Abstract

An extensive study for the optical beat noise in low-coherence two-beam interferometry under uniform spectrum illumination is presented. A new formula for the beat noise is derived. The analytical formulas for the signal-to-noise ratio and the resolution in the detection of small phase change and absolute optical path length are given. The effects of the beat noise on the achievable sensing performances in real sensing systems are numerically evaluated and discussed. The sensing performances are improved by use of a wider optical bandwidth of the illuminating light.

© 2005 Optical Society of America

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References

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  1. J. Dakin and B. Culshaw, eds., Optical Fiber Sensors (Artech House, Norwood, Mass., 1989), Vol. 1.
  2. T. Tanaka, Y. Igarashi, M. Nara, and T. Yoshino, "Automatic north sensor using fiber-optic gyroscope," Appl. Opt. 33, 120-123 (1993).
    [CrossRef]
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    [CrossRef] [PubMed]
  4. Y. J. Rao and D. A. Jackson, "Recent progress in fiber optic low coherence interferometry," Meas. Sci. Technol. 7, 981-999 (1996).
    [CrossRef]
  5. D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
    [CrossRef]
  6. F. A. Swanson, D. Huang, M. R. Hee, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
    [CrossRef] [PubMed]
  7. M. D. Ali and T. Yoshino, "Performance analysis of two-beam interferometry employing optical amplifier," in Optical Fiber Sensors , Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 500-503.
  8. K. Takada, "Noise in optical low coherence interferom-etry," IEEE J. Quantum Electron. 34, 1098-1108 (1998).
    [CrossRef]
  9. M. Rollins and J. A. Izatt, "Optical interferometer designs for optical coherence tomography," Opt. Lett. 24, 1484-1486 (1999).
    [CrossRef]
  10. A. Gh. Poodeleanu, "Unbalanced versus balanced operation in an optical coherence, tomography system," Appl. Opt. 39, 173-182 (2000).
    [CrossRef]
  11. M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 18, 2183-2189 (2003), http://www.opticsexpress.org.
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics (Pergamon, London, UK, 1964), pp. 256-369.
  13. P. G. Sinha and T. Yoshino, "Acoustically scanned low-coherence integrated simultaneous measurement of absolute strain and temperature using highly birefringent fiber," J. Lightwave Technol. 16, 2010-2015 (1998).
    [CrossRef]
  14. P. Sanzoz, G. Tribillion, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701-708 (1996).
    [CrossRef]
  15. M. N. Inci and T. Yoshino, "A fiber optic wavelength modulation sensor based on Tantalum pentoxide coatings for absolute temperature measurements," Opt. Rev. 7, 205-209 (2000).
    [CrossRef]

2003 (1)

2000 (2)

M. N. Inci and T. Yoshino, "A fiber optic wavelength modulation sensor based on Tantalum pentoxide coatings for absolute temperature measurements," Opt. Rev. 7, 205-209 (2000).
[CrossRef]

A. Gh. Poodeleanu, "Unbalanced versus balanced operation in an optical coherence, tomography system," Appl. Opt. 39, 173-182 (2000).
[CrossRef]

1999 (1)

1998 (2)

1996 (2)

P. Sanzoz, G. Tribillion, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701-708 (1996).
[CrossRef]

Y. J. Rao and D. A. Jackson, "Recent progress in fiber optic low coherence interferometry," Meas. Sci. Technol. 7, 981-999 (1996).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Choma, M. A.

de Groot, P.

Deck , L.

Foltte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Fujimoto, J. G.

F. A. Swanson, D. Huang, M. R. Hee, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Hee, M. R.

F. A. Swanson, D. Huang, M. R. Hee, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Huang, D.

F. A. Swanson, D. Huang, M. R. Hee, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Igarashi, Y.

Inci , M. N.

M. N. Inci and T. Yoshino, "A fiber optic wavelength modulation sensor based on Tantalum pentoxide coatings for absolute temperature measurements," Opt. Rev. 7, 205-209 (2000).
[CrossRef]

Izatt, J. A.

Jackson, D. A.

Y. J. Rao and D. A. Jackson, "Recent progress in fiber optic low coherence interferometry," Meas. Sci. Technol. 7, 981-999 (1996).
[CrossRef]

Lin, C. P.

F. A. Swanson, D. Huang, M. R. Hee, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Nara, M.

Perrin, H.

P. Sanzoz, G. Tribillion, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701-708 (1996).
[CrossRef]

Poodeleanu, A. Gh.

Puliafito, C. A.

F. A. Swanson, D. Huang, M. R. Hee, J. G. Fujimoto, C. P. Lin, and C. A. Puliafito, "High-speed optical coherence domain reflectometry," Opt. Lett. 17, 151-153 (1992).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Rao , Y. J.

Y. J. Rao and D. A. Jackson, "Recent progress in fiber optic low coherence interferometry," Meas. Sci. Technol. 7, 981-999 (1996).
[CrossRef]

Rollins , M.

Sanzoz, P.

P. Sanzoz, G. Tribillion, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701-708 (1996).
[CrossRef]

Sarunic, M. V.

Shuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Sinha , P. G.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Swanson, F. A.

Takada, K.

K. Takada, "Noise in optical low coherence interferom-etry," IEEE J. Quantum Electron. 34, 1098-1108 (1998).
[CrossRef]

Tanaka, T.

Tribillion, G.

P. Sanzoz, G. Tribillion, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701-708 (1996).
[CrossRef]

Yang, C.

Yoshino, T.

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

K. Takada, "Noise in optical low coherence interferom-etry," IEEE J. Quantum Electron. 34, 1098-1108 (1998).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

P. Sanzoz, G. Tribillion, and H. Perrin, "High-resolution profilometry by using phase calculation algorithms for spectroscopic analysis of white-light interferograms," J. Mod. Opt. 43, 701-708 (1996).
[CrossRef]

Meas. Sci. Technol. (1)

Y. J. Rao and D. A. Jackson, "Recent progress in fiber optic low coherence interferometry," Meas. Sci. Technol. 7, 981-999 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opt. Rev. (1)

M. N. Inci and T. Yoshino, "A fiber optic wavelength modulation sensor based on Tantalum pentoxide coatings for absolute temperature measurements," Opt. Rev. 7, 205-209 (2000).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Foltte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 919-925 (1991).
[CrossRef]

Other (3)

M. D. Ali and T. Yoshino, "Performance analysis of two-beam interferometry employing optical amplifier," in Optical Fiber Sensors , Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 500-503.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, UK, 1964), pp. 256-369.

J. Dakin and B. Culshaw, eds., Optical Fiber Sensors (Artech House, Norwood, Mass., 1989), Vol. 1.

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Figures (3)

Fig. 1
Fig. 1

Schematic of two-beam low-coherence interferometer.

Fig. 2
Fig. 2

Illuminating spontaneous emission power PSE dependence of signal-to-noise ratio S for small phase detection in low-coherence interferometry as a parameter of optical bandwidth Bo in comparison with laser interferometry of the same illuminating power.

Fig. 3
Fig. 3

Dependence of optical bandwidth Bo of illuminating light on beat-noise limited signal-to-noise ratio SSAT in low-coherence interferometry.

Equations (50)

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e(t)=l=-MM(2Noδν)1/2 cos[(νo+2πlδν)t+ϕl],
PSE=NoBo,
pSE=PSE/(2M)=Noδν.
e1(t)=l=-MM[(2pSE)1/2/2]cos[(ω0+2πlδν)t+ϕl],
e2(t-τ)=l=-MM[(2pSE)1/2/2]×cos[(ω0+2πlδν)(t-τ)+ϕl],
i(t)=R[e1(t)+e2(t-τ)]2,
i(t)=Rl=-MM(pSE/2)1/2 cos[(ωo+2πlδν)t+ϕl]+l=-MM(pSE/2)1/2 cos[(ωo+2πlδν)(t-τ)+ϕl]2=Rl=-MM(pSE/2)1/2 cos[(ωo+2πlδν)t+ϕl]2+Rl=-MM(pSE/2)1/2 cos[(fωo+2πlδν)(t-τ)+ϕl]2+2Rl=-MM(pSE/2)1/2 cos[(ωo+2πlδν)t+ϕl]l=-MM(pSE/2)1/2 cos[(ωo+2πlδν)(t-τ)+ϕl]=(RpSE/2)(1/2)k=l=-MM cos[2π(k-l)δνt+ϕk-ϕl]+kl=-MM cos[2π(k-l)δνt+ϕk-ϕl]+(RpSE/2)(1/2)k=l=-MM cos[2π(k-l)δν(t-τ)+ϕk-ϕl]+kl=-MM cos[2π(k-l)δν(t-τ)+ϕk-ϕl]+2(RpSE/2)(1/2)k=l=-MM cos[2π(k-l)δνt+(ωo+2πlδν)τ+ϕk-ϕl]+2(RpSE/2)(1/2)k=l=-MM cos[2π(k-l)δνt+(ωo+2πlδν)τ+ϕk-ϕl]=idc+iac(t),
idc=(RpSE/2)(1/2)k=l=-MM1+(RpSE/2)(1/2)×k=l=-MM1+(RpSE/2)l=-MM×cos[(ωo+2πlδν)τ],
iac(t)=(RpSE/2)(1/2)kl=-MM cos[2π(k-l)×δνt+ϕk-ϕl]+(RpSE/2)(1/2)kl=-MM cos[2π(k-l)×δν(t-τ)+ϕk-ϕl]+(RpSE/2)kl=-MM cos[2π(k-l)×δνt+(ωo+2πlδν)τ+ϕk-ϕl].
idc=l=-MM(RpSE/2){1+cos[(ω0+2πlδν)τ]},
=RPSE/2+(RPSE/2)×l=-MM(1/2M)cos(2πτlδν)cos(ω0τ).
l=-MM(1/2M)cos(2πτlδν)=sinc(πBoτ),(M)
idc=(RPSE/2)[1+sinc(πBoτ)cos(ωoτ)],
V(τ)=sinc(πBoτ),
iE(τ)=RPSEV(τ).
iac2(t)=[(RpSE/2)/2)]2kl=-MM cos[2π(k-l)δνt+ϕk-ϕl]kl=-MM cos[2π(k-l)δνt+ϕk-ϕl]+[(RpSE/2)/2)]2kl=-MM cos[2π(k-l)δν(t-τ)+ϕk-ϕl]×kl=-MM cos[2π(k-l)δν(t-τ)+ϕk-ϕl]+[(RpSE/2)]2kl=-MM cos[2π(k-l)δνt+(ωo+2πlδν)τ+ϕk-ϕl]×kl=-MM cos[2π(k-l)δνt+(ωo+2πlδν)τ+ϕk-ϕl]+2[(RpSE/2)/2][(RpSE/2)/2]kl=-MM cos[2π(k-l)δνt+ϕk-ϕl]×kl=-MM cos[2π(k-l)δν(t-τ)+ϕk-ϕl]+2[(RpSE/2)/2][(RpSE/2)]kl=-MM cos[2π(k-l)δν(t-τ)+ϕk-ϕl]×kl=-MM cos[2π(k-l)δνt+(ωo+2πlδν)τ+ϕk-ϕl]+2[(RpSE/2)/2][(RpSE/2)]×kl=-MM cos[2π(k-l)δνt+ϕk-ϕl]×kl=-MM cos[2π(k-l)δνt+(ωo+2πlδν)τ+ϕk-ϕl],
iac2(t)=(RpSE/2)2kl=-MM[(1/4)×(1/2)+(1/4)×(1/2)+(1/2)]+(RpSE/2)2/2×kl=-MM(1/2)cos[2π(k-l)τ]+(RpSE/2)2×kl=-MM(1/2)cos[(ωo+2πkδν)τ]+(RpSE/2)2kl=-MM(1/2)cos[(ωo+2πlδν)τ]=(RpSE/2)2kl=-MM(3/4)+(RpSE/2)2(1/4)×kl=-MM cos[2π(k-l)τ]+(RpSE/2)2×kl=-MM cos[(ωo+2πkδν)τ].
kl=-MM1=k,l=-MM1-k=l=-MM1=(2M+1)2-(2M+1),
kl=-MM cos[2π(k-l)δντ]
=k,l=-MM cos[2π(k-l)δντ]-k=l=-MM cos[2π(k-l)δντ]=k,l=-MM cos(2πkδντ)cos(2πlδντ)+k,l=-MM sin(2πkδντ)sin(2πlδντ)-(2M+1)=k=-MM cos(2πkδντ)l=-MM cos(2πlδντ)+k=-MM sin(2πkδντ)l=-MM sin(2πlδντ)-(2M+1)=k=-MM cos(2πkδντ)2-k=-MM sin(2πkδντ)2-(2M+1),=(2M)2 sinc2 (πB0τ)-(2M+1)[Eq.(11)].
kl=-MM cos[(ωo+2πlδν)τ]
=k,l=-MM cos[(ωo+2πlδν)τ]-k=l=-MM cos[(ωo+2πlδν)τ]=(2M+1)l=-MM cos[(ωo+2πlδν)τ]-l=-MM cos[(ωo+2πlδν)τ]=2Ml=-MM cos[(ωo+2πlδν)τ]=(2M)2 sinc(πBoτ)cos(ωoτ)[Eq.(11)].
iac(t)2=(RPSE/2)2(3/4){[1+1/(2M)]2-(2M+1)/(2M)2}+(RPSE/2)2(1/4)[sinc2(πBoτ)-(2M+1)/(2M)2]+(RPSE/2)2[sinc(πBoτ)cos(ω0τ)]=(RPSE/2)2(1/4)[3+sinc2(πBoτ)+4sinc(πBoτ)cos(ωoτ)],(M)
σˆSESE2(f)=σˆSESE2(0)(1-f/Bo).
σSESE2=0BeσˆSESE2(f)df=σˆSESE2(0)[1-Be/(2Bo)]Be.
iac(t)2=0BoσˆSESE2(f)df=σˆSESE2(0)Bo/2.
σSESE2=(1/8)(RPSE)2[3+sinc2(πBoτ)+4sinc(πBoτ)cos(ωoτ)(Be/Bo)].
ISIG=Δϑ[didc/d(ωoτ)]=RΔϑ(PSE/2){sinc(πBoτ)sin(ωoτ)-cos(ωoτ)d[sinc(πBoτ)]/d(ωoτ)}.
σT2=σsh2+σSESE2+σth2,
σsh2=2q(RPSE/2)[1+sinc(πBoτ)cos(ωoτ)]Be,
σth2=(4KBTe/RL)Be,
SNR=|ISIG|/(σsh2+σASEASE2+σth2)1/2.
ωoτ=π/2+nπ,(n:integers),
ISIG(π/2)=(RPSE/2)|Δϑ sinc(πBoτ)|.
σSESE2(π/2)=(1/8)(RPSE)2[3+sinc2(πBoτ)](Be/Bo),
σsh2(π/2)=qRPSEBe.
SNR(π/2)=(RPSE/2)|Δϑsinc(πBoτ)|{qRPSE+(1/8)(RPSE)2[3+sinc2(πBoτ)]/Bo+4KBTe/RL}1/2Be.
S=SNR(π/2)Be|Δϑ|=(RPSE/2)|sinc(πBoτ)|{qRPSE+(1/8)(RPSE)2[3+sinc2(πBoτ)]/Bo+4KBTe/RL}1/2,
S=(RNoBo/2)|sinc(πBoτ)|{qRNoBo+(1/8)R2No2[3+sinc2(πBoτ)]Bo+4KBTe/RL}1/2.
SSAT={|sinc(πBoτ)|/[3+sinc2(πBoτ)]1/2}2Bo.
SNR(π/2)
=(RPSE/2)|Δϑ|[qRPSE+(1/2)(RPSE)2/Bo+4KBTe/RL]1/2Be,
SSAT=Bo/2.
S(Laser)=(RPL/2)/(qRPL+4KBTe/RL)1/2,
|Δϑmin|=Be/S.
V(τ)=1-(πBoτ)2/6.
ISIG=iE(τ)-iE(0)=(RPSE)[V(τ)-V(0)]=-(RPSE)(πBoτ)2/6.
SNR=|ISIG|σT=[RPSE(πBo)2/6]τ2[2qRPSE+(RPSE)2/Bo+4KBTe/RL]1/2Be,
|Δl|min=c|τ|min=6c[2qRPSE+(RPSE)2/Bo+4KBTe/RL]1/4Be1/4/[(RPSE)1/2(πBo)],
|Δl|min=[6c/(πBo5/4)]Be1/4,

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