Abstract

This work provides a simple model for Residual Amplitude Modulation observed in Lithium Niobate phase modulators. It operates under two key assumptions: the optical field incident on the modulator is not perfectly aligned to the preferred axis, and the two linear polarizations become spatially separated while travelling down the waveguide. These assumptions allow for a straight forward transfer matrix based approach. The effects of chromatic dispersion present in the optical fiber following the modulator are included, as they become important for modulation frequencies over 20 GHz. The result is a closed form expression for the intensity modulated signal seen by the photodetector in a phase modulated system. The model describes a near-instantaneous control mechanism, which is useful in minimizing the residual amplitude modulation in fielded systems by offering over 40 dB of suppression. The model is compared to direct measurements, validating the polarization effects and control mechanism proposed. Furthermore, etalon effects are ruled out by doing a course temperature dependence measurement.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. A. Pogorelaya, M. A. Smolovik, V. E. Strigalev, A. S. Aleynik, and G. Deyneka, “An investigation of the influence of residual amplitude modulation in phase electro-optic modulator on the signal of fiber-optic gyroscope,” J. Phys. Conf. Ser. 735, 012040 (2016).
    [Crossref]
  2. E. Jaatinen, D. J. Hopper, and J. Back, “Residual amplitude modulation mechanisms in modulation transfer spectroscopy that uses electro-optic modulators,” Meas. Sci. Technol. 20(2), 025302 (2009).
    [Crossref]
  3. K. Kokeyama, K. Izumi, W. Z. Korth, N. Smith-Lefebvre, K. Aria, and R. X. Adhikari, “Residual amplitude modulation in interferometric gravitational wave detectors,” arXiv:1309.4522v1 (2013).
  4. E. A. Whittaker, M. Gehrtz, and G. C. Bjorklund, “Residual amplitude modulation in laser electro-optic phase modulation,” J. Opt. Soc. Am. B 2(8), 1320–1326 (1985).
    [Crossref]
  5. J. Sathian and E. Jaatinen, “Intensity dependent residual amplitude modulation in electro-optic phase modulators,” Appl. Opt. 51(16), 3684–3691 (2012).
    [Crossref] [PubMed]
  6. N. C. Wong and J. L. Hall, “Servo control of amplitude modulation in frequency-modulation spectroscopy: demonstration of shot-noise-limited detection,” J. Opt. Soc. Am. B 2(9), 1527–1533 (1985).
    [Crossref]
  7. J. Sathian and E. Jaatinen, “Polarization dependent photorefractive amplitude modulation production in MgO:LiNbO3 phase modulators,” in IQEC/CLEO Pac. Rim 2011 (2011).
  8. D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium Niobate,” Appl. Phys. Lett. 44(9), 847–849 (1984).
    [Crossref]
  9. L. Li, F. Liu, C. Wang, and L. Chen, “Measurement and control of residual amplitude modulation in optical phase modulation,” Rev. Sci. Instrum. 83(4), 043111 (2012).
    [Crossref] [PubMed]
  10. J. F. Diehl and V. J. Urick, “Chromatic Dispersion Induced Second-Order Distortion in Long-Haul Photonic Links,” J. Lightwave Tech. 34(20), 4646–4651 (2016).
    [Crossref]
  11. V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
    [Crossref]

2016 (2)

D. A. Pogorelaya, M. A. Smolovik, V. E. Strigalev, A. S. Aleynik, and G. Deyneka, “An investigation of the influence of residual amplitude modulation in phase electro-optic modulator on the signal of fiber-optic gyroscope,” J. Phys. Conf. Ser. 735, 012040 (2016).
[Crossref]

J. F. Diehl and V. J. Urick, “Chromatic Dispersion Induced Second-Order Distortion in Long-Haul Photonic Links,” J. Lightwave Tech. 34(20), 4646–4651 (2016).
[Crossref]

2012 (2)

L. Li, F. Liu, C. Wang, and L. Chen, “Measurement and control of residual amplitude modulation in optical phase modulation,” Rev. Sci. Instrum. 83(4), 043111 (2012).
[Crossref] [PubMed]

J. Sathian and E. Jaatinen, “Intensity dependent residual amplitude modulation in electro-optic phase modulators,” Appl. Opt. 51(16), 3684–3691 (2012).
[Crossref] [PubMed]

2011 (1)

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
[Crossref]

2009 (1)

E. Jaatinen, D. J. Hopper, and J. Back, “Residual amplitude modulation mechanisms in modulation transfer spectroscopy that uses electro-optic modulators,” Meas. Sci. Technol. 20(2), 025302 (2009).
[Crossref]

1985 (2)

1984 (1)

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium Niobate,” Appl. Phys. Lett. 44(9), 847–849 (1984).
[Crossref]

Aleynik, A. S.

D. A. Pogorelaya, M. A. Smolovik, V. E. Strigalev, A. S. Aleynik, and G. Deyneka, “An investigation of the influence of residual amplitude modulation in phase electro-optic modulator on the signal of fiber-optic gyroscope,” J. Phys. Conf. Ser. 735, 012040 (2016).
[Crossref]

Back, J.

E. Jaatinen, D. J. Hopper, and J. Back, “Residual amplitude modulation mechanisms in modulation transfer spectroscopy that uses electro-optic modulators,” Meas. Sci. Technol. 20(2), 025302 (2009).
[Crossref]

Bjorklund, G. C.

Bryan, D. A.

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium Niobate,” Appl. Phys. Lett. 44(9), 847–849 (1984).
[Crossref]

Bucholtz, F.

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
[Crossref]

Campillo, A. L.

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
[Crossref]

Chen, L.

L. Li, F. Liu, C. Wang, and L. Chen, “Measurement and control of residual amplitude modulation in optical phase modulation,” Rev. Sci. Instrum. 83(4), 043111 (2012).
[Crossref] [PubMed]

Devgan, P. S.

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
[Crossref]

Dexter, J. L.

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
[Crossref]

Deyneka, G.

D. A. Pogorelaya, M. A. Smolovik, V. E. Strigalev, A. S. Aleynik, and G. Deyneka, “An investigation of the influence of residual amplitude modulation in phase electro-optic modulator on the signal of fiber-optic gyroscope,” J. Phys. Conf. Ser. 735, 012040 (2016).
[Crossref]

Diehl, J. F.

J. F. Diehl and V. J. Urick, “Chromatic Dispersion Induced Second-Order Distortion in Long-Haul Photonic Links,” J. Lightwave Tech. 34(20), 4646–4651 (2016).
[Crossref]

Gehrtz, M.

Gerson, R.

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium Niobate,” Appl. Phys. Lett. 44(9), 847–849 (1984).
[Crossref]

Hall, J. L.

Hopper, D. J.

E. Jaatinen, D. J. Hopper, and J. Back, “Residual amplitude modulation mechanisms in modulation transfer spectroscopy that uses electro-optic modulators,” Meas. Sci. Technol. 20(2), 025302 (2009).
[Crossref]

Jaatinen, E.

J. Sathian and E. Jaatinen, “Intensity dependent residual amplitude modulation in electro-optic phase modulators,” Appl. Opt. 51(16), 3684–3691 (2012).
[Crossref] [PubMed]

E. Jaatinen, D. J. Hopper, and J. Back, “Residual amplitude modulation mechanisms in modulation transfer spectroscopy that uses electro-optic modulators,” Meas. Sci. Technol. 20(2), 025302 (2009).
[Crossref]

J. Sathian and E. Jaatinen, “Polarization dependent photorefractive amplitude modulation production in MgO:LiNbO3 phase modulators,” in IQEC/CLEO Pac. Rim 2011 (2011).

Li, L.

L. Li, F. Liu, C. Wang, and L. Chen, “Measurement and control of residual amplitude modulation in optical phase modulation,” Rev. Sci. Instrum. 83(4), 043111 (2012).
[Crossref] [PubMed]

Liu, F.

L. Li, F. Liu, C. Wang, and L. Chen, “Measurement and control of residual amplitude modulation in optical phase modulation,” Rev. Sci. Instrum. 83(4), 043111 (2012).
[Crossref] [PubMed]

McKinney, J. D.

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
[Crossref]

Pogorelaya, D. A.

D. A. Pogorelaya, M. A. Smolovik, V. E. Strigalev, A. S. Aleynik, and G. Deyneka, “An investigation of the influence of residual amplitude modulation in phase electro-optic modulator on the signal of fiber-optic gyroscope,” J. Phys. Conf. Ser. 735, 012040 (2016).
[Crossref]

Sathian, J.

J. Sathian and E. Jaatinen, “Intensity dependent residual amplitude modulation in electro-optic phase modulators,” Appl. Opt. 51(16), 3684–3691 (2012).
[Crossref] [PubMed]

J. Sathian and E. Jaatinen, “Polarization dependent photorefractive amplitude modulation production in MgO:LiNbO3 phase modulators,” in IQEC/CLEO Pac. Rim 2011 (2011).

Smolovik, M. A.

D. A. Pogorelaya, M. A. Smolovik, V. E. Strigalev, A. S. Aleynik, and G. Deyneka, “An investigation of the influence of residual amplitude modulation in phase electro-optic modulator on the signal of fiber-optic gyroscope,” J. Phys. Conf. Ser. 735, 012040 (2016).
[Crossref]

Strigalev, V. E.

D. A. Pogorelaya, M. A. Smolovik, V. E. Strigalev, A. S. Aleynik, and G. Deyneka, “An investigation of the influence of residual amplitude modulation in phase electro-optic modulator on the signal of fiber-optic gyroscope,” J. Phys. Conf. Ser. 735, 012040 (2016).
[Crossref]

Tomaschke, H. E.

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium Niobate,” Appl. Phys. Lett. 44(9), 847–849 (1984).
[Crossref]

Urick, V. J.

J. F. Diehl and V. J. Urick, “Chromatic Dispersion Induced Second-Order Distortion in Long-Haul Photonic Links,” J. Lightwave Tech. 34(20), 4646–4651 (2016).
[Crossref]

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
[Crossref]

Wang, C.

L. Li, F. Liu, C. Wang, and L. Chen, “Measurement and control of residual amplitude modulation in optical phase modulation,” Rev. Sci. Instrum. 83(4), 043111 (2012).
[Crossref] [PubMed]

Whittaker, E. A.

Williams, K. J.

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
[Crossref]

Wong, N. C.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium Niobate,” Appl. Phys. Lett. 44(9), 847–849 (1984).
[Crossref]

J. Lightwave Tech (1)

V. J. Urick, F. Bucholtz, J. D. McKinney, P. S. Devgan, A. L. Campillo, J. L. Dexter, and K. J. Williams, “Long-Haul Analog Photonics,” J. Lightwave Tech 29(8), 1182–1205 (2011).
[Crossref]

J. Lightwave Tech. (1)

J. F. Diehl and V. J. Urick, “Chromatic Dispersion Induced Second-Order Distortion in Long-Haul Photonic Links,” J. Lightwave Tech. 34(20), 4646–4651 (2016).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. Conf. Ser. (1)

D. A. Pogorelaya, M. A. Smolovik, V. E. Strigalev, A. S. Aleynik, and G. Deyneka, “An investigation of the influence of residual amplitude modulation in phase electro-optic modulator on the signal of fiber-optic gyroscope,” J. Phys. Conf. Ser. 735, 012040 (2016).
[Crossref]

Meas. Sci. Technol. (1)

E. Jaatinen, D. J. Hopper, and J. Back, “Residual amplitude modulation mechanisms in modulation transfer spectroscopy that uses electro-optic modulators,” Meas. Sci. Technol. 20(2), 025302 (2009).
[Crossref]

Rev. Sci. Instrum. (1)

L. Li, F. Liu, C. Wang, and L. Chen, “Measurement and control of residual amplitude modulation in optical phase modulation,” Rev. Sci. Instrum. 83(4), 043111 (2012).
[Crossref] [PubMed]

Other (2)

K. Kokeyama, K. Izumi, W. Z. Korth, N. Smith-Lefebvre, K. Aria, and R. X. Adhikari, “Residual amplitude modulation in interferometric gravitational wave detectors,” arXiv:1309.4522v1 (2013).

J. Sathian and E. Jaatinen, “Polarization dependent photorefractive amplitude modulation production in MgO:LiNbO3 phase modulators,” in IQEC/CLEO Pac. Rim 2011 (2011).

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Figures (5)

Fig. 1
Fig. 1 Experimental Setup for RAM measurements. VNA calibrated with RF cables and bias tee inline to isolate the photonic link for measurement accuracy.
Fig. 2
Fig. 2 Measurement results for the a) ordinary mode, and b) extraordinary mode at two random bias voltages. Measured data in black, theory in red dashes. The measurement noise floor prevented the system from resolving the null shown in the model.
Fig. 3
Fig. 3 Measurement results for both contributions at two random bias voltages. Measured data in black, theory in red dashes. Lower noise floor is achievable due to higher total gain (summed photocurrent).
Fig. 4
Fig. 4 Predicted temperature change during a heating (red) and cooling (blue dashed) cycle. These data were extracted from measured gain data using Eqs. (13), (15), and (16).
Fig. 5
Fig. 5 Measured RAM level as a function of a sweeping bias control voltage (black), fit to model (red dashed).

Tables (1)

Tables Icon

Table 1 Table of parameters used in model.

Equations (16)

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E in =[ 1 0 ] E 0 e i ω 0 t
[ E out,o E out,e ]= 1 2 [ cos( θ 2 ) isin( θ 2 ) isin( θ 2 ) cos( θ 2 ) ][ α 1 T( τ ) 0 0 α 2 ] ×[ e i φ dc,1 e i φ rf,1 sin( Ωt ) 0 0 e i φ dc,2 e i φ rf,2 sin( Ωt ) ][ cos( θ 1 ) isin( θ 1 ) isin( θ 1 ) cos( θ 1 ) ][ E 0 e i ω 0 t 0 ]
φ dc,i = V dc V π,dc,i π
φ rf,i = V rf V π,rf,i π
E out,o = 1 2 α 1 E 0 e i φ dc,1 e i ω 0 t ×[ cos( θ 1 )cos( θ 2 ) e i ω 0 τ e i φ rf,1 sin( Ωt ) α ¯ sin( θ 1 )sin( θ 2 ) e i φ dc ¯ e i φ rf,2 sin( Ωt ) ]+c.c.
E out,e = i 2 α 1 E 0 e i φ dc,1 e i ω 0 t ×[ cos( θ 1 )sin( θ 2 ) e i ω 0 τ e i φ rf,1 sin( Ωt ) + α ¯ sin( θ 1 )cos( θ 2 ) e i φ dc ¯ e i φ rf,2 sin( Ωt ) ]+c.c.
e i φ rf,i sin( Ωt ) = n= J n ( φ rf,i ) e inΩt ,
β( ω )= β 0 + β 1 ( ω ω 0 )+ 1 2 β 2 ( ω ω 0 ) 2 +
E out,o =c.c.+ 1 2 α 1 E 0 e i φ dc,1 e i ω 0 t × [ cos( θ 1 )cos( θ 2 ) e i ω 0 τ n= J n ( φ rf,1 ) e inΩt e i 2 β 2 ( nΩ ) 2 L α ¯ sin( θ 1 )sin( θ 2 ) e i φ dc ¯ n= J n ( φ rf,2 ) e inΩt e i 2 β 2 ( nΩ ) 2 L ]
E out,e =c.c.+ i 2 α 1 E 0 e i φ dc,1 e i ω 0 t × [ cos( θ 1 )sin( θ 2 ) e i ω 0 τ n= J n ( φ rf,1 ) e inΩt e i 2 β 2 ( nΩ ) 2 L + α ¯ sin( θ 1 )cos( θ 2 ) e i φ dc ¯ n= J n ( φ rf,2 ) e inΩt e i 2 β 2 ( nΩ ) 2 L ]
I Ω,o 1 2 α 1 2 E 0 2 sin( Ωt ) [ cos 2 ( θ 1 ) cos 2 ( θ 2 ) φ rf,1 sin( 1 2 β 2 Ω 2 L ) + α ¯ 2 sin 2 ( θ 1 ) sin 2 ( θ 2 ) φ rf,2 sin( 1 2 β 2 Ω 2 L ) α ¯ sin( θ 1 )cos( θ 1 )sin( θ 2 )cos( θ 2 ) × [ ( φ rf,1 + φ rf,2 )cos( ω 0 τ+ φ dc ¯ )sin( 1 2 β 2 Ω 2 L ) +( φ rf,1 φ rf,2 )sin( ω 0 τ+ φ dc ¯ )cos( 1 2 β 2 Ω 2 L ) ] ]
I Ω,e 1 2 α 1 2 E 0 2 sin( Ωt ) [ cos 2 ( θ 1 ) sin 2 ( θ 2 ) φ rf,1 sin( 1 2 β 2 Ω 2 L ) + α ¯ 2 sin 2 ( θ 1 ) cos 2 ( θ 2 ) φ rf,2 sin( 1 2 β 2 Ω 2 L ) + α ¯ sin( θ 1 )cos( θ 1 )sin( θ 2 )cos( θ 2 ) × [ ( φ rf,1 + φ rf,2 )cos( ω 0 τ+ φ dc ¯ )sin( 1 2 β 2 Ω 2 L ) +( φ rf,1 φ rf,2 )sin( ω 0 τ+ φ dc ¯ )cos( 1 2 β 2 Ω 2 L ) ] ]
G= P out P in = I 2 Z out Z in V rf 2
φ rf,1 sin( ω 0 τ+ 1 2 β 2 Ω 2 L ) φ rf,2 sin( ω 0 τ 1 2 β 2 Ω 2 L ) =( G V rf 2 Z out Z in 1 2 α 1 2 E 0 2 ( cos 2 ( θ 1 ) cos 2 ( θ 2 ) φ rf,1 sin( 1 2 β 2 Ω 2 L ) + α ¯ 2 sin 2 ( θ 1 ) sin 2 ( θ 2 ) φ rf,2 sin( 1 2 β 2 Ω 2 L ) ) ) ÷( 1 2 α 1 2 E 0 2 α ¯ sin( θ 1 )cos( θ 1 )sin( θ 2 )cos( θ 2 ) )
sin( ω 0 τ )= V π,rf,1 V π,rf,2 π( V π,rf,2 V π,rf,1 ) ( G Z out Z in π α 1 2 E 0 2 2 ×( cos 2 ( θ 1 ) cos 2 ( θ 2 ) V π,rf,1 sin( 1 2 β 2 Ω 2 L ) + α ¯ 2 sin 2 ( θ 1 ) sin 2 ( θ 2 ) V π,rf,2 sin( 1 2 β 2 Ω 2 L ) ) ÷( 1 2 α 1 2 E 0 2 α ¯ sin( θ 1 )cos( θ 1 )sin( θ 2 )cos( θ 2 ) )
ω 0 τ= ω 0 c ( Δn T L mod +Δn L mod T )T

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