Abstract

We propose and demonstrate a novel method to enhance the visibility of an optical interferometer when measuring low reflective materials. Because of scattering from a rough surface or its own low reflectivity, the visibility of the obtained interference signal is seriously deteriorated. By amplifying the weak light coming from the sample based on an injection-locking technique, the visibility can be enhanced. As a feasibility test, even with a sample having a reflectivity of 0.6%, we obtained almost the same visibility as a metal coated mirror. The suggested visibility enhanced interferometer can be widely used for measuring low reflective materials.

© 2010 OSA

1. Introduction

An optical interferometer with stabilized lasers can realize a standard ‘meter’ based on the speed of light, which is traceable to the length standard [13]. Because it also has a sub-nm measurement resolution, it has been exploited for various applications that need ultraprecision measurements. Although the optical interferometer is considered to be one of the most accurate measurement techniques, practical difficulties remain especially for low reflective materials. Low reflective materials are quite common in real measurements, which include bare glass, color filters, samples with rough surfaces as well as samples being soaked in liquid such as living cells, molecules, etc.

Of such measurements, precision calibration of accelerometers based on optical interferometry is one of the most important examples [46]. Accelerometers have been widely used is such applications as monitoring mechanical vibrations and earthquakes. They have to be calibrated repeatedly for reliable measurements because they are connected to human safety. For that purpose, national metrology institutes have installed and operated calibration systems composed of a vibration exciter and a laser interferometer system that is traceable to the international length and time standards. While modulating the accelerometer attached to the exciter, its displacement is measured using a laser interferometer. In this case, so as to obtain high reflectivity, an additional mirror has to be attached to the top surface of the accelerometers because it does not have a polished surface like a mirror but might have damages and scratches caused by mechanical contact with buildings, structures, and machines. However, the additional mass of the mirror attached to the top surface of the accelerometer distorts its own sensitivity.

For reliable measurements, the visibility of the interference signals needs to be sufficiently high to obtain and analyze the interference signals accurately. When measuring low reflective materials, however, the visibility deteriorates seriously due to the weak light reflected from the sample. This is because visibility is greatest when the intensities of the light from both arms of the interferometer are exactly the same. To avoid this degradation of visibility, the intensity of light coming from the measurement arm needs to be amplified to the same level as that of the reference beam. Alternatively, the intensity of light reflected from the reference mirror could be reduced to the same level as that of the measurement beam. In the latter case, interference signals are sometimes not observable if they are too weak to be detected.

In this paper, we suggest a novel method to enhance the visibility of optical interferometers to overcome these practical problems occurring in the measurement of low reflectivity materials. The visibility enhanced interferometer (VEI) can increase visibility by amplifying the reflected light from the sample with the help of an injection-locking technique. The seed of injection locking, even if it is very weak, can be amplified while maintaining its frequency and phase information. This is a first trial of using a secondary light source in only one arm of the interferometer to enhance visibility for real applications. The suggested VEI is expected to expand applicability of optical interferometers from standard tasks to industrial applications.

2. Basic principle and realization of VEI

The strength of interference is measured by the visibility, V. It is also called the modulation depth of the contrast of the interference signal. It depends on the intensities reflected from the reference mirror and the sample (or the measurement mirror), Ir and Is, which can be expressed by

V=2IrIsIr+Is.

When the intensities from the reference mirror and the sample are exactly the same, Ir=Is, the visibility is 1, the largest value. If the intensity from the sample, Is, is weak, the visibility approaches 0, the smallest value. When the visibility approaches 1, the interference signal is clear, which is required for reliable measurements. Therefore, to enhance visibility, the light reflected from the low reflective samples should be amplified to have the same intensity or at least a similar intensity as the reference. Alternatively, the intensity of the reference beam could be reduced to match the intensity of the light reflected from the sample. In the latter case, however, the interference signals sometimes cannot be detected because of the low optical power. In addition, whenever a sample having variable reflectivity is measured, the intensity of each arm needs to be adjusted carefully for enhanced visibility.

To realize this, the first concept based on optical amplification was adopted in this work. One optical amplification method useful for this purpose is an injection-locking technique, whose advantages are a high amplification factor, low noise, and easy configuration. Injection locking makes the frequency of an additional laser, referred to as the slave laser, the same as that of a seed laser [710]. The weak seed laser can be amplified to the optical power of the slave laser.

Figure 1 shows an optical layout of VEI. The light source is a distributed feedback laser having a center wavelength of 1542.295 nm with an optical power of 10 mW (FRL15DCWD-A81-19435-C, Furukawa Electronic). The frequency stability was estimated ~10−8 at 1 s. The light is split into two parts: one for reference, the other for measurement. In the measurement arm, an ND filter was used to simulate changing reflectivity of the sample. The transmittance of the ND filter was 0.08, which gives a reflectance of 0.006. A similar model of light source without an isolator (FRL15DCWA-A81-19435, Furukawa Electronic) was adopted as the slave laser. The mirror in the measurement arm was modulated by a ramp input having a travel of 20 μm with a rate of 0.3 Hz. The optical spectrum analyzer (OSA, 86142B, Agilent) gives the spectrum of the seed laser or the slave laser to check the state of injection locking. The wavelengths of the DFB laser and the slave laser were determined by using the OSA having an uncertainty for wavelength detection of 0.01 nm. The interference signals were observed by a photo-detector.

 

Fig. 1 Optical layout of VEI (DFB laser: distributed feedback laser, CL: collimation lens, M: mirror, C: circulator, PD: photo detector, OSA: optical spectrum analyzer, ND filter: neutral density filter).

Download Full Size | PPT Slide | PDF

To confirm the state of injection locking, the frequency shift was observed in the spectral domain. Figure 2 shows two optical spectra: free running of the slave laser, and the injection-locked light. The injection-locked light has a wavelength of 1542.295 nm, which is exactly the same wavelength as the light source in use. The wavelength difference between the seed laser and the slave laser before injection locking was ~0.025 nm, corresponding to ~3 GHz frequency difference. The wavelength difference was large enough to detect using the OSA. When light reflected from the sample was injected into the slave laser, a wavelength shift from that of the slave laser to that of the seed laser could be observed as shown in Fig. 2. By blocking the measurement arm to remove the seed laser, the optical spectrum of the slave laser was observed. Therefore, it could be ascertained that the wavelengths of the two lasers become exactly the same after injection locking.

 

Fig. 2 Spectrums of the master laser, the slave laser, and the injection-locked light. The intensity of the spectrum of the master laser was adjusted for easy comparison.

Download Full Size | PPT Slide | PDF

Figure 3 shows the obtained interference signals with and without the ND filter whose transmittance of 0.08 was measured using VEI. The visibility obtained by the interferometer without the injection-locking technique was reduced to 10% when inserting the ND filter as shown in Fig. 3 (d). However, the visibility obtained by VEI was maintained and kept at almost the same level as that before inserting the ND filter. In other words, VEI could enhance the visibility of the interference signals even if the reflected light of the sample is only 0.6%. To avoid unwanted uncertainties related to environmental parameters and optical fiber components, the interference signals were obtained with a low-pass filter having a cutoff frequency of 30 Hz. This is for evaluating the VEI only from the viewpoint of visibility enhancement [11].

 

Fig. 3 Interference signals obtained by VEI and a normal interferometer; (a) VEI without the ND filter (V=0.65), (b) VEI with the ND filter (V=0.65), (c) The normal interferometer without the ND filter (V=0.69), (d) The normal interferometer with the ND filter (V=0.06). Figures (a) and (c) were measured with two metal coated mirrors, and figures (b) and (d) were measured with the ND filter having a transmittance of 0.08. For clear recognition, the two-dimensional figures were generated numerically from the obtained interference signal profiles.

Download Full Size | PPT Slide | PDF

3. Discussion and summary

VEI has been demonstrated with the help of an injection-locking technique. It is the first trial to adopt a secondary light source in one arm of interferometers to enhance visibility for real applications. In the feasibility test, the visibility obtained by VEI was enhanced successfully for a sample having a reflectance of 0.6%. The visibility was almost the same as that obtained by the same interferometer when measuring a metal coated mirror having a reflectance of 0.99. However, the visibility obtained by the interferometer without using the injection-locking technique was substantially reduced to 0.06 (90% reduction) under the same conditions.

For injection locking, the phase difference between the seed laser and the slave laser should be considered carefully because the amplification is performed only in the measurement arm. A deviation in the phase difference might cause substantial measurement error. It could even lose the phase information of the samples. According to the previously reported work [12], the phase difference, Δφ, is expressed as

Δφ=sin1(Δν/ΔνL)tan1α,
where Δν is the frequency difference between the seed laser and the slave laser before injection locking, ΔνL is the injection-locking range, and α is the line-width enhancement factor, which is a constant in the range of 3 to 7 for semiconductor lasers [13,14]. A constant phase difference is not a considering factor for optical interferometers because it just corresponds to a fixed additional optical path. However, a time varying phase difference can badly affect measurement results. The fluctuation of phase difference, δΔφ, is proportional to the control noise of the driving current for the slave laser, δI, under the following conditions: (1) the frequency difference between the seed laser and the slave laser is much less than the injection-locking range, Δν/ΔνL<<1; (2) the frequency of the slave laser is proportional to the driving current, I, and (3) the temperature is stabilized. Here, the long-term drift of temperature was ignored considering the short measurement time. From Eq. (2), the fluctuation of phase difference is given by [711]
δΔφ=γδI/ΔνL,
where γ is a proportional constant. The injection-locking range was measured as ~8.3 GHz in frequency, corresponding to ~0.07 nm in wavelength. Because of the ~3 GHz frequency difference between them, the phase difference between the slave laser and the seed laser is nonzero according to Eq. (2). However, it makes for a constant phase delay, which is not critical for optical interferometers as mentioned before. Actually, in our experiments the frequency difference can be made much less than 3 GHz in practice, but it made for easy checking of the state of injection locking. The spectra obtained by the OSA in Fig. 2 shows the state of injection locking clearly. The ratio of the frequency difference and the locking range, Δν/ΔνL, was 0.36, which does not satisfy the condition of Δν/ΔνL<<1 required for using Eq. (3). Therefore, Eq. (3) should be modified by

δΔφ=γδI/ΔνL1cos(Δν/ΔνL).

The proportionality constant of the slave laser, γ, in Eq. (3) and Eq. (4) was estimated to be 0.57 MHz/mA, which equals 4.5 pm/mA in wavelength terms. It was determined by monitoring the frequency shift of the slave laser while increasing the driving current up to 160 mA. The additional term of Eq. (4), 1/cos(Δν/ΔνL), was 1.07 at these experimental conditions. The resolution of the driving current controller in use (ITC502, Thorlabs) was 1 μA with a typical noise less than 1.5 μA. Therefore, the critical factor, time varying phase shift, was estimated to be ~10−4 rad from Eq. (4), corresponding to less than 0.1 nm in length at the center wavelength of 1542.295 nm. Because the time varying phase difference term is very small, it could be neglected. Even if it is larger, it also can be easily removed using a low pass filter because the frequency of the time varying phase shift term is relatively high, depending on the feedback control rate of the driving current according to Eq. (4). Therefore, the effect of phase difference, Δφ, can be ignored in general use.

In conclusion, the suggested VEI works successfully even if the reflectance of a sample is very small, less than 1%. It is expected to expand applicable areas of VEI especially for length measurements of roughly surfaced and transparent materials. In the near future, we have a plan to use VEI for constructing a new calibration system of accelerometers for monitoring earthquake waves and leveling ultraprecision machines for high-technology applications. Vibration detection and length measurement of glass and color filters in the field of flat panel displays will also be considered as an application of this technique.

Acknowledgment

This work was supported in part by the National Program: Development of Application Technologies of Physical Measurement Standards (GP2010-0001), KRISS.

References and links

1. V. M. Khavinson, “Ring interferometer for two-sided measurement of the absolute lengths of end standards,” Appl. Opt. 38(1), 126–135 (1999). [CrossRef]  

2. J. Jin, Y.-J. Kim, Y. Kim, S.-W. Kim, and C.-S. Kang, “Absolute length calibration of gauge blocks using optical comb of a femtosecond pulse laser,” Opt. Express 14(13), 5968–5974 (2006). [CrossRef]   [PubMed]  

3. J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and T. B. Eom, “Thickness and refractive index measurement of a silicon wafer based on an optical comb,” Opt. Express 18(17), 18339–18346 (2010). [CrossRef]   [PubMed]  

4. “Methods for the calibration of vibration and shock transducers: 1. Basic concepts,” International Standard ISO 16063–1 (1998).

5. “Methods for the calibration of vibration and shock transducers: II. Primary vibration calibration by laser interferometry,” International Standard ISO 16063–11 (1999).

6. Y.-B. Lee, H. C. Kim, and S.-W. Kim, “Determination of the sensitivity phase of an accelerometer based on an analysis of the harmonic components of the interference signal,” Meas. Sci. Technol. 19(4), 045204 (2008). [CrossRef]  

7. G. R. Hadley, “Injection locking of diode lasers,” IEEE J. Quantum Electron. 22(3), 419–426 (1986). [CrossRef]  

8. R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982). [CrossRef]  

9. G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7(2), 87–143 (1995). [CrossRef]  

10. L. Zhang, R. Dou, and J. Chen, “Characteristics of the injection-locked master-slave lasers,” Appl. Opt. 47(14), 2648–2654 (2008). [CrossRef]   [PubMed]  

11. J. Jin, H.-G. Rhee, and S.-W. Kim, “Metrological atomic force microscopy integrated with a modified two-point diffraction interferometer,” Meas. Sci. Technol. 20(10), 105302 (2009). [CrossRef]  

12. F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985). [CrossRef]  

13. J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos (Springer, 2008) 2nd edition, pp. 32–33.

14. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. V. M. Khavinson, “Ring interferometer for two-sided measurement of the absolute lengths of end standards,” Appl. Opt. 38(1), 126–135 (1999).
    [Crossref]
  2. J. Jin, Y.-J. Kim, Y. Kim, S.-W. Kim, and C.-S. Kang, “Absolute length calibration of gauge blocks using optical comb of a femtosecond pulse laser,” Opt. Express 14(13), 5968–5974 (2006).
    [Crossref] [PubMed]
  3. J. Jin, J. W. Kim, C.-S. Kang, J.-A. Kim, and T. B. Eom, “Thickness and refractive index measurement of a silicon wafer based on an optical comb,” Opt. Express 18(17), 18339–18346 (2010).
    [Crossref] [PubMed]
  4. “Methods for the calibration of vibration and shock transducers: 1. Basic concepts,” International Standard ISO 16063–1 (1998).
  5. “Methods for the calibration of vibration and shock transducers: II. Primary vibration calibration by laser interferometry,” International Standard ISO 16063–11 (1999).
  6. Y.-B. Lee, H. C. Kim, and S.-W. Kim, “Determination of the sensitivity phase of an accelerometer based on an analysis of the harmonic components of the interference signal,” Meas. Sci. Technol. 19(4), 045204 (2008).
    [Crossref]
  7. G. R. Hadley, “Injection locking of diode lasers,” IEEE J. Quantum Electron. 22(3), 419–426 (1986).
    [Crossref]
  8. R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982).
    [Crossref]
  9. G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7(2), 87–143 (1995).
    [Crossref]
  10. L. Zhang, R. Dou, and J. Chen, “Characteristics of the injection-locked master-slave lasers,” Appl. Opt. 47(14), 2648–2654 (2008).
    [Crossref] [PubMed]
  11. J. Jin, H.-G. Rhee, and S.-W. Kim, “Metrological atomic force microscopy integrated with a modified two-point diffraction interferometer,” Meas. Sci. Technol. 20(10), 105302 (2009).
    [Crossref]
  12. F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
    [Crossref]
  13. J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos (Springer, 2008) 2nd edition, pp. 32–33.
  14. Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004).
    [Crossref]

2010 (1)

2009 (1)

J. Jin, H.-G. Rhee, and S.-W. Kim, “Metrological atomic force microscopy integrated with a modified two-point diffraction interferometer,” Meas. Sci. Technol. 20(10), 105302 (2009).
[Crossref]

2008 (2)

Y.-B. Lee, H. C. Kim, and S.-W. Kim, “Determination of the sensitivity phase of an accelerometer based on an analysis of the harmonic components of the interference signal,” Meas. Sci. Technol. 19(4), 045204 (2008).
[Crossref]

L. Zhang, R. Dou, and J. Chen, “Characteristics of the injection-locked master-slave lasers,” Appl. Opt. 47(14), 2648–2654 (2008).
[Crossref] [PubMed]

2006 (1)

2004 (1)

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

1999 (1)

1995 (1)

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7(2), 87–143 (1995).
[Crossref]

1986 (1)

G. R. Hadley, “Injection locking of diode lasers,” IEEE J. Quantum Electron. 22(3), 419–426 (1986).
[Crossref]

1985 (1)

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

1982 (1)

R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982).
[Crossref]

Chen, J.

Donati, S.

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

Dou, R.

Eom, T. B.

Giuliani, G.

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

Hadley, G. R.

G. R. Hadley, “Injection locking of diode lasers,” IEEE J. Quantum Electron. 22(3), 419–426 (1986).
[Crossref]

Jacobsen, G.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Jin, J.

Kang, C.-S.

Khavinson, V. M.

Kim, H. C.

Y.-B. Lee, H. C. Kim, and S.-W. Kim, “Determination of the sensitivity phase of an accelerometer based on an analysis of the harmonic components of the interference signal,” Meas. Sci. Technol. 19(4), 045204 (2008).
[Crossref]

Kim, J. W.

Kim, J.-A.

Kim, S.-W.

J. Jin, H.-G. Rhee, and S.-W. Kim, “Metrological atomic force microscopy integrated with a modified two-point diffraction interferometer,” Meas. Sci. Technol. 20(10), 105302 (2009).
[Crossref]

Y.-B. Lee, H. C. Kim, and S.-W. Kim, “Determination of the sensitivity phase of an accelerometer based on an analysis of the harmonic components of the interference signal,” Meas. Sci. Technol. 19(4), 045204 (2008).
[Crossref]

J. Jin, Y.-J. Kim, Y. Kim, S.-W. Kim, and C.-S. Kang, “Absolute length calibration of gauge blocks using optical comb of a femtosecond pulse laser,” Opt. Express 14(13), 5968–5974 (2006).
[Crossref] [PubMed]

Kim, Y.

Kim, Y.-J.

Lang, R.

R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982).
[Crossref]

Lee, Y.-B.

Y.-B. Lee, H. C. Kim, and S.-W. Kim, “Determination of the sensitivity phase of an accelerometer based on an analysis of the harmonic components of the interference signal,” Meas. Sci. Technol. 19(4), 045204 (2008).
[Crossref]

Lenstra, D.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7(2), 87–143 (1995).
[Crossref]

Mogensen, F.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Olesen, H.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Rhee, H.-G.

J. Jin, H.-G. Rhee, and S.-W. Kim, “Metrological atomic force microscopy integrated with a modified two-point diffraction interferometer,” Meas. Sci. Technol. 20(10), 105302 (2009).
[Crossref]

van Tartwijk, G. H. M.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7(2), 87–143 (1995).
[Crossref]

Yu, Y.

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

Zhang, L.

Appl. Opt. (2)

IEEE J. Quantum Electron. (3)

G. R. Hadley, “Injection locking of diode lasers,” IEEE J. Quantum Electron. 22(3), 419–426 (1986).
[Crossref]

R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982).
[Crossref]

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Y. Yu, G. Giuliani, and S. Donati, “Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect,” IEEE Photon. Technol. Lett. 16(4), 990–992 (2004).
[Crossref]

Meas. Sci. Technol. (2)

Y.-B. Lee, H. C. Kim, and S.-W. Kim, “Determination of the sensitivity phase of an accelerometer based on an analysis of the harmonic components of the interference signal,” Meas. Sci. Technol. 19(4), 045204 (2008).
[Crossref]

J. Jin, H.-G. Rhee, and S.-W. Kim, “Metrological atomic force microscopy integrated with a modified two-point diffraction interferometer,” Meas. Sci. Technol. 20(10), 105302 (2009).
[Crossref]

Opt. Express (2)

Quantum Semiclassic. Opt. (1)

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semiclassic. Opt. 7(2), 87–143 (1995).
[Crossref]

Other (3)

“Methods for the calibration of vibration and shock transducers: 1. Basic concepts,” International Standard ISO 16063–1 (1998).

“Methods for the calibration of vibration and shock transducers: II. Primary vibration calibration by laser interferometry,” International Standard ISO 16063–11 (1999).

J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos (Springer, 2008) 2nd edition, pp. 32–33.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Optical layout of VEI (DFB laser: distributed feedback laser, CL: collimation lens, M: mirror, C: circulator, PD: photo detector, OSA: optical spectrum analyzer, ND filter: neutral density filter).

Fig. 2
Fig. 2

Spectrums of the master laser, the slave laser, and the injection-locked light. The intensity of the spectrum of the master laser was adjusted for easy comparison.

Fig. 3
Fig. 3

Interference signals obtained by VEI and a normal interferometer; (a) VEI without the ND filter (V=0.65), (b) VEI with the ND filter (V=0.65), (c) The normal interferometer without the ND filter (V=0.69), (d) The normal interferometer with the ND filter (V=0.06). Figures (a) and (c) were measured with two metal coated mirrors, and figures (b) and (d) were measured with the ND filter having a transmittance of 0.08. For clear recognition, the two-dimensional figures were generated numerically from the obtained interference signal profiles.

Equations (4)

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

V = 2 I r I s I r + I s .
Δ φ = sin 1 ( Δ ν/ Δ ν L ) tan 1 α,
δΔφ = γ δI/Δν L ,
δΔφ = γ δI/Δν L 1 cos ( Δν/Δν L ) .

Metrics