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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 23206–23219
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Frequency-domain acquisition of fourth-order correlation by spectral intensity interferometry

Sucbei Moon, Heeso Noh, and Dug Young Kim  »View Author Affiliations


Optics Express, Vol. 21, Issue 20, pp. 23206-23219 (2013)
http://dx.doi.org/10.1364/OE.21.023206


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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

Interferometry is an optical method of analyzing the interference phenomena which are given as a consequence of combining optical signals [1

1. P. Hariharan, Optical Interferometry, 2nd Ed. (Academic Press, 2003).

]. The correlation feature of optical signals can be quantitatively evaluated when they originate from the same light source but pass through different paths in an optical interferometer. The acquired interference signal or the interferogram carries useful information on the characteristics of the interferometric paths and the light source. It has been one of the central subjects that lead advances in the optical science for understanding the fundamental nature of light. As well, the interferometry has been utilized as practical measurement tools in various applications [1

1. P. Hariharan, Optical Interferometry, 2nd Ed. (Academic Press, 2003).

11

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]

]. To cope with such diverse demands, a number of different interferometric schemes have been introduced and used in a variety of configurations so far.

Intensity interferometers or the fourth-order interferometry schemes extract the intensity-involved correlation information from the light fields whereas the second-order interferometers sense the field-level interference effect [1

1. P. Hariharan, Optical Interferometry, 2nd Ed. (Academic Press, 2003).

4

4. 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.

]. The fourth-order interference signal is generated by intensity-intensity products in the fourth power of the fields through a multi-photon process. It can be done either by nonlinear-optic signal generation or simply through photoelectric mixing, using electric signal processors such as solid-state mixers or coincidence photon counters. Since the first experimental demonstration reported by Hanbury Brown and Twiss [2

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

], the intensity interferometry has played an important role in studying the quantum nature of optics [1

1. P. Hariharan, Optical Interferometry, 2nd Ed. (Academic Press, 2003).

3

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]

]. In the practical measurement and metrology applications, the intensity interferometry species have been used widely in the short-pulse characterizations [4

4. 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.

]. However, they have been considered as less attractive in more typical interferometry applications such as length measurements. Their sensitivity and precision performances are much less competitive when compared with the well-established schemes of the second-order interferometry.

In this report, we systematically investigated the frequency-domain acquisition technique of the intensity correlation with emphasis on the practical applicability. In so-called the 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

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]

]. The intensity-involved interference signal is acquired in the SII without varying the relative time delay of the signals. Various characteristics of the SII have been studied in this research through theoretical and experimental investigations. The practical advantage of the SII has been found in the fixed delay that can extend to several kilometers or longer without any limit. This scalability was achieved without concern of annoying field-sensitive disturbances which are vulnerable to minute variations of the optical paths. It also allows us to use a single photodetector for the fourth-order interferometry without field-level interference effect. Because of those attractive features, the SII turns out to be very promising in large-scale interferometry applications where uncontrollable fluctuations of the long optical paths inevitably obscure the interference signal. To verify the feasibility, we demonstrated a precision time-of-flight measurement by using a fiber-optic SII with an amplified spontaneous emission (ASE) light source. The group velocity dispersion (GVD) measurement was successfully performed by our simple SII system with an impressive delay measurement precision of tens of femtoseconds.

2. Theory

In this section, a theoretical overview in a classical approach is presented on the principle of various interferometry schemes including that of the spectral intensity interferometry that we introduce, here. Throughout the discussion in the below, the physical constants associated with dimensional conversions are omitted for simplicity.

2.1 Correlation and spectrum

The correlation function is a measure of similarity between the two functions under investigation. For complex functions of a(t) and b(t), it will be denoted by Cτ{a, b} as a function of the relatively time delay τ, as defined by
G(τ)=Cτ{a(t),b(t)}+a*(t)b(tτ)dt
(1)
where the complex conjugate is denoted by the superscript of *. The field correlation measures the similarity between two fields of E1 and E2, expressed by Cτ{E1, E2}. The intensity correlation is similarly expressed by Cτ{I1, I2}, where I1 and I2 are the intensities of E1 and E2, respectively. The field correlation and the intensity correlation will be referred as the second-order and the fourth-order correlations.

For a given field of E(t), its spectrum is a projection to the reciprocal domain of frequency, ω, as defined by
E˜(ω)Fω{E(t)}=+E(t)eiωtdt
(2)
which 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 by
S(2)(ω)|E˜(ω)|2=|Fω{E(t)}|2.
(3)
By Wiener-Khinchin-Einstein theorem, it is widely acknowledged that the second-order spectrum is the Fourier conjugate of the auto-correlation function of Cτ{E, E}, which has established the principle of the interferometric spectroscopy techniques such as Fourier transform infrared (FTIR) spectroscopy [12

12. 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.

]. It also suggests that the cross-correlation can be analyzed with the power spectrum of the combined field.

We can generalize the spectrum to a higher order beyond the second, hoping to connect the high-order spectrum to the high-order correlation. The optical signal is carried also by the intensity fluctuations in time. The fourth-order spectrum, S(4) is defined by
S(4)(ω)|I˜(ω)|2=|Fω{|E(t)|2}|2
(4)
where I˜(ω) is the Fourier transform of the intensity I(t)E*E. It has the fourth-order power of the optical field as the name suggests.

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

A generalized description can be made to represent the second- and fourth-order interferometry techniques at once. Let us suppose that two optical signals of any kind are combined and interfered with each other. Their correlation function can be acquired in time delay domain by simply integrating the power of the combined signal as a function of the relative time delay τ. A general expression for the resulted interference signal is made by the integrated power for two signals of any form, f1 and f2,
H(τ)+|f1(t)+f2(tτ)+λg(t)|2dt,
(5)
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) to
H(τ)=+(f1*(t)f1(t)+f1*(t)f2(tτ)+λf1*(t)g(t)+f1(t)f2*(tτ)+f2*(tτ)f2(tτ)+λf2*(tτ)g(t)+λf1(t)g*(t)+λf2(tτ)g*(t)+λ2g*(t)g(t))dt
(6)
which 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.

The power spectrum of the combined signal contains the correlation information as well, which enables the frequency-domain interferometry. Varying 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 as
S(ω)|Fω{f1(t)+f2(tτ)+λg(t)}|2=|f˜1(ω)+f˜2(ω)eiωτ+λg˜(ω)|2
(7)
by the properties of Fourier transform. S(ω) is also expanded to nine terms as
S(ω)=f˜1*f˜1+f˜1*f˜2eiωτ+λf˜1*g˜+f˜1f˜2*e+iωτ+f˜2*f˜2+λf˜2*g˜e+iωτ+λf˜1g˜*+λf˜2g˜*eiωτ+λ2g˜*g˜
(8)
from Eq. (7). A multiplication product in Eq. (8) corresponds to the correlation function found in time domain, given by Eq. (6). The inverse Fourier transform of S(ω), denoted by H′(t), is found as
H(t)=X11(t)+X12(t)δ(tτ)+λY1(t)+X12*(t)δ(t+τ)+X22(t)+λY2*(t)δ(t+τ)+λY1*(t)+λY2*(t)δ(tτ)+λ2g*(t)g(t)
(9)
where δ is the Dirac delta function. Here, Xnm and Yn are the correlation functions as defined, respectively, by
Xnm(t)fn*(t)fn(t)=Ct{fn(t),fm(t)},
(10)
and
Yn(t)fn*(t)g(t)=Ct{fn(t),g(t)}
(11)
with the convolution operation of ⊗, defined by
a(ζ)b(ζ)+a(ν)b(ζν)dν.
(12)
Note that we have Xnm(t)δ(t±τ)=Xnm(t±τ) for the terms of Eq. (9), which suggests the function convolved with the delta function is shifted in time. Thus, the cross-correlation functions of Eq. (9) have their peaks at 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

An interferometer consists of multiple light paths from the source to the detector. For simplicity, we focus on a simple Mach-Zehnder interferometer with two optical paths or two arms of the interferometer, reference and test arms. The typical purpose of the interferometric system is to acquire the information carried by the probe light that passes through the test arm. It is done by analyzing the correlation of the probe light to the reference through the process of interference. Since the two fields originate from the same light source, each field of interference is characterized by the relative differences in amplitude and phase as well as in time delay. The acquired interferogram of any domain can reveal those parameters with its fringed waveform.

In a field-sensitive interferometer, the signal of interest is the combined fields at the photodetector. In Eq. (5), the signals of f1 and f2 are substituted by light fields of E1 and E2, 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 to
H(2)(τ)=|E1(t)+E2(tτ)|2dt=I1dt+I2dt+γX12(τ)+γX12*(τ)
(13)
where I1 and I2 are the intensities of E1 and E2, respectively, while X12 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.

The spectral interferometry can extract the same correlation function. Taking f1 = E1, f2 = E2 and λ = 0, the spectral interferogram of the second order, S(2), is found from Eq. (8) to be
S(2)(ω)=|E˜1|2+|E˜2|2+γ(E˜1*E˜2eiωτ+E˜1E˜2*e+iωτ)
(14)
for 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.

The intensity interferometry is a different class of correlation measurements. The signal of interest is, now, carried by the intensity function which is represented by the square of the absolutized field. The acquisition on the power of the intensity (I2) is performed by a nonlinear process, whether optical or electrical. The combined field in an interferometer makes its fourth-order interferogram, H(4), as
H(4)(τ)=|E1(t)+E2(tτ)|4dt=|I1(t)+I2(tτ)+γ(E1*(t)E2(tτ)+E1(t)E2*(tτ))|2dt
(15)
for the light fields of E1 and E2 with a degree of interference, γ. The expansion of the integral is found from the general formulae of Eqs. (5) and (6) by substituting f1 = I1, f2 = I2 and λ = γ. Here, the by-product of the field combination is given by
g(t)=E1*(t)E2(tτ)+E1(t)E2*(tτ)
(16)
which still carries the field interference content. The product of I1(t)⋅I2(t) in the integrand of Eq. (15) makes the desired cross-correlation function of Cτ{I1, I2} 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

The spectral intensity interferometry (SII) acquires the intensity correlation information in frequency domain from the fourth-order spectrum of the combined field. The signals under investigation in our SII are the optical intensity functions. By substituting f1 = I1, f2 = I2 and λ = γ in Eqs. (7) and (8), the fourth-order spectral interferogram, S(4), is obtained as
S(4)(ω)=I˜1*I˜1+I˜1*I˜2eiωτ+γI˜1*g˜+I˜1I˜2*e+iωτ+I˜2*I˜2+γI˜2*g˜e+iωτ+γI˜1g˜*+γI˜2g˜*eiωτ+γ2g˜*g˜
(17)
where g˜(ω) is the Fourier transform of g(t), given in Eq. (16). It is clear that the second and the fourth terms in Eq. (17) are the Fourier conjugates of Ct{I1, I2}⊗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.

To find the spectral characteristic of each term of Eq. (17), let us assume that the field of the source light has a finite spectral bandwidth of 2⋅Δω. The light field can be expressed by a modulated carrier wave with a center frequency ω0 as
E(t)=E0(t)ei(ω0t+ϕ(t))=A(t)eiω0t
(18)
with slowly-varying amplitude E0(t) and phase ϕ(t) or with a complex modulation function of A(t)≡E0⋅exp(iϕ). 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 as
g˜(ω)=[A˜1*(ω)(A˜2(ω)eiωτ)]δ(ω+ω0)+[A˜1(ω)(A˜2*(ω)e+iωτ)]δ(ωω0)
(19)
by the convolution theorem. The intensity of I(t) has its spectrum of I˜=A˜*(ω)A˜(ω) which resides in the base band with a full bandwidth of 4⋅Δω. In contrast, g˜(ω) occupies the two upper bands centered at ω = ± ω0 with the same full bandwidth of 4⋅Δω. In the case that 4⋅Δω<ω0 holds, I˜(ω)has no overlap with g˜(ω) in frequency and their multiplication product is automatically zero everywhere. Therefore, the four γ-dependent terms of Eq. (17) are zero.

It should be noted that the requirement of the narrow bandwidth (Δω<ω0/4) is not so essential for making the measurement independent of γ. Partial overlapping of I˜ and g˜ does not make any noticeable effect as long as the involved time delay is sufficiently large. The nature of convolution gives
g˜(ω)0   or   A˜1*(ω)(A˜2(ω)eiωτ)0   as    τΔω±
(21)
because the oscillating function of exp(−iωτ) is summed out to be nearly zero by the convolution for a large value of τ. The measure of the largeness, here, is made in comparison with the characteristic time of 1/Δω, which is the coherence time of the light source or the coherence length divided by the speed of light. On the other hand, a high-order coherence length can be defined by the fourth-order auto-correlation function, Cτ{I, I}. For a time delay longer than the coherence length of intensity, Cτ{I, I}(τ) = 0 holds effectively. Replicas of the source intensity function, I1 and I2, are mutually uncorrelated and their spectra have no measurable relation. Then, the multiplication product of the intensity spectra becomes invariant. Assuming the intensity spectrum of the source uniform over a certain measurement band, we can express
I˜1*(ω)I˜2(ω)=I02
(22)
within the measurement band for constant I0 if the time delay between them exceeds the coherence length of intensity fluctuations. Then, Eq. (20) may reduce to a simple form as
S(4)(ω)=S0+2I02cosωτ
(23)
where S0 is the background spectrum. The spectral fringe of Eq. (23) directly visualizes the degree of correlation by the relative fringe amplitude and the relative time delay by the period of the spectral fringe. After all, the SII makes a sinusoidal fringe of a simple form if the path length difference involved is sufficiently larger than the coherence length of the intensity auto-correlation.

The conclusion of Eq. (23) may look no more than an intuitive explanation. The intensity fluctuations of the source light are repeated temporally in a Mach-Zehnder interferometer so that the fourth-order spectrum appears to be a cosine function of frequency. The emphasis should be made rather on the lack of the field-sensitive interference in the fourth-order spectrum. Even when the band-limited source light is partially coherent, the fourth-order spectral interferogram is independent of the field-sensitive process and allows a reliable measurement in the presence of the minute variations. The consideration of the coherence length was found to be useful in practical SII but does not directly restrict the applicability.

3. Experiment

The ASE light automatically produces a randomly varying field. The spectrum of the ASE light is described by random phases with nearly uniform amplitudes within a band. The fourth-order spectrum of the source is found by the convolution of the modulation spectra. When the optical bandwidth is 2⋅Δω, the fourth-order spectrum occupies a band of 4⋅Δω 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 Δf = Δω/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.

A fourth-order spectrum of interference was acquired with our Mach-Zehnder intensity interferometer when the relative time delay was made by a 4.1-km long fiber placed in the test arm. Figure 3
Fig. 3 Fourth-order spectrum measured at fc = 2 GHz and a time delay of τ≈20 μs, made by 4.1-km long fiber.
shows the acquired fourth-order spectrum. The spectrum analyzer was set to have an acquisition center frequency of fc = 2 GHz and a frequency span of Δf = 1,500 kHz. The resolution bandwidth of the spectrum analyzer was 15 kHz with an acquisition time of Ta = 0.1 second. The spectrum data were averaged with an average number of Na = 5 for five consecutive spectrum acquisitions so that the net acquisition time of Na⋅Ta = 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.

The inverse Fourier transform of the spectral fringe visualized a clear peak of a shifted intensity correlation function. Figure 4
Fig. 4 Inverse Fourier transform of the measured data shown in Fig. 3.
shows the inverse transform of the spectrum given in Fig. 3 with the constant background being removed. A Hanning window was applied before the numerical process of the fast Fourier transform (FFT). The peak found at 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
Fig. 5 FFT-filtered spectral interferogram (black solid line) and its sine function fit (orange dotted line)
. 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.

In order to make an application example of the SII, fiber dispersion measurements were demonstrated by utilizing the SII setup described in Fig. 2. The band-pass filter in the light source had a tunable center wavelength to provide a wavelength-variable ASE source in a tuning range of 1,542~1,561 nm. The time-of-flight measurement could be performed by the fourth-order interferometry with various wavelengths. The wavelength-dependent time delay for a sample fiber placed in the test arm could reveal the GVD property of the sample fiber. Note that the measured time delay is interpreted as the wavelength-dependent group delay of the fiber which varies by a very little fraction of ~10−6. 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 Δϕ between the fourth-order spectra was evaluated at fc for the modulation frequency. The time delay difference, Δτ, is found at a wavelength of λ to be
Δτ(λ)=Δϕ(λ)2πfc
(24)
with respect to the reference time delay of the reference wavelength λ0.

Figure 6
Fig. 6 Measured relative group delays (square dots) and the 2nd-order polynomial fit (red line) for the 4.1-km long fiber.
shows the measured relative group delays (square dots) and the 2nd-order polynomial fit curve (solid line) for the sample SMF of nominal length 4.1 km. The measurement was performed at fc = 2 GHz with Δf = 1.5 MHz, Ta = 0.1 s and Na = 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 ng = 1.4682. The wavelength-dependent group delays were analyzed from the phase shifts with respect to the reference of λ0 = 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/nm2⋅km, respectively, and well matched the typical values of the SMF.

The GVD measurement was repeated for a short section of the same SMF. The length was physically measured to be 3.66 m. The GVD were analyzed in the same manner as described earlier but at a different spectral acquisition band for a better precision. The acquisition center frequency was increased to fc = 11.0 GHz with a span of Δf = 500 MHz. A longer time was spent with an acquisition time of Ta = 0.1 s and an average number of Na = 50. Thus, the net acquisition time was NaTa = 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.

By our SII GVD analyzer, the fiber length was measured to be 3.68 m under the assumption of ng = 1.4682 with a measurement error of 0.54%. The dispersion coefficient was found to be 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

16. P. Hernday, “Dispersion measurements,” in Fiber Optic Test and Measurement, D. Derickson ed. (Prentice Hall PTR, 1998), pp.475–518.

]. All of the measured results well agreed with each other as summarized in Table 1

Table 1. Summary of the GVD measurement result

table-icon
View This Table
. Those experimental observations verified the practical feasibility of the SII scheme in GVD measurements.

The precision of the group delay determinations was also tested in experiment when the 3.66-m fiber was placed in the test arm. The phase of the fourth-order spectrum at 11.0 GHz was repeatedly measured to find the variance of the measured delays. From six independent measurements, the phase or delay precision was evaluated by the standard deviation and found to be 21.6 fs when a net acquisition time of NaTa = 5.0 s is spent for each. And it was enhanced to 7.9 fs as a longer time of NaTa = 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.

4. Discussion

If the full information of the correlation function is demanded, very high-speed electronics are essential for the SII when it is based on the photoelectric mixing for the fourth-order signal generation. The measurement bandwidth of the SII system is mainly limited by those of the photodetector and the electronic components of the power spectrum analysis. In our experiment, frequencies up to 11.0 GHz were utilized, which was insufficient to obtain the full correlation information but more than enough for determining the time delay. Even though the maximum acquisition frequency does not solely limit the delay precision, use of a higher frequency naturally enhances the precision performance of the SII for a given SNR. The demand on a high frequency for a better precision can be a factor that makes our SII scheme relatively attractive when compared to the time-domain intensity interferometry. In the analog analysis, the state-of-the-art electronics and optoelectronics technologies can support very high frequencies up to hundreds of GHz’s [13

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

15

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]

]. A practical and economic range may be still bounded below 100 GHz for the majority. But even for them, it is worth to be noted that reliable signal delivery and combining are still difficult, especially when the frequency is above tens of GHz’s and the fractional signal bandwidth is large. Use of a single photodetector must be highly preferable for this reason as long as field-sensitive disturbances are not concerned. The unwanted field-sensitive interference may bring the needs for two separate photodetectors as found in typical time-domain intensity interferometers. The common electric path of the photo-current in our SII configuration eliminates the issues of possible signal unbalances or temperature-dependent variations of the electronic components. It also reduces the implementation costs involved with the high-frequency electronics as well.

The SII scheme exhibits attractive features that make it very suitable for precision time-of-flight measurements with a large-scale interferometer. The fourth-order spectrum measurement can provide an extremely fine spectral resolution. This enables long-range measurements with large delays. Our SII scheme is truly scalable and supports a wide range of path lengths, not only due to the fine spectral resolution but also owing to the fixed interferometer arms. The path length difference keeps unchanged during the acquisition, which greatly relieves the implementation difficulty in practical interferometry systems. Notice that varying the delay mostly complicates the interferometer implementation with mechanical parts. Sometimes, it is hardly possible to match the lengths of the two arms, particularly when the involved delay is very large e.g. for the case of the GVD measurement of long fibers.

As our experiment demonstrated, the SII is very suitable for fiber GVD measurement. It provides a simple and inexpensive way of fiber GVD analysis with a good precision performance. In the conventional measurement techniques, the GVD is evaluated by time-of-flight or phase measurements performed in time domain by utilizing intrinsically or extrinsically modulated light sources [16

16. P. Hernday, “Dispersion measurements,” in Fiber Optic Test and Measurement, D. Derickson ed. (Prentice Hall PTR, 1998), pp.475–518.

18

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]

]. The delay precision of such a method is mostly insufficient for analyzing short fiber samples where femtosecond precisions are desired. For fibers shorter than a meter, a white-light interferometer of the second-order interferometry has been used for its excellent delay precision [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

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]

]. This technique relies on the sensitive field-involved interference so that a very low level of dispersion can be successfully measured. At the same time, however, it suffers from the field sensitivity when the length of the sample fiber exceeds a few meters. Minute temperature-involved variations of the path length and the polarization state in a long sample fiber produce rapidly fluctuating unpredictable phase-sensitive signals that cause measurement errors or wash out the interferogram fringes. Even in a stable condition, this scheme is hardly applicable to a kilometer-long fiber because it requires an unacceptably long reference arm or a very fine spectral resolution of the optical spectrometer. Our SII-based GVD measurement method covers a wide range of sample fiber length from a few meters to kilometers. Along with a good delay precision useful for short fibers, our method can analyze long fibers owing to the field-insensitive interferograms acquired with a fine spectral resolution that can hardly obtained in the optical frequency domain. Furthermore, those attractive features are obtained with a simple configuration of the SII that consists of inexpensive optical components.

5. Conclusion

We reported a useful interferometry scheme of the spectral intensity interferometer as a frequency-domain variant of the fourth-order interferometry. The power spectrum of the intensity for a combined light field produces a spectral interferogram in the modulation frequency domain which can be acquired by means of the electric spectrum analysis. A theoretical description was presented to show the principle and the advantageous characteristics of our SII. The experimental observations agreed that the SII provides the intensity correlation function by the fringed spectrum with no field-sensitive disturbance associated with a polarization mismatch or minute optical path variation. In the SII, the signal acquisition is performed with a fixed time delay or a constant interferometric path which simplifies the interferometer configuration. These features are attractive in the applications of the SII such as delay determinations or time-of-flight measurements. A fiber GVD measurement was demonstrated with an SII system that utilizes an ASE light source of a tunable wavelength. Both kilometer-long and meter-long fibers could be successfully tested for sample fibers of the GVD measurement. The precision of the delay determinations was found to be tens of femtoseconds with a reasonable acquisition time less than seconds. Further technical advances may enable sub-wavelength precisions that have been believed to be obtained only by field-sensitive interferometers.

References and links

1.

P. Hariharan, Optical Interferometry, 2nd Ed. (Academic Press, 2003).

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]

4.

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.

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]

6.

R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003). [CrossRef] [PubMed]

7.

M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (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]

9.

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

12.

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.

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. 27(3), 343–355 (2009). [CrossRef]

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]

16.

P. Hernday, “Dispersion measurements,” in Fiber Optic Test and Measurement, D. Derickson ed. (Prentice Hall PTR, 1998), pp.475–518.

17.

TIA Standard TIA-455–175-B, Meausrement Methods and Test Procedures – Chromatic Dispersion, 2003.

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]

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

  1. P. Hariharan, Optical Interferometry, 2nd Ed. (Academic Press, 2003).
  2. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature177(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]
  4. 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.
  5. 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]
  6. 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]
  7. 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]
  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]
  9. W. P. Alford and A. Gold, “Laboratory measurement of the velocity of light,” Am. J. Phys.26(7), 481–484 (1958). [CrossRef]
  10. S. Diddams and J.-C. Diels, “Dispersion measurements with white light interferometry,” J. Opt. Soc. Am. B13(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. Express14(24), 11608–11615 (2006). [CrossRef] [PubMed]
  12. 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.
  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.27(3), 343–355 (2009). [CrossRef]
  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]
  16. P. Hernday, “Dispersion measurements,” in Fiber Optic Test and Measurement, D. Derickson ed. (Prentice Hall PTR, 1998), pp.475–518.
  17. TIA Standard TIA-455–175-B, Meausrement Methods and Test Procedures – Chromatic Dispersion, 2003.
  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]

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