The fourier spectrum analysis of optical feedback self-mixing signal under weak and moderate feedback

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1 University o Wollongong Research Online Faculty o Inormatics - Papers (Archive) Faculty o Engineering and Inormation Sciences 8 The ourier spectrum analysis o optical eedback sel-mixing signal under weak and moderate eedback Xiaojun Zhang University o Wollongong, xjz787@uowmail.edu.au Jiangtao Xi University o Wollongong, jiangtao@uow.edu.au Yanguang Yu University o Wollongong, yanguang@uow.edu.au Joe F. Chicharo University o Wollongong, chicharo@uow.edu.au Publication Details Zhang, X., Xi, J., Yu, Y. & Chicharo, J. F. (8). The ourier spectrum analysis o optical eedback sel-mixing signal under weak and moderate eedback. Fourth IEEE International Symposium on Electronic Design, Test and Applications (pp ). Los Alamitos: IEEE Computer Society. Research Online is the open access institutional repository or the University o Wollongong. For urther inormation contact the UOW Library: research-pubs@uow.edu.au

2 The ourier spectrum analysis o optical eedback sel-mixing signal under weak and moderate eedback Abstract The spectrum characteristics o sel-mixing signals observed in optical eedback sel-mixing intererometry (OFSMI) is studied in this paper. The purpose is to provide guidance or the design o pre-processing techniques or eliminating noise or disturbance. The inluence o two important parameters o the OFSMI system on the spectrum, that is, the optical eedback actor and the linewidth enhancment actor, are measured by means o Discrete Fourier transorm (DFT). The simulated results show that, at a weak eedback, the OFSMI signals are strictly band limited in nature, with the cut-o requencies being the vibration requency and the ringe requency respectively. Also increasing broadens the bandwidth but has little inluence on the bandwidth o the signal. At moderate eedback, the OFSMI signals are still band limited with much higher cut-o requencies, and there is a considerable spectrum spread over a very wide range in the high requency end. The presented analysis provides suicient inormation or applying signal processing to OFSMI signals such as the ilter design. Keywords ourier, spectrum, analysis, optical, eedback, sel, mixing, signal, under, weak, moderate, eedback Disciplines Physical Sciences and Mathematics Publication Details Zhang, X., Xi, J., Yu, Y. & Chicharo, J. F. (8). The ourier spectrum analysis o optical eedback sel-mixing signal under weak and moderate eedback. Fourth IEEE International Symposium on Electronic Design, Test and Applications (pp ). Los Alamitos: IEEE Computer Society. This journal article is available at Research Online:

3 4th IEEE International Symposium on Electronic Design, Test & Applications The Fourier Spectrum Analysis o Optical Feedback Sel-Mixing Signal under Weak and Moderate Feedback Xiaojun Zhang, Jiangtao Xi, Yanguang Yu and Joe.Chichero School o Electrical, Computer & Telecommunications Engineering Faculty o Inormatics University Wollongong, Wollongong, Australia xjz787@uow.edu.au, jiangtao@uow.edu.au, yanguang@uow.edu.au Abstract The spectrum characteristics o sel-mixing signals observed in optical eedback sel-mixing intererometry (OFSMI) is studied in this paper. The purpose is to provide guidance or the design o pre-processing techniques or eliminating noise or disturbance. The inluence o two important parameters o the OFSMI system on the spectrum, that is, the optical eedback actor and the linewidth enhancment actor, are measured by means o Discrete Fourier transorm (DFT). The simulated results show that, at a weak eedback, the OFSMI signals are strictly band limited in nature, with the cut-o requencies being the vibration requency and the ringe requency respectively. Also increasing broadens the bandwidth but has little inluence on the bandwidth o the signal. At moderate eedback, the OFSMI signals are still band limited with much higher cut-o requencies, and there is a considerable spectrum spread over a very wide range in the high requency end. The presented analysis provides suicient inormation or applying signal processing to OFSMI signals such as the ilter design. 1. Introduction In the last decade, optical eedback sel-mixing intererometry (OFSMI) has attracted many researchers and extensive studies o using OFSMI have been conducted or various applications [1-3]. An OFSMI system employs a single semiconductor laser diode with a build-in photodiode and thus is much simpler in contrast to conventional intererometry techniques. A signiicant conclusion regarding the perormance o OFSMI systems is that resolution o λ is considered to be achievable or displacement measurement by means o ringe-counting method under weak or moderate eedback regime, where λ is the operating wavelength o the laser [4]. To achieve a higher resolution, other more advanced techniques must be utilized [5-9]. Signal processing plays an important role in the OFSMI systems. Due to various actors signal waveorms observed in OFSMI systems always contain noise, sparkle disturbance and amplitude luctuation. In order to achieve accurate measurement, the raw signals must be pre-processed beore any measurement technique is applied. As the signal waveorm carries useul inormation, it is always desired that preprocessing is able to eliminate the noise and disturbance and at the same time to keep the waveorm shape unchanged. Out o other signal processing techniques, iltering is always an eective tool or noise elimination or OFSMI signals. Design and implementation o suitable ilters is an important issue which requires the knowledge and characteristics o the signals. However, to the best knowledge o the authors this issue has not been thoroughly addressed by other researchers so ar. The main purpose o this paper is to study characteristics o OFSMI signals in requency domain. To achieve this, a theoretical model o OFSMI is described in Sections and the spectral properties o simulated OFSMI signals are presented in Section 3. The discussion and conclusion are given in the Section 4.. The Background o OFSMI The Optical Feedback Sel-Mixing intererence is introduced by intentionally allowing a laser beam to be relected back into laser cavity, which induces output power luctuation to the laser light. This luctuation is monitored by a photodiode within the laser package. The optical output power with optical eedback can be expressed through ollowing ormulas [1, 11]: /8 $5. 8 IEEE DOI 1.119/DELTA

4 φ ( t) = φ ( t) C sin[ φ ( t) arctan( α )] (1) g( t) = cos[ φ ( t)] () P ( t) = P [1 + mg( t)] (3) where φ ( t) and φ ( t) are the laser phase shit with and without eedback respectively, α is Linewidth Enhancement actor, C is optical eedback level actor. In case o C 1, power luctuations at laser output are similar to the sinusoidal oscillation, and the laser diode is working at weak eedback regime. When C becomes greater, the luctuation becomes more saw-tooth-like. P is the laser emitted intensity without external eedback. m is modulation index depending with the relection coeicient o the relector. In the Equation (1), the phase shit φ ( t) o external target is determined by: 4 π L( t) φ ( t) = (4) λ where L( t) is the external cavity length and λ is the wavelength o the laser without optical eedback. In case that the external cavity is subject to sinusoidal vibration, L( t) is expressed as: L( t) = L + Lsin( π vt) (5) where L is the length external cavity when the object is at the equilibrium position o the vibration and L is the amplitude o vibration respectively, v is the vibration requency. Thus φ ( t) can be rewritten as: 4π 4π φ ( t) = L L sin( π vt) λ + λ (6) Obviously, asφ is periodic with a undamental requency v, the OFSMI waveorm g ( t) will also be periodic with the same undamental requency. 1 iπ kn 1 G ( k ) = g( n) e (7) n= ote that the irst + 1 DFT components correspond to requency components equally spaced between zero and s. In addition, as g( n) is periodic, we should choose the length o DFT to be an integer multiple o K, e.g. = KM. In such a way the samples cover M undamental periods o the signal waveorm g( n ). The spectrum resolution associated with the DFT in this case is. ote that longer DFT should be used i we want to achieve higher requency resolution. 3.. Spectrum Analysis o OFSMI Signals under Weak Feedback In order to analyze the spectrum o OFSMI signal, we consider a case where the external target is subject to a simple harmonic vibration with L λ = and L λ = respectively. The signal waveorm is sampled at the rate o 56 times higher than the s vibration requency, that is, K = = 56. In this case v we have: π n φ ( n) = 8π + 8π sin( ) (8) 56 ote that we used the normalized magnitude and requencies or convenient o manipulation. The length o DFT is chosen as = 496, and hence M = 16 implying that the data covers 16 vibration cycles o the external target and that the requency resolution is v 16. s 3. Spectrum Analysis o OFMSI Signals 3.1. Basic Considerations Consider the case that the OFSMI signal g( t) is sampled at the rate o s = 1 T sample per second and t = nt = n s, a discrete sequence g( n) = g( nt) is s obtained. ote that g( n) should be periodic i is an v s integer, that is = K and a signal undamental period v has K samples. The spectrum o above can be obtained by DFTs as ollows: Figure 1. The simulated OFSMI signal with C =.7 and α = 3 Under the weak eedback regime, the waveorm o OFSMI signal is similar to a distorted sinusoidal waveorm. As an example, the irst 3 samples o simulated signal g( n) are plotted in Figure 1. 49

5 The corresponding spectrum o OFSMI signal is shown in Figure. It is seen that the spectrum exhibits a harmonic structure where there is a peak component at the undamental vibration requency plus many harmonic components. It is noticed that there is a group o harmonic components between 14 th and 4 th order o vibration requency. In time domain, each intererometric ringe appears when the displacement varies over a range o λ, or equivalently, φ ( t) varies over a π. As the result, the number o ringes in the hal vibration period o 4 L OFSMI signal g( n) is determined by. For the λ example being considered, there are about 16 ringes in each period o g( n) as shown in Figure 1. In requency domain, these ringes will appear as harmonic components with the requencies o around 16 times higher than the vibration requency, that is, the 16 th order harmonics o the vibration requency. Figure shows that the strongest peak (at 18 v ) is close to ringe requency, which is consistent to what are expected. requency v and the ringe requency respectively. I we want to eliminate noise by means o a band-pass ilter, the ilter should have the pass-band covering the requency band o the signal. However, the perormance o iltering will depend on the distribution o spectrum. We wish that the signal to be strictly band-limited in that little power spreads outside the pass-band. In order to check the inluence o band-pass ilters, we assume that an ideal band pass ilter is applied which has the cut-o requencies o 1 ( 1 < v ) and respectively. The iltered output is obtained by setting all components outside the range [ 1, ] into zeros, and then calculating the inverse DFT. Figure 3 shows the comparison between the original signal (the solid line) and the iltered one (the dotted line) or the case when 1=.686 and =.773 respectively. In order to urther check the eect o bandpass iltering, we use means square error (MSE) to measure the dierence between the iltered output and original signals as ollows: 1 σ = [ g( n) g( n) ] ɶ (9) n= 1 where gɶ ( n) and g( n) are the iltered and raw signal respectively. By keeping the above MSE a constant value, we work out the minimal which is considered as the upper cut-o requency o the ilter. Table 1 shows the results or the case o σ =.4 with C varying rom.1 to.9. Figure. Fourier spectrum o the signal Table 1. Spectrum analysis o OFSMI signal weak eedback regime with α = 3 C Upper requency MSE o error or band pass ilter residualσ.1.194( 8 v.7.773( 71 v (1 v Figure 3. The raw signal g( n) with solid line and iltered signal gɶ ( n) with cycle dot line The spectrum in Figure also shows that the OFSMI signal is band-limited with most o the energy alling within the range between the vibration Table. Spectrum analysis o OFSMI signal at weak eedback regime with C =.7 α Upper requency MSE o error or band pass ilter residualσ.7.773( 71 v.773( 71 v 4.773( 71 v 6.773( 71 v 493

6 It shows an increasing trend in bandwidth when increasing C rom.1 to.9. On the other hand, when C is ixed, the inluence o α to the power spectrum distribution is revealed by Table. An interesting eature observed is that with the increasing oα, the bandwidth o simulated signal remains almost unchanged. It was noticed that some small peaks, appeared in higher requency area in Figure 5. This is because sharp edges appear in the waveorm. As these sharp edges contain useul inormation, the stopband and cuto requency should be chosen with more care Spectrum Analysis at Moderate Feedback Regime In case o moderate eedback, the signal exhibits a sawtooth-like waveorm [1-1]. Figure 4 illustrates a single period o simulated signal g( n ) with C =.6 andα = 3. The corresponding spectrum is plotted in Figure 5. Compared with the spectrum represented in Figure, the undamental requency becomes the dominant component. On the other hand, ringe requency components become much weaker. Figure 4. First 56 samples o simulated OFSMI signal with C =.6 andα = 3 Figure 5. Fourier spectrum o the OFSMI signal Figure 7. The error residual o iltered signal gɶ ( n) We also checked the inluences o band-pass iltering to the signals using the same approach presented above or the cases o weak optical eedback. For this case being studied, the pass band o ilter is chosen to cover the requency range between 1=.7 and = Figure 6 shows the comparison between the iltered signal (in dotted lines) and uniltered signals (in solid lines). The error residual is plotted in Figure 7. As it can be seen, althoughσ is quite small with the value o.4, signiicant dierence between two signals can still be ound at the sharp edge locations. This implies that in practice it is diicult to use a band-pass ilter that is able to eliminate noise while keeping the waveorm shape unchanged at the same time. Table 3. Spectrum analysis o OFSMI signal moderate eedback regime with α = 3 C Upper requency MSE o error or band pass ilter residualσ (118 v (113 v (14 v ( 99 v Figure 6. The raw signal g( n) with solid line and iltered signal gɶ ( n) with cycle dot line Table 4. Spectrum analysis o OFSMI signal at moderate eedback regime with C =.6 α Upper requency MSE o error or band pass ilter residualσ (19 v.4453 (114 v (113 v (115 v 494

7 Table 3 shows the variance o the upper requency o ideal band-pass ilter with respect to C value when we keep the MSE constant (i.e..4). It is seen that is much higher than that in case o weak optical eedback. When C is ixed, the variation o bandwidth with respect toα is shown in Table 4. It shows thatα has a slight increase in the pass band o the ilter. 4. Conclusion The spectrum analysis was carried out or OFSMI signals based on simulation with respect to the values o C andα. In case o weak eedback regime, the eedback parameter C has signiicant inluence to the bandwidth o OFSMI signal in that increasing C will broaden the bandwidth, butα has little inluence on the bandwidth o the signals. Also as the signals are strictly band-limited, we can always design a band-pass ilter or pre-processing the signal in order to eliminate noise while keeping the signal waveorm. However at moderate eedback regime, although the signals are still close to be band limited, the spectrum spreads over a much wider range at high requency area. In this situation use o linear bandpass ilters are problematic in that it always results in signal waveorm change. In other words, it is a challenging issue to extract OFSMI signal waveorm buried in noise in the case o moderate optical eedback. intererometer," Optical Engineering, vol. 38, March. 1999, pp [7] J. Kato,. Kikuchi, I. Yamaguchi, and S. Ozono, "Optical eedback displacement sensor using a laser diode and its perormance improvement," Measurement Science and Technology, vol. 6, Jan. 1995, pp [8] M. Wang and G. Lai, "Displacement measurement based on Fourier transorm method with external laser cavity modulation," Review o Scientiic Instruments, vol. 7, Aug. 1, pp [9] M. Wang, "Fourier transorm method or sel-mixing intererence signal analysis," Optics and Laser Technology, vol. 33, Sept. 1, pp [1] W. M. Wang, K. T. V. Grattan, A. W. Palmer, and W. J. O. Boyle, "Sel-mixing intererence inside a single-mode diode laser or optical sensing applications," IEEE Journal o Lightwave Technology, vol. 1, Sept. 1994, pp [11] Jiangtao Xi, Yanguang Yu, J. F. Chicharo, and T. Bosch, "Estimating the parameters o semiconductor lasers based on weak optical eedback sel-mixing intererometry," IEEE Journal o Quantum Electronics, vol. 41, Aug. 5, pp [1] G. Giuliani, M. orgia, S. Donati, and T. Bosch, "Laser diode sel-mixing technique or sensing applications," Journal o Optics A: Pure and Applied Optics, vol. 6, ov., pp. S83-S Reerences [1] R. Lang and K. Kobayashi, "External optical eedback eects on semiconductor injection laser properties," IEEE Journal o Quantum Electronics, vol. 16, March. 198, pp [] H. W. Jentink, F. F. M. de Mul, J. Greve, H. E. Suichies, and J. G. Aarnoudse, "Small laser Doppler velocimeter based on the sel-mixing eect in a diode laser," Applied Optics, vol. 7, 1988, pp [3]. Paone and L. Scalise, "Advances in sel-mixing vibrometry," Fiber Optic Sensor Technology, vol. 44, March. 1, pp [4] M. J. Rudd, "A laser Doppler velocimeter employing the laser as a mixer-oscillator," Journal o Physics E:Scientiic Instruments, vol. 1, July. 1968, pp [5] B. Ovryn and J. H. Andrews, "Phase-shited laser eedback intererometry," Optics Letters, vol. 3, 1998, pp [6] T. Suzuki, S. Hirabayashi, O. Sasaki, and T. Maruyama, "Sel-mixing type o phase-locked laser diode 495