulated laser beams. Alongside the phase modulation the laser beam also acquires a residual amplitude modulation (RAM) [5], resulting in systematic and fluctuating frequency shifts that degrade the laser frequency stability and distort the spectroscopic line profile.
Apart from the spatially averaged RAM, an early investigation showed that there is also a nonuniform distribution of RAM across a transverse plane of the laser beam emerging from the electro-optic (EO) crystal [6], resulting in a non-zero signal at the modulation frequency when the laser beam is partially blocked. This spatial inhomogeneity leads to instability in measurements adopting frequency modulation techniques in that an iris or some equivalent in the setup partially limiting the passage of the light beam will produce an extra amount of RAM.
An important experimental clue for the source of the inhomogeneity comes from a related work [7], which showed that if a spatial separation on the order of 0.35 µm between the sideband and carrier is assumed, then the calculated spatial distribution agrees with the experimental pattern. In an early investigation, two representative patterns of the spatial inhomogeneity were observed, namely a symmetric bell-shaped and an anti-symmetric S-shaped patterns [8]. Later, the interference arising from the crystal birefringence was excluded as the source of the spatial distribution [9], and it was found that the spatial patterns vary as the EO crystal is displaced or tilted. However, further identification of the underlying mechanism was prevented by the irregular evolution of the spatial patterns.
We carry out detailed measurement on the spatial distribution of RAM, revealing previously unexplored properties. We show that the spatial distribution consists of two distinct components whose amplitudes evolve periodically as the laser beam is transversally displaced inside the crystal and the patterns exhibit unique features when probed in different directions. The mechanism underlying the spatial distribution is then analyzed with additional experimental verification. Moreover, preliminary result of suppressing the spatial inhomogeneity is given, showing the prospect of further improvement on the stability of RAM in a variety of applications to which frequency modulation technique is applied.
2. Experimental setup for measuring spatial distribution of RAM
Figure 1(a) shows the experimental setup for probing the transverse spatial distribution of RAM. A 633 nm Helium Neon laser is used as light source. The beam waist is located approximately at the middle of the EO crystal and different spot sizes (w0 = 0.2-0.5 mm) are used in the measurement. Before entering the crystal, the light first passes through a Glan-Taylor (GT) prism to ensure a linear polarization with high extinction ratio (∼50 dB). The modulated laser beam is focused by a convex lens (focusing length, 11 mm) onto the active detection area of a photodetector (PD) with a 3-dB bandwidth of 180 MHz. To suppress the interference-related instability caused by the back reflection, an optical isolator, which provides about 30-dB isolation, is inserted between the crystal and the focusing lens. The output of PD at radio frequency (RF) is band-pass filtered and mixed with the local signal in a doubly balanced mixer. Totally five EO crystals [9–12] are used and their geometric configurations are shown in Fig. 1(b). Dimensions are 35(z) × 3(y) × 5(x) mm3 for crystals 1-3 and 30(z) × 5(y) × 5(x) mm3 for crystals 4 and 5.
Fig. 1. (a) Experimental setup. (b) Electro-optic crystals used in the measurement. The dashed frames denote two different configurations of the electrodes. Crystals 1-4, MgO:LiNbO3 (LN); Crystal 5, KTiOPO4 (KTP); GT, Glan-Taylor prism; ISO, optical isolator; PD, photodetector; BPF, band-pass filter; LO, local oscillator; PS, phase shifter; DBM, doubly balanced mixer.
As illustrated in Fig. 1(b), for crystals with wedged entrance and exit facets, the ordinary (o) and extraordinary (e) lights inside the crystals are spatially separated due to the birefringence of the crystal and only e light is used in the measurement. In case of crystal 2, the separation is not available due to nearly normal incidence, and the Glan-Taylor prism before the crystal is adjusted to minimize the o light. These measures greatly reduce the instability caused by the interference of o and e lights, which is susceptible to changes in polarization and ambient temperature.
Several measures are adopted to avoid the interference arising from etalon effect. Except for crystal 4, all other four crystals are anti-reflection (AR) coated on their entrance and exit facets. The light enters and leaves crystal 4 at Brewster angle and hence AR coating is not needed for this crystal. Being more prone to parasitic etalons, crystal 2 is slightly tilted with respect to the incoming laser beam. Prior to probing the spatial inhomogeneous RAM, the temperature of EO crystal in experimental setup is deliberately varied and the measurement proceeds if no fluctuation of the demodulated signal is observed.
Here, the RAM is quantitatively evaluated as the modulation index ɛ of an amplitude-modulated signal, which can be expressed as
where I0 is the average of the laser intensity, Ω is the modulation frequency, and ϕ is a phase shift. The spatial distribution of RAM is probed by transversely scanning a knife edge across a modulated laser beam and simultaneously recording the transmitted optical power as well as the signal at the modulation frequency.3. Two distinct components of spatial patterns of RAM
Unlike our previous investigation [9], the phase angle between the modulation and local signals is continuously varied and the resulting intermediate states are also measured in current analysis. Unless specifically stated, a small modulation index (β = 0.23-0.25) in EO modulation is chosen such that sidebands other than the lowest ±1 orders are negligible. At relatively high modulation frequency of 11 MHz, the spatial pattern varies with the demodulation phase, but two distinct components with 90° phase difference can be identified. Figure 2 shows these two components that are obtained with crystal 1 and can be described as a symmetric, bell-shaped and an anti-symmetric, S-shaped patterns. The spatial pattern at an arbitrary demodulation phase is the composition of these two intrinsic components, which, in general, are neither in-phase nor quadrature with respect to the modulation signal.
Fig. 2. Two characteristic components of the transverse spatial distribution measured 30 mm downstream the crystal at modulation frequencies of (a) 11 MHz and (b) 120 kHz. Crystal 1 is used in the measurement. Solid circles are measurements of the symmetric component (black), the anti-symmetric component (red), and an intermediate (purple) state with a demodulation phase of 20° relative to that of the anti-symmetric component. Green solid curve is the sum of the projections of the symmetric and anti-symmetric components onto the direction 20° from the anti-symmetric component in the phasor diagram. Gray solid curves are calculations of symmetric components at 11 MHz and 120 kHz using carrier-sideband separations of c = 31.7 µm and 0.34 µm, respectively. The horizontal scale is normalized by the beam spot size w at the location of the measurement.
Table 1 lists the measured peak-to-peak amplitudes of the symmetric and anti-symmetric patterns for all five crystals shown in Fig. 1(b). For either of the symmetric or anti-symmetric component, the spatial patterns produced by five crystals show similar profiles and their amplitudes are approximately same when measured at same modulation frequency. Both the symmetric and anti-symmetric patterns are observed at 11 MHz, but the latter is absent at low frequency of 120 kHz at which the symmetric, bell-shaped pattern is observed as a quadrature component.
Table 1. Peak-to-peak amplitudes of the symmetric and anti-symmetric patterns measured with five crystals shown in Fig. 1(b). The modulation indices are 0.25 and 0.23 for modulation frequencies of 11 MHz and 120 kHz, respectively.
The EO crystal is translated and tilted to examine the two compositing components found in the spatial distribution of RAM. The translation is in x (crystals 1-5) and y (crystals 4, 5) directions, and the rotation of crystals is operated around an axis approximately located at the center of the crystal and pointing towards y direction. At each position or angle, the demodulation phase is adjusted to extract each of the two components. The amplitude of the symmetric pattern recorded at modulation frequency of 11 MHz is shown in Fig. 3. The amplitude of anti-symmetric component shows similar dependences on the displacement and tilting angle. Notably, the amplitudes of both two components show periodical variation and the sinusoidal fit of experimental data gives a spatial period of 0.62(1) mm, which is in consistent with the wavelength (0.6 mm) of the 11-MHz ultrasound excited inside the crystal. For both two components, the amplitudes of spatial oscillations show a clear decreasing trend as the tilting angle is increased, but the spatial period does not change. These features provide strong evidence that the spatial inhomogeneity of RAM originates from acousto-optic (AO) interaction. In particular, the spatial separation of the two sidebands from the carrier, proposed in an early analysis [7], is caused by the diffraction of light passing through a refractive-index grating built up by sound wave inside the crystal [13].
Fig. 3. The variation of the peak-to-peak amplitude of the symmetric component as the crystal is translated in x direction and tilted. A LN crystal (crystal 3) is used at modulation frequency of 11 MHz. A spatial period of 0.62(1) mm is determined by sinusoidal fit of the data with tilting angle of 1°.
As shown in Fig. 3, in addition to the spatial periodicity, the amplitude of the spatial pattern sensitively depends on the tilting angle of the crystal. This phenomenon can be well explained by the diffraction of light arising from AO interaction. An intuitive geometric consideration [14] shows that as the angle formed by the laser beam and the wave plane of the sound deviates from zero, the laser beam experiences a reduced corrugation of the refractive index that is produced by the sound wave and responsible for diffraction of light, resulting in weakened sidebands and hence the decreased oscillating amplitude of the spatial pattern as the crystal is translated.
4. Spatial separation of the sidebands
4.1 Spatial patterns probed in different directions
In previous section, the spatial distribution of RAM is probed in horizontal and vertical directions, and it is related to the scattering of the light by the sound wave inside the EO crystal. To reveal the two-dimensional distribution of diffraction sidebands in a transverse plane behind the EO crystal, a large number of spatial patterns are recorded by scanning the knife edge across the laser beam in different directions covering all azimuthal angles. The measurements are performed at 11 MHz (β = 0.25) and 120 kHz (β = 0.23). Figure 4 illustrates the evolution of spatial patterns as probing direction (cutting angle) is varied for two representative cases of symmetric and anti-symmetric dominated patterns.
Fig. 4. The evolution of spatial patterns with probing direction (cutting angle) for two representative cases of (a) symmetric and (b) anti-symmetric dominated patterns. In each case, two illustrations with 90° phase difference are given. Arrows indicate the moving direction of knife edge. The transverse patterns are measured at two modulation frequencies of 11 MHz and 120 kHz. The S-shaped anti-symmetric pattern is not observed at 120 kHz.
The symmetric component is always observed at both 11 MHz and 120 kHz. Its anti-symmetric counterpart is present at 11 MHz but disappears in all directions when modulation frequency is changed to 120 kHz. As shown in Fig. 4, when the knife edge cuts through the beam in the opposite direction, the bell-shaped, symmetric pattern flips sign, but the sign of the S-shaped, anti-symmetric one does not change. For each of the two components, the amplitude of the spatial pattern reaches its maximum in horizontal (x) and vertical (y) directions with a 90° phase difference involved in the demodulation process.
Because of the weakness of the sidebands and the much smaller carrier-sideband separation compared with the beam spot size, the diffraction sidebands cannot be directly observed. However, their locations in a transverse plane behind the crystal can be indirectly deduced from the map of spatial patterns shown in Fig. 4. As an example, Fig. 5 sketches the diffraction sidebands responsible for the symmetric component. In this case the light is diffracted in the horizontal and vertical, and the relative phases between the sidebands and the central carrier are also determined and indicated in Fig. 5.
Fig. 5. The diffraction sidebands responsible for the symmetric component of the inhomogeneous RAM in a transverse plane behind the EO crystal. In either horizontal (x) or vertical (y) direction, there is a pair of first-order sidebands. The sideband with plus (minus) sign is in phase (antiphase) with the central carrier. The spatial separation of the sidebands from the carrier is greatly exaggerated.
The map shown in Fig. 4 provides important information for reconstructing the diffraction sidebands responsible for the anti-symmetric component. The diffraction sidebands shown in Fig. 5 cannot explain the anti-symmetric pattern. Nevertheless, the evolution of the spatial patterns shown in Fig. 4 reveals interesting connections between the symmetric and anti-symmetric components. For example, the amplitudes of two components reach maximum in two orthogonal directions with the corresponding signals being in phase with each other. The two peak amplitudes can also occur in the same direction, but their phases are in quadrature. These regularities imply that two different ultrasonic modes may be simultaneously excited by the EO modulation because of the coexisting adverse piezoelectric and photoelastic effects.
4.2 Calculation of the symmetric pattern
The symmetric, bell-shaped pattern can now be well understood in terms of the spatial separation of diffraction sidebands. With an unblocked laser beam, in the output of PD the two cross terms, each of which is the product of one sideband and the carrier, cancel each other out because the two sidebands are in antiphase. However, when the beam is partially blocked, the perfect cancellation of the two terms no longer holds, resulting in a non-zero component at modulation frequency. For example, when the knife edge cuts through the laser beam in x direction, the resulting spatial pattern can be calculated by
Here, only ±1 sidebands are considered for both EO and AO modulations, an approximation that is suffice for small modulation index. In Eq. (2), Jn(β) and Jn(βa) are the nth-order Bessel functions with the EO and AO modulation indices of $\beta = \pi l{\gamma _e}n_e^3V/(\lambda h)$ and ${\beta _a} = 2\pi l\Delta {n_a}/\lambda$, respectively, where l is the length of the crystal, ne and γe are the refractive index and EO coefficient for e light, respectively, V is the amplitude of the voltage signal applied to the electrodes, λ is the optical wavelength, h is the height of the crystal, and Δna is the sound-induced change in refractive index, which is determined by
where E (0, 0, V/h) is the electric field inside crystal, and the numerical values of the matrix elements of photoelastic tensor P and piezoelectric tensor d are taken from Refs. 15 and 16, respectively. U is the normalized field distribution of a Gaussian beam, which is where w is the beam spot size of the Gaussian beam at the location of measurement. U± = U (x ± c, y) are two spatially shifted distributions with carrier-sideband separation of c = Dλ/Λ, where D is the distance from crystal (exit facet) to knife edge, and λ and Λ are the wavelengths of the optical and sound waves, respectively. As shown in Fig. 2, good agreements between the calculated and experimental patterns are achieved at modulation frequencies of 11 MHz and 120 kHz. The source of the anti-symmetric pattern is presently not clear. Its spatial periodicity and evolution with the probing direction (Fig. 4) imply that the anti-symmetric component is of the same acoustic origin but may relate to a different ultrasonic mode.At modulation frequency of 120 kHz, the amplitude of the symmetric pattern remains unchanged when the crystal is translated. This different behavior at low frequency can well be understood by the much longer ultrasonic wavelength, which is about 60 mm, 12 times larger than the dimensions of the cross section of the crystal, meaning that the sound field inside crystal is nearly uniform and hence there is no spatial dependence of the AO interaction. Furthermore, the calculation shows that when switching from 11 MHz to 120 kHz the amplitude of the symmetric pattern will drop by two orders of magnitude because of the nearly two-order reduction of the diffraction angle, a difference that is in consistent with the experimental observations at the two frequencies (Fig. 2).
4.3 Spatial patterns with increased modulation depth
Thus far a small modulation index β is used such that only first-order sidebands appear in the EO modulation. The spatial patterns are also measured with β gradually increased from 0 to 0.72. Figure 6 displays spatial patterns of two symmetries at five modulation indices, capturing the major evolutional trend of the spatially inhomogeneous RAM as the modulation strength is increased. Here, crystal 3 is used in measurement and the modulation frequency is 11 MHz.
Fig. 6. Spatial patterns of the symmetric and anti-symmetric components with five modulation indices of (a) 0.22, (b) 0.34, (c) 0.45, (d) 0.58, and (e) 0.72. The horizontal scale is normalized by the beam spot size w at the location of the measurement.
As β is continuously varied from zero to relatively large value of 0.72, the spatial patterns of RAM evolve smoothly and no abrupt change is observed. When β is increased from zero to 0.22, the spatial patterns shown in Fig. 6(a) emerge from a flat background and their amplitudes gradually increase. With β approaching 0.34 two additional peaks gradually appear at the two sides of the central feature. The two emerging peaks are of same and opposite signs for symmetric and anti-symmetric components, respectively. This process of developing new peaks in a pairwise fashion repeats as the modulation index is further increased, a feature that is in consistent with the scattering of the light by a sound-induced refractive-index grating with increasing AO interaction.
5. Suppressing the spatial inhomogeneity of RAM
The angular and translation dependences of spatial inhomogeneity are employed to further suppress RAM and improve its stability. By manually translating and rotating the EO crystal, the two components of the spatial distributions can be minimized, and RAM is also reduced and shows improved stability. Figures 7(a), 7(b), and 7(c) are measurements on the spatial distributions before and after the optimization. Noticeable improvements are achieved at both low (120 kHz) and high (11 MHz) modulation frequencies, and the spatial distribution can be reduced to close to the noise background at low frequency of 120 kHz. Figures 7(d) and 7(e) show the effect on the magnitude and stability of RAM. The RF signal at modulation frequency, which is a direct consequence of RAM, is reduced from -72 dBm (-87 dBm) to -85 dBm (-90 dBm) at frequency of 11 MHz (120 kHz). After minimizing the two spatial components of RAM, the 1000-s stability of RAM, in terms of Allan deviation, is 3×10−7 (8×10−8) at 11 MHz (120 kHz), corresponding to a frequency instability of 1×10−17 (3×10−18) in a cavity stabilized laser, estimated by a Pound-Drever-Hall frequency discrimination slope of 1×10−4 V/Hz [17,18].
Fig. 7. Reducing spatial distribution by optimizing the crystal orientation. (a), (b), and (c) Experimental spatial patterns of the symmetric and anti-symmetric components. The horizontal scale is normalized by the beam spot size w at the location of the measurement. (d) RF signals around the modulation frequencies. (e) The instabilities of RAM and the converted laser frequency instabilities using a PDH discrimination coefficient of 1 × 10−4 V/Hz. RBW, resolution bandwidth.
6. Conclusion
In summary, the mechanism of the spatially inhomogeneous RAM generated by bulk EO crystals is investigated. At modulation frequency of 11 MHz, the measured spatial pattern can be decomposed into two distinct components, and only one component is observed at 120 kHz. In addition to a spatial periodicity that agrees with the ultrasonic wavelength, the amplitudes and phases of the two components show regular variations with the relative orientation between the crystal and the laser beam. These observations can be explained by sound-generated diffraction of light into various sidebands. These sidebands form unique diffraction patterns that no longer perfectly overlap with the carrier, resulting in a nonuniform distribution of RAM in a transverse plane of the modulated light. By fine adjustment of the orientation of the EO crystal, the spatial inhomogeneity can be reduced, resulting in improved stability of RAM, which is 8×10−8 (1000 s) at 120-kHz modulation frequency and showing encouraging prospect on high-purity phase modulation for a variety of precision experiments based on frequency-modulation technique. Moreover, the rich information revealed by these spatial patterns provide a sensitive probe into the acoustic field inside the crystal.
Funding
National Key Research and Development Program of China (2020YFC2200300, 2021YFC2201800); National Natural Science Foundation of China (11327407, 11235004); The Strategic Priority Research Program of the Chinese Academy of Sciences (XDB21010300, XDB23030202).
Acknowledgments
Insightful suggestions from John L. Hall (JILA, NIST and University of Colorado) are gratefully acknowledged. One of authors (L. Chen) benefits from a stimulating discussion with Haifeng Jiang at National Time Service Center, CAS. Qunfeng Chen at APM provided one of the EOMs used in the experiment.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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