Method for simultaneously calibrating peak retardation and static retardation of a photoelastic modulator

Wei Quan; Qinghua Wang; Yueyang Zhai; Liwei Jiang; Lihong Duan; Ye Wu

doi:10.1364/AO.56.004491

1. INTRODUCTION

In recent years, with the rapid development of quantum manipulation techniques, the high-precision instruments using atoms as sensitive media continue to emerge, such as the atomic clock, the atomic magnetometer, and the nuclear magnetic resonance analyzer. The atomic sensors based on the spin exchange relaxation free (SERF) theory [1–5] have ultrahigh precision and sensitivity, and the measurement results of magnetic field and inertia are usually obtained by measuring the small optical rotation angle of a linearly polarized light [6,7]. The photoelastic modulator (PEM) has some unique optical characteristics, such as a large acceptance angle, high modulation efficiency and purity, good modulation stability, and low noise in the low-frequency range [8–10]. So, the atomic sensors, such as the atomic magnetometer, are developed with a PEM to modulate the phase of probe laser beam [7].

A PEM based on the photoelastic effect can be used to generate sinusoidal phase retardation of an incident light beam. The peak retardation is the main parameter of a PEM, and the accuracy of the retardation can directly affect the performance of the atomic sensors. The peak retardation may change due to the external environment during the modulation process, resulting in a slight deviation between the actual values and the set values [11]. However, the PEM in the atomic sensors is required to work with a modulation amplitude of less than 1 rad, and a slight peak retardation deviation of the PEM may lead to a large deviation of the photoelastic modulation detection results. Therefore, it is essential to precisely calibrate the peak retardation of the PEM in practical use. In addition, the static retardation [12] of the PEM is one of the fundamental frequency component deviation sources of the photoelastic modulation detection method [13]. Thus, the static retardation of the PEM also needs to be calibrated to correct the atomic spin precession detection results. In summary, both accurately calibrating the peak retardation and the static retardation are essential to ensure the optimum performance of the PEM.

The main calibration methods of the PEM include oscilloscope waveform method [14], Bessel function zero method [15], single-photon counting method [16], and multiple harmonic intensity ratio method [11,12]. The oscilloscope waveform method has large calibration errors because it depends on the subjective judgment of the experimenter. The zero-order Bessel function zero method can be greatly influenced by the intensity fluctuations of the incident light. Both the oscilloscope waveform method and the Bessel function zero method are only applicable to the calibration with a fixed large peak retardation, and they cannot calibrate the peak retardation of the PEM when the peak retardation is less than 1 rad. Moreover, the two calibration methods do not consider the existence of the static retardation of the PEM. Although the single-photon counting method and the multiple harmonic intensity ratio method can calibrate the PEM with any peak retardation, the single-photon counting method is complicated and also does not take the static retardation into account. The multiple harmonic intensity ratio method in [12] mentioned calibrating the static retardation, but relevant experiments are still missing.

An accurate calibration method based on normalized fundamental frequency component for PEM is proposed in this paper, which can measure the peak retardation and static retardation at the same time. The calibration method has a high sensitivity, and its accuracy is not affected by the fluctuations of incident light intensity.

2. CALIBRATION PRINCIPLE

The intensity function of the modulation signal can be expanded with the first-kind Bessel function. The first-kind Bessel function curve (Fig. 1) shows that the slope of the first-order Bessel function curve is the largest compared to other orders of Bessel function, when the modulation peak retardation of the PEM is varying from 0 to 1 rad. That means the first-order Bessel function has the highest sensitivity in this modulation range. Thus, the peak retardation of a PEM working with a modulation amplitude of less than 1 rad can be calculated using the normalized fundamental frequency component, which contains the first-order Bessel function. In Fig. 1, $W_{1}$ and $W_{2}$ are the deformations of experimental results of the normalized fundamental frequency components $W$ and $W^{'}$ , respectively (the details can be seen in Section 3).

Fig. 1. Plots of Bessel functions versus the peak retardation.

Download Full Size | PDF

The schematic (Fig. 2) includes a distributed feedback (DFB) laser, a polarizer, a quarter-wave plate, a PEM, a half-wave plate, a polarization beam splitter (PBS), two photodetectors (PDs), a signal conditioning unit, a lock-in amplifier, a PEM controller, and a data acquisition (DAQ) system. The light propagates along the $z$ axis, and the modulation axis of the PEM is parallel to the $x$ axis (the 0° position is defined by the modulation axis of the PEM). The azimuth of the transmission axis of the polarizer is 45°; the azimuths of fast axis of the quarter-wave plate and the half-wave plate are 0° and 22.5°, respectively. The collimated laser light becomes circularly polarized light after passing through the polarizer and the quarter-wave plate. The circularly polarized light is modulated by the PEM with the modulation frequency $ω$ ; then the modulated light vector is rotated 45° by the half-wave plate, so that the modulated light can be split into two subbeams whose polarization direction is perpendicular to each other after passing through the PBS. The optical signals of the two subbeams are converted into electrical signals by two PDs, respectively. The electrical signals are sent to the signal conditioning unit. The direct current components output from the signal conditioning unit are directly sent to the DAQ, while the alternating current components signals output from the signal conditioning unit are sent to the lock-in amplifier first to get the fundamental frequency components, and then the fundamental frequency components are sent to the DAQ. The DAQ processes the direct current components and the fundamental frequency components of the two signals to get the normalized fundamental frequency component. Two normalized fundamental frequency components containing the first-order Bessel function are obtained before and after taking away the quarter-wave plate from the optical setup. Finally, the peak retardation and static retardation of the PEM can be calculated at the same time.

Fig. 2. Schematic for the calibration of the PEM.

Download Full Size | PDF

The intensities of the two subbeams split by the PBS can be obtained from the Jones matrices. In the schematic, the laser beam passing through the polarizer becomes linearly polarized light. Its Jones vector $G_{0}$ can be written as

G_{0} = E_{0} \frac{1}{\sqrt{2}} [\begin{matrix} 1 \\ 1 \end{matrix}],

where

E_{0}

is the electric field amplitude of the incident light, and the incident light intensity can be expressed as

I_{0} = E_{0}^{2}

.

The Jones matrix of the quarter-wave plate with its fast axis oriented at 0° is

G_{λ / 4} = [\begin{matrix} 1 & 0 \\ 0 & i \end{matrix}] .

The Jones matrix of the PEM with its modulation axis oriented at 0° is

G_{PEM} = [\begin{matrix} \cos \frac{δ (t)}{2} - i \sin \frac{δ (t)}{2} & 0 \\ 0 & \cos \frac{δ (t)}{2} + i \sin \frac{δ (t)}{2} \end{matrix}],

where

δ (t)

is the retardation of the PEM, and it can be expressed as

δ (t) = δ_{0} \sin (ω t) + δ_{s},

where

δ_{0}

and

δ_{s}

are the peak retardation and the static retardation, respectively.

ω

is the modulation angular frequency.

The Jones matrix of the half-wave plate with its fast axis oriented at 22.5° is

G_{λ / 2} = (- \frac{i}{\sqrt{2}}) [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}] .

The PBS can be seen as a pair of analyzers whose azimuths of the transmission axes are perpendicular to each other. So, the Jones matrices of the two analyzers with one transmission axis in parallel with the $y$ axis and the other transmission axis in parallel with the $x$ axis are

G_{PBS 1} = [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}],

G_{PBS 2} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] .

Jones vectors $G_{1}$ and $G_{2}$ of the beams on PD1 and PD2 can be given, respectively, by

G_{1} = G_{PBS 1} G_{λ / 2} G_{PEM} G_{λ / 4} G_{0},

G_{2} = G_{PBS 2} G_{λ / 2} G_{PEM} G_{λ / 4} G_{0} .

The intensities $I_{1}$ and $I_{2}$ on PD1 and PD2 can be obtained by

I_{1} = G_{1}^{*} G_{1} = \frac{I_{0}}{2} [1 + \sin (δ_{0} \sin ω t + δ_{s})] = \frac{I_{0}}{2} [1 + \sin (δ_{0} \sin ω t) \cos δ_{s} + \cos (δ_{0} \sin ω t) \sin δ_{s}],

I_{2} = G_{2}^{*} G_{2} = \frac{I_{0}}{2} [1 - \sin (δ_{0} \sin ω t + δ_{s})] = \frac{I_{0}}{2} [1 - \sin (δ_{0} \sin ω t) \cos δ_{s} - \cos (δ_{0} \sin ω t) \sin δ_{s}],

where the function of

\sin (δ_{0} \sin ω t)

and

\cos (δ_{0} \sin ω t)

can be expanded with the Bessel function of the first kind [17], as the following:

\sin (δ_{0} \sin ω t) = 2 \sum_{k = 1}^{\infty} J_{2 k - 1} (δ_{0}) \sin [(2 k - 1) ω t],

\cos (δ_{0} \sin ω t) = J_{0} (δ_{0}) + 2 \sum_{k = 1}^{\infty} J_{2 k} (δ_{0}) \cos [(2 k) ω t],

where

J_{0}

is the zeroth order of the Bessel function,

J_{2 k - 1}

and

J_{2 k}

are the

(2 k - 1)

th order and

(2 k)

th order of the Bessel function, respectively. Substituting Eqs. (4), (12), and (13) into Eqs. (10) and (11), we obtain

I_{1} = \frac{I_{0}}{2} [1 + J_{0} (δ_{0}) \sin δ_{s} + 2 J_{1} (δ_{0}) \cos δ_{s} \sin (ω t) + 2 J_{2} (δ_{0}) \sin δ_{s} \cos (2 ω t) + \dots],

I_{2} = \frac{I_{0}}{2} [1 - J_{0} (δ_{0}) \sin δ_{s} - 2 J_{1} (δ_{0}) \cos δ_{s} \sin (ω t) - 2 J_{2} (δ_{0}) \sin δ_{s} \cos (2 ω t) - \dots] .

Equations (14) and (15) show that the detection signals contain the direct current components, the fundamental frequency components, the second-harmonic components, and the higher-order harmonic components. The direct current components $V_{dc 1}$ , $V_{dc 2}$ of intensities $I_{1}$ , $I_{2}$ can be obtained, respectively, by the signal conditioning unit

V_{dc 1} = η \frac{I_{0}}{2} [1 + J_{0} (δ_{0}) \sin δ_{s}],

V_{dc 2} = η \frac{I_{0}}{2} [1 - J_{0} (δ_{0}) \sin δ_{s}],

where

η

is the photoelectric conversion coefficient; the gains of the direct current components and alternating current components are 1. At the same time, the fundamental frequency components

V_{1 f 1}

,

V_{1 f 2}

of intensities

I_{1}

,

I_{2}

can be obtained, respectively, by the lock-in amplifier

V_{1 f 1} = η I_{0} J_{1} (δ_{0}) \cos δ_{s},

V_{1 f 2} = η I_{0} J_{1} (δ_{0}) \cos δ_{s} .

According to Eqs. (16) to (19), the normalized fundamental frequency component can be written as

W = \frac{V_{1 f 1} + V_{1 f 2}}{V_{dc 1} + V_{dc 2}} = 2 J_{1} (δ_{0}) \cos δ_{s} .

Then the quarter-wave plate is taken away from the optical setup shown in Fig. 2, and the placements of other instruments remain unchanged. Making use of the same method, we can get the Jones vectors $G_{1}^{'}$ and $G_{2}^{'}$ of the beams on PD1 and PD2:

G_{1}^{'} = G_{PBS 1} G_{λ / 2} G_{PEM} G_{0},

G_{2}^{'} = G_{PBS 2} G_{λ / 2} G_{PEM} G_{0} .

The intensities $I_{1}^{'}$ and $I_{2}^{'}$ on PD1 and PD2 can be obtained by

I_{1}^{'} = G_{1}^{' *} G_{1}^{'} = \frac{I_{0}}{2} [1 - \cos δ (t)] = \frac{I_{0}}{2} [1 - \cos (δ_{0} \sin ω t + δ_{s})],

I_{2}^{'} = G_{2}^{' *} G_{2}^{'} = \frac{I_{0}}{2} [1 + \cos δ (t)] = \frac{I_{0}}{2} [1 + \cos (δ_{0} \sin ω t + δ_{s})] .

With the Bessel function, Eqs. (23) and (24) can be rewritten as

I_{1}^{'} = \frac{I_{0}}{2} [1 - J_{0} (δ_{0}) \cos δ_{s} + 2 J_{1} (δ_{0}) \sin δ_{s} \sin (ω t) - 2 J_{2} (δ_{0}) \cos δ_{s} \cos (2 ω t) + \dots],

I_{2}^{'} = \frac{I_{0}}{2} [1 + J_{0} (δ_{0}) \cos δ_{s} - 2 J_{1} (δ_{0}) \sin δ_{s} \sin (ω t) + 2 J_{2} (δ_{0}) \cos δ_{s} \cos (2 ω t) + \dots] .

The direct current components and fundamental frequency components of intensities $I_{1}^{'}$ and $I_{2}^{'}$ are

V_{dc 1}^{'} = η \frac{I_{0}}{2} [1 - J_{0} (δ_{0}) \cos δ_{s}],

V_{dc 2}^{'} = η \frac{I_{0}}{2} [1 + J_{0} (δ_{0}) \cos δ_{s}],

V_{1 f 1}^{'} = η I_{0} J_{1} (δ_{0}) \sin δ_{s},

V_{1 f 2}^{'} = η I_{0} J_{1} (δ_{0}) \sin δ_{s} .

According to Eqs. (27) to (30), the normalized fundamental frequency component can be written as

W^{'} = \frac{V_{1 f 1}^{'} + V_{1 f 2}^{'}}{V_{dc 1}^{'} + V_{dc 2}^{'}} = 2 J_{1} (δ_{0}) \sin δ_{s} .

Therefore, we can get the values of the first Bessel function and the $δ_{s}$ simultaneously according to Eqs. (20) and (31)

J_{1} (δ_{0}) = \frac{\sqrt{W^{2} + W^{' 2}}}{2},

δ_{s} = \tan^{- 1} \frac{W^{'}}{W} .

In order to resolve the peak retardation directly from Eq. (32), we can use the function fitting tool of MATLAB to fit the approximate function relationship between $J_{1} (δ_{0})$ and $δ_{0}$ . Equation (34) is the best fitting result, whose root mean square error is $8.382 \times 10^{- 6}$ and approximate R-square is 1:

J_{1} (δ_{0}) = 0.5787 \sin (0.8638 δ_{0} + 0.000026) .

With Eqs. (32) and (34), we can get

δ_{0} = [\sin^{- 1} (\sqrt{W^{2} + W^{' 2}} / 1.1574) - 0.000026] / 0.8638 .

Thus, the peak retardation and static retardation can be acquired simultaneously with Eqs. (35) and (33). The calculation of $δ_{0}$ and $δ_{s}$ are independent of each other. Besides, this method can effectively eliminate the measurement error caused by the instability of initial intensity.

3. EXPERIMENTS AND RESULTS ANALYSIS

The experimental setup is shown in Fig. 2. The light source was a DFB laser (model DL DFB, Toptica) with a collimator lens, and its center wavelength was 795 nm. The polarizer was a Glan–Taylor prism with an extinction ratio greater than $100, 000 ∶ 1$ . The wave plates were a zero-order quarter-wave plate and a zero-order half-wave plate. The PEM to be calibrated was a Hinds Instruments model II/FS42 PEM with a modulation frequency of 42 kHz, and its peak retardation was controlled by a Hinds Instruments model PEM-100 controller. The signal conditioning unit was a Hinds Instruments model SCU-100 with its gains of direct current component and alternating current component both set to 1. An HF2LI type lock-in amplifier from Zurich Instruments was used to get the fundamental frequency components, and its reference signal was supplied by the PEM-100 controller. The DAQ unit included a National Instruments model Pxle-1082 DAQ card and a LabVIEW data processing program.

In order to simultaneously measure the peak retardation and static retardation of the PEM, the experiment was first carried out according to the schematic. The peak retardation of the PEM was adjusted by the PEM-100 controller. When the peak retardation of the PEM is gradually increased from 0.06 rad to 1 rad with an interval of 0.02 rad, the normalized fundamental frequency components $W$ in Eq. (20) of each setting were obtained. Then the quarter-wave plate was taken away, and the placements of other instruments remained unchanged. The above experiment processes were repeated to obtain the normalized fundamental frequency components $W^{'}$ in Eq. (31). Finally, the peak retardation and static retardation of the PEM were calculated by using Eqs. (35) and (33). In order to facilitate the analysis, we processed the experimental results $W$ and $W^{'}$ to get the corresponding values $W_{1}$ and $W_{2}$ , respectively. The values of $W_{1}$ and $W_{2}$ , which both only contain the first-order Bessel function $J_{1} (δ_{0})$ , were shown in Fig. 1. It indicates that the experimental results and the theoretical values conform well with Fig. 1.

The peak retardation and static retardation of the PEM measured by experiments were plotted against the PEM-100 controller set values, and the results are shown in Figs. 3 and 4, respectively.

Fig. 3. Calibration results of peak retardation.

Download Full Size | PDF

Fig. 4. Calibration results of static retardation.

Download Full Size | PDF

The measured peak retardation is linearly related to those set by PEM-100, as shown in Fig. 3, and their fitted curve is

δ_{0 M} = 0.9983 δ_{0 set} + 0.0339,

where

δ_{0 M}

is the measured peak retardation and

δ_{0 set}

is the set value.

The goodness of the fitted result is high, since the root mean square error of Eq. (36) is 0.0004 and the R-square is 0.9999. The slope of $δ_{0 M} / δ_{0 set}$ is close to 1. The average deviation between measured values and set values is 0.0339 rad. There are two reasons for this deviation: One is that the peak retardation can be influenced by modulation environment; The other one is that there is a static retardation in the PEM. The deviation between measured values and set values can be ignored when the PEM is working in a large modulation amplitude state. However, when the modulation amplitude is less than 1 rad, the minimum error of the photoelastic modulation detection results induced by the deviation is about 3.22% [13]. The error cannot be ignored in our conditions of magnetic field and inertia measurements. So, the measured peak retardation results are useful to avoid the error in our practical applications.

One can consider that the static retardation with an average value of 0.0161 rad (Fig. 4) is constant during the experiments. The standard deviation between the measured values and the average value is 0.0003 rad, while the maximum deviation is 0.0019 rad. The static retardation of the PEM does not change with the peak retardation set by PEM-100, because its value only depends on the structure of the PEM itself. According to our experimental results, the ratio of the static retardation influence is 47.46%, and the modulation environmental influence is 52.54%, which are two deviation sources of measured peak retardations and the set values. As mentioned in [13], the static retardation will induce the fundamental frequency component deviation of the photoelastic modulation detection results. The measured small optical rotation angle (less than 1 rad) error induced by the static retardation is 0.0081 rad [13]. Although the static retardation can be ignored in other practical applications, it should not be ignored in such a situation, where the modulation amplitude of the PEM is less than 1 rad. So, the static retardation measured by experiments is suited for correcting the fundamental frequency component deviation of the photoelastic modulation detection results.

4. CONCLUSION

In this paper, we proposed a method for simultaneously calibrating the peak retardation and static retardation of a PEM. Two normalized fundamental frequency components obtained by optimizing the polarization modulation system are used to calculate the peak retardation and static retardation. The method specially suits the calibration of the PEM, whose peak retardation is less than 1 rad, since the normalized fundamental frequency component containing the first-order Bessel function has high sensitivity. And the calibration result is immune to the fluctuations of the initial intensity. The average deviation between measured peak retardations and corresponding set values is 0.0339 rad. The average static retardation measured in our experiments is 0.0161 rad, and the standard deviation between the measured static retardations and the average value is 0.0003 rad. The results prove this method conforms well with the theory. This calibration method can be used to correct the drift of the peak retardation, which is caused by the influence of modulation environment. Moreover, the static retardation measured by this method can be used to correct the fundamental frequency component deviation of the photoelastic modulation detection results, resulting in improving the detection accuracy of magnetic field and inertia measurement results.

Funding

National High Technology Research and Development Program of China (863 Program); National Natural Science Foundation of China (NSFC) (61227902, 61374210, 61421063); National Key R&D Program of China (2016YFB0501600).

Acknowledgment

We thank Associate Professor Yueyang Zhai for useful discussions.

REFERENCES

1. J. C. Fang, J. Qin, S. A. Wan, and R. J. Li, “Atomic spin gyroscope based on 129Xe-Cs comagnetometer,” Chin. Sci. Bull. 58, 1512–1515 (2013). [CrossRef]

2. J. C. Fang, T. Wang, W. Quan, H. Yuan, H. Zhang, Y. Li, and S. Zou, “In situ magnetic compensation for potassium spin-exchange relaxation-free magnetometer considering probe beam pumping effect,” Rev. Sci. Instrum. 85, 063108 (2014). [CrossRef]

3. J. C. Fang, T. Wang, H. Zhang, Y. Li, and S. Zou, “Optimizations of spin-exchange relaxation-free magnetometer based on potassium and rubidium hybrid optical pumping,” Rev. Sci. Instrum. 85, 123104 (2014). [CrossRef]

4. J. C. Fang, T. Wang, H. Zhang, Y. Li, and H. W. Cai, “In situ measurement of magnetic field gradient in a magnetic shield by spin-exchange relaxation-free magnetometer,” Chin. Phys. B 24, 060702 (2015). [CrossRef]

5. W. Quan, Y. Li, and B. Q. Liu, “Simultaneous measurement of magnetic field and inertia based on hybrid optical pumping,” Europhys. Lett. 110, 60002–60006 (2015). [CrossRef]

6. J. M. Brown, “A new limit on Lorentz-and CPT-violating neutron spin interactions using a K-3He comagnetometer,” Ph.D. dissertation (Princeton University, 2011).

7. L. H. Duan, J. C. Fang, R. J. Li, L. W. Jiang, M. Ding, and W. Wang, “Light intensity stabilization based on the second harmonic of the photoelastic modulator detection in the atomic magnetometer,” Opt. Express 23, 32481–32489 (2015). [CrossRef]

8. A. J. Zeng, F. Y. Li, L. L. Zhu, and H. J. Huang, “Simultaneous measurement of retardance and fast axis angle of a quarter-wave plate using one photoelastic modulator,” Appl. Opt. 50, 4347–4352 (2011). [CrossRef]

9. B. Wang and J. List, “Basic optical properties of the photoelastic modulator: Part I. Useful aperture and acceptance angle,” Proc. SPIE 5888, 58881I (2005). [CrossRef]

10. B. Wang, E. Hinds, and E. Krivoy, “Basic optical properties of the photoelastic modulator part II: residual birefringence in the optical element,” Proc. SPIE 7461, 746110 (2009). [CrossRef]

11. M. W. Wang, Y. F. Chao, K. C. Leou, and F. H. Tsai, “Calibrations of phase modulation amplitude of photoelastic modulator,” Jpn. J. Appl. Phys. 43, 827–832 (2004). [CrossRef]

12. M. W. Wang, F. H. Tsai, and Y. F. Chao, “In situ calibration technique for photoelastic modulator in ellipsometry,” Thin Solid Films 455, 78–83 (2004). [CrossRef]

13. S. A. Wan, “Study on experiment of error analysis and suppression methods for SERF atomic spin gyroscope,” Ph.D. dissertation (Beihang University, 2014).

14. T. C. Oakberg, J. Trunk, and J. C. Sutherland, “Calibration of photoelastic modulators in the vacuum UV,” Proc. SPIE 4133, 101–111 (2000). [CrossRef]

15. K. Li, “High sensitive measurement of optical rotation based on photo-elastic modulation,” Acta Phys. Sinica 64, 5837–5843 (2015). [CrossRef]

16. R. A. Cline, W. B. Westerveld, and J. S. Risley, “A new method for measuring the retardation of a photoelastic modulator using single photon counting techniques,” Rev. Sci. Instrum. 64, 1169–1174 (1993). [CrossRef]

17. Y. Zhang, F. J. Song, H. Y. Li, and X. G. Yang, “Precise measurement of optical phase retardation of a wave plate using modulated-polarized light,” Appl. Opt. 49, 5837–5843 (2010). [CrossRef]

Method for simultaneously calibrating peak retardation and static retardation of a photoelastic modulator

Abstract

1. INTRODUCTION

2. CALIBRATION PRINCIPLE

3. EXPERIMENTS AND RESULTS ANALYSIS

4. CONCLUSION

Funding

Acknowledgment

REFERENCES

Cited By

Figures (4)

Equations (36)

Applied Optics