Abstract

Digitally enhanced heterodyne interferometry (DEHI) combines the sub-wavelength displacement measurements of conventional laser interferometry with the multiplexing capabilities of spread-spectrum modulation techniques to discriminate between multiple electric fields at a single photodetector. Technologies that benefit from DEHI include optical phased arrays, which require the simultaneous phase measurement of a large number of electric fields. A consequence of measuring the phase of multiple electric fields is the introduction of crosstalk, which can degrade measurement precision. This work analytically and experimentally investigates the crosstalk when using DEHI to measure the phase of an arbitrarily large number of electric fields at a single photodetector. Also considered is the practical limit the dynamic range of the photodetector and shot noise imposes on the number of electric fields that can be discriminated. We describe how to minimize crosstalk by design. Experimental results demonstrate up to 55 dB suppression of crosstalk between two electric fields with a phase measurement bandwidth of 20 kHz and 1-10 pm$/\sqrt {\textrm {Hz}}$ displacement sensitivity for audio frequencies. Additionally, we demonstrate scaling of crosstalk proportional to the square-root of the number of electric fields when using an M-sequence modulation. Based on this analysis, we estimate that digitally enhanced heterodyne interferometry should be capable of measuring the phase of several hundreds of electric fields at a single photodetector while maintaining the same measurement bandwidth.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  26. L. E. Roberts, “Internally sensed optical phased arrays,” Thesis (2016).
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    [Crossref]
  28. K.-S. Isleif, O. Gerberding, S. Köhlenbeck, A. Sutton, B. Sheard, S. Goßler, D. Shaddock, G. Heinzel, and K. Danzmann, “Highspeed multiplexed heterodyne interferometry,” Opt. Express 22(20), 24689–24696 (2014).
    [Crossref]
  29. T. S. Schwarze, O. Gerberding, F. G. Cervantes, G. Heinzel, and K. Danzmann, “Advanced phasemeter for deep phase modulation interferometry,” Opt. Express 22(15), 18214–18223 (2014).
    [Crossref]

2019 (1)

2018 (2)

A. Klenke, M. Müller, H. Stark, A. Tünnermann, and J. Limpert, “Sequential phase locking scheme for a filled aperture intensity coherent combination of beam arrays,” Opt. Express 26(9), 12072–12080 (2018).
[Crossref]

S. P. Francis, T. T. Y. Lam, D. E. McClelland, and D. A. Shaddock, “Multi-link laser interferometry architecture for interspacecraft displacement metrology,” J. Geod. 92(3), 241–251 (2018).
[Crossref]

2017 (2)

2016 (2)

2015 (2)

2014 (5)

2013 (2)

2011 (3)

2007 (2)

K. Fouli and M. Maier, “Ocdma and optical coding: Principles, applications, and challenges [topics in optical communications],” IEEE Commun. Mag. 45(8), 27–34 (2007).
[Crossref]

D. A. Shaddock, “Digitally enhanced heterodyne interferometry,” Opt. Lett. 32(22), 3355–3357 (2007).
[Crossref]

2006 (2)

T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14(25), 12188–12195 (2006).
[Crossref]

D. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the lisa phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

1996 (1)

R. N. Mutagi, “Pseudo noise sequences for engineers,” Electron. Commun. Eng. J. 8(2), 79–87 (1996).
[Crossref]

1982 (1)

R. Scholtz, “The origins of spread-spectrum communications,” IEEE Trans. Commun. 30(5), 822–854 (1982).
[Crossref]

Ahn, H. K.

Altin, P. A.

Azarian, A.

B. Glebov, L.

C. A. Lu, A. Flores, E. Bochove, W. P. Roach, V. Smirnov, and L. B. Glebov, “Active coherent superposition of five fiber amplifiers at 670w using multiplexed volume bragg gratings,” pp. 86011A–86011A–6.

Bandutunga, C. P.

Bochove, E.

C. A. Lu, A. Flores, E. Bochove, W. P. Roach, V. Smirnov, and L. B. Glebov, “Active coherent superposition of five fiber amplifiers at 670w using multiplexed volume bragg gratings,” pp. 86011A–86011A–6.

Bourdon, P.

Canuel, B.

Cervantes, F. G.

Charrett, T. O. H.

Chow, J. H.

Danzmann, K.

de Vine, G.

Dong, X.

Du, W.

Fleddermann, R.

Flores, A.

C. A. Lu, A. Flores, E. Bochove, W. P. Roach, V. Smirnov, and L. B. Glebov, “Active coherent superposition of five fiber amplifiers at 670w using multiplexed volume bragg gratings,” pp. 86011A–86011A–6.

Fouli, K.

K. Fouli and M. Maier, “Ocdma and optical coding: Principles, applications, and challenges [topics in optical communications],” IEEE Commun. Mag. 45(8), 27–34 (2007).
[Crossref]

Francis, S. P.

S. P. Francis, T. T. Y. Lam, D. E. McClelland, and D. A. Shaddock, “Multi-link laser interferometry architecture for interspacecraft displacement metrology,” J. Geod. 92(3), 241–251 (2018).
[Crossref]

L. E. Roberts, R. L. Ward, S. P. Francis, P. G. Sibley, R. Fleddermann, A. J. Sutton, C. Smith, D. E. McClelland, and D. A. Shaddock, “High power compatible internally sensed optical phased array,” Opt. Express 24(12), 13467–13479 (2016).
[Crossref]

Genin, E.

Gerberding, O.

Goßler, S.

Gray, M. B.

Halverson, P. G.

D. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the lisa phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Heinzel, G.

Isleif, K.-S.

Jaouën, Y.

Jiang, M.

Kissinger, T.

Klenke, A.

Klipstein, B.

D. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the lisa phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Köhlenbeck, S.

Kong, H. J.

Lam, T. T. Y.

Leng, J.

Limpert, J.

Liu, Z.

Lombard, L.

Lu, C. A.

C. A. Lu, A. Flores, E. Bochove, W. P. Roach, V. Smirnov, and L. B. Glebov, “Active coherent superposition of five fiber amplifiers at 670w using multiplexed volume bragg gratings,” pp. 86011A–86011A–6.

Ma, H.

Ma, Y.

Maier, M.

K. Fouli and M. Maier, “Ocdma and optical coding: Principles, applications, and challenges [topics in optical communications],” IEEE Commun. Mag. 45(8), 27–34 (2007).
[Crossref]

Malikides, E. A.

Marque, J.

McClelland, D. E.

McRae, T. G.

Müller, M.

Mutagi, R. N.

R. N. Mutagi, “Pseudo noise sequences for engineers,” Electron. Commun. Eng. J. 8(2), 79–87 (1996).
[Crossref]

Ngo, S.

P. Roach, W.

C. A. Lu, A. Flores, E. Bochove, W. P. Roach, V. Smirnov, and L. B. Glebov, “Active coherent superposition of five fiber amplifiers at 670w using multiplexed volume bragg gratings,” pp. 86011A–86011A–6.

Pulford, B.

B. Pulford, “Locset phase locking: operation, diagnoistics, and applications,” Thesis (2012).

Rabeling, D. S.

Riesen, N.

Roberts, L. E.

Scholtz, R.

R. Scholtz, “The origins of spread-spectrum communications,” IEEE Trans. Commun. 30(5), 822–854 (1982).
[Crossref]

Schwarze, T. S.

Shaddock, D.

Shaddock, D. A.

S. P. Francis, T. T. Y. Lam, D. E. McClelland, and D. A. Shaddock, “Multi-link laser interferometry architecture for interspacecraft displacement metrology,” J. Geod. 92(3), 241–251 (2018).
[Crossref]

D. Tarquin Ralph, P. A. Altin, D. S. Rabeling, D. E. McClelland, and D. A. Shaddock, “Interferometric wavefront sensing with a single diode using spatial light modulation,” Appl. Opt. 56(8), 2353–2358 (2017).
[Crossref]

L. E. Roberts, R. L. Ward, S. P. Francis, P. G. Sibley, R. Fleddermann, A. J. Sutton, C. Smith, D. E. McClelland, and D. A. Shaddock, “High power compatible internally sensed optical phased array,” Opt. Express 24(12), 13467–13479 (2016).
[Crossref]

S. Ngo, D. A. Shaddock, T. G. McRae, T. T. Y. Lam, J. H. Chow, and M. B. Gray, “Suppressing rayleigh backscatter and code noise from all-fiber digital interferometers,” Opt. Lett. 41(1), 84–87 (2016).
[Crossref]

L. E. Roberts, R. L. Ward, A. J. Sutton, R. Fleddermann, G. de Vine, E. A. Malikides, D. M. R. Wuchenich, D. E. McClelland, and D. A. Shaddock, “Coherent beam combining using a 2d internally sensed optical phased array,” Appl. Opt. 53(22), 4881–4885 (2014).
[Crossref]

S. Ngo, T. G. McRae, M. B. Gray, and D. A. Shaddock, “Homodyne digital interferometry for a sensitive fiber frequency reference,” Opt. Express 22(15), 18168–18176 (2014).
[Crossref]

D. M. R. Wuchenich, T. T. Y. Lam, J. H. Chow, D. E. McClelland, and D. A. Shaddock, “Laser frequency noise immunity in multiplexed displacement sensing,” Opt. Lett. 36(5), 672–674 (2011).
[Crossref]

D. A. Shaddock, “Digitally enhanced heterodyne interferometry,” Opt. Lett. 32(22), 3355–3357 (2007).
[Crossref]

Shay, T. M.

Sheard, B.

Si, L.

Sibley, P. G.

Smirnov, V.

C. A. Lu, A. Flores, E. Bochove, W. P. Roach, V. Smirnov, and L. B. Glebov, “Active coherent superposition of five fiber amplifiers at 670w using multiplexed volume bragg gratings,” pp. 86011A–86011A–6.

Smith, C.

Spero, R. E.

D. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the lisa phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Stark, H.

Su, R.

Sutton, A.

Sutton, A. J.

Tarquin Ralph, D.

Tatam, R. P.

Tünnermann, A.

Vajente, G.

Vasseur, O.

Wang, X.

Ward, R. L.

Ware, B.

D. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the lisa phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Wuchenich, D. M. R.

Xiao, H.

Xu, X.

Zhang, Y.

Zhang, Z.

Zhao, Y.

Zhou, P.

Zhu, J.

AIP Conf. Proc. (1)

D. Shaddock, B. Ware, P. G. Halverson, R. E. Spero, and B. Klipstein, “Overview of the lisa phasemeter,” AIP Conf. Proc. 873, 654–660 (2006).
[Crossref]

Appl. Opt. (5)

Electron. Commun. Eng. J. (1)

R. N. Mutagi, “Pseudo noise sequences for engineers,” Electron. Commun. Eng. J. 8(2), 79–87 (1996).
[Crossref]

IEEE Commun. Mag. (1)

K. Fouli and M. Maier, “Ocdma and optical coding: Principles, applications, and challenges [topics in optical communications],” IEEE Commun. Mag. 45(8), 27–34 (2007).
[Crossref]

IEEE Trans. Commun. (1)

R. Scholtz, “The origins of spread-spectrum communications,” IEEE Trans. Commun. 30(5), 822–854 (1982).
[Crossref]

J. Geod. (1)

S. P. Francis, T. T. Y. Lam, D. E. McClelland, and D. A. Shaddock, “Multi-link laser interferometry architecture for interspacecraft displacement metrology,” J. Geod. 92(3), 241–251 (2018).
[Crossref]

Opt. Express (11)

S. Ngo, T. G. McRae, M. B. Gray, and D. A. Shaddock, “Homodyne digital interferometry for a sensitive fiber frequency reference,” Opt. Express 22(15), 18168–18176 (2014).
[Crossref]

T. S. Schwarze, O. Gerberding, F. G. Cervantes, G. Heinzel, and K. Danzmann, “Advanced phasemeter for deep phase modulation interferometry,” Opt. Express 22(15), 18214–18223 (2014).
[Crossref]

N. Riesen, T. T. Y. Lam, and J. H. Chow, “Bandwidth-division in digitally enhanced optical frequency domain reflectometry,” Opt. Express 21(4), 4017–4026 (2013).
[Crossref]

B. Canuel, E. Genin, G. Vajente, and J. Marque, “Displacement noise from back scattering and specular reflection of input optics in advanced gravitational wave detectors,” Opt. Express 21(9), 10546–10562 (2013).
[Crossref]

T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14(25), 12188–12195 (2006).
[Crossref]

L. E. Roberts, R. L. Ward, S. P. Francis, P. G. Sibley, R. Fleddermann, A. J. Sutton, C. Smith, D. E. McClelland, and D. A. Shaddock, “High power compatible internally sensed optical phased array,” Opt. Express 24(12), 13467–13479 (2016).
[Crossref]

A. Klenke, M. Müller, H. Stark, A. Tünnermann, and J. Limpert, “Sequential phase locking scheme for a filled aperture intensity coherent combination of beam arrays,” Opt. Express 26(9), 12072–12080 (2018).
[Crossref]

Y. Zhang, C. P. Bandutunga, M. B. Gray, and J. H. Chow, “Multi-target cw interferometric acoustic measurements on a single optical beam,” Opt. Express 27(13), 18477–18483 (2019).
[Crossref]

K.-S. Isleif, O. Gerberding, S. Köhlenbeck, A. Sutton, B. Sheard, S. Goßler, D. Shaddock, G. Heinzel, and K. Danzmann, “Highspeed multiplexed heterodyne interferometry,” Opt. Express 22(20), 24689–24696 (2014).
[Crossref]

T. Kissinger, T. O. H. Charrett, and R. P. Tatam, “Range-resolved interferometric signal processing using sinusoidal optical frequency modulation,” Opt. Express 23(7), 9415–9431 (2015).
[Crossref]

H. K. Ahn and H. J. Kong, “Cascaded multi-dithering theory for coherent beam combining of multiplexed beam elements,” Opt. Express 23(9), 12407–12413 (2015).
[Crossref]

Opt. Lett. (4)

Other (4)

L. E. Roberts, “Internally sensed optical phased arrays,” Thesis (2016).

B. Pulford, “Locset phase locking: operation, diagnoistics, and applications,” Thesis (2012).

C. A. Lu, A. Flores, E. Bochove, W. P. Roach, V. Smirnov, and L. B. Glebov, “Active coherent superposition of five fiber amplifiers at 670w using multiplexed volume bragg gratings,” pp. 86011A–86011A–6.

A. Goldsmith, ed., Spread Spectrum (Cambridge University Press, Cambridge, 2005), pp. 403–451.

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Figures (14)

Fig. 1.
Fig. 1. Layout of an optical setup and key signal processing steps for a N-channel optical phase sensor using DEHI. Each step: 2.1, 2.2, 2.3, 2.4, 2.5 and 2.6 correspond to the electric fields interference, signal detection, PRBS decoding, Quadrature demodulation (using numerically controlled oscillator (NCO), Signal filtering/decimation and Phase readout sections. The signal arm shown (highlighted by the red box), separates channels based on fiber delays.
Fig. 2.
Fig. 2. a) Frequency spectrum of a signal encoded with 15 chip length PRBS and b) Magnitude Response of a 2nd order CIC filter with decimation set to the period of the PRBS. When the filter decimation is set to the code frequency, the spacing of filter nulls matches PRBS harmonic spacing.
Fig. 3.
Fig. 3. Frequency spectra around DC representing the three categories of crosstalk signals when the heterodyne frequency is equal to the chip frequency and a 4-bit PRBS. This results in a PRBS spectrum null centred in the shaded measurement band for the $s_{het}\uparrow$ and $s_{hom}$ signals which should remove crosstalk from these signals. The harmonic in the measurement band for $s_{het}\downarrow$ signals is equal to $1/L$ which is the DC average of an M-sequence PRBS.
Fig. 4.
Fig. 4. The experimental heterodyne Mach-Zehnder interferometer used for the investigation of crosstalk when multiplexing optical phase signals using DEHI. Three different signal arm configurations are outlined, used for different measurements.
Fig. 5.
Fig. 5. Bandwidth effects on an experimental PRBS modulation generated by the FPGA’s digital outputs. a) A time domain measurement of the 20 MHz PRBS voltage signal applied to the electro-optic modulators showing the chip transition overshoots characteristic of bandwidth limitations. b) A section of the power spectrum of the PRBS encoded optical heterodyne beatnote measured on the photodetector showing a spike positioned at $f_{het}+f_{chip}$ not present in the simulated frequency spectrum of a "pristine" PRBS (e.g. Figure 2).
Fig. 6.
Fig. 6. Simulated frequency spectra of PRBS encoded sinusoids representing of $s_{hom}$ (left) and $s_{het}\uparrow$ (right) signals with four, two and one samples per chip (SPC). With a reduced number of samples per chip, the spectra loses the characteristic sinc shape as well as the null centred at DC.
Fig. 7.
Fig. 7. Multiplication of two differently delayed PRBS. a) Aligned chip transitions where the PRBS are separated by an integer chip delay ($\tau _{\textrm {chip}}$), producing another PRBS with the same chip frequency. b) Mis-aligned chip transitions where the two PRBS are separated by a fractional delay, resulting in the appearance of shorter, higher frequency chips.
Fig. 8.
Fig. 8. Simulated frequency spectra of the product of subchip delayed decoding PRBS with four samples per chip. a) Two PRBS are separated by integer chip delays producing the ideal spectrum. b) Two PRBS separated by a fractional delay. This changes the height of many of the PRBS harmonics which could cause a varying amount of crosstalk for configurations where $f_{het}$ is not equal to the chip frequency. c) Two PRBS separated by less than a chip displays a combination of a decoded signal (large spike at DC) and a spread spectrum (across the entire frequency range). There are also spikes at multiples of the chip frequency which could reduce suppression of $s_{het}\uparrow$ if $f_{het}=f_{chip}$. d) Shows the same PRBS but modulating a signal at twice the chip frequency (representing $s_{het}\uparrow$) which shows a spike positioned in the measurement band.
Fig. 9.
Fig. 9. Phase amplitude spectral density measurements of a 20 MHz heterodyne beatnote isolated with 7-bit PRBS demonstrating suppression of a 2 kHz tone acting as crosstalk. In green a single heterodyne beatnote without PRBS as a baseline showing no signal at 2 kHz. In red a measurement of channel 2 showing the unattenuated amplitude of the injected phase tone at 2 kHz. In blue the measurement of channel 1 while suppressing channel showing the reduced amplitude of the 2 kHz phase tone.
Fig. 10.
Fig. 10. a) Suppression of the 2 kHz tone when changing cable length between photo-detector and ADC for the same heterodyne frequencies encoded with a 9-bit PRBS, showing an increase crosstalk and sensitivity to the subchip delay changes when $f_{het}=f_{chip}$. b) Phase amplitude spectral density of channel one’s phase signal for different heterodyne frequencies $f_{het}=f_{chip}$ and $f_{het}=4.5 ~\textrm {MHz}$. When the heterodyne frequency is not aligned to a multiple of the code frequency, multiple harmonics appear in the measurement band.
Fig. 11.
Fig. 11. Overlay of the PRBS harmonics near the measurement bandwidth with CIC filter’s sinc like magnitude response for different frequency signals. a) A DC PRBS representative of $s_{het}\downarrow$ signals. b) A PRBS shifted in frequency by a fraction of code frequency (0.1 $f_{code}$) illustrating misalignment of the filter nulls with PRBS harmonics leading to an increase in crosstalk.
Fig. 12.
Fig. 12. Dependence of crosstalk suppression on signal frequency ($f_{sig}$) for different codelengths. Compared are the analytically expected crosstalk (dashed) and the mean experimentally measured crosstalk (dots).
Fig. 13.
Fig. 13. Scaling of crosstalk with an increasing number of channels. The dot represents the mean crosstalk across all randomized tone phases. The bars represent the measured range of crosstalk as the phase difference between the injected tones is changed. a) Shows the increase in crosstalk with number of channels when setting the low pass filter decimation equal to the code length and heterodyne frequency to $f_{het}=f_{chip}-f_{code}$. b) Shows a reduced increase in crosstalk when averaging over four codelengths and setting the heterodyne frequency to $f_{het}=f_{chip}-1.25f_{code}$.
Fig. 14.
Fig. 14. a) Predicted number supported channels before reaching 1% crosstalk depending on ability to suppress crosstalk for different local oscillator, signal arm power ratios. Case 1 is crosstalk solely from $s_{het}\downarrow$ signals, case 2a is crosstalk from all signals with $1/N$ signal arm power scaling and case 2b is the crosstalk from all signals with no power scaling. b) Simulated detector noise and shot noise when changing the signal arm power to accommodate an increased number of channels according to $ P_n = \frac{P_LO}{N}$.

Tables (3)

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Table 1. Parameters for each signal comprising M Q . The first row contains the correctly decoded signal and the remaining rows are the signals contributing to crosstalk

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Table 2. Measured crosstalk of a 2 kHz tone for different heterodyne frequencies and a 7-bit PRBS. See Fig. 10(a), for sensitivity to subchip delays. See Fig. 10(b), where crosstalk is not confined to 2 kHz frequency bin

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Table 3. Scaling of key DEHI parameters with number of channels, N and optical powers P L O and P n . A change in notation is used for brevity, where Λ h e t , Λ h e t and Λ h o m represent the crosstalk fraction from a single suppressed signal

Equations (29)

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E P D = E L O e i ( ω L O t + ϕ L O ) + n = 1 N E n e i ( ω n t + ϕ n + β c ( t τ n ) )
s P D = s D C + s h e t + s h o m
s D C = A L O 2 + n = 1 N A n 2
s h e t = n = 1 N 2 A L O A n cos ( ω h e t t + ϕ L O ϕ n β c ( t τ n ) )
s h o m = m = 1 N 1 n = 1 N m 2 A n A n + m cos ( ( ϕ n ϕ n + m ) + β c ( t τ n ) β c ( t τ n + m ) )
cos ( ϕ + β c ( t τ ) ) = sin ( β 2 ) sin ( ϕ + β 2 ) p ( t τ ) Encoded Component + cos ( β 2 ) cos ( ϕ + β 2 ) Unencoded Component
p ( t τ i ) p ( t τ j ) = { 1 if i = j : Correctly Decoded p ( t τ k ) if i j : Incorrectly Decoded
d m i n = c n f i b e r f c h i p
M I = s P D p i sin ( ω h e t t + ϕ N C O )
M Q = s P D p i cos ( ω h e t t + ϕ N C O )
A sin ( ω t + ϕ ( t ) ) P
R = L f s f c h i p
H C I C : M t h ( f ) ( f s π f sin 2 ( π R f f s ) ) M
Measurement Bandwidth = f s 2 R = 1 2 f c o d e = 1 2 f c h i p L (Hz)
Φ = tan 1 ( Q I ) = tan 1 ( L A L O A i sin ( Φ i ) + A L O n = 1 ; n i N A n η n sin ( Φ n ) L A L O A i cos ( Φ i ) + A L O n = 1 ; n i N A n η n cos ( Φ n ) )
Φ = Φ i + tan 1 ( n = 1 , n i N 1 η n L A n A i sin ( ϕ i ( t ) ϕ n ( t ) + ϕ p n ) 1 + n = 1 , n i N 1 η n L A n A i cos ( ϕ i ( t ) ϕ n ( t ) + ϕ p n ) )
Φ = Φ i + n = 1 , n i N 1 η n L A n A i sin ( ϕ i ( t ) ϕ n ( t ) + ϕ p n )
Φ 1 = ϕ L O ϕ 1 π 2 + n = 1 ; n 1 N η n L A n A 1 sin ( ϕ 1 ϕ n ξ sin ( ω ξ t + ϕ ξ , n ) + ϕ p n )
Φ 2 = ϕ L O ϕ 2 ξ sin ( ω ξ t + ϕ ξ , 2 ) π 2 + n = 1 ; n 2 N η n L A n A 2 sin ( ϕ 2 + ξ sin ( ω ξ t + ϕ ξ , n ) ϕ n + ϕ p n )
sin ( ξ sin ( ω t + ϕ ) ) = 2 n = 1 J 2 n 1 ( ξ ) sin ( ( 2 n 1 ) ( ω t + ϕ ) )
ξ 1 ¯ ξ 2 ¯ = n = 1 ; n 1 N η n L A n A 1 cos ( ϕ 1 ( t ) ϕ n ( t ) + ϕ p n ) 2 J 1 ( ξ ) ξ + n = 1 ; n 1 N η n L A n A 2 cos ( ϕ 2 ( t ) ϕ n ( t ) + ϕ p n ) 2 J 1 ( ξ ) n = 1 ; n 1 N η n L A n A 1 cos ( ϕ 1 ( t ) ϕ n ( t ) + ϕ p n )
ξ 1 ¯ ξ 2 ¯ η 2 L A 2 A 1 cos ( ϕ 1 ( t ) ϕ 2 ( t ) + ϕ p 2 )
Crosstalk = 20 log 10 ( ξ 1 ¯ ξ 2 ¯ ) [ dB ]
f c r o s s t a l k = { f s i g + 0 for  s h e t f s i g + m i n ( | H × f c h i p L f h e t | ) for  s h o m f s i g + m i n ( | H × f c h i p L 2 f h e t | ) for  s h e t Where H is an integer
h 0 = 1 L ( f s π f s i g sin 2 ( π L f s i g f c h i p ) ) M
h 1 a = 2 k / 2 L ( f s π ( f c o d e f s i g ) sin 2 ( π L ( f c o d e f s i g ) f c h i p ) ) M
h 1 b = 2 k / 2 L ( f s π ( f c o d e + f s i g ) sin 2 ( π L ( f c o d e + f s i g ) f c h i p ) ) M
η ( f s i g ) = h 0 ( f s i g ) 2 + h 1 a ( f s i g ) 2 + h 1 b ( f s i g ) 2
Crosstalk ( N ) = 20 log 10 ( ξ 1 ¯ ( n = 1 ; n 1 N ξ n ¯ ) / ( N 1 ) ) [ d B ]