#
A Novel Method for Detecting Fe^{2+} at a Micromolar Concentration Based on Multiple Self-Mixing Interference Using a Fiber Laser

^{*}

## Abstract

**:**

^{2+}indicator as the electrolyte sample at a micromolar concentration. The theoretical expressions were derived based on the lasing amplitude condition in the presence of the reflected lights considering the concentration of the Fe

^{2+}indicator via the absorption decay according to Beer’s law. The experimental setup was built to observe MSMI waveform using a green laser whose wavelength was located in the extent of the Fe

^{2+}indicator’s absorption spectrum. The waveforms of the multiple self-mixing interference were simulated and observed at different concentrations. The simulated and experimental waveforms both contained the main and parasitic fringes whose amplitudes varied at different concentrations with different degrees, as the reflected lights participated in the lasing gain after absorption decay by the Fe

^{2+}indicator. The experimental results and the simulated results showed a nonlinear logarithmic distribution of the amplitude ratio, the defined parameter estimating the waveform variations, versus the concentration of the Fe

^{2+}indicator via numerical fitting.

## 1. Introduction

^{2+}. The MSMI waveform contained the main fringes and parasitic fringes that captured the absorption decay, as every reflected light’s intensity corresponded to the amplitudes of the fringes. The main fringe and parasitic fringe usually experienced different decrements, and the difference was related to the material concentration.

^{2+}at a micromolar concentration based on multiple self-mixing interference using a fiber laser. The laser intensity equations were derived considering the multiple reflected lights absorbed by the absorption material in the external cavity based on lasing conditions, and the waveforms were simulated according to the derived equation at different concentrations of the Fe

^{2+}indicator. The experimental setup was constructed to observe the MSMI waveform using a green laser whose wavelength was located in the extent of the Fe

^{2+}indicator’s absorption spectrum. The MSMI waveforms around the micromolar concentrations of the Fe

^{2+}indicator were obtained in the time domain experimentally. Both the simulated and experimental waveforms contained the main and parasitic fringes, and a parameter A

_{2}/A

_{1}indicating the amplitude ratio of the parasitic fringes versus the main fringes was defined to determine the distribution between the concentration and the MSMI waveforms using the numerical fitting, specifically, by estimating the R

^{2}of the fitting.

## 2. Materials and Methods

^{2+}indicator sample (FE), an optical splitter (OS), a photoelectric detector (D), a loudspeaker (LS), and a signal generator (SG). A cuvette, a type of highly transparent container that guaranteed the laser can pass through without decay, was employed to change the absorption decay in the external cavity by diluting the Fe

^{2+}indicator. The loudspeaker with a high-level reflector was employed to continuously generate low-frequency displacement to modulate the laser wave phase carrying the concentration of the Fe

^{2+}indicator sample in the time domain. The high-reflecting plane mirror was employed to act as one side of the external cavity. Furthermore, the plane mirror could guarantee the homocentricity of the reflected wave, and the rays of the laser could be considered in the same direction, with the reflected rays persisting collinear to the optical axis of the laser propagation direction. Although the sample was diluted with water, there is only the absorption of Fe

^{2+}since the absorption spectrum of Fe

^{2+}lay in the green band, within which there was nearly no absorption of the water [42].

^{2+}indicator, a chemical complex Fe(C

_{12}H

_{8}N

_{2})

_{3}

^{2+}that can be obtained from the ferrous ion and the phenanthroline, served as a type of absorption sample in the experiment. The standard concentration of the solution was 0.02 mol/L and was diluted with pure water, and we determined the concentration with a photometer until the Fe

^{2+}in the dilution reached the micromolar concentration.

^{2+}absorption spectrum. The loudspeaker was driven by a continuous sine function generated by a signal generator, whose amplitude and frequency were selected as 400 mVpp (in terms of μm magnitude) and 2 KHz, respectively. The absorption decay varied with the Fe

^{2+}concentration of the indicator sample in the rectangular cuvette with a width of 1 cm, which was equal to the optical depth of the diluting sample. The laser was split into two beams by the optical splitter. One forward beam irradiated the mirror on the loudspeaker surface perpendicularly, and the backward reflected E

_{1}and E

_{2}waves after one round trip and two round trips turned back into the laser through the original path to generate the multiple self-mixing interference signal. Meanwhile, the alternating current (AC) photoelectric detector was used to monitor the other beam after magnification to obtain the multiple self-mixing interference signal. Because the laser wave phase was modulated versus time, the detected multiple self-mixing interference was also a time-domain signal, which could be obtained via an oscilloscope.

_{0}(t) accounts for the laser free amplitude without multiple self-mixing interference, the wave amplitude E

_{L}in the laser cavity after one round trip can be expressed as

_{L}= r

_{1}r

_{2}·exp[−j·4πL/λ) + gL]·E

_{0},

_{1}and r

_{2}of the laser cavity consist of the input and output facets, v denotes the light frequency, φ

_{0}is the initial phase, L is the laser cavity length, and λ is the light wavelength.

_{1}and E

_{2}were treated as a part of the laser cavity after multiple reflections. Therefore, the space between the output facet and the external reflector was treated as an external cavity according to the lasing condition. The absorption dilution in the external cavity would decrease the first and second reflected lights if the wave expression was multiplied by the amplitude parameter, indicating that the wave amplitude would be scaled up and down according to the value of this parameter. The wave amplitude E

_{1}of the first reflected light after one round trip in the external cavity can be expressed as

_{1}= r

_{1}(t

_{2})

^{2}r

_{3}f

^{2}·exp[−j·4π(L + l)/λ) + gL]·E

_{0},

_{3}is the amplitude reflectance of the external reflector, t

_{2}is the amplitude transmittance of the output facet, and f is the amplitude transmittance of the absorption dilution in the external cavity.

_{2}of the second reflected light can be expressed by

_{2}= r

_{1}r

_{2}(t

_{2})

^{2}(r

_{3})

^{2}f

^{4}·exp[−j·4π(L + 2l)/λ + gL]·E

_{0},

_{L}+ E

_{1}+ E

_{2}= r

_{1}r

_{2}·exp[−j·4πL/λ) + gL]·E

_{0}+

r

_{1}(t

_{2})

^{2}r

_{3}f

^{2}·exp[−j·4π(L + l)/λ) + gL]·E

_{0}+ r

_{1}r

_{2}(t

_{2})

^{2}(r

_{3})

^{2}f

^{4}·exp[−j·4π(L + 2l)/λ + gL]·E

_{0}.

_{0}, yielding

_{1}r

_{2}+ r

_{1}(t

_{2})

^{2}r

_{3}f

^{2}·exp[−j·4πl/λ) + r

_{1}r

_{2}(t

_{2})

^{2}(r

_{3})

^{2}f

^{4}·exp[−j·4π(2l)/λ]} = 1.

_{1}(t

_{2})

^{2}r

_{3}f

^{2}and β = r

_{1}r

_{2}(t

_{2})

^{2}(r

_{3})

^{2}f

^{4}and obtain the amplitude condition

_{1}r

_{2}·exp[−j(4πL/λ) + gL][1 + α/r

_{1}r

_{2}·exp(−jφ

_{1}) + β/r

_{1}r

_{2}·exp(−jφ

_{2})] = 1,

_{1 =}φ

_{0}+ φ and φ

_{2 =}φ

_{0}+ 2φ denote the phases of E

_{1}and E

_{2}, respectively. Because the external cavity will usually be tilted in the experiment to generate the MSMI waveform, φ

_{2}will not be precisely equal to φ

_{1}. Therefore, the parameter φ

_{0}was introduced to denote the phase resulting from the tilted external cavity, whose value would be assigned in the simulation to satisfy the experiment results.

_{1}r

_{2}·exp(−jφ

_{1}) + β/r

_{1}r

_{2}·exp(−jφ

_{2})},

_{1}r

_{2}(1 + α·exp(−jφ

_{1}) + β·exp(−jφ

_{2}))|/2.

_{1}) + β·exp(−jφ

_{2})] = 2π−2πL/λ,

_{1}r

_{2}|z|·exp(gL)·exp{−j(4πL/λ + θ)} = 1.

_{1}) + β·sin(−φ

_{2})/(r

_{1}r

_{2}),

_{1}r

_{2}≈ 1, α << 1, and β << 1, as the output facet was a high-level reflection mirror.

_{1}) + β·sin(−φ

_{2})/(r

_{1}r

_{2}).

_{1}r

_{2}+ r

_{1}r

_{2}α·exp(−jφ

_{1}) + r

_{1}r

_{2}β·exp(−jφ

_{2}))/2.

_{1}r

_{2}|z|)/2= −ln(r

_{1}r

_{2}) /2 −ln|z|/2,

_{1}r

_{2})/2 − ln|z|/2= −ln(r

_{1}r

_{2}) /2 − [Re(z)]/2,

^{2}+ [Re(z)]

^{2}}

^{1/2}≈Re(z) if Re(z) >> Im(z) under the condition of α << 1 and β << 1, as mentioned above. Thus,

_{1}r

_{2})/2 − ln[Re(z)]/2= −ln(r

_{1}r

_{2}) /2 − [α·cosφ

_{1}/(r

_{1}r

_{2}) + β·cosφ

_{2}/(r

_{1}r

_{2})]/2,

_{1}/(r

_{1}r

_{2}) + β·cosφ

_{2}/(r

_{1}r

_{2})] = α·cosφ

_{1}/(r

_{1}r

_{2}) + β·cosφ

_{2}/(r

_{1}r

_{2}).

_{1}r

_{2})/2L − [α·cosφ

_{1}/(r

_{1}r

_{2}) + β·cosφ

_{2}/(r

_{1}r

_{2})]/2L.

_{0}could be obtained as g

_{0}= −ln(r

_{1}r

_{2})/2L without cosine feedback terms. Therefore, the influence of multiple self-mixing interference on the gain coefficient was estimated by Δg, according to

_{0}= −[α·cosφ

_{1}/(r

_{1}r

_{2}) + β·cosφ

_{2}/(r

_{1}r

_{2})]/2L.

_{0}with Δg in the laser cavity could be expressed as

_{0}= 1 − Δg·2L = 1 + α·cosφ

_{1}/(r

_{1}r

_{2}) + β·cosφ

_{2}/(r

_{1}r

_{2}).

^{2+}indicator sample in the external cavity once the experimental setup was complete. Since the feedback beams passed through the Fe

^{2+}indicator sample at different times during their round trips in the external cavity, the amplitude coefficients in the front of E

_{1}and E

_{2}had to depend on the absorption decay and the reflection decay in the external cavity. Therefore,

_{1}(t

_{2})

^{2}r

_{3}f

^{2},

_{1}r

_{2}(t

_{2})

^{2}(r

_{3})

^{2}f

^{4},

^{−}

^{ε}

^{bc},

^{2+}indicator sample, respectively.

_{0}= −π/2 to make the simulated waveform close to the experimental waveform. Because the external cavity length varied slightly following the sine function Asin(wt) versus time t with amplitude A = 2 μm and frequency w = 2 kHz, the length of the external cavity l = 1.8 m for the phase φ was also modulated periodically by a sine-like signal in the time domain carrying the concentration c of the Fe

^{2+}indicator sample.

## 3. Results

#### 3.1. Simulated Results

_{0}at different Fe

^{2+}concentrations c were simulated with the assigned parameters of r

_{1}= 1.0, r

_{2}= 0.97 [43], r

_{3}= 0.99, b = 1 cm, and ε = 7955 L/(mol·cm). These parameters were assigned according to both the theoretical model and the experimental setup. The molar absorption coefficient of the diluent at the laser wavelength was estimated by measuring the diluent absorbance at different concentrations. For the laser cavity consisting of the input and output facets with reflectivity values of r

_{1}and r

_{2}, we considered the input facet as a totally reflective mirror (r

_{1}= 1) corresponding to the theoretical model without intensity loss in the inner cavity, while the output facet was a partially reflective mirror with a specific value of 0.97. The external reflector was a highly reflective mirror with a reflectivity of 0.99. If the absorption sample in the external cavity was the Fe

^{2+}indicator whose concentration c ranged from 4.8 × 10

^{−6}mol/L to 1.4 × 10

^{−5}mol/L, the absorption decay depended on its molar absorption coefficient ε = 7955 L/(mol·cm) at the experimental laser wavelength and effective optical depth b = 1 cm, neglecting the intensity decay of the highly transparent container.

_{1}and E

_{2}, which were decayed by the absorption in the external cavity. Thus, the feedback wave E

_{1}caused some main fringes, while E

_{2}caused some low parasitic fringes. Moreover, if the Fe

^{2+}indicator was used as the absorption sample; it enabled the absorption decay for both E

_{1}and E

_{2}when the waves passed through the sample with different round trips. The peak values A

_{1,p}and A

_{2,p}and valley values A

_{1,v}and A

_{2,v}of the main and parasitic fringes were selected to obtain the amplitudes A

_{1}and A

_{2}. The main and parasitic fringes were both decayed by the absorption of Fe

^{2+}according to Beer’s law, and the parasitic fringe amplitude A

_{2,s}= (A

_{2,p}− A

_{2,v})/2 decayed much more dramatically than the main fringe amplitude A

_{1,s}= (A

_{1,p}− A

_{1,v})/2 with increasing concentration c of Fe

^{2+}due to the difference in absorption decay between E

_{1}and E

_{2}in the external cavity.

#### 3.2. Experimental Results

_{1}and E

_{2}from the external cavity. The reflected lights after one and two round trips entered back into the laser cavity and mixed with the wave field in the inner laser cavity. The waveform of the output laser consisted of the superposition of the two-fold signals, as the optical phase of E

_{2}was approximately twice that of E

_{1}after two round trips in the external cavity. As mentioned in Section 3.1, the self-mixing interference signal was affected by the Fe

^{2+}indicator of the external cavity via the first and second reflected lights. Thus, the experimental signal at different absorption decays could be obtained by varying the Fe

^{2+}concentration.

^{2+}concentrations from 4.8 × 10

^{−6}mol/L to 1.4 × 10

^{−5}mol/L, as shown in Figure 4. When the concentration was quite low, such as approximately 10

^{−7}mol/L, a change in A

_{2,e}/A

_{1,e}could barely be obtained from the waveform. However, when the concentration reached a high value of more than 10

^{−5}mol/L, the parasitic fringes became very imperceptible, and A

_{2,e}/A

_{1,e}was nearly zero.

_{1,e}and A

_{2,e}of the main and parasitic fringes. If the optical decay of the external cavity changed with the Fe

^{2+}concentration, the amplitudes of the main fringes and parasitic fringes in the waveform line scaled up and down at different degrees as E

_{1}and E

_{2}underwent one and two round trips through the Fe

^{2+}indicator, as the simulation results predicted.

_{1,e}and A

_{2,e}were the products of the actual intensity and photoelectric conversion coefficient. Therefore, the amplitude ratio A

_{2,e}/A

_{1,e}was employed to estimate the relationship between the waveforms and the concentration, as A

_{2,e}/A

_{1,e}could eliminate the photoelectric conversion coefficient.

## 4. Discussion

_{2,s}/A

_{1,s}, which gradually decreased with the Fe

^{2+}concentration, as shown in Figure 5. Although the points in the figure show a linear trend, the distribution was not entirely linear. After multiple attempts of numerical fitting with the linear formula and the nonlinear logarithmic formula, we determined that a nonlinear formula could better fit the simulated A

_{2,s}/A

_{1,s}, based on the R

^{2}of the fitting. The amplitude ratio A

_{2,s}/A

_{1,s}was employed to fit with a concentration c, following the fitting logarithmic formula in the 95% confidence band and 95% prediction band, and the fitting result was A

_{2,s}/A

_{1,s}= −5.15 − 0.52 ln (c + 3.77 × 10

^{−5}) with a goodness of fit R

^{2}= 0.9998.

^{2+}concentration, the points of the amplitude ratio A

_{2,e}/A

_{1,e}around micromolar Fe

^{2+}concentration are presented in Figure 6, with the simulated A

_{2,s}/A

_{1,s}points at the same concentration also presented for comparison. The distribution was not entirely linear and was not as it appeared, similar to the fitting result in the simulation section. After the comparison of the fitting results using the linear and logarithmic formulas based on the fitting parameter R

^{2}, we determined that a nonlinear logarithmic formula might better fit the simulated A

_{2,s}/A

_{1,s}since R

^{2}= 0.9966 for linear fitting and R

^{2}= 0.9979 for logarithmic fitting. The amplitude ratio A

_{2,s}/A

_{1,s}was employed to fit with the concentration c following the fitting logarithmic formula in the 95% confidence band and 95% prediction band, and the fitting result was A

_{2,e}/A

_{1,e}= −5.08 − 0.52 ln (c + 4.34 × 10

^{−5}) with a goodness of fit R

^{2}= 0.9979. The goodness of fit indicated that the fitting logarithmic formula was suitable for both simulated and experimental results. In addition, the simulated results located in the prediction band calculated from the fitting results of the experimental results are shown in Figure 6.

^{2+}indicator. We measured the output power to calculate the absorbance, as shown in Figure 7, before and after the laser passed through the Fe

^{2+}indicator at different concentrations with the same laser in the experimental setup and with an optical power meter. Under the condition of b = 1 cm, the slope of the absorbance versus the Fe

^{2+}concentration was estimated as the molar absorption coefficient ε = 7955 L/(mol·cm) at the wavelength of the laser. We noticed that the value of the optical power meter could barely be read when the concentration was less than 10

^{−4}mol/L, as the meter was not sufficiently sensitive in this concentration regime. Furthermore, this made observations more difficult around 10

^{−6}mol/L. We considered that the attributes of the reflected light carrying the absorption decay could be magnified in the laser cavity.

## 5. Conclusions

^{2+}at a micromolar concentration based on multiple self-mixing interference. The absorption decay by the Fe

^{2+}indicator in multiple self-mixing interference was observed, and we obtained its experimental waveforms around a micromolar concentration in an external cavity modulated by a periodic signal carrying the amplitude decay in the periodic form. The theoretical waveforms were simulated based on the lasing amplitude condition, considering multiple reflected lights. The experiment was conducted under the condition of Fe

^{2+}indicator dilution at different concentrations in the external cavity. The waveform consisted of the main and parasitic fringes in the time domain and the amplitude decay of the fringes with different degrees, which was due to the absorption of Fe

^{2+}indicator dilution. The experimental and simulated waveforms exhibited a similar trend when compared with the Fe

^{2+}indicator concentration. Therefore, it was the absorption of the Fe

^{2+}indicator that resulted in amplitude decay in multiple self-mixing interference.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Scheme of the experimental setup (L: laser, IA: iris aperture, FE: a cuvette of Fe

^{2+}indicator, OS: optical splitter, D: detector, SC: oscilloscope, LS: loudspeaker with a reflector, and SG: signal generator).

**Figure 3.**Simulated waveform at specific Fe

^{2+}indicator concentrations: (

**a**) 1.1 × 10

^{−5}mol/L, (

**b**) 7.7 × 10

^{−6}mol/L, and (

**c**) and 4.8 × 10

^{−6}mol/L.

**Figure 4.**Experimental waveform at specific Fe

^{2+}indicator concentrations: (

**a**) 1.1×10

^{−5}mol/L, (

**b**) 7.7×10

^{−6}mol/L, and (

**c**) 4.8×10

^{−6}mol /L.

**Figure 6.**Numerical fitting of the experimental results around the micromolar Fe

^{2+}concentration.

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## Share and Cite

**MDPI and ACS Style**

Sun, W.; Yang, Z.; Feng, G.; Chen, Z.; Chang, Q.; Hai, L.; Guo, Z.
A Novel Method for Detecting Fe^{2+} at a Micromolar Concentration Based on Multiple Self-Mixing Interference Using a Fiber Laser. *Sensors* **2023**, *23*, 2838.
https://doi.org/10.3390/s23052838

**AMA Style**

Sun W, Yang Z, Feng G, Chen Z, Chang Q, Hai L, Guo Z.
A Novel Method for Detecting Fe^{2+} at a Micromolar Concentration Based on Multiple Self-Mixing Interference Using a Fiber Laser. *Sensors*. 2023; 23(5):2838.
https://doi.org/10.3390/s23052838

**Chicago/Turabian Style**

Sun, Wu, Zhuo Yang, Guo Feng, Zhou Chen, Qiaoyun Chang, Lan Hai, and Zeqing Guo.
2023. "A Novel Method for Detecting Fe^{2+} at a Micromolar Concentration Based on Multiple Self-Mixing Interference Using a Fiber Laser" *Sensors* 23, no. 5: 2838.
https://doi.org/10.3390/s23052838