Abstract

We report several signal reconstruction algorithms for processing phase separated homodyne interferometric signals. Methods that take advantage of the phase of the signal are experimentally shown to achieve a signal-to-noise ratio (SNR) improvement of up to 5 dB over commonly used algorithms. To begin, we present a derivation of the SNR resulting from five image reconstruction algorithms in the context of a 3×3 fiber-coupler based homodyne optical coherence tomography (OCT) system, and clearly show the improvement in SNR associated with phase-based algorithms. Finally, we experimentally verify this improvement and demonstrate the enhancement in contrast and improved image quality afforded by these algorithms through homodyne OCT imaging of a Xenopus laevis tadpole. These algorithms can be generally applied in signal extraction processing where multiple phase separated measurements are available.

© 2007 Optical Society of America

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2007 (1)

J. G. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Q. Cui and C. H. Yang, "Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer," Appl. Phys. Lett. 90, (2007).
[CrossRef]

2006 (3)

2005 (1)

2004 (1)

D. Erdogmus, R. Yan, E. G. Larsson, J. C. Principe and J. R. Fitzsimmons, "Image construction methods for phased array magnetic resonance imaging," J. Magn. Reson. Imaging 20,306-314 (2004).
[CrossRef] [PubMed]

2003 (1)

2001 (1)

N. Aydin and H. S. Markus, "Time-scale analysis of quadrature Doppler ultrasound signals," IEE P-Sci.Meas. Tech. 148,15-22 (2001).
[CrossRef]

1997 (2)

C. D. Constantinides, E. Atalar and E. R. McVeigh, "Signal-to-noise measurements in magnitude images from NMR phased array," Mag. Res. Med. 38,852-857 (1997).
[CrossRef]

I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22,1268-1270 (1997).
[CrossRef] [PubMed]

1996 (2)

M. E. Smith and J. H. Strange, "NMR techniques in materials physics: A review," Meas. Sci. Technol. 7,449-475 (1996).
[CrossRef]

A. H. Andersen and J. E. Kirsch, "Analysis of noise in phase contrast MR imaging," Med. Phys 23,857-869 (1996).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

J. R. Barry and E. A. Lee, "Performance of coherent optical receivers," Proc. IEEE 78,1369-1394 (1990).
[CrossRef]

1985 (1)

L. G. Kazovsky, "Optical heterodyning versus optical homodyning: A comparison," J. Opt. Commun. 6,18-24 (1985).

1984 (1)

D. A. Jackson, A. D. Kersey and A. C. Lewin, "Fiber gyroscope with passive quadrature detection," Electron. Lett. 20,399-401 (1984).
[CrossRef]

1981 (1)

S. K. Sheem, "Optical fiber interferometers with 3x3 directional couplers - analysis," J. Appl. Phys. 52,3865-3872 (1981).
[CrossRef]

1980 (1)

S. K. Sheem, "Fiberoptic gyroscope with 3x3 directional coupler," Appl. Phys. Lett. 37,869-871 (1980).
[CrossRef]

1972 (1)

1971 (1)

S. D. Personic, "Image band interpretation of optical heterodyne noise," AT&T Tech. J. 50, 213-& (1971).

1967 (1)

E. J. Post, "Sagnac effect," Rev. Mod.Phys. 39,475 (1967).
[CrossRef]

Appl. Phys. Lett. (2)

S. K. Sheem, "Fiberoptic gyroscope with 3x3 directional coupler," Appl. Phys. Lett. 37,869-871 (1980).
[CrossRef]

J. G. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Q. Cui and C. H. Yang, "Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer," Appl. Phys. Lett. 90, (2007).
[CrossRef]

AT&T Tech. J. (1)

S. D. Personic, "Image band interpretation of optical heterodyne noise," AT&T Tech. J. 50, 213-& (1971).

Electron. Lett. (1)

D. A. Jackson, A. D. Kersey and A. C. Lewin, "Fiber gyroscope with passive quadrature detection," Electron. Lett. 20,399-401 (1984).
[CrossRef]

J. Appl. Phys. (1)

S. K. Sheem, "Optical fiber interferometers with 3x3 directional couplers - analysis," J. Appl. Phys. 52,3865-3872 (1981).
[CrossRef]

J. Magn. Reson. Imaging (1)

D. Erdogmus, R. Yan, E. G. Larsson, J. C. Principe and J. R. Fitzsimmons, "Image construction methods for phased array magnetic resonance imaging," J. Magn. Reson. Imaging 20,306-314 (2004).
[CrossRef] [PubMed]

J. Opt. Commun. (1)

L. G. Kazovsky, "Optical heterodyning versus optical homodyning: A comparison," J. Opt. Commun. 6,18-24 (1985).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Mag. Res. Med. (1)

C. D. Constantinides, E. Atalar and E. R. McVeigh, "Signal-to-noise measurements in magnitude images from NMR phased array," Mag. Res. Med. 38,852-857 (1997).
[CrossRef]

Meas. Sci. Technol. (1)

M. E. Smith and J. H. Strange, "NMR techniques in materials physics: A review," Meas. Sci. Technol. 7,449-475 (1996).
[CrossRef]

Meas. Tech. (1)

N. Aydin and H. S. Markus, "Time-scale analysis of quadrature Doppler ultrasound signals," IEE P-Sci.Meas. Tech. 148,15-22 (2001).
[CrossRef]

Med. Phys (1)

A. H. Andersen and J. E. Kirsch, "Analysis of noise in phase contrast MR imaging," Med. Phys 23,857-869 (1996).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (4)

Proc. IEEE (1)

J. R. Barry and E. A. Lee, "Performance of coherent optical receivers," Proc. IEEE 78,1369-1394 (1990).
[CrossRef]

Rev. Mod.Phys. (1)

E. J. Post, "Sagnac effect," Rev. Mod.Phys. 39,475 (1967).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Experimental setup for 3×3 fiber coupler based homodyne optical coherence tomography. SLD: superluminescent diode, Dn: nth photodetector, M: mirror, X-Y: x-y scanner, OBJ: 20× microscope objective. (b) In this homodyne system the reference mirror (M) is stationary. We can think of the measured signal as a single point (black arrow) on the modulated coherence function that would be obtained if the reference arm was swept. (c) These points are the projections of a complex value onto axes separated by 120°.

Fig. 2.
Fig. 2.

2×2 (50/50) interferometric setups utilizing a) homodyne and b) heterodyne detection. In (a) the reference mirror is stationary, while it is translated in (b). The 180° phase shifts of the fiber coupler are evident in the acquired signals at the two output ports.

Fig. 3.
Fig. 3.

(a) Reconstructed signals from an attenuated mirror. A beam chopper was used to make measurements of both signal and background noise, which were used to experimentally determine the SNR of the five methods. (b) A magnified view of the noise from (a) depicting experimentally determined values for the mean and variance of the noise.

Fig. 4.
Fig. 4.

These images show a portion of a highly attenuated Air Force test target, representing a very low signal situation. The three images were reconstructed from a single data set and reconstructed using Methods 1–5 (described above). Methods 3 and 5 clearly perform better than the others, showing a notable increase in contrast between the bars of the test target and the background.

Fig. 5.
Fig. 5.

In the first column the image reconstruction algorithms were evaluated on images from a stage 54 Xenopus tadpole. Again, Methods 3 and 5 produced images with improved SNR, more clearly distinguishing biological features such as cell nuclei from background noise. In the second column of images the DC noise has been subtracted from the image. The increase noise variance is now visible in the background of the images corresponding to Methods 1 and 2 in a blown up portion of the background (third column)

Fig. 6.
Fig. 6.

SNR is plotted verses phase error for the five reconstruction methods. Only Methods 3 and 5 are phase dependent. Here, we see that these methods are relatively robust to phase error, only dropping below the other methods for fairly large errors in phase.

Tables (1)

Tables Icon

Table 1. Comparison of theoretical and experimental results. Notably, the phase dependent methods (3 and 5) show superior SNR and noise performance with respect to the others.

Equations (45)

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P j ( z ) = P r , j + P s , j + 2 ( 1 s j ) α 41 α 4 j α 51 α 5 j P r ( P S ( z ) γ ( z ) ) cos ( θ ( z ) + φ j ) .
S i = 2 n P R P S ε τ h ν cos ( θ + φ i ) ± N i ,
σ N i = P r ε τ n h υ
M Optimal = 1 4 ( S 1 S 2 ) 2 ,
Sig M Optimal = P R P S ( ε τ h υ ) 2 .
E [ M Optimal ( N i ) ] = E [ 1 4 ( N 1 N 2 ) 2 ] = 1 4 ( E [ N 1 2 ] + E [ N 2 2 ] ) = 1 2 σ 2 .
σ M Optimal 2 = E [ 1 16 ( N 1 N 2 ) 4 ] ( 1 2 σ 2 ) 2 =
= 1 16 E [ N 1 4 ] + 6 16 E [ N 1 2 ] E [ N 2 2 ] + 1 16 E [ N 2 4 ] 1 4 σ 4 .
= 3 16 σ 4 + 6 16 σ 4 + 3 16 σ 4 1 4 σ 4 2 = 1 2 σ 4
SNR optimal = ( Sig M Optimal σ M Optimal ) P R P S ( ε τ h υ ) 2 P R ε τ 2 2 h υ = 2 2 P S ε τ h υ .
M heterodyne = ( i = 1 X ( S i , 1 S i , 2 ) cos ( Δ ω i ) ) 2 + ( i = 1 X ( S i , 1 S i , 2 ) sin ( Δ ω i ) ) 2 ,
E [ M heterodyne ( N i ) ] = 2 X σ 2
σ heterodyne 2 = 4 X 2 σ 4 .
SNR heterodyne = P R P S ( ε τ h υ ) 2 2 X P R ε τ 2 X h υ = P S ε τ h υ .
M heterodyne with phase knowledge = ( i = 1 X ( S i , 1 S i , 2 ) cos ( Δ ω i + θ ) ) 2 .
E [ M heterodyne with phase knowledge ] = X σ 2
σ heterodyne with phase knowledge 2 = 2 X 2 σ 4
SNR heterodyne with phase knowledge = P R P S ( ε τ h υ ) 2 2 X P R ε τ 2 X h υ = 2 P S ε τ h υ .
M 1 = 3 2 ( S 1 2 + S 2 2 + S 3 2 ) .
E [ M 1 ( N i ) ] = 9 2 σ 2
σ M 1 2 = 27 2 σ 4 ,
SNR M 1 = P R P S ( ε τ h υ ) 2 3 3 P R ε τ 2 3 h υ = 6 3 P S ε τ h υ .
S IM = S 1 cos φ 2 S 2 β sin φ 2 β = α 41 α 51 α 42 α 52 .
M 2 = 9 4 ( S RE 2 + S IM 2 ) = 3 ( S 1 2 + S 2 2 + S 1 S 2 ) .
E [ M 2 ( N i ) ] = 6 σ 2
σ M 2 2 = 45 σ 4 ,
SNR M 2 = P R P S ( ε τ h υ ) 2 3 5 P R ε τ 3 h υ = 5 5 P S ε τ h υ .
M 3 = 9 4 [ a 1 S 1 cos ( θ + φ 1 ) + a 2 S 2 cos ( θ + φ 2 ) + a 3 S 3 cos ( θ + φ 3 ) ] 2 .
a i = 2 3 [ cos 2 ( θ + φ i ) ] .
E [ M 3 [ N i ] ] = 3 2 σ 2
σ M 3 2 = 9 2 σ 4
SNR M 3 = P R P S ( ε τ h υ ) 2 3 P R ε τ 2 3 h υ = 2 P S ε τ h υ .
n 2 4 ( i = 1 n a i S i cos ( θ + 2 π n ( i 1 ) ) ) 2 ,
a i = 2 n cos 2 ( θ + 2 π n ( i 1 ) ) .
E [ M 3 , n ports ] = n 2 σ 2
σ M 3 , n ports 2 = 1 2 n 2 σ 4 ,
SNR M 3 , n ports = P R P S ( ε τ h υ ) 2 n P R ε τ 2 n h υ = 2 P S ε τ h υ .
M 4 = ( i = 1 n S i cos ( ϕ i ) ) 2 + ( i = 1 n S i sin ( ϕ i ) ) 2 ,
E [ M 4 ( N i ) ] = n σ 2 .
σ M 4 2 = n 2 σ 4 .
SNR M 4 = ( P R P S ( ε τ h υ ) 2 n P R ε τ n h υ ) = P S ε τ h ν .
M 5 = ( i = 1 n S i cos ( ϕ i + θ ) ) 2
E [ M 5 ( N i ) ] = n 2 σ 2
σ M 5 2 = 1 2 n 2 σ 4 .
SNR M 5 = ( P R P S ( ε τ h υ ) 2 n 2 P R ε τ n h υ ) = 2 P S ε τ h v .

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