## Frequency-domain acquisition of fourth-order correlation by spectral intensity interferometry |

Optics Express, Vol. 21, Issue 20, pp. 23206-23219 (2013)

http://dx.doi.org/10.1364/OE.21.023206

Acrobat PDF (1213 KB)

### Abstract

We report on the spectral intensity interferometer (SII) which is a frequency-domain variant of the fourth-order interferometry. In the SII, the power spectrum of the intensity is acquired for light fields of an interferometer. It produces a fringed spectral interferogram which can be acquired by means of an electric spectrum analyzer in keeping the relative time delay constant during the acquisition. Through both theoretical and experimental investigations, we have found that the SII interferogram provides the intensity correlation information without concern of field-sensitive disturbances which are vulnerable to minute variations of the optical paths. As an application example, a precision time-of-flight measurement was demonstrated by using a fiber-optic SII with an amplified spontaneous emission (ASE) light source. A large delay of 4.1-km long fiber was successfully analyzed from the fringe period. Its wavelength-dependent group delay or the group velocity dispersion (GVD) was also measured from the phase shift of the cosine fringe with a sub-picosecond delay precision.

© 2013 OSA

## 1. Introduction

11. J. Y. Lee and D. Y. Kim, “Versatile chromatic dispersion measurement of a single mode fiber using spectral white light interferometry,” Opt. Express **14**(24), 11608–11615 (2006). [CrossRef] [PubMed]

2. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature **177**(4497), 27–29 (1956). [CrossRef]

3. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. **59**(18), 2044–2046 (1987). [CrossRef] [PubMed]

5. S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express **11**(22), 2953–2963 (2003). [CrossRef] [PubMed]

8. J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. **28**(21), 2067–2069 (2003). [CrossRef] [PubMed]

5. S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express **11**(22), 2953–2963 (2003). [CrossRef] [PubMed]

9. W. P. Alford and A. Gold, “Laboratory measurement of the velocity of light,” Am. J. Phys. **26**(7), 481–484 (1958). [CrossRef]

*spectral intensity interferometry*(SII), the fourth-order correlation information is obtained in the modulation frequency domain by using a photoelectric spectrum acquisition means. It can be understood as a fourth-order variant of the spectral white-light interferometry [10

10. S. Diddams and J.-C. Diels, “Dispersion measurements with white light interferometry,” J. Opt. Soc. Am. B **13**(6), 1120–1129 (1996). [CrossRef]

11. J. Y. Lee and D. Y. Kim, “Versatile chromatic dispersion measurement of a single mode fiber using spectral white light interferometry,” Opt. Express **14**(24), 11608–11615 (2006). [CrossRef] [PubMed]

## 2. Theory

### 2.1 Correlation and spectrum

*a*(

*t*) and

*b*(

*t*), it will be denoted by

*C*{

_{τ}*a*,

*b*} as a function of the relatively time delay

*τ*, as defined bywhere the complex conjugate is denoted by the superscript of *. The field correlation measures the similarity between two fields of

*E*

_{1}and

*E*

_{2}, expressed by

*C*{

_{τ}*E*

_{1},

*E*

_{2}}. The intensity correlation is similarly expressed by

*C*{

_{τ}*I*

_{1},

*I*

_{2}}, where

*I*

_{1}and

*I*

_{2}are the intensities of

*E*

_{1}and

*E*

_{2}, respectively. The field correlation and the intensity correlation will be referred as the second-order and the fourth-order correlations.

*E*(

*t*), its spectrum is a projection to the reciprocal domain of frequency,

*ω*, as defined bywhich is the first-order spectrum of the field. The power spectrum of a higher order is rather observable in practice. It is defined by the square of the absolute spectrum. Because an optical signal is carried primarily by the electromagnetic field, the optical power spectrum of the first kind is associated with the second-order power, so called the

*second-order spectrum*of the field,

*S*

^{(2)}, which is found byBy Wiener-Khinchin-Einstein theorem, it is widely acknowledged that the second-order spectrum is the Fourier conjugate of the auto-correlation function of

*C*{

_{τ}*E*,

*E*}, which has established the principle of the interferometric spectroscopy techniques such as Fourier transform infrared (FTIR) spectroscopy [12]. It also suggests that the cross-correlation can be analyzed with the power spectrum of the combined field.

*fourth-order spectrum*,

*S*

^{(4)}is defined bywhere

### 2.2 General description of time-domain and frequency-domain interferometry

*τ*. A general expression for the resulted interference signal is made by the integrated power for two signals of any form,

*f*

_{1}and

*f*

_{2},given as a delay-dependent power. Here,

*g*(

*t*) is the by-product of combining the two signals, being multiplied by a real-valued coefficient

*λ*. If the nature of the signals make no combinational by-product, we can set

*λ*= 0 simply. We can expand Eq. (5) towhich includes the cross-correlation function terms in the second and fourth places of the integrand. This presents a classical way to acquire the correlation function in the time-domain interferometry.

*τ*is unnecessary in this case as the spectrum measurement is performed for a constant time delay. The generalized power spectrum of the combined signal,

*S*, is expressed asby the properties of Fourier transform.

*S*(

*ω*) is also expanded to nine terms as

*S*(

*ω*), denoted by

*H*′(

*t*), is found as

*δ*is the Dirac delta function. Here,

*X*and

_{nm}*Y*are the correlation functions as defined, respectively, byandwith the convolution operation of ⊗, defined byNote that we have

_{n}*t*= ±

*τ*after the inverse transform. This proves that the whole contents of the correlation are acquired in the frequency-domain interferometry as well as the relative time delay between the two signals.

### 2.3 Field-sensitive and intensity-sensitive interferometry schemes

*f*

_{1}and

*f*

_{2}are substituted by light fields of

*E*

_{1}and

*E*

_{2}, respectively. It has no by-product of

*g*(

*t*) so that we have

*λ*= 0 for the case of the second-order interferometry. Then, Eq. (6) gives the second-order interferogram,

*H*

^{(2)}, to be acquired in experiment. A more apprepriate expression can be made by introducing the degree of interference,

*γ*, which leads towhere

*I*

_{1}and

*I*

_{2}are the intensities of

*E*

_{1}and

*E*

_{2}, respectively, while

*X*

_{12}is their field cross-correlation. Due to the vectorial nature of the optical fields, we have a finite degree of interference,

*γ*, which ranges from zero to unity. It can be understood as the ratio of the vectorial inner product to the magnitude multiplication. The maximum value of

*γ*= 1is obtained when the two fields have the same state of polarization as well as the same spatial properties of propagation. The integrals of Eq. (13) is performed from −∞ to + ∞ to make an exact correlation function. But a finite integration time makes no significant difference as long as it is much longer than the time delay

*τ*.

*f*

_{1}=

*E*

_{1},

*f*

_{2}=

*E*

_{2}and

*λ*= 0, the spectral interferogram of the second order,

*S*

^{(2)}, is found from Eq. (8) to befor a degree of interference,

*γ*. It is obvious that the spectral fringe of

*S*

^{(2)}depends on the degree of interference so that careful alignments such as polarization matching are necessary for reliable measurement results. The acquired power spectrum can be inversely transformed, if needed, through a numerical signal processing means and gives the correlation functions shifted by ±

*τ*as suggested by Eq. (9). This scheme of the frequency-domain interferometry provides the same contents of the correlation function except for the inevitable ambiguity on the sign of the time delay, caused by the shifts of ±

*τ*. The Fourier transform of the real-valued power spectrum is Hermitian symmetric and produces a fictitious mirror image on the other side in time.

*I*

^{2}) is performed by a nonlinear process, whether optical or electrical. The combined field in an interferometer makes its fourth-order interferogram,

*H*

^{(4)}, asfor the light fields of

*E*

_{1}and

*E*

_{2}with a degree of interference,

*γ*. The expansion of the integral is found from the general formulae of Eqs. (5) and (6) by substituting

*f*

_{1}=

*I*

_{1},

*f*

_{2}=

*I*

_{2}and

*λ*=

*γ*. Here, the by-product of the field combination is given bywhich still carries the field interference content. The product of

*I*

_{1}(

*t*)⋅

*I*

_{2}(

*t*) in the integrand of Eq. (15) makes the desired cross-correlation function of

*C*{

_{τ}*I*

_{1},

*I*

_{2}} by the integral. This intensity correlation information appears together with the field-sensitive interference signal unless the degree of interference is zero.

### 2.4 Spectral intensity interferometry

*f*

_{1}=

*I*

_{1},

*f*

_{2}=

*I*

_{2}and

*λ*=

*γ*in Eqs. (7) and (8), the fourth-order spectral interferogram,

*S*

^{(4)}, is obtained as

*g*(

*t*), given in Eq. (16). It is clear that the second and the fourth terms in Eq. (17) are the Fourier conjugates of

*C*{

_{t}*I*

_{1},

*I*

_{2}}⊗

*δ*(

*t*±

*τ*),

*i.e.*the shifted intensity correlation functions as suggested by Eq. (9). The effect of the first and the fifth terms in Eq. (17) are trivial because the fourth-order spectrum of the source can be assumed to be uniform or calibrated separately. The other five

*γ*-dependent terms are carrying the field correlation information which is desired to be distinguished from that of the intensity correlation.

*ω*. The light field can be expressed by a modulated carrier wave with a center frequency

*ω*

_{0}aswith slowly-varying amplitude

*E*

_{0}(

*t*) and phase

*ϕ*(

*t*) or with a complex modulation function of

*A*(

*t*)≡

*E*

_{0}⋅exp(i

*ϕ*). Note that the square of the absolutized

*A*(

*t*) is the optical intensity and its power spectrum is the fourth-order spectrum of the light field. The photodetection of photon-to-electron conversion can be understood as a down-conversion process for extracting this function from the optical field. The spectrum of the modulation function

*i.e.*the modulation spectrum occupies the base band in the vicinity of

*ω*= 0 whereas the spectrum of the field still resides in the vicinity of the carrier frequency. From Eq. (16), the Fourier transform of the combinational by-product is found asby the convolution theorem. The intensity of

*I*(

*t*) has its spectrum of

*ω*. In contrast,

*ω*= ±

*ω*

_{0}with the same full bandwidth of 4⋅Δ

*ω*. In the case that 4⋅Δ

*ω*<

*ω*

_{0}holds,

*γ*-dependent terms of Eq. (17) are zero.

## 3. Experiment

13. Agilent Technologies Inc, “Spectrum Analysis Basics,” http://www.home.agilent.com/upload/cmc_upload/All/5952-0292EN.pdf.

*i.e.*the fourth-order spectrum of the light. This coherent-detection technique is a popular method in the electric signal analysis for its fine and adjustable spectral resolutions. This kind of SII system can cover a very range of modulation frequency up to a few hundred GHz’s in the current states of the electronics and optoelectronics technologies [13

13. Agilent Technologies Inc, “Spectrum Analysis Basics,” http://www.home.agilent.com/upload/cmc_upload/All/5952-0292EN.pdf.

15. E. Rouvalis, M. J. Fice, C. C. Renaud, and A. J. Seeds, “Millimeter-wave optoelectronic mixers based on uni-traveling carrier photodiodes,” IEEE Trans. Microw. Theory Tech. **60**(3), 686–691 (2012). [CrossRef]

2. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature **177**(4497), 27–29 (1956). [CrossRef]

3. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. **59**(18), 2044–2046 (1987). [CrossRef] [PubMed]

*ω*, the fourth-order spectrum occupies a band of 4⋅Δ

*ω*in full width. As a consequence, narrower the bandwidth is, higher the spectral power density we can get for the fourth-order spectrum. To take a high signal-to-noise ratio (SNR), the bandwidth of the source was reduced to 0.3 nm in 3-dB wavelength bandwidth by using the band-pass filter before the amplification stage. The spectral half width was Δ

*f*= Δ

*ω/*2π = 20 GHz, given by the 3-dB bandwidth of 0.3 nm centered at 1,550 nm in wavelength. Taking a simple approximation of the rectangular spectrum, the fourth-order spectrum formed a base band of 0 to 40 GHz, roughly. Because the photo-receiver used in experiment had a detection bandwidth of ~10 GHz, the fourth-order spectra must have had a nearly uniform power within the measurable range of frequency. The average optical power of the light source was ~20 μW when measured before the first coupler. It gave a sufficiently high SNR to overcome the noise of the measurement system.

*f*= 2 GHz and a frequency span of Δ

_{c}*f*= 1,500 kHz. The resolution bandwidth of the spectrum analyzer was 15 kHz with an acquisition time of

*T*= 0.1 second. The spectrum data were averaged with an average number of

_{a}*N*= 5 for five consecutive spectrum acquisitions so that the net acquisition time of

_{a}*N*= 0.5 s was spent for the final spectrum. The actual acquisition time was 20% longer including the short intervals between the consecutive acquisitions. As seen in Fig. 3, the measured fourth-order spectrum exhibited a clear fringed characteristic of the SII. The sinusoidal interferogram was the experimental evidence of the fourth-order correlation information being present in the form suggested by Eq. (23). No considerable polarization dependence was observed while varying the state of polarization of the test arm with a fiber-optic polarization controller. It suggests that the field-sensitive interference gave no considerable contribution to the acquired interferogram as anticipated by the theory.

_{a}⋅T_{a}*t*≈20 μs was interpreted as the shifted correlation function. Due to the finite measurement bandwidth of the system, it was convolved with the 1.5-MHz wide window function. This time-domain plot tells that the spectral fringe of Fig. 3 had been contaminated by the random noise distributed widely in time. A numerical filtering was performed for the data set of Fig. 4 to suppress the noise. A 20-μs wide slot centered at the observed peak was selected so that the noise out of this region was masked out to be zero. The inverse FFT of the filtered data set is shown in Fig. 5. The solid black line is the FFT-filtered fringe while the dotted orange line shows its sine function fit. A nearly perfect sine curve was recovered with a significantly reduced amount of noise.

^{−6}. In the data processing, the raw data of the spectra were filtered by the similar manner as described about Fig. 5. The length of the fiber was measured at first by the period of the spectral fringe. As well, a small variation of the group delay was evaluated by the phase shift of the cosine fringe observed at a fixed modulation frequency. The phase shift was tracked while the wavelength of the source changed step by step. For two different center wavelengths of the source, a phase difference of Δ

*ϕ*between the fourth-order spectra was evaluated at

*f*for the modulation frequency. The time delay difference, Δ

_{c}*τ*, is found at a wavelength of

*λ*to bewith respect to the reference time delay of the reference wavelength

*λ*

_{0}.

*f*= 2 GHz with Δ

_{c}*f*= 1.5 MHz,

*T*= 0.1 s and

_{a}*N*= 5. The instrumental group delays of the measurement system were calibrated in the absence of the sample fiber. The group delay of the sample fiber was measured to be 20,371 ns from the fringe period. It corresponded to 4,148 m for the group index of the fiber being assumed to be

_{a}*n*= 1.4682. The wavelength-dependent group delays were analyzed from the phase shifts with respect to the reference of

_{g}*λ*

_{0}= 1542.8 nm. The first-order and the second-order derivatives of the delay curve were taken as the GVD properties. By normalizing them with the measured fiber length, the dispersion coefficient,

*D*, and the dispersion slope,

*K*, were evaluated at a wavelength of 1,550 nm. They were

*D*= 16.45 ps/nm⋅km and

*K*= 0.0515 ps/nm

^{2}⋅km, respectively, and well matched the typical values of the SMF.

*f*= 11.0 GHz with a span of Δ

_{c}*f*= 500 MHz. A longer time was spent with an acquisition time of

*T*= 0.1 s and an average number of

_{a}*N*= 50. Thus, the net acquisition time was

_{a}*N*⋅

_{a}*T*= 5.0 s for a wavelength. Note that such a short SMF section makes a very small group delay difference of ~1 ps for the full range of 1,542~1,561 nm. Sub-picosecond precisions were necessary for reliable measurement results with such a short fiber sample.

_{a}*n*= 1.4682 with a measurement error of 0.54%. The dispersion coefficient was found to be

_{g}*D*= 16.59 ps/nm⋅km through a 1st-order linear fit of the measured delays. The standard deviation of the fitting errors was 0.037 ps or 37 fs, which was small enough for a line fit but it was too high for the high-order fitting. The obtained dispersion coefficient was nearly the same as that of the long fiber sample (4.1 km long) with a little difference of 0.85%. This result proves that our SII can provide a measurement precision enough to analyze meter-long fiber samples. For comparison, the same sample fiber (4.1 km long) was analyzed by a commercial measurement system of fiber dispersion (ODA, Agilent Technologies Inc.), which uses the conventional modulation phase shift method based on a tunable laser source [16]. All of the measured results well agreed with each other as summarized in Table 1. Those experimental observations verified the practical feasibility of the SII scheme in GVD measurements.

*N*⋅

_{a}*T*= 5.0 s is spent for each. And it was enhanced to 7.9 fs as a longer time of

_{a}*N*⋅

_{a}*T*= 25 s was spent for acquisition. It was impressive that femtosecond-level precisions were easily taken by the SII. Further improvements on the source power and optimization of the measurement techniques might enable sub-wavelength precisions in the group delay measurements, being still performed in a reasonable acquisition time.

_{a}## 4. Discussion

13. Agilent Technologies Inc, “Spectrum Analysis Basics,” http://www.home.agilent.com/upload/cmc_upload/All/5952-0292EN.pdf.

15. E. Rouvalis, M. J. Fice, C. C. Renaud, and A. J. Seeds, “Millimeter-wave optoelectronic mixers based on uni-traveling carrier photodiodes,” IEEE Trans. Microw. Theory Tech. **60**(3), 686–691 (2012). [CrossRef]

*e.g.*for the case of the GVD measurement of long fibers.

18. S. Moon and D. Y. Kim, “Reflectometric fiber dispersion measurement using a supercontinuum pulse source,” IEEE Photon. Technol. Lett. **21**(17), 1262–1264 (2009). [CrossRef]

10. S. Diddams and J.-C. Diels, “Dispersion measurements with white light interferometry,” J. Opt. Soc. Am. B **13**(6), 1120–1129 (1996). [CrossRef]

11. J. Y. Lee and D. Y. Kim, “Versatile chromatic dispersion measurement of a single mode fiber using spectral white light interferometry,” Opt. Express **14**(24), 11608–11615 (2006). [CrossRef] [PubMed]

## 5. Conclusion

## References and links

1. | P. Hariharan, |

2. | R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature |

3. | C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. |

4. | L. Sarger and J. Oberlé, “How to measure the characteristics of laser pulses,” in |

5. | S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express |

6. | R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express |

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

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

9. | W. P. Alford and A. Gold, “Laboratory measurement of the velocity of light,” Am. J. Phys. |

10. | S. Diddams and J.-C. Diels, “Dispersion measurements with white light interferometry,” J. Opt. Soc. Am. B |

11. | J. Y. Lee and D. Y. Kim, “Versatile chromatic dispersion measurement of a single mode fiber using spectral white light interferometry,” Opt. Express |

12. | P. Griffiths and J. A. de Haseth, “Chapter 2. Theoretical Background,” in |

13. | Agilent Technologies Inc, “Spectrum Analysis Basics,” http://www.home.agilent.com/upload/cmc_upload/All/5952-0292EN.pdf. |

14. | A. Beling and J. C. Campbell, “InP-based high-speed photodetectors,” J. Lightwave Technol. |

15. | E. Rouvalis, M. J. Fice, C. C. Renaud, and A. J. Seeds, “Millimeter-wave optoelectronic mixers based on uni-traveling carrier photodiodes,” IEEE Trans. Microw. Theory Tech. |

16. | P. Hernday, “Dispersion measurements,” in |

17. | TIA Standard TIA-455–175-B, |

18. | S. Moon and D. Y. Kim, “Reflectometric fiber dispersion measurement using a supercontinuum pulse source,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(060.2300) Fiber optics and optical communications : Fiber measurements

(070.4550) Fourier optics and signal processing : Correlators

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: July 18, 2013

Revised Manuscript: August 30, 2013

Manuscript Accepted: August 30, 2013

Published: September 24, 2013

**Citation**

Sucbei Moon, Heeso Noh, and Dug Young Kim, "Frequency-domain acquisition of fourth-order correlation by spectral intensity interferometry," Opt. Express **21**, 23206-23219 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23206

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### References

- P. Hariharan, Optical Interferometry, 2nd Ed. (Academic Press, 2003).
- R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature177(4497), 27–29 (1956). [CrossRef]
- C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59(18), 2044–2046 (1987). [CrossRef] [PubMed]
- L. Sarger and J. Oberlé, “How to measure the characteristics of laser pulses,” in Femtosecond Laser Pulses, Claude Rullière ed. (Springer, 1998), pp. 177–202.
- S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express11(22), 2953–2963 (2003). [CrossRef] [PubMed]
- R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express11(8), 889–894 (2003). [CrossRef] [PubMed]
- M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express11(18), 2183–2189 (2003). [CrossRef] [PubMed]
- J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett.28(21), 2067–2069 (2003). [CrossRef] [PubMed]
- W. P. Alford and A. Gold, “Laboratory measurement of the velocity of light,” Am. J. Phys.26(7), 481–484 (1958). [CrossRef]
- S. Diddams and J.-C. Diels, “Dispersion measurements with white light interferometry,” J. Opt. Soc. Am. B13(6), 1120–1129 (1996). [CrossRef]
- J. Y. Lee and D. Y. Kim, “Versatile chromatic dispersion measurement of a single mode fiber using spectral white light interferometry,” Opt. Express14(24), 11608–11615 (2006). [CrossRef] [PubMed]
- P. Griffiths and J. A. de Haseth, “Chapter 2. Theoretical Background,” in Fourier Transform Infrared Spectrometry, 2nd Ed. (John Wiley & Sons, 2007), pp. 19–56.
- Agilent Technologies Inc, “Spectrum Analysis Basics,” http://www.home.agilent.com/upload/cmc_upload/All/5952-0292EN.pdf .
- A. Beling and J. C. Campbell, “InP-based high-speed photodetectors,” J. Lightwave Technol.27(3), 343–355 (2009). [CrossRef]
- E. Rouvalis, M. J. Fice, C. C. Renaud, and A. J. Seeds, “Millimeter-wave optoelectronic mixers based on uni-traveling carrier photodiodes,” IEEE Trans. Microw. Theory Tech.60(3), 686–691 (2012). [CrossRef]
- P. Hernday, “Dispersion measurements,” in Fiber Optic Test and Measurement, D. Derickson ed. (Prentice Hall PTR, 1998), pp.475–518.
- TIA Standard TIA-455–175-B, Meausrement Methods and Test Procedures – Chromatic Dispersion, 2003.
- S. Moon and D. Y. Kim, “Reflectometric fiber dispersion measurement using a supercontinuum pulse source,” IEEE Photon. Technol. Lett.21(17), 1262–1264 (2009). [CrossRef]

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