## SNR enhancement through phase dependent signal reconstruction algorithms for phase separated interferometric signals

Optics Express, Vol. 15, Issue 16, pp. 10103-10122 (2007)

http://dx.doi.org/10.1364/OE.15.010103

Acrobat PDF (962 KB)

### 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

## 1. Introduction

8. M. A. Choma, C. H. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3×3 fiber-optic couplers,” Opt. Lett. **28**, 2162–2164 (2003). [CrossRef] [PubMed]

9. Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne *en face* optical coherence tomography,” Opt. Lett. **31**, 1815–1817 (2006). [CrossRef] [PubMed]

10. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. **62**, 1267–1277 (1972). [CrossRef]

11. S. K. Sheem, “Optical fiber interferometers with 3×3 directional couplers - analysis,” J. Appl. Phys. **52**, 3865–3872 (1981). [CrossRef]

8. M. A. Choma, C. H. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3×3 fiber-optic couplers,” Opt. Lett. **28**, 2162–2164 (2003). [CrossRef] [PubMed]

*en face*OCT images of biological samples [9

9. Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne *en face* optical coherence tomography,” Opt. Lett. **31**, 1815–1817 (2006). [CrossRef] [PubMed]

12. M. V. Sarunic, M. A. Choma, C. H. Yang, and J. A. Izatt, “Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3×3 fiber couplers,” Opt. Express **13**, 957–967 (2005). [CrossRef] [PubMed]

13. M. V. Sarunic, B. E. Applegate, and J. A. Izatt, “Real-time quadrature projection complex conjugate resolved Fourier domain optical coherence tomography,” Opt. Lett. **31**, 2426–2428 (2006). [CrossRef] [PubMed]

14. J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE **78**, 1369–1394 (1990). [CrossRef]

17. E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, “A generalized noise variance analysis model and its application to the characterization of 1/f noise in homodyne interferometry,” Opt. Express **15**, 3833–3848. [PubMed]

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

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

21. 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]

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

## 2. 3×3 homodyne OCT theory

*en face*imaging of biological samples (following Ref. 1). This scheme has the ability to decouple amplitude and phase information without the need for complex rapid scanning optical delay mechanisms used in heterodyne systems, or expensive components such as spectrometers or swept laser sources. Figure 1(a) shows the experimental setup utilized in this study. Broadband light from an SLD (λ0=1300nm, Δλ=85nm) enters a 2×2 fiber coupler, followed by a 3×3 fiber coupler. Backscattered light from the sample is mixed with reference light to create an interference pattern at detectors 1–3. Detector 4 is used to monitor and correct for source fluctuations. Figure 1(b,c) diagrams the type of data that we are collecting. Using a stationary reference arm, we are essentially measuring a single point on the interferogram (represented by the thick black arrow, Fig. 1(b)). Thus, we measure three interferometric signals that can be thought of as the projections of a complex signal onto axes separated by 120° (Fig. 1(c)). The optical signal at the j

^{th}detector is given by:

*P*and

_{r,j}*P*represent the total DC power returning from the reference and sample arms, respectively;

_{s,j}*1/s*is a scaling factor that accounts for both coupler and detector loss;

_{j}*P*is the returning reference power;

_{r}*P*is the returning coherent light from a depth

_{s}(z)*z*within the sample;

*γ(z)*is the source autocorrelation function;

*θ(z)=2k*, is the phase associated with each depth in the sample, where

_{0}z+ψ(z)*k*is the optical wavenumber corresponding to the center wavelength of the source and

_{0}*Ψ(z)*is the intrinsic reflection phase shift of the sample at depth

*z*; Finally,

*φ*represents the phase shifts between each of the three detectors, attributable to the intrinsic phase shifts of the 3×3 fiber coupler. The signal of interest, which describes the reflectivity profile of the sample, is the coefficient of the cosine term, which can be isolated in several ways following removal of the DC terms. Below we describe several techniques to reconstruct the coefficient of the cosine term.

_{j}## 3. Theoretical SNR corresponding to image reconstruction algorithms

_{R}and P

_{S}are the power returning from the reference and sample arms, respectively, n is the number of detection ports (n≥2), ε is the detector quantum efficiency, τ is the integration time, h is Planck’s constant, and υ is the optical frequency. N

_{i}represents a fluctuating noise term that is zero mean, and assumed to be Gaussian distributed with standard deviation as expected given shot noise limited detection:

_{R}≫P

_{S}), which is typical when imaging highly scattering biological samples. In Eq. 2 we have assumed that the terms

*P*and

_{r,j}*P*from Eq. 1 have been subtracted. This can be accomplished in a practical setting by alternately blocking the sample and reference arms to measure their individual contributions.

_{s,j}### 3.1 Optimal SNR in common interferometric topologies

14. J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE **78**, 1369–1394 (1990). [CrossRef]

_{i}

^{2}]. For some of the following methods we will have terms of the form E[N

_{i}

^{4}] as well. We can evaluate this simplified expression based on a knowledge of the variance at a single detection port: E[N

_{i}

^{2}]=σ

^{2}and E[N

_{i}

^{4}]=3σ

^{4}(where σ is given by Eq. 3). These substitutions can be made since the noise at each port is assumed to be Gaussian. The expectation of odd powers of N

_{i}is zero since the noise is zero mean. In a similar manner we can now evaluate the variance of this method:

### 3.2 Method 1

*α*=1/3),

_{ij}*φ*=120°, and

_{j}*s*=1, and the signal and noise at each point is given by Eqs. 2 and 3 where n=3.

_{i}9. Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne *en face* optical coherence tomography,” Opt. Lett. **31**, 1815–1817 (2006). [CrossRef] [PubMed]

### 3.3 Method 2

*et al*[8

8. M. A. Choma, C. H. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3×3 fiber-optic couplers,” Opt. Lett. **28**, 2162–2164 (2003). [CrossRef] [PubMed]

*S*, as our real signal, the imaginary signal is reconstructed as:

_{1}*φ*=120° and

_{i}*α*=1/3. We are then able to reconstruct our image as the magnitude of this complex signal, (

_{ij}*S*+

_{RE}^{2}*S*). We can simplify the expression as follows:

_{IM}^{2}### 3.4 Method 3

*θ*=tan

^{-1}(

*S*). The estimated phase is then used to divide out the cosine terms present in Eq. 2. Finally, we scale and sum the three signals using scaling factors,

_{IM}/S_{RE}*a*, constrained to sum to 1. In this way, we isolate the desired signal as follows:

_{i}*φ*, and minimizing the resulting noise. For example, if the values of

_{i}*θ*and

*φ*for a given channel produce a cosine value close to zero, then the noise would increase greatly after dividing it by this small number. Hence, this channel would be weighted the least compared to the others. And conversely, maximally interfering signals (large cosine value) are weighted more heavily than others. Since the noise in each channel is equivalent, larger interferometric signals should lead to an increase in SNR. The values of the scaling factors can be expressed as a function of the phase as well as the phase shifts between subsequent ports:

_{i}*a*is:

_{i}*a*values is 1. The above expressions can be used to determine the expected value and variance of the noise for this generalized reconstruction method:

### 3.5 Methods 4 and 5

*a*converts Method 3 into a form identical to Method 5. However similar, we note that these methods differ in the case where the phase shifts at the ports of the fiber coupler are not equally spaced (i.e.

_{i}*φ*≠2

_{i}*π*(i-1)/n). In this case, the

*a*’s can be determined through a minimization, and the method will produce an image with a different SNR than that derived above. Method 5 requires that the phase shifts be equally spaced, and will not perform well under these conditions.

_{i}## 4. Experimental methods

17. E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, “A generalized noise variance analysis model and its application to the characterization of 1/f noise in homodyne interferometry,” Opt. Express **15**, 3833–3848. [PubMed]

*Xenopus laevis*tadpoles. Each data set was processed using the five image reconstruction algorithms described above, and displayed on equivalent color scales. The improved image contrast obtained using reconstruction Methods 3 and 5 confirms our theoretical findings in biological samples.

## 5. Results and discussion

### 5.2 Imaging results

*Xenopus laevis*tadpole. Again, Methods 3 and 5 produced images that more clearly distinguish biological structure from background noise. The nuclei of the cellular structures at the bottom of the image are more visible. The ability to achieve superior SNR based only on reconstruction algorithm implies that, to achieve the same SNR as through commonly used reconstruction algorithms, the optical power incident on fragile biological tissues can be reduced. In Fig. 5 (second column) we have subtracted the DC value of the noise in each image in order to compare the noise variance between images. When a portion of the background noise is magnified (Fig. 5, column 3), there is significantly more background fluctuation in images corresponding to Methods 1 and 2 than in the other images.

### 5.3 Robustness to phase error

_{i}, and the angles between adjacent ports of the fiber coupler,

*φ*. Uncertainty in these values leads to uncertainly in the phase at various points in the image, and additionally leads to an improper choice of noise minimization coefficients,

_{ij}*a*, in Method 3. To reduce the effects of this potential problem, we calibrated the 3×3 system immediately before image acquisition, making the assumption that drifts in the system calibration parameters are slow.

_{i}*π*/2. The results are plotted in Fig. 6 in terms of the SNR coefficient (i.e. the coefficient of P

_{S}ετ/hυ). The computation shows that Method 3 and 5 are surprisingly robust in the presence of phase error. Very large phase errors may be incorporated before these methods drop below the others in terms of SNR. These results imply that the phase-dependent methods not only provide improved SNR, but are relatively insensitive to errors in system calibration.

## 6. Conclusions

## Appendix

## A.1 Derivation of variance for heterodyne detection

_{i,1-Ni,2})

^{4}]=12σ

^{4}and E[(N

_{i,1-Ni,2})

^{2}]=4σ

^{4}.

## A.2 Derivation of variance for heterodyne detection with phase knowledge

## A.3 Derivation of variance for Method 1

_{i})

^{4}]=3σ4 and E[(N

_{i},)

^{2}]=σ

^{2}.

## A.4 Derivation of variance for Method 2

## A.5 Derivation of variance for Method 3

_{1}=2/3, a

_{2}=1/6, a

_{3}=1/6.

## A.6 Derivation of variance for Method 3, n ports

## A.7 Derivation of variance for Methods 4 and 5

## Acknowledgements

## References and links

1. | G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A |

2. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

3. | E. J. Post, “Sagnac effect,” Rev. Mod.Phys. |

4. | D. A. Jackson, A. D. Kersey, and A. C. Lewin, “Fiber gyroscope with passive quadrature detection,” Electron. Lett. |

5. | S. K. Sheem, “Fiberoptic gyroscope with 3×3 directional coupler,” Appl. Phys. Lett. |

6. | Z. Yaqoob, J. G. Wu, X. Q. Cui, X. Heng, and C. H. Yang, “Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements,” Opt. Express |

7. | 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. |

8. | M. A. Choma, C. H. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3×3 fiber-optic couplers,” Opt. Lett. |

9. | Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne |

10. | A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. |

11. | S. K. Sheem, “Optical fiber interferometers with 3×3 directional couplers - analysis,” J. Appl. Phys. |

12. | M. V. Sarunic, M. A. Choma, C. H. Yang, and J. A. Izatt, “Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3×3 fiber couplers,” Opt. Express |

13. | M. V. Sarunic, B. E. Applegate, and J. A. Izatt, “Real-time quadrature projection complex conjugate resolved Fourier domain optical coherence tomography,” Opt. Lett. |

14. | J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE |

15. | L. G. Kazovsky, “Optical heterodyning versus optical homodyning: A comparison,” J. Opt. Commun. |

16. | S. D. Personic, “Image band interpretation of optical heterodyne noise,” AT&T Tech. J. |

17. | E. J. McDowell, X. Cui, Y. Yaqoob, and C. Yang, “A generalized noise variance analysis model and its application to the characterization of 1/f noise in homodyne interferometry,” Opt. Express |

18. | M. E. Smith and J. H. Strange, “NMR techniques in materials physics: A review,” Meas. Sci. Technol. |

19. | A. H. Andersen and J. E. Kirsch, “Analysis of noise in phase contrast MR imaging,” Med. Phys |

20. | C. D. Constantinides, E. Atalar, and E. R. McVeigh, “Signal-to-noise measurements in magnitude images from NMR phased array,” Mag. Res. Med. |

21. | 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 |

22. | N. Aydin and H. S. Markus, “Time-scale analysis of quadrature Doppler ultrasound signals,” IEE P-Sci. Meas. Tech. |

**OCIS Codes**

(110.4280) Imaging systems : Noise in imaging systems

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.4500) Medical optics and biotechnology : Optical coherence tomography

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: June 19, 2007

Revised Manuscript: July 20, 2007

Manuscript Accepted: July 24, 2007

Published: July 26, 2007

**Virtual Issues**

Vol. 2, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Emily J. McDowell, Marinko V. Sarunic, Zahid Yaqoob, and Changhuei Yang, "SNR enhancement through phase dependent signal reconstruction algorithms for phase separated interferometric signals," Opt. Express **15**, 10103-10122 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-16-10103

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### References

- G. Lai and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8,822-827 (1991). [CrossRef]
- I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22,1268-1270 (1997). [CrossRef] [PubMed]
- E. J. Post, "Sagnac effect," Rev. Mod.Phys. 39,475 (1967). [CrossRef]
- D. A. Jackson, A. D. Kersey and A. C. Lewin, "Fiber gyroscope with passive quadrature detection," Electron. Lett. 20,399-401 (1984). [CrossRef]
- S. K. Sheem, "Fiberoptic gyroscope with 3x3 directional coupler," Appl. Phys. Lett. 37,869-871 (1980). [CrossRef]
- Z. Yaqoob, J. G. Wu, X. Q. Cui, X. Heng and C. H. Yang, "Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements," Opt. Express 14,8127-8137 (2006). [CrossRef] [PubMed]
- 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]
- M. A. Choma, C. H. Yang and J. A. Izatt, "Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers," Opt. Lett. 28,2162-2164 (2003). [CrossRef] [PubMed]
- Z. Yaqoob, J. Fingler, X. Heng and C. Yang, "Homodyne en face optical coherence tomography," Opt. Lett. 31,1815-1817 (2006). [CrossRef] [PubMed]
- A. W. Snyder, "Coupled-mode theory for optical fibers," J. Opt. Soc. Am. 62,1267-1277 (1972). [CrossRef]
- S. K. Sheem, "Optical fiber interferometers with 3x3 directional couplers - analysis," J. Appl. Phys. 52,3865-3872 (1981). [CrossRef]
- M. V. Sarunic, M. A. Choma, C. H. Yang and J. A. Izatt, "Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3x3 fiber couplers," Opt. Express 13,957-967 (2005). [CrossRef] [PubMed]
- M. V. Sarunic, B. E. Applegate and J. A. Izatt, "Real-time quadrature projection complex conjugate resolved Fourier domain optical coherence tomography," Opt. Lett. 31,2426-2428 (2006). [CrossRef] [PubMed]
- J. R. Barry and E. A. Lee, "Performance of coherent optical receivers," Proc. IEEE 78,1369-1394 (1990). [CrossRef]
- L. G. Kazovsky, "Optical heterodyning versus optical homodyning: A comparison," J. Opt. Commun. 6,18-24 (1985).
- S. D. Personic, "Image band interpretation of optical heterodyne noise," AT&T Tech. J. 50, 213-& (1971).
- E. J. McDowell, X. Cui, Y. Yaqoob and C. Yang, "A generalized noise variance analysis model and its application to the characterization of 1/f noise in homodyne interferometry," Opt. Express 15, 3833-3848(2007). [PubMed]
- 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]
- 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]
- 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]
- N. Aydin and H. S. Markus, "Time-scale analysis of quadrature Doppler ultrasound signals," IEE P-Sci.Meas. Tech. 148,15-22 (2001). [CrossRef]

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