Photoelastic modulator modulation amplitude frequency-response-free calibration device and method

Abstract

The invention provides a photoelastic modulator modulation amplitude frequency-response-free calibration device and method, and relates to the technical field of polarization optical detection. As a phase modulation device, the phase modulation amplitude of the photoelastic modulator is influenced by modulation voltage and dispersion, so that accurate calibration must be carried out before the photoelastic modulator is applied to precise optical measurement. A calibration device is formed by a light source, a polarizer, a wave plate, a photoelastic modulator, a photoelastic modulator controller, an analyzer, a detector and a digital lock-in amplifier, first and second frequency multiplication signals are synchronously scanned, zero value points of first and second frequency multiplication components are accurately positioned, phase modulation amplitudes are calibrated by respectively utilizing inverse solution first and second order Bessel functions, and the phase modulation amplitude is calculated. And high-resolution phase modulation amplitude calibration without the influence of the frequency response characteristic in a full calibration interval is obtained in a segmented recombination form. The device and the method have no special numerical value and precision requirements on the azimuth angle of the device and the phase retardation of the wave plate, and have robustness.

Description

Technical Field

[0001] The invention relates to a device and method for frequency-free calibration of modulation amplitude of a photoelastic modulator, belonging to the technical field of polarization optical detection. Background Art

[0002] The photoelastic modulator is a phase modulation device. Its basic working principle is to generate stress standing waves by applying periodic stress inside the originally optically isotropic optical material (such as fused quartz, single crystal silicon, calcium fluoride, etc.). Due to the photoelastic effect, high-frequency refractive index difference fluctuations are generated in two orthogonal directions, thereby phase modulating the polarized light passing through the photoelastic modulator. It is a relatively common phase modulation device with the characteristics of high modulation frequency, pure modulation frequency, large aperture, low power consumption, and wide applicable band. In the actual phase modulation use of the photoelastic modulator, we need to know the actual phase delay amplitude A generated by the photoelastic modulator at a specific wavelength. m The curve of the relationship between the nominal amplitude and the nominal amplitude on the controller panel is also called the amplitude calibration curve of the photoelastic modulator. Usually, commercial photoelastic modulators are calibrated at the factory, and their calibration accuracy is better than 5%. This accuracy basically meets most phase modulation usage scenarios, but for situations that require high-precision phase modulation amplitude (such as ellipsometers, Mueller matrix ellipsometers, etc.), the photoelastic modulator needs to be calibrated with higher accuracy.

[0003] The common methods for calibrating photoelastic modulators at present mainly include the half-wave flat-top method and the Bessel function zero-point method. Among them, the half-wave flat-top method uses the flat-top structure of the output light intensity time domain waveform when the phase delay amplitude is half a wavelength. This method can only calibrate a specific phase delay amplitude (πrad). The Bessel function zero-point method uses the point where the Bessel function takes a zero value to calibrate the phase delay amplitude of 2.405rad, 3.832rad, 5.136rad, etc., and cannot calibrate the phase delay amplitude within a continuous interval. In recent years, compound Bessel function calibration methods have been invented, such as the calibration method using the ratio of first- and second-order Bessel functions, which realizes the calibration of the phase modulation amplitude of the photoelastic modulator within a certain range. The reported calibration accuracy is better than 1%, but these compound Bessel function calibration methods cannot eliminate the influence of the frequency response of the detection system on the calibration accuracy. The detection system consists of detectors, signal transmission lines and phase-locked amplifiers. Since the detection system generally has frequency response characteristics, its response to different frequency-multiplied signals of the same amplitude is different. A sophisticated calibration system must try to eliminate the influence of this frequency response characteristic.

[0004] To solve the above problems, the present invention proposes a device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response, which is completely unaffected by the frequency response characteristics of the detection system and has a large calibration range. The calibration accuracy of the 0.1-0.9λ calibration range at a wavelength of 515nm is better than 0.25%, and the calibration accuracy of the 0.2-0.9λ calibration range at a wavelength of 653nm is better than 0.12%. At the same time, the calibration device and method also have the characteristics of no requirements for the orientation accuracy of devices such as polarizers, analyzers, and wave plates, and strong robustness. Summary of the invention

[0005] Problems to be solved and objectives of the present invention: To provide a device and method for frequency-response-free calibration of the modulation amplitude of a photoelastic modulator, so as to overcome the shortcoming of the existing composite Bessel function calibration method that the calibration accuracy is affected by the frequency response characteristics of the detector, and to achieve precise calibration of the photoelastic modulator with outstanding robustness and without being affected by the frequency response characteristics within a wider phase modulation amplitude range.

[0006] The solution of the present invention is as follows: like Figure 1 As shown in the figure, the light beam emitted by the light source passes through the polarizer, wave plate, photoelastic modulator, analyzer, and detector after being collimated. The detector converts the light signal into an electrical signal and outputs it to the digital phase-locked amplifier. The phase-locked amplifier uses the modulation frequency of the photoelastic modulator as the reference frequency, extracts the first and second frequency components, and outputs them to the computer. With the direction of the external stress of the photoelastic modulator as the x-axis and the direction of light propagation as the z-axis, a coordinate system is established as follows: Figure 1 As shown in the figure, the polarizer transmission azimuth is P, the analyzer transmission azimuth is A, the wave plate fast axis azimuth is θ, the phase difference between the fast and slow axes is Δ, and the photoelastic modulator phase delay is δ, δ=A m sinωt+δ s , δ s is the static phase delay caused by the static residual stress of the photoelastic modulator, ω is the modulation frequency of the photoelastic modulator, A m is the phase delay amplitude, then the light intensity I incident on the detector can be expressed as (1) Where I0 is the intensity of linearly polarized light after passing through the polarizer. Perform Bessel series expansion on the light intensity I to obtain the amplitude V of the one-fold frequency component in the output signal of the detector. 1f and the amplitude of the double frequency component V 2f They are (2) (3) Among them, T1 and T2 correspond to the product of the frequency response coefficient and the amplification coefficient of the detector's first and second frequency respectively. From the frequency-doubled component amplitude formulas 2 and 3, it can be seen that the frequency-doubled component amplitude V 1f、V 2f Respectively with the first and second order Bessel function values ​​J1(A m )、J2(A m ), that is, different phase delay amplitudes A m1 , A m2 There is a relationship between the corresponding frequency-octave component amplitudes: (4) (5) according to Figure 2 The Bessel function curve shown in the figure changes when the modulation voltage amplitude V of the photoelastic modulator is increased. m The actual phase delay amplitude A produced by the photoelastic modulator m When it increases gradually, the corresponding first-order and second-order Bessel functions increase first, reach the extreme value, and then gradually decrease. m ) in A m =3.832rad(0.61λ)is zero, J2(A m ) in A m =5.136rad (0.82λ) is zero. Therefore, from formulas (2) and (3), we can know that the amplitude of the first and second frequency signals separated by the digital lock-in amplifier is V 1f 、V 2f It will also increase first and then decrease, and decrease to zero when the first-order and second-order Bessel functions are zero. Thanks to the fact that the first-order and second-order Bessel functions have large curve slopes near the zero point, when adjusting V m In the process of finding the zero point, a higher positioning accuracy can be obtained. Based on the precise positioning of the zero points of the first and second harmonic components, we can use formula (4) or (5) to achieve the A by inverse solution of the Bessel function. m This method is not affected by the frequency response characteristics of the detection system.

[0007] According to formulas (4) and (5), the numerical changes of A, P, θ, and Δ do not affect the calibration results. However, according to the amplitude formulas (2) and (3) of the double frequency components, the numerical values ​​of A, P, θ, and Δ affect the numerical values ​​of the double frequency components. Therefore, by adjusting their numerical values, the signal-to-noise ratio of the double frequency signal can be enhanced. The adjustment of Δ can be achieved by selecting different wave plates. Furthermore, a combination of wave plates can be used instead of a single wave plate, and a more convenient equivalent Δ numerical adjustment can be achieved by adjusting the azimuth angle between the wave plates.

[0008] Since the first-order Bessel function curve changes smoothly around 0.29λ, the resolution of the inverse solution of the first-order Bessel function is low; the second-order Bessel function changes smoothly around 0.49λ, and the resolution of the inverse solution of the second-order Bessel function is low. In order to obtain the best resolution of the inverse solution of the Bessel function in the entire calibration interval, we analyze the first-order derivatives of the first- and second-order Bessel functions, such as Figure 3 As shown, a combined calibration method of first- and second-order Bessel function segmented inverse solution is creatively proposed. In the ranges of 0-0.2λ and 0.37-0.69λ, the first-order Bessel function inverse solution is used, and in other ranges, the second-order Bessel function inverse solution is used. This achieves good Bessel function inverse solution resolution in the entire calibration range, thereby obtaining high-precision calibration in the entire photoelastic modulator calibration range. For an incident wavelength of 653nm, in the calibration range of 0.2-0.9λ, we achieved a calibration accuracy better than 0.12%.

[0009] It should be noted that the calibration device and method have no special numerical limit requirements for the polarizer azimuth P, the analyzer azimuth A, and the wave plate fast axis azimuth θ, and do not need to be limited to 0. o , ±45 o For standard angles, only the amplitude of the frequency component V 1f , the amplitude of the double frequency component V 2f Not zero, that is, A≠0 o , A≠±90 o , that is, the transmittance axis of the polarizer is not parallel or perpendicular to the stress axis of the photoelastic modulator. Therefore, the value range of the polarizer azimuth angle A is (-90 o ,0 o ) and (0 o , 90 o ), excluding 0 o , ±90 o The value range of the polarizer azimuth angle P is [-90 o ,90 o ], the wave plate fast axis azimuth angle θ ranges from [-90 o ,90 o ]; The calibration device and method have no special numerical limit requirements for the phase delay ∆ of the wave plate, and do not need to be limited to standard wave plates such as λ / 4 and λ / 2. They are not affected by dispersion and can calibrate the entire working band of the photoelastic modulator. It can be proved by formulas (2), (3), (4), and (5) that the static phase delay δ of the photoelastic modulator is s It does not affect the calibration of the phase modulation amplitude of the photoelastic modulator of the present invention. Therefore, in addition to being unaffected by the frequency response characteristics of the detection system and having high resolution in the full calibration interval, the calibration device and method also have the characteristics of simple optical path adjustment, strong robustness, large calibration interval, and being unaffected by residual birefringence. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 Schematic diagram of a device for calibrating the phase modulation amplitude of a photoelastic modulator; Figure 2 1. Graph of the second-order Bessel function; Figure 3 1. The first derivative curve of the second-order Bessel function; Figure 4 The first and second harmonic frequency signal test curves when the incident wavelength is 515nm; Figure 5 Phase modulation amplitude calibration curve based on inverse first-order Bessel function at 515nm; Figure 6 Phase modulation amplitude calibration curve based on inverse solution of second-order Bessel function at 515nm; Figure 7 Phase modulation amplitude calibration curve of the combination at 515nm Figure 8 The test curve of the first and second harmonic frequency signals when the incident wavelength is 653nm; Fig. 9 Phase modulation amplitude calibration curve based on the inverse solution of the first-order Bessel function at 653nm; Fig.10 Phase modulation amplitude calibration curve based on inverse solution of second-order Bessel function at 653nm; Fig.11 Combined phase modulation amplitude calibration curve at 653nm.

[0011] Below we further describe the present invention in conjunction with specific embodiments: Example

[0012] like Figure 1 As shown, the 515nm wavelength laser of the semiconductor is incident on the photodetector after passing through the polarizer, wave plate, photoelastic modulator, and analyzer. The stress direction of the photoelastic modulator is taken as the x-axis, the azimuth angle of the polarizer transmission axis is randomly set, and the wave plate is randomly selected. The phase delay Δ corresponding to the 515nm wavelength is about 2.7rad, and the fast axis azimuth θ of the wave plate is roughly set to about 5 o The azimuth angle A of the polarizer transmission axis is roughly set to 50 o .

[0013] To increase the V divided by the digital lock-in amplifier 1f 、V 2f With the goal of increasing signal strength and thus the signal-to-noise ratio, the values ​​of angles A, P, and θ are adjusted respectively.

[0014] Through the controller of the photoelastic modulator, the nominal phase delay amplitude of the photoelastic modulator is gradually increased from 0.1λ to 0.9λ in steps of 0.02λ, and the V obtained by the digital lock-in amplifier is recorded at the same time. 1f 、V 2f Signal, using the phase value of the first and second frequency signals to normalize V 1f 、V 2f To the corresponding order Bessel function form, we get Figure 4 The curves of the first and second harmonic frequency signals changing with the nominal phase delay are shown. Corresponding to V 2f The zero point of the actual phase delay is A m It should be 0.82λ. Using the standard double frequency signal V corresponding to this point 1f According to formula (4), A can be obtained by inversely solving the first-order Bessel function. m Perform calibration, the calibration curve is as follows Figure 5 As shown. Figure 4 For V 1f The zero point of the actual phase delay is A m It should be 0.61λ, and the double frequency signal V corresponding to this point is used. 2f According to formula (5), A can be obtained by inversely solving the second-order Bessel function. m Perform calibration, the calibration curve is as follows Figure 6 shown.

[0015] Pick Figure 5 The calibration curve segments in the 0.1-0.2λ (0.63-1.25rad) and 0.37-0.7λ (2.3-4.35rad) intervals are Figure 6 The calibration curve segments in the range of 0.2-0.37λ (1.25-2.3rad) and 0.7-0.9λ (4.35-5.65rad) are combined to form the final calibration curve. Figure 7 As shown in the figure, the calibration curve is a straight line, indicating that the actual phase delay amplitude is linearly related to the nominal phase delay amplitude of the controller, and the nominal phase delay of the controller is linearly related to the amplitude of the photoelastic modulator control voltage. Therefore, the calibration result shows that the actual phase delay amplitude is linearly proportional to the amplitude of the photoelastic modulator control voltage, which is consistent with physical reality. This combined calibration curve eliminates Figure 5 , Figure 6 The problem of low resolution near the extreme points of the Bessel function was solved. Multiple measurements showed that the maximum uncertainty in the entire calibration range of 0.1-0.9λ was only 0.25%.

[0016] Example 2 like Figure 1As shown, the 653.5nm wavelength laser of the semiconductor is incident on the photodetector after passing through the polarizer, wave plate, photoelastic modulator, and analyzer. The stress direction of the photoelastic modulator is the x-axis, the azimuth angle of the polarizer transmission axis is randomly set, the wave plate is a combination of two randomly selected different wave plates, and the azimuth angle A of the analyzer transmission axis is roughly set to 40 o .

[0017] To increase the V divided by the digital lock-in amplifier 1f 、V 2f With the goal of improving signal strength and thus signal-to-noise ratio, the analyzer azimuth angle A, the polarizer azimuth angle P, and the angle between the fast axes of the dual wave plates are adjusted respectively.

[0018] After the adjustment is completed, the nominal phase delay amplitude of the photoelastic modulator is gradually increased from 0.2λ to 0.9λ in steps of 0.02λ through the controller of the photoelastic modulator, and the V obtained by the digital lock-in amplifier is recorded at the same time. 1f 、V 2f Signal, using the phase value of the first and second frequency signals to normalize V 1f 、V 2f To the corresponding order Bessel function form, we get Figure 8 The curves of the first and second harmonic frequency signals changing with the nominal phase delay are shown. 2f The zero point of the actual phase delay is A m It should be 0.82λ (5.136rad), using the standard double frequency signal V corresponding to this point 1f According to formula (4), A can be obtained by inversely solving the first-order Bessel function. m Perform calibration, the calibration curve is as follows Fig. 9 As shown. Figure 8 For V 1f The zero point of the actual phase delay is A m It should be 0.61λ (3.832rad), and the double frequency signal V corresponding to this point is used. 2f According to formula (5), A can be obtained by inversely solving the second-order Bessel function. m Perform calibration, the calibration curve is as follows Fig.10 shown.

[0019] Pick Fig. 9 The calibration curve segment in the 0.37-0.7λ (2.3-4.35rad) interval is Fig.10 The calibration curve segments in the range of 0.2-0.37λ (1.25-2.3rad) and 0.7-0.9λ (4.35-5.65rad) are combined to form the final calibration curve. Fig.11 The results of multiple measurements show that the phase delay amplitude A in the entire calibration range of 0.2-0.9λm The maximum uncertainty is only 0.12%.

Claims

1. A device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response, the device comprising a light source, a polarizer, a photoelastic modulator, a photoelastic modulator controller, a wave plate, an analyzer, a detector, a digital phase-locked amplifier, and a computer. The light emitted by the light source is collimated and then incident on the subsequent polarizer, wave plate, photoelastic modulator, analyzer, and detector. The front and rear surfaces of the polarizer and analyzer are perpendicular to the incident light. The detector converts the light intensity signal into a voltage signal. The signal output end of the detector is connected to the signal input end of the digital phase-locked amplifier. The modulation frequency signal output end of the photoelastic modulator controller is connected to the reference signal input end of the digital phase-locked amplifier. The photoelastic modulator and the wave plate are placed between the polarizer and the analyzer. The azimuth angles of the polarizer, the photoelastic modulator, the wave plate, and the analyzer are independently adjustable. The light source can use a laser diode and a wide spectrum light source with a monochromator. The invention is characterized in that The calibration device described above is used and the calibration method comprises the following calibration steps: a. Make the light beam emitted by the light source pass through the surface of the vertical polarizer, wave plate, photoelastic modulator and analyzer after collimation, and finally enter the detector. The azimuth angles of the polarizer, photoelastic modulator, wave plate and analyzer are randomly set. The digital lock-in amplifier outputs the amplitude and phase of the first and second frequency components of the detector voltage signal for computer display and calculation analysis; b. Adjust the nominal modulation amplitude A of the photoelastic modulator from small to large through the photoelastic modulator controller m , synchronously scan and record the amplitude and phase change curves of the first and second frequency components in the detector output voltage signal; c. standardizing the amplitude change curves of the one-fold and two-fold frequency components into the corresponding Bessel function curve form according to the phases of the one-fold and two-fold frequency components; d. Using the second-order Bessel function J2(A m )=0 corresponding to the actual phase delay amplitude A m =0.82λ, inversely solve the ratio of the first-order Bessel function, and obtain the calibration curve based on the inverse first-order Bessel function; e. Using the first-order Bessel function J1(A m )=0 corresponding to the actual phase delay amplitude A m =0.61λ, inversely solve the ratio of the second-order Bessel function, and obtain the calibration curve based on the inverse second-order Bessel function; f. Select the calibration curve segments in the intervals of 0.1-0.2λ and 0.37-0.7λ in the calibration curve based on the inverse solution of the first-order Bessel function, and combine them with the calibration curve segments in the intervals of 0.2-0.37λ and 0.7-0.9λ in the calibration curve based on the inverse solution of the second-order Bessel function to form a high-resolution calibration curve for the entire interval of 0.1-0.9λ.

2. The device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response according to claim 1, characterized in that There is no fixed angle setting requirement for the azimuth angles of the polarizer, photoelastic modulator, wave plate, and analyzer. In step a, the signal-to-noise ratio is increased by adjusting the azimuth angles of any one or more of the four.

3. The device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response according to claim 1, characterized in that There is no special numerical value or numerical accuracy requirement for the phase delay Δ of the wave plate. In step a, the signal-to-noise ratio can be enhanced by selecting different wave plates and a combination of multiple wave plates.

4. A device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response according to claim 3, characterized in that By adjusting the azimuth angle between the multi-wave plates and the value of the equivalent Δ, the signal-to-noise ratio can be enhanced.

5. The device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response according to claim 1, characterized in that The phase delay of the wave plate in the device has no special value and numerical accuracy requirements, and the influence of the dispersion of the phase delay of the wave plate is shielded, so that the entire working wavelength band of the photoelastic modulator can be calibrated.

6. The device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response according to claim 1, characterized in that The calibration device and method shield the static phase delay δ caused by the static residual stress of the photoelastic modulator. s The influence of the large static phase delay δ s The photoelastic modulator is calibrated.

7. The device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response according to claim 1, characterized in that Steps d and e do not include the calculation of the ratio of Bessel functions of different orders, thereby shielding the influence of the frequency response characteristics of the detection system.

8. The device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response according to claim 1, characterized in that After the optical path is adjusted, the position of the analyzer is fixed during the measurement process of step b to shield the influence of the polarization response characteristics of the detector.

9. The device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response according to claim 1, characterized in that After the optical path is adjusted, the polarizer is fixed in position during the measurement process of step b to shield the influence of the polarization characteristics of the light source.

10. The device and method for calibrating the modulation amplitude of a photoelastic modulator without frequency response according to claim 1, characterized in that The final calibration curve in step f is in the form of segmented combination, and the selection of segmented points is based on the method of comparing the first-order derivatives of the first-order and second-order Bessel functions.

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