Abstract

Wavelength scanning interferometry and swept-source optical coherence tomography require accurate measurement of time-varying laser wavenumber changes. We describe here a method based on recording interferograms of multiple wedges to provide simultaneously high wavenumber resolution and immunity to the ambiguities caused by large wavenumber jumps. All the data required to compute a wavenumber shift are provided in a single image, thereby allowing dynamic wavenumber monitoring. In addition, loss of coherence of the laser light is detected automatically. The paper gives details of the analysis algorithms that are based on phase detection by a two-dimensional Fourier transform method followed by temporal phase unwrapping and correction for optical dispersion in the wedges. A simple but robust method to determine the wedge thicknesses, which allows the use of low-cost optical components, is also described. The method is illustrated with experimental data from a Ti:sapphire tunable laser, including independent wavenumber measurements with a commercial wavemeter. A root mean square (rms) difference in measured wavenumber shift between the two of 4m1 has been achieved, equivalent to an rms wavelength shift error of 0.4pm.

© 2012 Optical Society of America

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References

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  1. A. Yamamoto and I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
    [CrossRef]
  2. A. Yamamoto, C. Kuo, K. Sunouchi, S. Wada, I. Yamaguchi, and H. Tashiro, “Surface shape measurement by wavelength scanning interferometry using an electronically tuned Ti:sapphire Laser,” Opt. Rev. 8, 59–63 (2001).
    [CrossRef]
  3. P. D. Ruiz, J. M. Huntley, and R. D. Wildman, “Depth-resolved whole-field displacement measurement by wavelength-scanning electronic speckle pattern interferometry,” Appl. Opt. 44, 3945–3953 (2005).
    [CrossRef]
  4. A. F. Fercher, W Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Reps. Prog. Phys. 66, 239–303 (2003).
  5. HSL-1000, Santec Corporation, Aichi, Japan
  6. J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
    [CrossRef]
  7. B. Alipieva, B. Stoykova, and V. Nikolovac, “Wavemeter with Fizean interferometer for CW lasers,” Proc. SPIE 4397, 129–133 (2001).
    [CrossRef]
  8. L. S. Lee and A. L. Schawlow, “Multiple-wedge wavemeter for pulsed lasers,” Opt. Lett. 6, 610–612 (1981).
    [CrossRef]
  9. M. Françon, Optical Interferometry (Academic, 1966).
  10. J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
    [CrossRef]
  11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
    [CrossRef]
  12. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965).
    [CrossRef]

2005

2003

A. F. Fercher, W Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Reps. Prog. Phys. 66, 239–303 (2003).

2002

A. Yamamoto and I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
[CrossRef]

2001

A. Yamamoto, C. Kuo, K. Sunouchi, S. Wada, I. Yamaguchi, and H. Tashiro, “Surface shape measurement by wavelength scanning interferometry using an electronically tuned Ti:sapphire Laser,” Opt. Rev. 8, 59–63 (2001).
[CrossRef]

B. Alipieva, B. Stoykova, and V. Nikolovac, “Wavemeter with Fizean interferometer for CW lasers,” Proc. SPIE 4397, 129–133 (2001).
[CrossRef]

1999

1997

J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[CrossRef]

1982

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[CrossRef]

1981

1965

Alipieva, B.

B. Alipieva, B. Stoykova, and V. Nikolovac, “Wavemeter with Fizean interferometer for CW lasers,” Proc. SPIE 4397, 129–133 (2001).
[CrossRef]

Drexler, W

A. F. Fercher, W Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Reps. Prog. Phys. 66, 239–303 (2003).

Fercher, A. F.

A. F. Fercher, W Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Reps. Prog. Phys. 66, 239–303 (2003).

Françon, M.

M. Françon, Optical Interferometry (Academic, 1966).

Hitzenberger, C. K.

A. F. Fercher, W Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Reps. Prog. Phys. 66, 239–303 (2003).

Huntley, J. M.

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[CrossRef]

Kaufmann, G. H.

Kerr, D.

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[CrossRef]

Kuo, C.

A. Yamamoto, C. Kuo, K. Sunouchi, S. Wada, I. Yamaguchi, and H. Tashiro, “Surface shape measurement by wavelength scanning interferometry using an electronically tuned Ti:sapphire Laser,” Opt. Rev. 8, 59–63 (2001).
[CrossRef]

Lasser, T.

A. F. Fercher, W Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Reps. Prog. Phys. 66, 239–303 (2003).

Lee, L. S.

Malitson, I. H.

Nikolovac, V.

B. Alipieva, B. Stoykova, and V. Nikolovac, “Wavemeter with Fizean interferometer for CW lasers,” Proc. SPIE 4397, 129–133 (2001).
[CrossRef]

Ruiz, P. D.

Saldner, H. O.

J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[CrossRef]

Schawlow, A. L.

Stoykova, B.

B. Alipieva, B. Stoykova, and V. Nikolovac, “Wavemeter with Fizean interferometer for CW lasers,” Proc. SPIE 4397, 129–133 (2001).
[CrossRef]

Sunouchi, K.

A. Yamamoto, C. Kuo, K. Sunouchi, S. Wada, I. Yamaguchi, and H. Tashiro, “Surface shape measurement by wavelength scanning interferometry using an electronically tuned Ti:sapphire Laser,” Opt. Rev. 8, 59–63 (2001).
[CrossRef]

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[CrossRef]

Tashiro, H.

A. Yamamoto, C. Kuo, K. Sunouchi, S. Wada, I. Yamaguchi, and H. Tashiro, “Surface shape measurement by wavelength scanning interferometry using an electronically tuned Ti:sapphire Laser,” Opt. Rev. 8, 59–63 (2001).
[CrossRef]

Wada, S.

A. Yamamoto, C. Kuo, K. Sunouchi, S. Wada, I. Yamaguchi, and H. Tashiro, “Surface shape measurement by wavelength scanning interferometry using an electronically tuned Ti:sapphire Laser,” Opt. Rev. 8, 59–63 (2001).
[CrossRef]

Wildman, R. D.

Yamaguchi, I.

A. Yamamoto and I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
[CrossRef]

A. Yamamoto, C. Kuo, K. Sunouchi, S. Wada, I. Yamaguchi, and H. Tashiro, “Surface shape measurement by wavelength scanning interferometry using an electronically tuned Ti:sapphire Laser,” Opt. Rev. 8, 59–63 (2001).
[CrossRef]

Yamamoto, A.

A. Yamamoto and I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
[CrossRef]

A. Yamamoto, C. Kuo, K. Sunouchi, S. Wada, I. Yamaguchi, and H. Tashiro, “Surface shape measurement by wavelength scanning interferometry using an electronically tuned Ti:sapphire Laser,” Opt. Rev. 8, 59–63 (2001).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am 72, 156–160 (1982).
[CrossRef]

J. Opt. Soc. Am.

Meas. Sci. Technol.

J. M. Huntley and H. O. Saldner, “Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithms,” Meas. Sci. Technol. 8, 986–992 (1997).
[CrossRef]

Opt. Lett.

Opt. Rev.

A. Yamamoto and I. Yamaguchi, “Profilometry of sloped plane surfaces by wavelength scanning interferometry,” Opt. Rev. 9, 112–121 (2002).
[CrossRef]

A. Yamamoto, C. Kuo, K. Sunouchi, S. Wada, I. Yamaguchi, and H. Tashiro, “Surface shape measurement by wavelength scanning interferometry using an electronically tuned Ti:sapphire Laser,” Opt. Rev. 8, 59–63 (2001).
[CrossRef]

Proc. SPIE

B. Alipieva, B. Stoykova, and V. Nikolovac, “Wavemeter with Fizean interferometer for CW lasers,” Proc. SPIE 4397, 129–133 (2001).
[CrossRef]

Reps. Prog. Phys.

A. F. Fercher, W Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Reps. Prog. Phys. 66, 239–303 (2003).

Other

HSL-1000, Santec Corporation, Aichi, Japan

M. Françon, Optical Interferometry (Academic, 1966).

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Figures (10)

Fig. 1.
Fig. 1.

Optical setup: LB: Ti:sapphire laser beam. MO: microscope objective and spatial filter. L1: collimating lens. θ: refracted angle of illuminating beam. W: set of four wedges with thickness increasing along x. L2 and L3: imaging lenses. CCD: camera.

Fig. 2.
Fig. 2.

(a) Wedge fringe patterns at a wavelength of 750 nm, where the thickest wedge is at the top and the thinnest at the bottom. (b) Blurring of the patterns when coherence is lost due to mechanical movement of the BRF.

Fig. 3.
Fig. 3.

(a), (b) 2D Fourier transforms of the fringe patterns from the top wedge shown in Figs. 2(a) and (b), respectively.

Fig. 4.
Fig. 4.

Measured phase change over a wavelength scan of 400 frames for step 1 (wedge pairs 1,2 and 2,3). (a) Temporally unwrapped phase signals ΔΦu1,2(t,0) (dotted) and ΔΦ2,3(t,0) (continuous line). (b) Scaled thin wedge phase signal R1ΔΦu1,2(t,0) (dotted) and the resultant unwrapped thick wedge signal ΔΦu2,3(t,0) (continuous line) after unwrapping according to Eq. (12).

Fig. 5.
Fig. 5.

Unwrapped phase change values over a wavelength scan of 400 frames. (a) Step 2 (in which wedges 2 and 3 are used to unwrap the difference between wedges 3 and 4) and (b) step 3 (in which wedges 3 and 4 are used to unwrap the phase of wedge 1).

Fig. 6.
Fig. 6.

(a) Previous Fig. 5(a) for comparison of unwrapping phase errors, where the optimal thickness ratio R=3.1242 has been used to unwrap the high-sensitivity phase data. (b) R=2, and (c) R=4.

Fig. 7.
Fig. 7.

Cost function S(Rm) [Eq. (23)] for step m=2.

Fig. 8.
Fig. 8.

Fringe patterns and phase at 764.81 nm: (a) Intensity distribution and (b) corresponding wrapped phase map from wedge 1. Rectangular boxes in (a) show the regions from which phase values were extracted, with their centers indicated as crosses in (b).

Fig. 9.
Fig. 9.

Fused silica refractive index dispersion curve for wavenumbers in the wavelength range from 750 to 850 nm.

Fig. 10.
Fig. 10.

Calculated wavenumber changes for the data shown in Figs. 46 (solid line), and the corresponding changes measured simultaneously by a commercial wavemeter (dotted line with an offset of 700m1). The rms deviation between the two curves was 3.86m1.

Equations (27)

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ψj(x)=k(2ndcos(θ)+λ/2),
f(θ,α)=cosα(cos2(θ+α)+1)/cos(θ+α).
Δψj(t,t1)=ndΔk(t,t1)f(θ,α),
dj<πnΔkmf(θ,α),
ΔΦi,j(t,t1)=Δψi(t,t1)Δψj(t,t1),
ΔΦwi,j(t,t1)=W{ΔΦi,j(t,t1)},
dsi,j=didj.
dsi,j<πnΔkmf(θ,α),
ΔΦi,j(t,0)=t=1tΔΦwi,j(t,t1)
ΔΦui,j(t,0)=ΔΦi,j(t,0).
R1=dsi2,j2dsi1,j1.
ΔΦui2,j2(t,0)=U{ΔΦi2,j2(t,0),R1ΔΦui1,j1(t,0)},
U{ΦA,ΦB}=ΦA2πNINT(ΦAΦB2π),
W{ΦA}=U{ΦA,0}.
Rm=dsim+1,jm+1dsim,jm,
ΔΦuim+1,jm+1(t,0)=U{ΔΦim+1,jm+1(t,0),RmΔΦuim,jm(t,0)}.
d1=d0Rs1dj=d1d0Rj2j=2,3,,s,
R=(dsmax/dsmin)1/Ns.
Rσϕπ.
Δk(t,0)=ΔΦuiNs+1,jNs+1(t,0)/ndsmaxf(θ,α),
α=Nfλ/2nW.
ΔΦui,j(t,0)=ΔΦui,j(t,tκ)+k=2κΔΦui,j(tk,tk1)+ΔΦui,j(t1,0).
S=t=1Nt[ΔΦuim+1,jm+1(t,0)RmΔΦuim,jm(t,0)]2N.
dsi0,j0=dsiNs+1,jNs+1R0×R1×R2××RNs.
ΔΨ=Ψ(xi0,yi0,t)Ψ(xj0,yj0,t),
λ=2πndsi0,j0f(θ,α)δψ.
Δk(t,0)=ΔΦuiNs+1,jNs+1(t,0)n(t)dsmaxf(θ,α)k(0)(n(0)n(t)1).

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