## A generalized noise variance analysis model and its application to the characterization of 1/f noise

Optics Express, Vol. 15, Issue 7, pp. 3833-3848 (2007)

http://dx.doi.org/10.1364/OE.15.003833

Acrobat PDF (314 KB)

### Abstract

We present a novel generalized model for the analysis of noise with a known spectral density. This model is particularly appropriate for the analysis of noise with a 1/f^{α} distribution. The noise model reveals that, for α > 1, 1/f^{α} noise significantly impacts the signal-to-noise ratio (SNR) for integration times that near a characteristic time, beyond which the SNR will no longer significantly improve with increasing integration time. We experimentally verify our theoretical findings with a set of experiments employing a quadrature homodyne optical coherence tomography (OCT) system and find good agreement. The characteristic integration time is measured to be approximately 2 ms for our system. Additionally, we find that the 1/f noise characteristics, including the exponent, α, as well as the characteristic integration time, are system and photodetector dependent.

© 2007 Optical Society of America

## 1. Introduction

01. P. Dutta and P. M. Horn, “Low-frequency fluctuations in solids – 1/f noise,” Rev. Mod. Phys. **53**,497516 (1981). [CrossRef]

05. R. F. Voss, “Linearity of 1/f noise mechanisms,” Phys. Rev. Lett. **40**,913–916 (1978). [CrossRef]

06. P. C. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos **11**,641–652 (2001). [CrossRef]

07. T. Musha and H. Higuchi, “1/f fluctuation of a traffic current on an expressway,” Jpn. J. Appl. Phys. **15**,1271–1275 (1976). [CrossRef]

^{α}, where α commonly ranges from 0.5 to 1.5 [8]. Despite significant effort in describing a universal model for the origin of 1/f noise [9

09. B. Kaulakys, V. Gontis, and M. Alaburda, “Point process model of 1/f noise vs a sum of lorentzians,” Phys. Rev. E **71**,051105 (2005). [CrossRef]

10. J. B. Johnson, “The schottky effect in low frequency circuits,” Phys. Rev. **26**,0071–0085 (1925). [CrossRef]

11. W. Schottky, “Small-shot effect and flicker effect,” Phys. Rev. **28**,74–103 (1926). [CrossRef]

12. Z. Siwy and A. Fulinski, “Origin of 1/f(alpha) noise in membrane channel currents,” Phys. Rev. Lett.89, (2002). [CrossRef] [PubMed]

13. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science **254**,1178–1181 (1991). [CrossRef] [PubMed]

14. A. M. Rollins and J. A. Izatt, “Optimal interferometer designs for optical coherence tomography,” Opt. Lett. **24**,1484–1486 (1999). [CrossRef]

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

19. X. Q. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, “Polarization interferometer for measuring small displacement,” IEEE Trans. Instrum. Meas. **50**,868–871 (2001). [CrossRef]

20. C. M. Wu, C. S. Su, G. S. Peng, and Y. J. Huang, “Polarimetric, nonlinearity-free, homodyne interferometer for vibration measurement,” Metrologia **33**,533–537 (1996). [CrossRef]

21. C. Chao, Z. H. Wang, W. G. Zhu, and O. K. Tan, “Scanning homodyne interferometer for characterization of piezoelectric films and microelectromechanical systems devices,” Rev. Sci. Instrum. **76**,063906 (2005). [CrossRef]

22. D. L. Mazzoni and C. C. Davis, “Trace detection of hydrazines by optical homodyne interferometry,” Appl. Opt. **30**,756–764 (1991). [CrossRef] [PubMed]

23. E. Beaurepaire, L. Moreaux, F. Amblard, and J. Mertz, “Combined scanning optical coherence and two-photon-excited fluorescence microscopy,” Opt. Lett. **24**,969–971 (1999). [CrossRef]

27. R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of Fourier domain vs. time domain optical coherence tomography,” Opt. Express **11**,889–894 (2003). [CrossRef] [PubMed]

30. Y. Salvade and R. Dandliker, “Limitations of interferometry due to the flicker noise of laser diodes,” J. Opt. Soc. Am. A **17**,927–932 (2000). [CrossRef]

31. R. H. Hamstra and P. Wendland, “Noise and frequency-response of silicon photodiode operational amplifier combination,” Appl. Opt. **11**,1539 (1972). [CrossRef] [PubMed]

_{white-to-1/f}) at which 1/f noise begins to dominate over white noise, such as shot noise. In Section 3, we describe our 3x3 fiber coupler based homodyne interferometer, as well as the experiments that were conducted to characterize 1/f noise in that system. In Section 4, we compare our theoretical results to experimental findings from the homodyne interferometer. In addition to validating our theoretical results, we also find that the 1/f noise characteristics are detector dependent. Finally, we summarize our findings in Section 5.

## 2. Theoretical noise model

*τ*, will yield two terms. The first term simply integrates over τ to give an expected value of

*X(τ)*=

*x*. For the second term we have the following:

_{0}τ*ΔX(τ):*

*δ*is varied in the interval [0, 2π].

_{i}*S(f)*=

*A*). In this situation, Eq. (7) can be rewritten as:

_{white}*A*is given by 2

_{white}*x*. This leads to σ

_{0}^{2}

*(τ) =*

_{X,shot noise}*x*; a result that is consistent with the Poissonian nature of shot noise.

_{o}τ*f*and α. Additionally, since there is no straightforward analytical solution to Eq. (10), we approximate the solution in order to show the form of the dependence on integration time (τ).

_{min}### 2.1 Choice of f_{min}

*f*=

_{min}*1/T*, where

*T*is the total experimental time frame. This is different from the integration time τ, which gives the time step over which the signal is sampled. The difference between these two time constants can be better appreciated in the following scenario. Suppose we have a light source with a known 1/f noise power spectrum,

*S(f)*, which we decide to amplitude modulate in order to send a message. The message length is

*T*in its entirety. The message is analog in nature but is band limited such that it does not contain frequency components beyond

*f*. The message can be received, with no information loss, by measuring the light intensity over a time frame of

_{signal}*T*and choosing a time step of τ=2/

*f*for signal integration. Intuitively, we can appreciate that this time step integration is useful for suppressing high frequency (

_{signal}*f*>

*f*) noise contributions in our measurements. The noise variance for this experiment can be calculated using Eq. (10) based on the abovementioned parameters. We can see that the message length

_{signal}*T*is relevant for noise variance consideration; as the length of

*T*is increased, more low frequency noise components will be incorporated, and the SNR will correspondingly deteriorate. Noise components of frequency lower than

*f*are present in the collected signal trace. However, these components are manifested as a net DC shift in the entire collected signal, and have no impact on the content of the sent message (see Fig. 1).

_{min}_{min}imposed by the detection system). In reality, this type of natural capping has only been seen in very few experimental situations [32

32. J. Clarke and T. Y. Hsiang, “Low-frequency noise in tin and lead films at superconducting transition,” Phys. Rev. B **13**,4790–4800 (1976). [CrossRef]

33. M. S. Keshner, “1/f noise,” Proc. IEEE **70**,212–218 (1982). [CrossRef]

34. B. Pellegrini, R. Saletti, P. Terreni, and M. Prudenziati, “1/f-gamma noise in thick-film resistors as an effect of tunnel and thermally activated emissions, from measures versus frequency and temperature,” Phys. Rev. B **27**,1233–1243 (1983). [CrossRef]

35. M. A. Caloyannides, “Microcycle spectral estimates of 1/f noise in semiconductors,” J. Appl. Phys. **45**,307–316 (1974). [CrossRef]

36. B. B. Mandelbrot and J. R. Wallis, “Some long-run properties of geophysical records,” Water Resources Research **5**,321 (1969). [CrossRef]

### 2.2 Influence of the noise exponent, α

*A*will change as α is varied. A

_{pink}*f*of 1.6 mHz, corresponding to a relatively long experimental time frame, was used in these simulations.

_{min}*τ*) at which 1/f noise begins to dominate over white noise.

_{white-to-l/f}*X*/

*σ*)

^{2}[37], where

*X*is the total number of photon counts, which increases linearly with τ, and σ is the standard deviation of the noise. As expected, the theoretical shot noise limited SNR increases linearly with integration time. For 0<α<1, the SNR transitions from a curve that is approximately linear (similar to white noise), to a curve that appears to taper towards a constant value. The linear dependence of the 1/f noise standard deviation in Fig. 2(b) implies that the corresponding SNR will be constant, since the total signal also increases linearly with increasing integration time. Figure 3(b) shows that this appears to be empirically true.

*;1/f*, Eq. (10) can be easily simplified. In the context of the thought experiment described above (Section 2.1), this constraint corresponds to a situation in which the message is long (

_{min}*f*small) and the maximum signal frequency is high (τ=2/

_{min}*f*is small) – a signal that can be expected to describe an overwhelmingly large fraction of practical situations. Under these conditions, we arrive at the following expressions for the noise variance:

_{signal}**t**

^{α-l}e

^{-t}over the interval [0,∞] for α>0. Z

^{+}refers to the set of positive integers. Eq. (11) shows that, for small

*f*and increasing τ, two distinct regimes exist. In these regimes, the SNR depends on τ as follows:

_{min}^{α+1}. This confirms our intuition of a transition from white noise to 1/f noise, as the SNR moves from a function with τ dependence to a function of constant value. This dependence implies that the SNR can still increase after crossing the characteristic integration time (τ

_{white-to-1/f}) at which 1/f noise begins to dominate, although the gain in SNR from further increases in τ occurs with diminishing returns as α approaches 1.

^{2}dependence. These approximations confirm our observations from Figs. 2 and 3 that, for α>1, the SNR should reach a constant value when 1/f noise is the dominant noise process. This is quite a remarkable fact, implying that once the integration time is increased past τ

*, there will be no further significant improvements in SNR. The t dependence of Eq. (10) and Eq. (11) is plotted in Fig. 4 versus the 1/f exponent, α. For integer values of α, Eq. (10) cannot be easily solved, and these locations are represented by open circles.*

_{white-to-1/f}## 3. Experimental methods

*et al*. [25

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

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

_{0}=1300nm, Δλ=85nm) enters a 2x2 fiber coupler, followed by a 3x3. Backscattered light from the sample is mixed with reference light to create an interference pattern that is detected at detectors 1-3. Detector 4 is used to monitor and correct for source fluctuations. Figure 5(b) diagrams the vector relationship between the signals at each arm of the 3x3 coupler, noting the dependence on the power transfer coefficients α

*of the fiber coupler. The interferometric signals are phase separated by nominally 120° (depending on the α*

_{mn}*), as required by conservation of energy. The optical signal at the*

_{mn}*j*detector is given by:

^{th}*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*+

_{0}z*Ψ(z)*, is the phase associated with each depth in the sample, where

*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*represent the phase shifts between each of the three detectors, attributable to the phase shifts inherent to the 3x3 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, after DC removal, by simply squaring and summing the signals from the three ports.

_{j}*φ*=116.6 ± 1.2°,

_{1}*φ*=120.7 ± 0.9°,

_{2}*φ*=122.5 ± 0.8°. The objective lens was a 20x, 0.4 NA IR lens, allowing for a measured lateral resolution of 9.4 μm. The broadband SLD source provided a measured axial resolution of 14 μm. A sample arm power of approximately 30 μW, measured at a single detector, was used in all following experiments. In order to experimentally measure the SNR of the homodyne interferometer, time traces of the OCT signal were acquired with a mirror in the sample arm. To measure the noise contribution, the sample arm was blocked. Varying integration times were used to bin the measured signal into integrated ‘blocks’. These ‘blocks’ are represented in Fig. 1 by dashed lines with spacing τ, and the integrated signal is proportional to the total number of photons detected over this time interval. The power spectrum of the noise was determined using only the integrated noise signal. The SNR was determined by taking the square of the mean value of the integrated signal divided by the standard deviation of the integrated noise signal

_{3}## 4. Results and discussion

### 4.1 Measured power spectrum

*A*, used in the above derivation. The constant value of the white noise determined

_{pink}*A*. Additionally, we note that the frequency at which white noise processes, shot noise in this case, became dominant was approximately 70 Hz.

_{white}### 4.2 Experimental SNR versus integration time

### 4.3 Characteristic time

_{min}. To verify our choice of

*f*=

_{min}*1/T*, we fit the data in Fig. 7, sampled at 30 kHz for 1 second to an expression for SNR, including both 1/f and shot noise terms:

*σ*and

^{2}_{X,pink}*σ*are given by Eq. (9) and Eq. (10), respectively. Amplitude values contained in these equations,

^{2}_{X,white}*A*and

_{white}*A*, were obtained from the power spectrum, as described in section 4.1. The only free variables in Eq. (14) are the total photon count rate,

_{pink}*x*, and

_{0}*f*, contained in

_{min}*σ*. The fit can be seen in blue in Fig. 7. From this fit we determined a

^{2}_{X,pink}*f*of 1.1 Hz, which is approximately equal to 1/(

_{min}*T*=1s). This result helps to confirm the validity of our model for making predictions about experimental results.

*f*, we used the theoretical noise model to predict the characteristic integration time at which 1/f noise became dominant. This time was determined as the time at which white noise and 1/f noise give an equivalent noise variance, and the following equation holds:

_{min}*=1.65 ms, and note that the SNR of the measurement should begin to be affected at shorter times when 1/f noise is less than, but not negligible compared to white noise. This time agrees fairly well with our experimentally determined time of 2.1 ms.*

_{white-to-1/f}### 4.4 1/f noise power dependence

*A*and

_{pink}*A*. For shot noise, the constant can be determined using knowledge of Poisson statistics. One question that arises concerns the form of the constant for 1/f noise,

_{white}*A*. Figure 9 shows the dependence of 1/f noise on reference arm power. These noise values were computed using an integration time of τ=10 ms, which falls well above the point at which 1/f noise becomes dominant in Figs. 7 and 8. The linear trend in Fig. 9 makes intuitive sense; like shot noise, the 1/f noise is directly proportional to the number of detected photons.

_{pink}### 4.5 Sources of 1/f noise

_{4}. We note that this type of correction, commonly used to reduce excess intensity noise [14

14. A. M. Rollins and J. A. Izatt, “Optimal interferometer designs for optical coherence tomography,” Opt. Lett. **24**,1484–1486 (1999). [CrossRef]

## 5. Conclusions

*=2.1 ms, beyond which increases in integration time did not produce corresponding increases in SNR. This time agrees fairly well with the theoretically determined value of τ*

_{white-to-1/f}*=1.65 ms based on the measured power spectral characteristics of system. This characteristic time depends on the 1/f characteristics of the optical system, and is both system and detector dependent. Finally, we note that careful photodetector selection and characterization is important in order to minimize 1/f noise in homodyne detection.*

_{white-to-1/f}## Acknowledgments

## References and links

01. | P. Dutta and P. M. Horn, “Low-frequency fluctuations in solids – 1/f noise,” Rev. Mod. Phys. |

02. | W. H. Press, “Flicker noises in astronomy and elsewhere,” Comments Astrophys . |

03. | M. B. Weissman, “1/f noise and other slow, nonexponential kinetics in condensed matter,” Rev. Mod. Phys. |

04. | W. T. Li and D. Holste, “Universal 1/f noise, crossovers of scaling exponents, and chromosome-specific patterns of guanine-cytosine content in DNA sequences of the human genome,” Phys. Rev. E |

05. | R. F. Voss, “Linearity of 1/f noise mechanisms,” Phys. Rev. Lett. |

06. | P. C. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos |

07. | T. Musha and H. Higuchi, “1/f fluctuation of a traffic current on an expressway,” Jpn. J. Appl. Phys. |

08. | E. Milotti, “1/f noise: A pedagogical review,” invited talk to E-GLEA-2 (2001). |

09. | B. Kaulakys, V. Gontis, and M. Alaburda, “Point process model of 1/f noise vs a sum of lorentzians,” Phys. Rev. E |

10. | J. B. Johnson, “The schottky effect in low frequency circuits,” Phys. Rev. |

11. | W. Schottky, “Small-shot effect and flicker effect,” Phys. Rev. |

12. | Z. Siwy and A. Fulinski, “Origin of 1/f(alpha) noise in membrane channel currents,” Phys. Rev. Lett.89, (2002). [CrossRef] [PubMed] |

13. | D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science |

14. | A. M. Rollins and J. A. Izatt, “Optimal interferometer designs for optical coherence tomography,” Opt. Lett. |

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

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

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

18. | V. Greco, C. Iemmi, S. Ledesma, A. Mannoni, G. Molesini, and F. Quercioli, “Multiphase homodyne displacement sensor,” Optik |

19. | X. Q. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, “Polarization interferometer for measuring small displacement,” IEEE Trans. Instrum. Meas. |

20. | C. M. Wu, C. S. Su, G. S. Peng, and Y. J. Huang, “Polarimetric, nonlinearity-free, homodyne interferometer for vibration measurement,” Metrologia |

21. | C. Chao, Z. H. Wang, W. G. Zhu, and O. K. Tan, “Scanning homodyne interferometer for characterization of piezoelectric films and microelectromechanical systems devices,” Rev. Sci. Instrum. |

22. | D. L. Mazzoni and C. C. Davis, “Trace detection of hydrazines by optical homodyne interferometry,” Appl. Opt. |

23. | E. Beaurepaire, L. Moreaux, F. Amblard, and J. Mertz, “Combined scanning optical coherence and two-photon-excited fluorescence microscopy,” Opt. Lett. |

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

25. | Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. |

26. | N. Choudhury, G. J. Song, F. Y. Chen, S. Matthews, T. Tschinkel, J. F. Zheng, S. L. Jacques, and A. L. Nuttall, “Low coherence interferometry of the cochlear partition,” Hearing Res. |

27. | R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of Fourier domain vs. time domain optical coherence tomography,” Opt. Express |

28. | M. A. Choma, M. Sarunic, C. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express |

29. | J. F. de Boer, C. B., B. Park, M. Pierce, G. Tearney, and B. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. |

30. | Y. Salvade and R. Dandliker, “Limitations of interferometry due to the flicker noise of laser diodes,” J. Opt. Soc. Am. A |

31. | R. H. Hamstra and P. Wendland, “Noise and frequency-response of silicon photodiode operational amplifier combination,” Appl. Opt. |

32. | J. Clarke and T. Y. Hsiang, “Low-frequency noise in tin and lead films at superconducting transition,” Phys. Rev. B |

33. | M. S. Keshner, “1/f noise,” Proc. IEEE |

34. | B. Pellegrini, R. Saletti, P. Terreni, and M. Prudenziati, “1/f-gamma noise in thick-film resistors as an effect of tunnel and thermally activated emissions, from measures versus frequency and temperature,” Phys. Rev. B |

35. | M. A. Caloyannides, “Microcycle spectral estimates of 1/f noise in semiconductors,” J. Appl. Phys. |

36. | B. B. Mandelbrot and J. R. Wallis, “Some long-run properties of geophysical records,” Water Resources Research |

37. | A. Yariv and P. Yeh, |

**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: December 18, 2006

Revised Manuscript: March 13, 2007

Manuscript Accepted: March 13, 2007

Published: April 2, 2007

**Virtual Issues**

Vol. 2, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Emily J. McDowell, Xiquan Cui, Zahid Yaqoob, and Changhuei Yang, "A generalized noise variance analysis model and its application to the characterization of 1/f noise," Opt. Express **15**, 3833-3848 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-7-3833

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### References

- P. Dutta and P. M. Horn, "Low-frequency fluctuations in solids - 1/f noise," Rev. Mod. Phys. 53, 497-516 (1981). [CrossRef]
- W. H. Press, "Flicker noises in astronomy and elsewhere," Comments Astrophys. 7, 103-119 (1978).
- M. B. Weissman, "1/f noise and other slow, nonexponential kinetics in condensed matter," Rev. Mod. Phys. 60, 537-571 (1988). [CrossRef]
- W. T. Li and D. Holste, "Universal 1/f noise, crossovers of scaling exponents, and chromosome-specific patterns of guanine-cytosine content in DNA sequences of the human genome," Phys. Rev. E 71, 041910 (2005). [CrossRef]
- R. F. Voss, "Linearity of 1/f noise mechanisms," Phys. Rev. Lett. 40, 913-916 (1978). [CrossRef]
- P. C. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001). [CrossRef]
- T. Musha and H. Higuchi, "1/f fluctuation of a traffic current on an expressway," Jpn. J. Appl. Phys. 15, 1271-1275 (1976). [CrossRef]
- E. Milotti, "1/f noise: A pedagogical review," invited talk to E-GLEA-2 (2001).
- B. Kaulakys, V. Gontis, and M. Alaburda, "Point process model of 1/f noise vs a sum of lorentzians," Phys. Rev. E 71, 051105 (2005). [CrossRef]
- J. B. Johnson, "The schottky effect in low frequency circuits," Phys. Rev. 26, 71-85 (1925). [CrossRef]
- W. Schottky, "Small-shot effect and flicker effect," Phys. Rev. 28, 74-103 (1926). [CrossRef]
- Z. Siwy and A. Fulinski, "Origin of 1/f(alpha) noise in membrane channel currents," Phys. Rev. Lett. 89, (2002). [CrossRef] [PubMed]
- D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991). [CrossRef] [PubMed]
- A. M. Rollins and J. A. Izatt, "Optimal interferometer designs for optical coherence tomography," Opt. Lett. 24, 1484-1486 (1999). [CrossRef]
- J. R. Barry and E. A. Lee, "Performance of coherent optical receivers," Proc. IEEE 78, 1369-1394 (1990). [CrossRef]
- S. D. Personic, "Image band interpretation of optical heterodyne noise," AT&T Tech. J. 50, 213 (1971).
- L. G. Kazovsky, "Optical heterodyning versus optical homodyning: A comparison," J. Opt. Commun. 6, 18-24 (1985).
- V. Greco, C. Iemmi, S. Ledesma, A. Mannoni, G. Molesini, and F. Quercioli, "Multiphase homodyne displacement sensor," Optik 97, 15-18 (1994).
- X. Q. Liu, W. Clegg, D. F. L. Jenkins, and B. Liu, "Polarization interferometer for measuring small displacement," IEEE Trans. Instrum. Meas. 50, 868-871 (2001). [CrossRef]
- C. M. Wu, C. S. Su, G. S. Peng, and Y. J. Huang, "Polarimetric, nonlinearity-free, homodyne interferometer for vibration measurement," Metrologia 33,533-537 (1996). [CrossRef]
- C. Chao, Z. H. Wang, W. G. Zhu, and O. K. Tan, "Scanning homodyne interferometer for characterization of piezoelectric films and microelectromechanical systems devices," Rev. Sci. Instrum. 76,063906 (2005). [CrossRef]
- D. L. Mazzoni and C. C. Davis, "Trace detection of hydrazines by optical homodyne interferometry," Appl. Opt. 30,756-764 (1991). [CrossRef] [PubMed]
- E. Beaurepaire, L. Moreaux, F. Amblard, and J. Mertz, "Combined scanning optical coherence and two-photon-excited fluorescence microscopy," Opt. Lett. 24,969-971 (1999). [CrossRef]
- M. A. Choma, C. H. Yang, and J. A. Izatt, "Instantaneous quadrature low-coherence interferometry with 3 x 3 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]
- N. Choudhury, G. J. Song, F. Y. Chen, S. Matthews, T. Tschinkel, J. F. Zheng, S. L. Jacques, and A. L. Nuttall, "Low coherence interferometry of the cochlear partition," Hearing Res. 220,1-9 (2006). [CrossRef]
- R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, "Performance of Fourier domain vs. time domain optical coherence tomography," Opt. Express 11,889-894 (2003). [CrossRef] [PubMed]
- M. A. Choma, M. Sarunic, C. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11,2183 - 2189 (2003). [CrossRef] [PubMed]
- J. F. de Boer, C. B., B. Park, M. Pierce, G. Tearney, and B. Bouma, "Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography," Opt. Lett. 28,2067 -2069 (2003). [CrossRef] [PubMed]
- Y. Salvade and R. Dandliker, "Limitations of interferometry due to the flicker noise of laser diodes," J. Opt. Soc. Am. A 17,927-932 (2000). [CrossRef]
- R. H. Hamstra and P. Wendland, "Noise and frequency-response of silicon photodiode operational amplifier combination," Appl. Opt. 11,1539 (1972). [CrossRef] [PubMed]
- J. Clarke and T. Y. Hsiang, "Low-frequency noise in tin and lead films at superconducting transition," Phys. Rev. B 13,4790-4800 (1976). [CrossRef]
- M. S. Keshner, "1/f noise," Proc. IEEE 70,212-218 (1982). [CrossRef]
- B. Pellegrini, R. Saletti, P. Terreni, and M. Prudenziati, "1/f-gamma noise in thick-film resistors as an effect of tunnel and thermally activated emissions, from measures versus frequency and temperature," Phys. Rev. B 27,1233-1243 (1983). [CrossRef]
- M. A. Caloyannides, "Microcycle spectral estimates of 1/f noise in semiconductors," J. Appl. Phys. 45,307-316 (1974). [CrossRef]
- B. B. Mandelbrot and J. R. Wallis, "Some long-run properties of geophysical records," Water Resources Research 5,321 (1969). [CrossRef]
- A. Yariv and P. Yeh, Photonics: Optical electronics in modern communications (Oxford University Press, New York, 2007).

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