Noise Decrease in a Balanced Self-Mixing Interferometer: Theory and Experiments

In a self-mixing interferometer built around a laser diode, the signals at the outputs of the two mirrors are in phase opposition, whereas noise fluctuations are partially correlated. Thus, on making the difference between the two outputs, the useful signal is doubled in amplitude and the signal-to-noise ratio is even more enhanced. Through a second-quantization model, the improvement is theoretically predicted to be dependent on laser facets reflectivity. The results are then validated by experimental measurements with different laser types that show very good agreement with theoretical results. The new technique is applicable to a number of already existent self-mixing sensors, potentially improving significantly their measurement performances.

Noise Decrease in a Balanced Self-Mixing Interferometer: Theory and Experiments Parisa Esmaili , Michele Norgia , Senior Member, IEEE, and Silvano Donati , Life Fellow, IEEE Abstract-In a self-mixing interferometer built around a laser diode, the signals at the outputs of the two mirrors are in phase opposition, whereas noise fluctuations are partially correlated. Thus, on making the difference between the two outputs, the useful signal is doubled in amplitude and the signal-to-noise ratio is even more enhanced. Through a second-quantization model, the improvement is theoretically predicted to be dependent on laser facets reflectivity. The results are then validated by experimental measurements with different laser types that show very good agreement with theoretical results. The new technique is applicable to a number of already existent self-mixing sensors, potentially improving significantly their measurement performances.
To access the AM self-mixing signal, it is customary to use the internal monitor photodiode, usually mounted in front of the rear mirror. We can process the AM signal either digital or analog, respectively, by fringes counting for measuring displacement up to several meters or by measuring vibrations taking advantage of the signal linearity, down to 100 pm and even much less with appropriate processing [2], [3] magnitude with respect to AM [27], [28], [29], [30], but it requires a more complex and expensive optical setup. The aim of this work is to propose an improvement in traditional AM SMI signal detection, while keeping the absolute simplicity of the optical setup and potential low cost of the sensor. As the SMI signal is a modulation present in the laser beam, it can be measured at the rear output, where we take advantage of the monitor photodiode PD2 (see Fig. 1), but also at the front-mirror output. Ideally, we can use a simple beamsplitter (BS) and photodiode PD1 combination to collect it or any equivalent technique to extract some power from the measurement beam.
As demonstrated in [31], the SMI signal in a laser diode (LD) well above threshold is in phase opposition between the two LD outputs. This effect can be exploited to achieve a sort of balanced detection, by taking the difference between PD1 and PD2, as introduced in [32]. Through this difference, all the signal contributions due to LD power variation are canceled, both linked to laser driver noise or disturbances, and also pump current modulations [13], [14], [15], [16].
In this article, we experimentally demonstrate that the balanced detection for SMI is not limited to disturbances reduction but also brings down the optical shot noise contribution, more or less depending on the type of LD.

II. THEORETICAL MODEL AND ANALYSIS
Let us now start the analysis by evaluating the signals of the front and rear outputs of the LD. Let the power reflectivity of mirrors M1 and M2 be R 1 and R 2 , with output powers P 1 and P 2 . These powers are converted into electrical current signals I 1 = σ P 1 and I 2 = σ P 2 by photodiodes placed in Second quantization model (bottom) of the LD cavity (top). E 0 is affected by the coherent-state fluctuation E coh , and the vacuum state fluctuations E vac1,2 come from the unused ports of the mirrors. front of the output mirrors. Noise analysis will be carried out in Section III.
Output power P is a quadratic function of the electric field amplitude E. The amplitudes of the two output fields E 1 and E 2 have been expressed in [31] as a function of the unperturbed cavity field E 0 , and the parameters of the SMI: target distance s, attenuation of the back injected field A, and optical phase shift of the round-trip path to the target φ = 2 ks, with the wavevector k = 2π /λ.
The calculated modulations m 1 and m 2 , defined as the ratio of the SMI signal and the unperturbed field superposed to them, are equal to [27] Hence, the ratio becomes where r 1,2 = √ R 1,2 and t 1,2 = √ T 1,2 are the field reflection and transmission of mirrors M1 and M2, respectively (see Fig. 2); 2γ L is the round-trip gain along the laser cavity of length L.
As reported in [31], the outputs are in phase (m 1 /m 2 close to 1) at small values of the gain minus loss parameter (2γ L + ln R 1 R 2 ); then, in the normal operating conditions for a LD above threshold, with 2γ L + ln R 1 R 2 ⟩ T 1 /R 1 , the outputs become in phase opposition with m 1 /m 2 negative, typically ≈ −0.8.
The average components carrying the SMI signals can obviously be brought to the same value by amplification, and in this case, the amplitudes of the two SMI signals measured by PD1 and PD2 are proportional to the modulation coefficients given by (1) and (2).

III. NOISE ANALYSIS
For a rigorous description of the noise properties of SMI signals, we shall resort to the semi-classical translation of second quantization of the optical field [33]. As shown in Fig. 2, we represent the total field as the superposition of an average E 0 and a fluctuation E coh known as the coherent state fluctuation. E coh is a Gaussian noise, and its amplitude is such that the power P 0 = aE 2 0 /2Z 0 of the field E 0 is the classical shot noise, σ 2 p = 2hν P 0 B, observed on a bandwidth B [6]. Z 0 is the vacuum impedance and a is the cross section area of the beam.
The fluctuation E coh has zero average, ⟨ E coh ⟩ = 0, and a quadratic mean value given by ⟨ E 2 coh ⟩ = (a/2Z 0 )1/2 hν B (factor a/2Z 0 is omitted for simplicity in the following). Equivalently, E 2 coh has a power spectral density d⟨ E 2 coh ⟩/d f = 1/2 hν equal to half photon per Hertz. In addition to the noise inherent of the oscillating field, there are other noise sources from BSs and mirrors. Even if any port is unused, indeed, the second quantization theory models it as a port open to the vacuum-state fluctuation, E vac (see Fig. 2). This theory states that the coherent state fluctuation, E coh , is superimposed to every field, and it does not depend on the field amplitude E 0 . It is also present when the amplitude E 0 = 0, such as in the case of an unused port [29].
Including, as shown in Fig. 2, E vac1 and E vac2 , the second quantization model is complete [33], and it is possible to evaluate the fluctuations of output fields E 1 and E 2 , and then, the variance of noises superposed to P 1 and P 2 . We will take account of correlation of terms originated by the same fluctuation and of their cancellation in a difference operation, as well as of the uncorrelation of contributions originating by different terms (e.g., E vac1 and E vac2 ) that instead will add up.
A preliminary account of the theory, in the special case of equal mirrors reflectivity, has been reported in [34]; here, we develop the general case.
With reference to Fig. 2, we can evaluate the mean value and variance of the power at output 1, P 1 = ⟨|E 1 | 2 ⟩. At mirror M1, we can write where t 1 = √ T 1 and r 1 = √ R 1 are the field transmission and reflection of the mirrors. E vac1 has the same distribution, but it is uncorrelated to E coh , entering from the unused port of the BSs. The properties of such fluctuations are P 1 has a mean value given by the classical expression ⟨P 1 ⟩ ∝ E 2 , but we have to subtract the vacuum field, because it cannot be observed, as pointed out in [33] Inserting (4) in (6), we get As the second, third, and last terms cancel out, we obtain Also, noting that the mean values of E coh and E vac1 are zero, E coh and E vac1 are uncorrelated, and E 2 0 = P 0 and t 2 1 = T 1 , we get that is, the expected result. Variance is given by the difference σ 2 Additionally, as T 1 E 2 0 = P 1 and ⟨ E 2 coh ⟩ = ⟨ E 2 vac1 ⟩ = 1/2 hν B, we finally obtain Let us note that as R 1 + T 1 = 1, (11) is also written as σ 2 P1 = 2P 1 hν B, corresponding to the variance of power P 1 following a Poisson statistic.
Upon repeating the same calculation for output 2, we get Before making the difference of SMI signals P 1 and P 2 , we have to equalize their amplitudes, as performed in the experiment's practical implementation by a (noiseless) amplification. For the equalization, in view of (9) and (12), we shall multiply P 1 by √ (T 2 /T 1 ) and P 2 by √ (T 1 /T 2 ), so that the new mean values become As we equalize the mean values, the variances of P 1 and P 2 are multiplied by T 2 /T 1 and T 1 /T 2 , respectively, so that we have and Now, comparing (15) and (16), we can see that the first terms of variances are the same, because they derive from the same fluctuation E coh . It means that they are canceled out by making the difference P = P 1eq − P 2eq . On the contrary, the second terms of (15) and (16) come from different fluctuations, E vac1 and E vac2 , and are totally uncorrelated.
Taking this into account, the variance of P = P 1eq − P 2eq is written as This result must be compared to the variances of the outputs P 1eq and P 2eq , which, in view of (15) and (16) Taking the monitor photodiode (rear output, P 2 ) as the reference value, because it is commonly used for self-mixing measurements, we get the ratio of variances equal to and the corresponding ratio of SNR, considering also that the differential self-mixing signal has double the amplitude, becomes As a particular case of equal mirrors, R 1 = R 2 , the above equations become For a Fabry-Perot (FP) laser, the typical reflectivity is R = 0.3, corresponding to F = 2/0.3 ∼ = 6.7.
If the electronic noise is negligible (in quantum regime [29]), we obtain the same SNR for the output voltage signal V = R tr σ P, given by the photodiode current I= σ P, on a resistance R tr .
This approach is not applicable to all kinds of lasers, for example, in a He-Ne laser, the two outputs are in phase [31], so the difference cancels out also the useful signal. About previous works, we report that a second-quantization approach was used in [35], for calculating the correlation factors of the two output beams in an FP laser, finding correlation values close to 0.8. With regard to other noise components, in [36], the 1/ f -noise correlation was considered, demonstrating an even stronger correlation between the two laser outputs.
In this discussion, we consider amplitude additive noise and its contribution to measurement in the main hypothesis of standard signal processing. For example, small vibration measurement working with the interferometer in quadrature, taking advantage of the signal linearity with target displacement: this is the typical case for evaluating the noise equivalent displacement (NED) of the interferometer [3], [4]. In that condition, phase variation is proportional to amplitude variation; therefore, both noise reduction and signal doubling contribute to improve the measurement. We did not consider phase noise contribution, due to LD limited coherence, because it is typically negligible for short distances.

IV. LOW-NOISE PLACEMENT OF THE FRONT PHOTODIODE
Using a BS to pick up power from the front output, as schematically shown in Fig. 1, introduces extra noise, in view of second quantization, because of the unused input port of the BS: it lets the vacuum fluctuation E 2 vac enter in the experiment and sum up to the cavity field. Fig. 3. Using a reflecting prism, to remove a fraction a' of the output beam, eliminates the vacuum fluctuation of the BS (see Fig. 1).
Indeed, letting R BS for the BS reflectivity, and following the guidelines of the previous section to calculate powers and associated variances, we find in the case R 1 = R 2 = R that power at the detector PD1by is given by while power at the other mirror is still P 1 = t 2 E 2 0 = T P 0 , a quantity larger than P 1BS . The variance of fluctuations associated with P 1BS is To equalize the amplitude of P 1BS and P 2 , we have to amplify by a factor 1/R BS , getting for the equalized variance σ P1BS(eq) to be compared with σ 2 P1 = 2TP 1 hν B + 2R P 1 hνB.. Also, in this case, the first terms are correlated, and the difference becomes and whereas the single-channel has SNR 2 P1 = P 1 /(2hν B). Thus, the final result is In conclusion, the insertion of a BS, to take a portion of the emitted light, severely reduces the improvement of the balanced detection. For example, with a 50/50 BS, we have F = 2/(R + 1).
In order to overcome this limit, it is more efficient to take a small fraction of the emitted beam by means of a mirror (or a prism, as shown in Fig. 3).
The power collected by PD1 is still a fraction of the emitted power, but in this case, there is no open port to vacuum fluctuations. Indeed, the partialization of the output beam can be modeled as a two-port device, instead of the four port of the BS. Therefore, expression (11) also holds for this setup, with P 1 replaced by P 1P , and the improvement in SNR is confirmed.
Another possible solution is placing the external photodiode directly to detect a fraction of the output beam. This solution allows to minimize part count and cost, and it is particularly indicated when the monitor photodiode intercepts a small fraction of the emitted power (a very common situation for visible and near-infrared LDs).

V. EXPERIMENTS
The purpose of this section is the experimental verification of the improvement obtainable through the balanced configuration of SMI in terms of SNR. As demonstrated in Section III, the achievable improvement depends on the reflectivity of the laser mirrors; therefore, it is expected to obtain different results depending on the type of laser tested. In this article, two different FP LDs, one distributed-feedback (DFB), and one vertical-cavity surface-emitting (VCSEL) LD have been tested. Table I reports LDs models with their main specifications: wavelength (λ), output power (P o ), threshold current (I th ), and biased dc current (I bias ).
For the verification, a special power supply and measurement electronics was developed, capable of switching from a single-PD configuration to a balanced configuration by acting on simple jumpers. Fig. 4 shows a simplified circuit diagram of the LD driver and transimpedance amplifier, realized for implementing the subtraction between internal and external photodiodes. As shown in this figure, both monitor and external photodiodes are fed to the same trans-impedance amplifier with a 100-k feedback resistor. The output signal of the transimpedance amplifier is proportional to P = P 1 − P 2 which is directly connected to a digital oscilloscope (model RTB2004). The employed external photodiodes are: a silicon photodiode (model BPW34) with a spectral range of 430-1100 nm and an InGaAs photodiode (model SD039-151-011) with a spectral range of 800-1700 nm. For balancing P 1 and P 2 , a 16-kHz triangular waveform with an amplitude of 1.5 mA is applied as a modulation signal to the LD bias current. The external photodiode is mounted on a slider to adjust its position for canceling the current modulation measured in transimpedance output, proportional to P. Once the equal amplitude condition is satisfied, the modulation signal can be easily removed using a jumper (J3 in Fig. 4). To evaluate the self-mixing signal, the laser beam is focused on white article, placed on a loudspeaker at about 10 cm from the LD, using a biconvex lens with a focal length of 25 mm. Fig. 5 shows a  photograph of the experimental setup. For noise measurement, the target is removed in order to avoid spurious SMI signals.

VI. RESULTS AND DISCUSSION
For every tested LD, after the balancing of the power measured by external and monitor PD, we acquired the transimpedance output, with and without the external photodiode, by acting on jumper J2 (see Fig. 4). Great care was taken to avoid any possible change in mechanical alignment or physical condition and to minimize electrical disturbances on LD supply, because as already demonstrated in [32], they cancel out in balanced detection and may appear to have a more marked improvement in SNR. The power spectrum in dBm is directly calculated by the digital oscilloscope, considering 50-load. With the same mechanical setup, a vibrating target is added to evaluate SMI signal with and without the external photodiode.
Self-mixing signal amplitude decreases with temperature, as shown in [37]. In this work, we did not consider that dependence and we made all the comparisons at the same temperature.
The first tested LD is a very-common FP red laser. This laser is often used for the demonstration of the SMI signal, because it is low cost, easy to find, and quite sensitive to optical back injection, even if it is not robust to moderate feedback level. Fig. 6 shows the power spectrum comparison: with both PDs connected (balanced detection) and with only monitor PD (single channel). In this case, the noise reduction is evident, about 6 dB, and can be appreciated also in the timedomain SMI signal (see Fig. 7). Here, we get about 12 dB of SNR improvement, considering the signal doubling, also evident in Fig. 7. This improvement is coherent with the theory of Section III, if considering the anti-reflection coating of the LD output facet, very common for red LD [38]. Indeed, (20) gives about 11 dB of SNR improvement for 1%-2% output mirror reflectivity.
The second tested laser is one of the most employed LDs for SMI interferometer [6], [7], [14], because of its very-good sensitivity and robustness to back injection: near-infrared FP LD, model HL7851G. For this LD, the noise reduction is exactly what is expected from theory: for cleaved facets (R = 0.3), it is about 2 dB (see Fig. 8). As shown in Fig. 9, the signal doubling leads to about 8 dB of SNR improvement for this laser.
It is worth noting that the monitor PD takes a small fraction of the emitted power (about 1%, typical for this kind of LD); therefore, it is easy to place the external photodiode without excessively choking the SMI measurement beam.
Next tested LD is a DFB model, also often used for SMI [8], [17], [37], [39]. The distributed feedback is not perfectly modeled by the proposed theory, but we can evaluate experimental results and consider a sort of equivalent mirror reflectivity. In   this case, noise reduction is similar to the FP laser without reflection coating, about 2 dB for shot noise contribution, as shown in Fig. 10 (for frequency higher than 30 kHz). It  is a little higher for 1/ f noise, about 4 dB. This behavior is not explained by the proposed theory, but it is already known that shot-noise and 1/ f noise can have different correlations between the two LD outputs [36]. Even the SNR improvement is similar to the one for HL7851G, considering that we still have a doubling in the SMI signal (see Fig. 11).
The last LD model considered is a single-mode VCSEL, with an internal monitor photodiode. Fig. 12 shows the noise spectrum with and without the external photodiode. In this case, the difference is negligible (∼0 dB). This indicates that there is a slight correlation; otherwise, it would be +3dB, but it is lower than the other LDs tested. From (20), this value suggests equivalent mirrors reflectivity of about 50%. Fig. 13 reports the time-domain SMI signals, showing an amplitude improvement of only ∼50% from the subtraction, as predicted by (5). Table II summarizes the experimental results for the different LDs. As evident in the time-domain signals, a strong improvement is obtained for low-cost red LD, while for the VCSEL, the advantages of the proposed technique are negligible with respect to the additional complexity.  Interferometric self-mixing signal waveforms photodiode using PS85F1P1U-KC: single channel (upper trace) and balanced detection (lower trace).

VII. CONCLUSION
In this article, we confirmed experimentally a rigorous theory based on second quantization developed to evaluate the signal-to-noise improvement of the balanced detector configuration used in an SMI. We have tested several specimens of LDs and found excellent agreement between theoretical and experimental values. In particular, we have shown that an improvement of up to 12 dB can be obtained in SNR of a self-mixing measurement with a red FP LD.
Another important finding is about the placement of the front-exit pickup of signal: both theory and experiment show that it should be done by partializing the beam rather than by an ordinary BS because the latter opens a port on the vacuum fluctuations.
The results obtained are relevant to several applications of SMI, like, for example, absolute distance measurement [15] vibration sensing, and all the measurements involving a small amplitude of returning signal [9], [10].