Abstract

A dual-wavelength erbium-doped fiber (EDF) ring laser was developed and its application to step-height measurement using two-wavelength self-mixing interferometry (SMI) was demonstrated. The fiber laser can emit two different wavelengths without any laser mode competition. It is composed of two EDF laser cavities and employs fiber Bragg gratings to determine which wavelengths are emitted. The step heights can be measured using SMI of the two wavelengths, and the maximum height that can be measured is half the synthetic wavelength of the two wavelengths. A step height of 1mm was constructed using two gauge blocks and then measured using the laser. The measurement was repeated ten times, and the standard deviation of the measurements was 2.4nm.

© 2016 Optical Society of America

1. Introduction

Self-mixing interference (SMI) occurs in lasers when the light emitted from a laser is reflected by an object outside the laser cavity, and the reflected light then re-enters the cavity where it mixes with the original light. The theory and applications of SMI have been discussed in [1].The concept of SMI is similar to that of two-beam interference (TBI), and both have the same phase sensitivity, i.e., the shift of one fringe corresponds to a variation in the displacement of a half wavelength (λ/2). However, compared to TBI, the configuration of SMI within the laser is considerably simple. As there are no additional optical components such as beam splitters and reflective mirrors in the system, it eliminates the need to adjust any optical components. In addition, because the two mixed beams of SMI have almost identical optical paths, the SMI system is sufficiently robust for on-line measurements.

Because of these advantages, SMI has been adopted for the measurement of distance [1,2], displacement [3], velocity [4], and vibration [5]. Several different kinds of lasers, such as a gas laser [5–9], laser diode [2], and fiber laser [10], have been employed for SMI. As there is some phase ambiguity in the SMI signal, when SMI is used to measure distance and displacement, the maximum step height must be limited to λ/2. Thus, conventional SMI cannot be used to measure large step heights.

In this study, we developed a novel dual-wavelength fiber ring laser that utilizes two-wavelength SMI, and demonstrated its ability to measure the step height. This fiber laser, which is composed of two erbium-doped fiber (EDF) laser cavities and employs fiber Bragg gratings (FBGs) to determine the emission wavelength of each cavity, can simultaneously emit two different wavelengths. As there is no laser mode competition between the two wavelengths, the power and frequency of each wavelength are stable during the operation of the fiber ring laser. By employing SMI, a step height larger than λ/2 can be measured using this laser. The maximum step height that can be measured using this technology is half the synthetic wavelength of the two wavelengths. As the values of the two wavelengths are determined using the FBGs, they can be arbitrarily chosen based on the value of the step height, and step heights ranging from zero to half the synthetic wavelength can be measured. In our experiments, we measured a step height of 1mm, which was constructed using two gauge blocks. We repeated this measurement ten times, and the standard deviation of the ten measurements was 2.4nm.

2. Principle of dual-wavelength fiber ring laser

The dual-wavelength fiber ring laser includes two laser cavities as shown in the schematic diagram of the laser in Fig. 1. The light from the 980nm laser is split into two beams as it passes through the 3-dB-coupler 1. One beam is coupled into one of the laser cavities through a wavelength-division-multiplexer(WDM), which includes WDM 1, a length of erbium-doped fiber (EDF 1), circulator 1 (C1), fiber Bragg grating (FBG11), 3-dB-coupler 2, circulator 3 (C3), fiber Bragg grating (FBG12), and coupler 3(1:9). The WDM has a power splitting ratio of 1:9. Simultaneously, the other beam is coupled into the other laser cavity through another WDM, which includes WDM 2, a length of erbium-doped fiber (EDF 2), circulator 2 (C2), fiber Bragg grating (FBG21), 3-dB-coupler 2, circulator 3 (C3), fiber Bragg grating (FBG12),circulator 4 (C4), fiber Bragg grating (FBG22), and coupler 4(1:9). This WDM also has a power splitting ratio of 1:9. The FBGs are used to select and stabilize the wavelengths emitted from the laser. The emitted wavelengths will be the same as the Bragg wavelength of the FBGs. FBG11 and FBG12 have the same Bragg wavelength λ1, which permits one laser cavity to emit light with a wavelength of λ1 from one port of the coupler 3 (1:9). Similarly, FBG21 and FBG22 have the same Bragg wavelength λ2, which permits the other laser cavity to emit light with a wavelength of λ2 from one port of the coupler 4 (1:9). As both the laser cavities possess a gain medium (the EDF), the two wavelengths emitted from the fiber ring laser do not compete with each other. The power and frequency of each wavelength were stable throughout the operation of the fiber ring laser. Moreover, because the value of each wavelength is determined by the Bragg wavelengths of the FBGs, the two emitted wavelengths can be arbitrarily chosen.

 figure: Fig. 1

Fig. 1 Schematic diagram of the dual-wavelength fiber ring laser

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3. Principle of step-height measurement using dual-wavelength SMI

In order to measure the step height using SMI of the dual-wavelength fiber ring laser, another circulator (C5) was added within the common length of the two fiber laser cavities, as shown in Fig. 2. One port of C5 is connected to a GRIN lens. The light emitted from the GRIN lensreflects off of the object to be measured and then re-enters the laser cavity, where it mixes with the original light to produce SMI. The GRIN lens end and measured surface comprise the SMI cavity, and the distance between the two is Lext. The two SMI signals of the two wavelengths are outputs from one port of coupler 3 and one port of coupler 4, and are detected by the photo detectors (PDs) PD1 and PD2, respectively. The signals detected by the two PDs are processed using the electronic processor and are converted into digital data by an analog-to-digital (A/D) converter. Finally, the digital data are processed using a software program running on a personal computer to obtain the final measurement results.

 figure: Fig. 2

Fig. 2 Principle of step-height measurement based on SMI of the dual-wavelength fiber ring laser

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As the concept of SMI is similar to that of TBI, the SMI signals detected by PD1 and PD2 can be expressed approximately as follows:

P1=P10+k1cos(2ω1Lextc)=P10+k1cos(4πLextλ1)
P2=P20+k2cos(2ω2Lextc)=P20+k2cos(4πLextλ2)
where ω1 and ω2 are the two frequencies emitted by the fiber laser, λ1 and λ2 are the two wavelengths emitted by the fiber laser, c is the velocity of light in vacuum, P10 and P20 are the dc components of the signals, k is a constant coefficient, and Lext is the distance between the GRIN lens end and measured surface.

To measure the step height, the beam is projected on the two parts of the step height, as shown in Fig. 3. The beam is first projected perpendicularly on point a on part 1 and then perpendicularly on point b on part 2.

 figure: Fig. 3

Fig. 3 Procedure for step-height measurement

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When the beam is projected on point a, the phases of the SMI signals of the two wavelengths can be expressed as follows:

φ1a=φ10a+2m1aπ=4πLext1λ1
φ2a=φ20a+2m2aπ=4πLext1λ2
where φa 1 and φa 2 are the phases of the SMI signals, φa 10 and φa 20 are the decimal parts of the two phases, ma 1 and ma 2 are integers, and Lext1 is the SMI cavity corresponding to point a on part 1.

When the beam is projected on point b, the phases of the SMI signals of the two wavelengths will have the same forms as Eqs. (3) and (4); however, the superscript “a” will be replaced by “b” and “Lext1” will be replaced by “Lext2.

When the step height Δhλ1λ22(λ1λ2), based on Eqs. (3) and (4) and the phases of the SMI signals when the beam is projected on point b, Δh can be calculated as follows:

Δh=Lext1Lext2=λ1λ24π(λ1λ2)[(φ10bφ10a)(φ20bφ20a)]
where φa 10, φa 20, φb 10, and φb 20 are the decimal parts of the two phases. It can be found from Eq. (5) that by measuring only the decimal parts of the phase, the step height Δh can be calculated.

4. Measurement of step height using dual-wavelength SMI

A dual-wavelength fiber ring laser based on the schematic diagram shown in Fig. 1 was constructed. The output power of the 980nm laser, which is used as the pump light source, is 320mW. The Bragg wavelengths of FBG11 and FBG12 are 1550.084nm with a full width at half maximum (FWHM) of 0.22nm. The Bragg wavelengths of FBG21 and FBG22 are 1551.064nm with a FWHM of 0.23nm. The output spectrum of the dual-wavelength fiber ring laser is monitored using an optical spectrum analyzer in which there are two wavelengths of 1550.084nm and 1551.064nm, as shown in Fig. 4. The power of wavelength 1551.064nm is lower than that of wavelength 1550.084nm because the reflectivity of FBG11 and FBG12 (95%) is higher than that of FBG21 and FBG22 (90%). In our experiments, the dual-wavelength fiber ring laser had been operating continuously for 4 hours, during which the wavelength shifting was recorded as shown in Fig. 5. From Fig. 5, it can be observed that the maximum shift value of the emitted wavelength is about 0.02nm, which indicates that the stability of the emitted wavelengths is considerably good.

 figure: Fig. 4

Fig. 4 Spectrum of the light emitted from the dual-wavelength fiber ring laser

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 figure: Fig. 5

Fig. 5 Fluctuations in the wavelengths emitted from the dual-wavelength fiber ring laser

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The step height measured in the experiments was constructed using two gauge blocks, which are shown in Fig. 6. Each block had a height of 1mm and was attached to the other to form a step with a nominal height of 1mm. The step height was measured using SMI of the dual-wavelength fiber ring laser, as shown in Fig. 2. First, the light emitted from the GRIN lens was projected perpendicularly on point a on the surface of gauge block 1. It was reflected back by the surface and then re-entered the system. A one-dimensional translational stage (TS1, P-625.1CD, Physik Instrumente(PI) GmbH & Co.KG) was used to modulate the length Lext linearly, and simultaneously, PD1 and PD2 were used to detect the two SMI signals of wavelength λ1 and λ2, respectively. The two detected signals were then processed using the electronic processor, and were converted into digital data using an A/D converter. Finally, thedata were processed using a software program running in a personal computer. The signals shown in Fig. 7(a) are the two signals detected by PD1 and PD2, respectively, when Lext was modulated linearly with a value of 14μm, which indicates that the two signals were varying periodically. By processing the detected signals shown in Fig. 7(a), the phase decimal parts, φa 10 and φa 20, of the SMI signals were obtained. Second, another translation stage (TS2) was used to move the two gauge blocks laterally and allow the beam to project on point b on the surface of gauge block 2. By repeating the measurement procedure used for gauge block 1, the phase decimal parts for point b, φb 10 and φb 20, were also obtained. The step height Δh was calculated with the phase decimal parts, φa 10, φa 20, φb 10, and φb 20, and the relationship shown in Eq. (5).

 figure: Fig. 6

Fig. 6 Step height constructed using two gauge blocks

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 figure: Fig. 7

Fig. 7 (a) SMI signals and (b) ten step height measurement results

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During these experiments, the step height was measured ten times. The nominal value of the step height of gauge block 1 was 1.00005mm, and the measurement results are shown in Fig. 7(b). The corresponding standard deviation of the ten measurements was 2.4nm.

Compared to the sine curve, the SMI signals shown in Fig. 7(a) are distorted; however, one period still corresponds to a displacement of λ/2. The level of distortion is determined by the optical feedback factor, which changes with the reflectivity of the two surfaces of the SMI cavity. The level of distortion will be lower as the optical feedback factor becomes smaller. When the reflectivity of the measured surface is lower, the optical feedback factor is also smaller. In these experiments, the reflectivity of the GRIN lens end is about 4% and that of the gauge block is about 60%. This indicates that a moderate feedback regime should be considered. We have experimentally measured surfaces with different amounts of reflectivity, and have discovered that when the reflectivity of the two cavity surfaces is about 4%, the SMI signal is precisely the same as a sine curve. Furthermore, these experiments have demonstrated that whether the reflectivity is high or low, the SMI signals will still have the same period.

In order to improve the measurement precision, when the two wavelengths are chosen, half the synthetic wavelength of the two wavelengths should be larger than, but almost equal to, the measured step height. If half the synthetic wavelength is much larger than the measured step height, it would be similar to using a large scale to measure a small dimension, making a high measurement precision difficult to achieve.

5. Conclusion

A dual-wavelength fiber ring laser was developed and used to measure step height using SMI of the two wavelengths emitted from the fiber ring laser. In this laser, there is no laser mode competition between the two wavelengths, and each wavelength has a stable power maintained at all times. Because FBGs are used to determine the wavelengths emitted by the laser, the wavelengths can be chosen arbitrarily. It has been proven experimentally that the step height can be measured effectively using SMI of the dual-wavelength fiber ring laser. The standard deviation when measuring a height of 1 mm is 2.4nm.

Acknowledgments

The authors thank the Beijing Natural Science Foundation (3132033) for supporting this research.

References and links

1. T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015). [CrossRef]  

2. G. Mourat, N. Servagent, and T. Bosch, “Distance measurement using the self-mixing effect in a three-electrode distributed Bragg reflector laser diode,” Opt. Eng. 39(3), 738–743 (2000). [CrossRef]  

3. L. Fei and S. Zhang, “The discovery of nanometer fringes in laser self-mixing interference,” Opt. Commun. 273(1), 226–230 (2007). [CrossRef]  

4. X. Cheng and S. Zhang, “Multiple self-mixing effect in VCSELs with asymmetric external cavity,” Opt. Commun. 260(1), 50–56 (2006). [CrossRef]  

5. J. Li, Y. Tan, and S. Zhang, “Generation of phase difference between self-mixing signals in a-cut Nd:YVO₄ laser with a waveplate in the external cavity,” Opt. Lett. 40(15), 3615–3618 (2015). [CrossRef]   [PubMed]  

6. W. Mao, S. Zhang, L. Cui, and Y. Tan, “Self-mixing interference effects with a folding feedback cavity in Zeeman-birefringence dual frequency laser,” Opt. Express 14(1), 182–189 (2006). [CrossRef]   [PubMed]  

7. Y. Tan, S. Zhang, and Y. Zhang, “Laser feedback interferometry based on phase difference of orthogonally polarized lights in external birefringence cavity,” Opt. Express 17(16), 13939–13945 (2009). [CrossRef]   [PubMed]  

8. S. Zhang, Y. Tan, and Y. Li, “Orthogonally polarized dual frequency lasers and applications in self-sensing metrology,” Meas. Sci. Technol. 21(5), 054016 (2010). [CrossRef]  

9. G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003). [CrossRef]  

10. X. Dai, M. Wang, Y. Zhao, and J. Zhou, “Self-mixing interference in fiber ring laser and its application for vibration measurement,” Opt. Express 17(19), 16543–16548 (2009). [CrossRef]   [PubMed]  

References

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  1. T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
    [Crossref]
  2. G. Mourat, N. Servagent, and T. Bosch, “Distance measurement using the self-mixing effect in a three-electrode distributed Bragg reflector laser diode,” Opt. Eng. 39(3), 738–743 (2000).
    [Crossref]
  3. L. Fei and S. Zhang, “The discovery of nanometer fringes in laser self-mixing interference,” Opt. Commun. 273(1), 226–230 (2007).
    [Crossref]
  4. X. Cheng and S. Zhang, “Multiple self-mixing effect in VCSELs with asymmetric external cavity,” Opt. Commun. 260(1), 50–56 (2006).
    [Crossref]
  5. J. Li, Y. Tan, and S. Zhang, “Generation of phase difference between self-mixing signals in a-cut Nd:YVO₄ laser with a waveplate in the external cavity,” Opt. Lett. 40(15), 3615–3618 (2015).
    [Crossref] [PubMed]
  6. W. Mao, S. Zhang, L. Cui, and Y. Tan, “Self-mixing interference effects with a folding feedback cavity in Zeeman-birefringence dual frequency laser,” Opt. Express 14(1), 182–189 (2006).
    [Crossref] [PubMed]
  7. Y. Tan, S. Zhang, and Y. Zhang, “Laser feedback interferometry based on phase difference of orthogonally polarized lights in external birefringence cavity,” Opt. Express 17(16), 13939–13945 (2009).
    [Crossref] [PubMed]
  8. S. Zhang, Y. Tan, and Y. Li, “Orthogonally polarized dual frequency lasers and applications in self-sensing metrology,” Meas. Sci. Technol. 21(5), 054016 (2010).
    [Crossref]
  9. G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
    [Crossref]
  10. X. Dai, M. Wang, Y. Zhao, and J. Zhou, “Self-mixing interference in fiber ring laser and its application for vibration measurement,” Opt. Express 17(19), 16543–16548 (2009).
    [Crossref] [PubMed]

2015 (2)

T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

J. Li, Y. Tan, and S. Zhang, “Generation of phase difference between self-mixing signals in a-cut Nd:YVO₄ laser with a waveplate in the external cavity,” Opt. Lett. 40(15), 3615–3618 (2015).
[Crossref] [PubMed]

2010 (1)

S. Zhang, Y. Tan, and Y. Li, “Orthogonally polarized dual frequency lasers and applications in self-sensing metrology,” Meas. Sci. Technol. 21(5), 054016 (2010).
[Crossref]

2009 (2)

2007 (1)

L. Fei and S. Zhang, “The discovery of nanometer fringes in laser self-mixing interference,” Opt. Commun. 273(1), 226–230 (2007).
[Crossref]

2006 (2)

2003 (1)

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

2000 (1)

G. Mourat, N. Servagent, and T. Bosch, “Distance measurement using the self-mixing effect in a three-electrode distributed Bragg reflector laser diode,” Opt. Eng. 39(3), 738–743 (2000).
[Crossref]

Bertling, K.

T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

Bosch, T.

T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

G. Mourat, N. Servagent, and T. Bosch, “Distance measurement using the self-mixing effect in a three-electrode distributed Bragg reflector laser diode,” Opt. Eng. 39(3), 738–743 (2000).
[Crossref]

Cheng, X.

X. Cheng and S. Zhang, “Multiple self-mixing effect in VCSELs with asymmetric external cavity,” Opt. Commun. 260(1), 50–56 (2006).
[Crossref]

Cui, L.

Dai, X.

Fei, L.

L. Fei and S. Zhang, “The discovery of nanometer fringes in laser self-mixing interference,” Opt. Commun. 273(1), 226–230 (2007).
[Crossref]

Li, J.

Li, Y.

S. Zhang, Y. Tan, and Y. Li, “Orthogonally polarized dual frequency lasers and applications in self-sensing metrology,” Meas. Sci. Technol. 21(5), 054016 (2010).
[Crossref]

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

Lim, Y. L.

T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

Liu, G.

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

Mao, W.

Mourat, G.

G. Mourat, N. Servagent, and T. Bosch, “Distance measurement using the self-mixing effect in a three-electrode distributed Bragg reflector laser diode,” Opt. Eng. 39(3), 738–743 (2000).
[Crossref]

Nikolic, M.

T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

Rakic, A.

T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

Servagent, N.

G. Mourat, N. Servagent, and T. Bosch, “Distance measurement using the self-mixing effect in a three-electrode distributed Bragg reflector laser diode,” Opt. Eng. 39(3), 738–743 (2000).
[Crossref]

Taimre, T.

T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

Tan, Y.

Wang, M.

Zhang, S.

J. Li, Y. Tan, and S. Zhang, “Generation of phase difference between self-mixing signals in a-cut Nd:YVO₄ laser with a waveplate in the external cavity,” Opt. Lett. 40(15), 3615–3618 (2015).
[Crossref] [PubMed]

S. Zhang, Y. Tan, and Y. Li, “Orthogonally polarized dual frequency lasers and applications in self-sensing metrology,” Meas. Sci. Technol. 21(5), 054016 (2010).
[Crossref]

Y. Tan, S. Zhang, and Y. Zhang, “Laser feedback interferometry based on phase difference of orthogonally polarized lights in external birefringence cavity,” Opt. Express 17(16), 13939–13945 (2009).
[Crossref] [PubMed]

L. Fei and S. Zhang, “The discovery of nanometer fringes in laser self-mixing interference,” Opt. Commun. 273(1), 226–230 (2007).
[Crossref]

X. Cheng and S. Zhang, “Multiple self-mixing effect in VCSELs with asymmetric external cavity,” Opt. Commun. 260(1), 50–56 (2006).
[Crossref]

W. Mao, S. Zhang, L. Cui, and Y. Tan, “Self-mixing interference effects with a folding feedback cavity in Zeeman-birefringence dual frequency laser,” Opt. Express 14(1), 182–189 (2006).
[Crossref] [PubMed]

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

Zhang, Y.

Zhao, Y.

Zhou, J.

Zhu, J.

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

Adv. Opt. Photonics (1)

T. Taimre, M. Nikolic, K. Bertling, Y. L. Lim, T. Bosch, and A. Rakic, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015).
[Crossref]

Meas. Sci. Technol. (1)

S. Zhang, Y. Tan, and Y. Li, “Orthogonally polarized dual frequency lasers and applications in self-sensing metrology,” Meas. Sci. Technol. 21(5), 054016 (2010).
[Crossref]

Opt. Commun. (3)

G. Liu, S. Zhang, J. Zhu, and Y. Li, “Theoretical and experimental study of intensity branch phenomena in self-mixing interference in a He-Ne laser,” Opt. Commun. 221(4–6), 387–393 (2003).
[Crossref]

L. Fei and S. Zhang, “The discovery of nanometer fringes in laser self-mixing interference,” Opt. Commun. 273(1), 226–230 (2007).
[Crossref]

X. Cheng and S. Zhang, “Multiple self-mixing effect in VCSELs with asymmetric external cavity,” Opt. Commun. 260(1), 50–56 (2006).
[Crossref]

Opt. Eng. (1)

G. Mourat, N. Servagent, and T. Bosch, “Distance measurement using the self-mixing effect in a three-electrode distributed Bragg reflector laser diode,” Opt. Eng. 39(3), 738–743 (2000).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

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Figures (7)

Fig. 1
Fig. 1 Schematic diagram of the dual-wavelength fiber ring laser
Fig. 2
Fig. 2 Principle of step-height measurement based on SMI of the dual-wavelength fiber ring laser
Fig. 3
Fig. 3 Procedure for step-height measurement
Fig. 4
Fig. 4 Spectrum of the light emitted from the dual-wavelength fiber ring laser
Fig. 5
Fig. 5 Fluctuations in the wavelengths emitted from the dual-wavelength fiber ring laser
Fig. 6
Fig. 6 Step height constructed using two gauge blocks
Fig. 7
Fig. 7 (a) SMI signals and (b) ten step height measurement results

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

P 1 = P 10 + k 1 cos( 2 ω 1 L ext c )= P 10 + k 1 cos( 4π L ext λ 1 )
P 2 = P 20 + k 2 cos( 2 ω 2 L ext c )= P 20 + k 2 cos( 4π L ext λ 2 )
φ 1 a = φ 10 a +2 m 1 a π=4π L ext1 λ 1
φ 2 a = φ 20 a +2 m 2 a π=4π L ext1 λ 2
Δh= L ext1 L ext2 = λ 1 λ 2 4π( λ 1 λ 2 ) [ ( φ 10 b φ 10 a )( φ 20 b φ 20 a ) ]

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