Abstract

We calculate the root mean square (rms) value of the spectral noise caused by optical path phase measurement errors in a spatial heterodyne spectrometer (SHS) featuring a complex Fourier transformation. In our calculation the deviated phases of each Mach–Zehnder interferometer in the in-phase and quadrature states are treated as statistically independent random variables. We show that the rms value is proportional to the rms error of the phase measurement and that the proportionality coefficient is given analytically. The relationship enables us to estimate the potential performance of the SHS such as the sidelobe suppression ratio for a given measurement error.

© 2013 Optical Society of America

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References

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  1. M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach–Zehnder interferometers,” Opt. Express 15, 18176–18189 (2007).
    [CrossRef]
  2. M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Spatial heterodyne planar waveguide spectrometer: theory and design” Proc. SPIE 7099, 70991L1 (2008).
    [CrossRef]
  3. T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15, 2107–2113 (1997).
    [CrossRef]
  4. K. Takada, T. Aoyagi, and K. Okamoto, “Complex-Fourier-transform integrated-optic spatial heterodyne spectrometer using phase shift technique,” Electron. Lett. 46, 1620–1621 (2010).
    [CrossRef]
  5. K. Takada, H. Aoyagi, and K. Okamoto, “Correction for phase-shift deviation in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer with an active phase-shift scheme,” Opt. Lett. 36, 1044–1046 (2011).
    [CrossRef]
  6. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, 1974), Chap. 6.
  7. K. Takada, M. Seino, A. Chiba, and K. Okamoto, “Spectral noise due to measurement errors of Mach–Zehnder interferometer optical path phases in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer,” Opt. Commun. (2013) (to be published).
    [CrossRef]
  8. K. Takada, H. Yamada, and Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 14, 1677–1689 (1996).
    [CrossRef]

2011 (1)

2010 (1)

K. Takada, T. Aoyagi, and K. Okamoto, “Complex-Fourier-transform integrated-optic spatial heterodyne spectrometer using phase shift technique,” Electron. Lett. 46, 1620–1621 (2010).
[CrossRef]

2008 (1)

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Spatial heterodyne planar waveguide spectrometer: theory and design” Proc. SPIE 7099, 70991L1 (2008).
[CrossRef]

2007 (1)

1997 (1)

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15, 2107–2113 (1997).
[CrossRef]

1996 (1)

K. Takada, H. Yamada, and Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 14, 1677–1689 (1996).
[CrossRef]

Aoyagi, H.

Aoyagi, T.

K. Takada, T. Aoyagi, and K. Okamoto, “Complex-Fourier-transform integrated-optic spatial heterodyne spectrometer using phase shift technique,” Electron. Lett. 46, 1620–1621 (2010).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, 1974), Chap. 6.

Cheben, P.

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Spatial heterodyne planar waveguide spectrometer: theory and design” Proc. SPIE 7099, 70991L1 (2008).
[CrossRef]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach–Zehnder interferometers,” Opt. Express 15, 18176–18189 (2007).
[CrossRef]

Chiba, A.

K. Takada, M. Seino, A. Chiba, and K. Okamoto, “Spectral noise due to measurement errors of Mach–Zehnder interferometer optical path phases in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer,” Opt. Commun. (2013) (to be published).
[CrossRef]

Florjanczyk, M.

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Spatial heterodyne planar waveguide spectrometer: theory and design” Proc. SPIE 7099, 70991L1 (2008).
[CrossRef]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach–Zehnder interferometers,” Opt. Express 15, 18176–18189 (2007).
[CrossRef]

Goh, T.

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15, 2107–2113 (1997).
[CrossRef]

Inoue, Y.

K. Takada, H. Yamada, and Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 14, 1677–1689 (1996).
[CrossRef]

Janz, S.

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Spatial heterodyne planar waveguide spectrometer: theory and design” Proc. SPIE 7099, 70991L1 (2008).
[CrossRef]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach–Zehnder interferometers,” Opt. Express 15, 18176–18189 (2007).
[CrossRef]

Okamoto, K.

K. Takada, H. Aoyagi, and K. Okamoto, “Correction for phase-shift deviation in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer with an active phase-shift scheme,” Opt. Lett. 36, 1044–1046 (2011).
[CrossRef]

K. Takada, T. Aoyagi, and K. Okamoto, “Complex-Fourier-transform integrated-optic spatial heterodyne spectrometer using phase shift technique,” Electron. Lett. 46, 1620–1621 (2010).
[CrossRef]

K. Takada, M. Seino, A. Chiba, and K. Okamoto, “Spectral noise due to measurement errors of Mach–Zehnder interferometer optical path phases in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer,” Opt. Commun. (2013) (to be published).
[CrossRef]

Scott, A.

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Spatial heterodyne planar waveguide spectrometer: theory and design” Proc. SPIE 7099, 70991L1 (2008).
[CrossRef]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach–Zehnder interferometers,” Opt. Express 15, 18176–18189 (2007).
[CrossRef]

Seino, M.

K. Takada, M. Seino, A. Chiba, and K. Okamoto, “Spectral noise due to measurement errors of Mach–Zehnder interferometer optical path phases in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer,” Opt. Commun. (2013) (to be published).
[CrossRef]

Solheim, B.

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Spatial heterodyne planar waveguide spectrometer: theory and design” Proc. SPIE 7099, 70991L1 (2008).
[CrossRef]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach–Zehnder interferometers,” Opt. Express 15, 18176–18189 (2007).
[CrossRef]

Sugita, A.

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15, 2107–2113 (1997).
[CrossRef]

Suzuki, S.

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15, 2107–2113 (1997).
[CrossRef]

Takada, K.

K. Takada, H. Aoyagi, and K. Okamoto, “Correction for phase-shift deviation in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer with an active phase-shift scheme,” Opt. Lett. 36, 1044–1046 (2011).
[CrossRef]

K. Takada, T. Aoyagi, and K. Okamoto, “Complex-Fourier-transform integrated-optic spatial heterodyne spectrometer using phase shift technique,” Electron. Lett. 46, 1620–1621 (2010).
[CrossRef]

K. Takada, H. Yamada, and Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 14, 1677–1689 (1996).
[CrossRef]

K. Takada, M. Seino, A. Chiba, and K. Okamoto, “Spectral noise due to measurement errors of Mach–Zehnder interferometer optical path phases in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer,” Opt. Commun. (2013) (to be published).
[CrossRef]

Xu, D. X.

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Spatial heterodyne planar waveguide spectrometer: theory and design” Proc. SPIE 7099, 70991L1 (2008).
[CrossRef]

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Multiaperture planar waveguide spectrometer formed by arrayed Mach–Zehnder interferometers,” Opt. Express 15, 18176–18189 (2007).
[CrossRef]

Yamada, H.

K. Takada, H. Yamada, and Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 14, 1677–1689 (1996).
[CrossRef]

Electron. Lett. (1)

K. Takada, T. Aoyagi, and K. Okamoto, “Complex-Fourier-transform integrated-optic spatial heterodyne spectrometer using phase shift technique,” Electron. Lett. 46, 1620–1621 (2010).
[CrossRef]

J. Lightwave Technol. (2)

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15, 2107–2113 (1997).
[CrossRef]

K. Takada, H. Yamada, and Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 14, 1677–1689 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D. X. Xu, “Spatial heterodyne planar waveguide spectrometer: theory and design” Proc. SPIE 7099, 70991L1 (2008).
[CrossRef]

Other (2)

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, 1974), Chap. 6.

K. Takada, M. Seino, A. Chiba, and K. Okamoto, “Spectral noise due to measurement errors of Mach–Zehnder interferometer optical path phases in a complex Fourier-transform integrated-optic spatial heterodyne spectrometer,” Opt. Commun. (2013) (to be published).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Configuration of silica-based planar waveguide SHS with N=32 interleaved MZIs. A small heater was placed on the MZI array to induce a phase shift.

Fig. 2.
Fig. 2.

Simulation of the spectral noise (i) generated when a Gaussian spectrum (ii) is retrieved by using phase values ϕk and εk that have the measurement errors of δϕk and δεk(k=1,2,31) shown in the inset.

Fig. 3.
Fig. 3.

Plot of the relative error defined as 11C as a function of Δ. Δ is the maximum value of the dataset {|Δk(t)|}(k=1,2,N1). The relative error is 0.3 at Δ=0.35rad.

Fig. 4.
Fig. 4.

Dependence of zeroth-order proportionality coefficient α(0) on w. The number of the MZIs is N=32 and α(0) is represented by Eq. (32). w is the relative width of a Gaussian spectrum defined by the ratio of the half-width at 1/e maximum to the width of the FSR.

Fig. 5.
Fig. 5.

Experimental setup of optical low coherence MZI for measuring the optical phases of lights that have passed through the two arms of each MZI in the array. PC1, PC2, and PC3 are polarization controllers.

Fig. 6.
Fig. 6.

Isolated two fringes of the interferogram produced for the third (k=2) MZI with the optical low coherence interferometer shown in Fig. 5.

Fig. 7.
Fig. 7.

Distribution of deviated optical phases {εk} (k=1 to 31) in the in-phase states of the MZIs in our SHS that were measured repeatedly with the optical low coherence interferometer.

Fig. 8.
Fig. 8.

Standard deviations of deviated phase values measured at individual MZIs from k=1 to 31. Their mean value is plotted by a dotted line.

Equations (50)

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ΔPk(I)=m=0M1Gm(t)cos(2πmkM+εk(t)).
ΔPk(Q)=m=0M1Gm(t)cos(2πmkM+ϕk(t)).
Uk(t)=m=0M1Gm(t)e2πimkM,
Uk(t)=i{ΔPk(I)eiϕk(t)ΔPk(Q)eiεk(t)sin(ϕk(t)εk(t))}.
Gm(t)=1Mk=0M1Uk(t)e2πikmM
Uk=i{ΔPk(I)eiϕkΔPk(Q)eiεksin(ϕkεk)}(0kN1)
Gm=1Mk=0M1Uke2πikmM.
Uk=Uk(t)+(Ukϕk)(t)(ϕkϕk(t))+(Ukεk)(t)(εkεk(t)).
(Ukϕk)(t)=ieiεk(t){ΔPk(I)+ΔPk(Q)cos(ϕk(t)εk(t))sin2(ϕk(t)εk(t))},
(Ukεk)(t)=ieiϕk(t){ΔPk(Q)+ΔPk(I)cos(ϕk(t)εk(t))sin2(ϕk(t)εk(t))},
(Ukϕk)(t)=ieiεk(t)|Uk(t)|Ak,(Ukεk)(t)=ieiϕk(t)|Uk(t)|Bk,
Ak=sin(ϕk(t)+φk)cosΔk(t),Bk=sin(εk(t)+φk)cosΔk(t),
Δk(t)=(ϕk(t)εk(t))π2,
Uk=Uk(t)+(Ukϕk)(t)δϕk+(Ukεk)(t)δεk.
δϕMkδϕk,δεMkδεk,
(UMkϕMk)(t)(Ukϕk)(t)*,(UMkεMk)(t)(Ukεk)(t)*,
Gm=Gm(t)+ΔGm
ΔGm=1Mk=1M1[(Ukϕk)(t)δϕk+(Ukεk)(t)δεk]e2πikmM.
(Ukϕk)(t)=iRe(Uk(t)),(Ukεk)(t)=Im(Uk(t)),
ΔGm2¯1Mm=0M1ΔGm2=1M2k=1M1|(Ukϕk)(t)δϕk+(Ukεk)(t)δεk|2,
m=0M1e2πipmM{e2πiqmM}*=M×δp,q
ΔGm2¯=2M2k=1N1{|(Ukϕk)(t)|2δϕk2+|(Ukεk)(t)|2δεk2+[(Ukϕk)(t)(Ukεk)(t)*δϕkδεk+c.c.]},
ΔGm2¯=2M2k=1N1|Uk(t)|2{Ak2δϕk2+Bk2δεk2},
(ΔG)rmsΔGm2¯=1M2k=1N1|Uk(t)|2(Ak2+Bk2)·δrms,
δϕk2=δεk2=δrms.
(ΔG)rms=1M2k=1N1|Uk(t)|2(1Ck)·δrms,
Ck=2sinΔk(t)cos(εk(t)+φk+Δk(t))sin(εk(t)+φk)cos2Δk(t).
(ΔG)rmsGm(t)=α·δrms,
α=δΩΩ2k=1N1|Uk(t)|2(1Ck)
δΩ=σLσRg(σ)g(σm)dσ,
Uk(t)=σLσRg(σ)exp[2πik(σσL)/Ω]dσσLσRg(σ)dσ.
α(0)=δΩΩ2k=1N1|Uk(t)|2.
αα(0)=1ηwithη=k=1N1|Uk(t)|2Ckk=1N1|Uk(t)|2.
Ck={sinΔk(t)}{sin(2εk(t)+2φk+Δk(t))sinΔk(t)}cos2Δk(t).
|Ck|=|sinΔk(t)|·|sin(2εk(t)+2φk+Δk(t))sinΔk(t)|cos2Δk(t)|sinΔk(t)|·{|sin(2εk(t)+2φk+Δk(t))|+|sinΔk(t)|}cos2Δk(t)|sinΔk(t)|{1+|sinΔk(t)|}cos2Δk(t)=|sinΔk(t)|1|sinΔk(t)|.
|Ck|CsinΔ1sinΔ.
|η|=|k=1N1|Uk(t)|2Ck|k=1N1|Uk(t)|2|k=1N1|Uk(t)|2|Ck=1N1|Uk(t)|2=C
|αα(0)1||1η1|11C.
α(0)=w2πk=1N1exp{2(πkw)2}.
ΔPk(I)=cosεk(t)m=0M1Gm(t)cos(2πmkM)sinεk(t)m=0M1Gm(t)sin(2πmkM),ΔPk(Q)=cosϕk(t)m=0M1Gm(t)cos(2πmkM)sinϕk(t)m=0M1Gm(t)sin(2πmkM).
Re(Uk(t))=m=0M1Gm(t)cos(2πmkM),Im(Uk(t))=m=0M1Gm(t)sin(2πmkM).
ΔPk(I)=cosεk(t)Re(Uk(t))sinεk(t)Im(Uk(t)),
ΔPk(Q)=cosϕk(t)Re(Uk(t))sinϕk(t)Im(Uk(t)).
Re(Uk(t))=|Uk(t)|cosφk,Im(Uk(t))=|Uk(t)|sinφk.
ΔPk(I)=cosεk(t)×|Uk(t)|cosφksinεk(t)×|Uk(t)|sinφk=|Uk(t)|cos(εk(t)+φk),
ΔPk(Q)=cosϕk(t)×|Uk(t)|cosφksinϕk(t)×|Uk(t)|sinφk=|Uk(t)|cos(ϕk(t)+φk).
ΔPk(I)+ΔPk(Q)cos(ϕk(t)εk(t))=|Uk(t)|{cos(εk(t)+φk)+cos(ϕk(t)+φk)cos(ϕk(t)εk(t))}=|Uk(t)|{cos(εk(t)+φk)+12cos(2ϕk(t)+φkεk(t))+12cos(εk(t)+φk)}=|Uk(t)|×12{cos(2ϕk(t)+φkεk(t))cos(εk(t)+φk)}=|Uk(t)|sin(ϕk(t)+φk)sin(ϕk(t)εk(t)).
ΔPk(Q)+ΔPk(I)cos(ϕk(t)εk(t))=|Uk(t)|{cos(ϕk(t)+φk)+cos(εk(t)+φk)cos(ϕk(t)εk(t))}=|Uk(t)|×12{cos(2εk(t)+φkϕk(t))cos(ϕk(t)+φk)}=|Uk(t)|sin(εk(t)+φk)sin(ϕk(t)εk(t)).
Uk(t)=exp{2πik(σcσL)/Ω}×σLσcσRσcg(σc+x)exp(2πikx/Ω)dx/σLσcσRσcg(σc+x)dx.
Uk(t)=exp{2πik(σcσL)/Ω}×+g(σc+x)exp(2πikx/Ω)dx/+g(σc+x)dx.

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