1.

Introduction

Technologies utilized in current laser gyroscopes, microelectromechanical systems, and cavity ring-down techniques require mirrors with a reflectance of $\sim 0.9999$.^1–3 Such reflectance can be achieved in multilayer dielectric mirrors manufactured with direct monitoring.⁴ During the process of manufacturing, in situ measurements are taken to calculate optical properties of the layer deposited on the substrate. This allows to compensate for the manufacturing errors in the current layer with the subsequent one. For the optical measurements, a major issue is to provide long-term output stability of the optical sensor despite fluctuations of light source intensity.⁵

To stabilize an output of the optical sensor, the feedback methods^6–14 and the ratio method^15–24 are used. The sensors based on these methods comprise a light source, a measuring channel, a reference channel, a compensation channel, and a beam splitter dividing the light source intensity between the reference channel and the measuring one. Detector-amplifier units of the measuring and reference channels convert the input intensities into analog signals. If the input intensity of the measuring channel is small, a modulator of the light source intensity and the lock-in-amplifier are optionally used to increase signal-to-noise ratio at the channel’s output. The reference channel tracks fluctuations of the light source intensity and controls the compensation channel to stabilize the output of the sensor. The compensation channel of the sensors based on the feedback methods controls either the intensity of the light source^6–10 or the sensitivity of the measuring channel.^11–14 In the sensors based on the ratio method, the compensation channel defines the output of the optical sensor as a ratio between the outputs of the measuring and reference channels. The low limit of the relative rms output instability achieved with the optical sensors based on the above methods was $>1.0\times {10}^{-4}$ for $\sim 1\mathrm{h}$.^9,15

There are two problems to overcome the instability limit achieved with the mentioned sensors. The first problem implies that the conversion of the input intensity into the analog output in the measuring channel and that in the reference one are not fully identical. Because of that, the same fluctuations of the light source intensity cause different variations of the analog outputs in both reference and measuring channels. The second problem relates to compensation of the output instability caused by the intensity fluctuations. It implies that the compensation should take into account the difference between the conversion of the light intensity into the analog output in the reference channel and that in the measuring one. In the feedback methods, the compensation should also take into account nonlinear dependence of the light source intensity and the measuring channel sensitivity on the output of the compensation channel.

To overcome the above stability limit, a new optical sensor was developed. This sensor comprises a light source generating smoothly modulated intensity, a measuring channel, a reference channel, and a beam splitter. The compensation channel is not utilized in the sensor. The beam splitter and detector-amplifier units of the measuring and reference channels operate in the same way as mentioned above. In the reference channel, the analog output and voltage of a high stability voltage reference are compared. At a moment of time when the analog output equals the reference voltage, the reference channel opens the measuring channel and the input intensity of the measuring channel is converted into the output of the sensor. Note that the output of the reference channel’s detector-amplifier unit is acquired at the same specific value of the light source intensity. Therefore the detector-amplifier unit of the reference channel is not required to be identical to that of the measuring channel.

Said specific value of the light source intensity depends only on the reflectance of the beam splitter, the conversion factor of the reference channel’s detector-amplifier unit, and the reference voltage. If the reflectance, the conversion factor, and the reference voltage are stable, the specific intensity should be stable too. Since the measuring channel converts the input intensity into the output of the sensor at the specific intensity, the intensity fluctuations of the light source should not influence the output stability of the sensor.

This paper describes the design of the developed sensor, the principle of the stabilization, and study of the sensor’s stability.

2.

Design of Sensor

A schematic diagram of the sensor is shown in Fig. 1. A light source comprises a semiconductor laser diode, a sinusoidal current driver, a lens (1), a diaphragm (2), and a Glan prism (3) stabilizing polarization of the light source’s beam. The beam is divided by a beam splitter (4) into two. The first beam passes through a neutral density filter (5), is reflected from a smooth steel surface (6), and falls onto a detector (7). The second beam falls onto a detector (8). Both outputs of the detectors go through preamplifiers and amplifiers. The leading edge of a pulse generated by a comparator occurs when the output of the reference channel amplifier equals a specific reference voltage. The pulse and an output of the measuring channel amplifier go to an analog-to-digital converter (ADC) comprising a gate, a capacitor, and a successive approximation register (SAR). The gate is opened by the leading edge of the pulse and the capacitor samples a part of the output of the measuring channel amplifier. Then SAR converts the capacitor charge into the digital output of the sensor.

Fig. 1

Schematic diagram of the sensor. (1) Lens. (2) Diaphragm. (3) Glan prism. (4) Beam splitter. (5) Neutral density filter. (6) Smooth steel surface. (7) and (8) Detectors.

Figure 2 shows the design of the sensor. A semiconductor laser diode LCU633541A ($\lambda =635\mathrm{nm}$) and the laser driver operating at the frequency $f=285\mathrm{Hz}$ were utilized in the light source. The laser diode, the driver, and the lens were mounted in a single unit. Precision Hamamatsu detectors S1226-8BK, high-precision and low-noise preamplifiers AD 8512, and amplifiers OPA 2228 were used in the reference and measuring channels. The S1226-8BK detectors have photosensitive area of $5.8\times 5.8\mathrm{mm}$, a photosensitivity of $\sim 0.33\mathrm{A}/\mathrm{W}$ at the wavelength of 635 nm, and a rise time of 2 μs. An ultraprecision and low-noise voltage reference ADR420, a comparator LMC7211, and a 16-bit ADC ADS8515 were also used in the sensor. The sampling time for the ADC capacitor was 2 μs. The output of the optical sensor was obtained by averaging the ADC outputs for time of $100/f$. This helped to reduce the dependence of the sensor’s stability on noise caused by the detectors and the electronics. To prevent temperature instability of the electronics, the computer was located as far away as possible.

Fig. 2

Design of the sensor.

3.

Principle of Stabilization and Estimation of Instability

In this section, the analysis of the stabilization principle is performed assuming that the contribution of the intrinsic noise of the detectors and electronics to the output of the sensor is negligible in comparison to that of the light source’s intensity fluctuations.

The principle of the stabilization is illustrated by Fig. 3 showing the output $U_1(t)$ of the reference channel amplifier, the output $U_2(t)$ of the measuring channel amplifier, the reference voltage $U_0$, the comparator pulses $U_p(t)$, and the charges $U_c(t_i)$ of the capacitor.

Fig. 3

Signals internal to the optical sensor.

The $U_1(t)$ and $U_2(t)$ outputs are described as

In Fig. 3, $t_i$ moments of time ($i=1$, 2, and 3) correspond to equality between the $U_1(t)$ outputs of the reference channel amplifier and the $U_0$ reference voltage.

At these moments, the leading edges of the comparator pulses $U_p(t)$ open the gate and the capacitor samples the $U_2(t)$ outputs of the measuring channel amplifier. The $U_c(t_i)$ charge of the capacitor is described as

However, as we can see from Fig. 3, the amplitude instability of the $U_1(t)$ outputs caused by the intensity fluctuations makes the $t_i$ moments drift against the $(i-0.5)/f$ ones. Therefore, behavior of the $U_2(t)$ output is different for the different $[t_i,t_i+{\tau }_s]$ intervals, and the $U_c(t_i)$ value is not stable. It means that the output amplitude instability $\delta U_2$ of the measuring channel amplifier affects the output instability $\mathrm{\Delta }{U}_I$ of the optical sensor.

The relative instabilities $\delta U_2$ and $\mathrm{\Delta }{U}_I$ were calculated from the equation²⁶

To calculate the $\delta U_2$ instability, the variables $\mathrm{\Delta }$, $U_{\max }$, and $U_{\min }$ in Eq. (7) were substituted with $\delta U_2$, $U_{2\max }=I_0{\xi }_{\mathrm{bs}}{\xi }_{d}R{\eta }_2[1+\alpha ]$, and $U_{2\min }=I_0{\xi }_{\mathrm{bs}}{\xi }_{d}R{\eta }_2[1-\alpha ]$, respectively, and $\delta U_2=\alpha$ was obtained. To calculate the $\mathrm{\Delta }{U}_I$ instability, the variables $\mathrm{\Delta }$, $U_{\max }$, and $U_{\min }$ in Eq. (7) were substituted with $\mathrm{\Delta }{U}_I$, $U_{c\max }$, and $U_{c\min }$, respectively, and the values of $U_{c\max }$ and $U_{c\min }$ were calculated using Eqs. (1) to (6). In these calculations, the $\alpha$ index was varied from 0.001 to 0.019 at $U_0=0.9U_1(t_1=0.5/f)$, the frequencies of 285 and 70 Hz, and the sampling time of 2 and 0.5 μs.

Figure 4 shows the dependence of the $\mathrm{\Delta }{U}_I$ instability on the $\delta U_2$ instability, the $f$ modulation frequency, and the ${\tau }_s$ sampling time. As we can see from Fig. 4, the $\mathrm{\Delta }{U}_I$ instability increases with increasing the $\delta U_2$ instability and decreases with decreasing the $f$ frequency and the ${\tau }_s$ sampling time. Such behavior of the $\mathrm{\Delta }{U}_I$ instability is caused by the following. Decrease of the output amplitude instability, the modulation frequency, and the sampling time leads to decrease of influence of the analog output’s shape on the charge of the ADC’s capacitor. Because of that, the output stability of the sensor increases. Note that the lower limit of the modulation frequency depends on the conditions of an experiment. The lower limit of the sampling time depends on the conversion sensitivity of the ADC.

Fig. 4

(a) The relative output instability of the optical sensor versus the relative output instability of the measuring channel amplifier. The dashed line and the solid one correspond to the frequencies $f=285\mathrm{Hz}$ and $f=70\mathrm{Hz}$, respectively, at the sampling time of 2 μs. The plots are satisfactorily described by the $\mathrm{\Delta }{U}_I=A[\exp (-\delta U_2/t)-1]$ function, where $A=6.98\times {10}^{-3}$ and $t=-0.757$ at $f=285\mathrm{Hz}$ and $A=1.4\times {10}^{-4}$ and $t=-0.067$ at $f=70\mathrm{Hz}$. (b) The relative output instability of the optical sensor versus the relative output instability of the measuring channel amplifier. The dashed line and the solid one correspond to the frequencies $f=285\mathrm{Hz}$ and $f=70\mathrm{Hz}$, respectively, at the sampling time of 0.5 μs. The plots are satisfactorily described by the $\mathrm{\Delta }{U}_I=A[\exp (-\delta U_2/t)-1]$ function, where $A=8.9\times {10}^{-4}$ and $t=-0.388$ at $f=285\mathrm{Hz}$ and $A=-5.4\times {10}^{-4}$ and $t=0.995$ at $f=70\mathrm{Hz}$.

4.

Experimental Study of Sensor

In this section, the output $U$ of the sensor, its relative instability $\mathrm{\Delta }U$, and its relative rms instability $\sigma U$ are considered as values that depend on both the intensity fluctuations and the noise of the detectors and electronics. The sensor’s instability was studied at $U_0\approx 0.9U_1(t=0.5/f)$ and two different light intensities on the detector of the measuring channel. Different neutral density filters were used to obtain these intensities.

4.2.

Analysis

Improvement in the output stability caused by reducing influence of the intensity fluctuations on the output of the sensor was defined as the ratio between the $\delta U_2$ output instability of the measuring channel amplifier and the $\mathrm{\Delta }U$ output instability of the sensor. The $\delta U_2$ values were determined using Fig. 4 for the $\mathrm{\Delta }U$ values obtained experimentally at $f=285\mathrm{Hz}$ and ${\tau }_s=2\mu \text{s}$. The improvement in the stability was $\sim 110$ for both intensities in the measuring channel.

The output instability described by the Allan variance ${\sigma }_y^2(M,T,\tau )$ was also estimated from the equation²⁸

Eq. (12)

$${\sigma }_y^2(M,T,\tau )=[1/(M-1)]\sum _{m=1}^M{[{\overline{y}}_m-(1/M)\sum _{z=1}^M{\overline{y}}_z]}^2,$$

where ${\overline{y}}_m=(Um-U)/U$, $T=1\min$, $\tau =100/f$, and $M$ is the number of $U_m$ measurements. The $M$ value was varied from 10 to 60.

Plots of the ${\sigma }_y(M,T,\tau )$ rms variance as function of the $M$ value are presented in Fig. 5 for $U=\mathrm{23,400.6}$ and $U=1308.05$. As we can see from the figure, the ${\sigma }_y(M,T,\tau )$ values of both plots tend to constant value at $M\ge 55$. We assume that this constant value is average rms variance $\langle {\sigma }_y(M,T,\tau )\rangle$, where the $\langle \rangle$ brackets denote infinite time averaging.

Fig. 5

Dependence of the rms variance on the number of measurements. Squares and triangles correspond to $U=23,400.6$ and $U=1308.05$, respectively.

The rms output instability $\sigma U$ of the optical sensor was also estimated at the modulation frequency of 70 Hz. The $\sigma U/\mathrm{\Delta }U$ experimental value and the $\mathrm{\Delta }U$ instability determined from Fig. 4 were used in the estimation. As follows from Fig. 4, change of the modulation frequency from 285 Hz down to 70 Hz reduces the $\mathrm{\Delta }U$ instability down to $2.73\times {10}^{-5}$. Taking into account the $\sigma U/\mathrm{\Delta }U$ experimental value, we obtained the $\sigma U$ estimate of $\sim 1.0\times {10}^{-5}$ at a frequency of 70 Hz and a sampling time of 2 μs. A similar result was obtained at a modulation frequency of 285 Hz and a sampling time of 0.5 μs.

The developed optical sensor belongs to the devices exhibiting long-term output stability. In Ref. 22, the rms repeatability of the measurement was $<3\times {10}^{-4}$ at the modulation frequency of 250 Hz. Measurements on Lambda 800 and 900 spectrophotometers²³ are performed at the modulation frequency of 46 Hz. The photometric peak-to-peak instability is $<2\times {10}^{-4}$. The same instability is obtained with Cary 4000, 5000, and $6000i$ spectrophotometers²⁴ at the modulation frequency of 30 Hz. In Ref. 15, the rms output instability of the order of ${10}^{-4}$ was achieved at the modulation frequency of 90 Hz. Therefore, we can assume that the low limit of the rms instability achieved by the commercially available products is of the order of ${10}^{-4}$ in the modulation frequency range of 30 to 250 Hz. We believe that the technical solutions suggested in our paper allow to obtain the rms instability of ${10}^{-5}$ at the modulation frequencies up to 285 Hz.

5.

Summary

A sensor for reducing the influence of light source’s intensity fluctuations on the output stability was developed, designed, and studied experimentally. The sensor comprises a light source generating smoothly modulated intensity, a measuring channel, a reference channel, and a beam splitter dividing the light source intensity between the reference channel and the measuring one. The sensor operated at a wavelength of 635 nm and a modulation frequency of 285 Hz. The reference channel opened the measuring channel when the light source intensity equals a specific value and the intensity in the measuring channel was converted into the output of the sensor for a sampling time of 2 μs.

The principle of stabilization was considered and influence of the intensity fluctuations on the output stability of the sensor was estimated for modulation frequencies of 285 and 70 Hz and sampling times of 2 and 0.5 μs. It was found that decrease of the modulation frequency and the sampling time leads to increase of the sensor’s output stability.

The experiments were performed for two different intensities at the input of the measuring channel. Although the outputs of the sensor were varied almost 18-fold for the intensities used, the dependence of the output stability on the intensity was not found. The relative rms instability of the outputs was $\sim 4.5\times {10}^{-5}$ for 1 h. The noise of the detectors and the electronics was also estimated experimentally. It was found that the contribution of the noise to the output instability was negligible.

The improvement in the output stability of the sensor was estimated as the ratio between the output instability of the measuring channel’s detector-amplifier unit and that of the sensor. It was $\sim 110$ for both intensities at the input of the measuring channel. Dependence of the rms variance on the number of measurements was considered. It was found that the measured output instabilities are close to the average rms variance. The experimental results were utilized to estimate the rms instability at (1) a modulation frequency of 70 Hz and a sampling time of 2 μs and (2) at 285 Hz and 0.5 μs. The obtained instability was $\sim 1.0\times {10}^{-5}$ for both conditions.