Abstract

The concept of filtered Microwave Photonic Links is proposed in order to provide the most general and versatile description of complex analog photonic systems. We develop a field propagation model where a global optical filter, characterized by its optical transfer function, embraces all the intermediate optical components in a linear link. We assume a non-monochromatic light source characterized by an arbitrary spectral distribution which has a finite linewidth spectrum and consider both intensity modulation and phase modulation with balanced and single detection. Expressions leading to the computation of the main figures of merit concerning the link gain, noise and intermodulation distortion are provided which, to our knowledge, are not available in the literature. The usefulness of this derivation resides in the capability to directly provide performance criteria results for complex links just by substituting in the overall closed-form formulas the numerical or measured optical transfer function characterizing the link. This theory is presented thus as a potential tool for a wide range of relevant microwave photonic application cases which is extendable to multiport radio over fiber systems.

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References

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  1. A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24(12), 4628–4641 (2006).
    [CrossRef]
  2. J. Capmany and D. Novak, “Microwave Photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
    [CrossRef]
  3. C. H. Cox III, Analog Photonic Links: Theory and Practice (Cambridge University Press, Cambridge, U.K., 2004).
  4. V. J. Urick, M. S. Rogge, F. Bucholtz, and K. J. Williams, “The performance of analog photonic links employing highly-compressed erbium-doped fiber amplifiers,” IEEE Trans. Microw. Theory Tech. 54(7), 3141–3145 (2006).
    [CrossRef]
  5. V. J. Urick, F. Bucholtz, P. S. Devgan, J. D. McKinney, and K. J. Williams, “Phase Modulation With Interferometric Detection as an Alternative to Intensity Modulation With Direct Detection for Analog-Photonic Links,” IEEE Trans. Microw. Theory Tech. 55(9), 1978–1985 (2007).
    [CrossRef]
  6. J. M. Wyrwas and M. C. Wu“Dynamic Range of Frequency Modulated Direct-Detection Analog Fiber Optic Links,” J. Lightwave Technol. 27(24), 5552–5562 (2009).
    [CrossRef]
  7. T. E. Darcie and P. F. Driessen, “Class-AB techniques for high-dynamic-range microwave-photonic links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006).
    [CrossRef]
  8. J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microw. Theory Tech. 57(8), 2093–2099 (2009).
    [CrossRef]
  9. V. J. Urick, M. Godinez, P. S. Devgan, J. D. McKinney, and F. Bucholtz, “Analysis of an analog fiber-optic link employing a low-biased mach-zehnder modulator followed by an erbium-doped fiber amplifier,” J. Lightwave Technol. 27(12), 2013–2019 (2009).
    [CrossRef]
  10. T. E. Darcie, J. Zhang, P. F. Driessen, and J.-J. Eun, “Class-B Microwave-Photonic Link Using Optical Frequency Modulation and Linear Frequency Discriminators,” J. Lightwave Technol. 25(1), 157–164 (2007).
    [CrossRef]
  11. H. Chi, X. Zou, and J. Yao, “Analytical Models for Phase-Modulation-Based Microwave Photonic Systems With Phase Modulation to Intensity Modulation Conversion Using a Dispersive Device,” J. Lightwave Technol. 27(5), 511–521 (2009).
    [CrossRef]
  12. D. Marpaung, C. Roeloffzen, A. Leinse, and M. Hoekman, “A photonic chip based frequency discriminator for a high performance microwave photonic link,” Opt. Express 18(26), 27359–27370 (2010).
    [CrossRef] [PubMed]
  13. B. Moslehi, “Analysis of optical phase noise in fiber-optic systems employing a laser source with arbitrary coherence time,” J. Lightwave Technol. 4(9), 1334–1351 (1986).
    [CrossRef]
  14. D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19(10), 1653–1660 (1980).
    [CrossRef] [PubMed]
  15. F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
    [CrossRef]
  16. G. Yabre, “Comprehensive Theory of Dispersion in Graded-Index Optical Fibers,” J. Lightwave Technol. 18(2), 166–177 (2000).
    [CrossRef]
  17. I. Gasulla and J. Capmany, “Transfer function of multimode fiber links using an electric field propagation model: Application to Radio over Fibre Systems,” Opt. Express 14(20), 9051–9070 (2006).
    [CrossRef] [PubMed]
  18. I. Gasulla and J. Capmany, “Principal mode coefficients for multimode fibers,” in Proceedings of 34th European Conference on Optical Communication ECOC, (Brussels, Belgium, 2008), pp. 1–2.
  19. I. Gasulla and J. Capmany, “Analysis of the harmonic and intermodulation distortion in a multimode fiber optic link,” Opt. Express 15(15), 9366–9371 (2007).
    [CrossRef] [PubMed]

2010

2009

2007

J. Capmany and D. Novak, “Microwave Photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[CrossRef]

V. J. Urick, F. Bucholtz, P. S. Devgan, J. D. McKinney, and K. J. Williams, “Phase Modulation With Interferometric Detection as an Alternative to Intensity Modulation With Direct Detection for Analog-Photonic Links,” IEEE Trans. Microw. Theory Tech. 55(9), 1978–1985 (2007).
[CrossRef]

T. E. Darcie, J. Zhang, P. F. Driessen, and J.-J. Eun, “Class-B Microwave-Photonic Link Using Optical Frequency Modulation and Linear Frequency Discriminators,” J. Lightwave Technol. 25(1), 157–164 (2007).
[CrossRef]

I. Gasulla and J. Capmany, “Analysis of the harmonic and intermodulation distortion in a multimode fiber optic link,” Opt. Express 15(15), 9366–9371 (2007).
[CrossRef] [PubMed]

2006

I. Gasulla and J. Capmany, “Transfer function of multimode fiber links using an electric field propagation model: Application to Radio over Fibre Systems,” Opt. Express 14(20), 9051–9070 (2006).
[CrossRef] [PubMed]

A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24(12), 4628–4641 (2006).
[CrossRef]

T. E. Darcie and P. F. Driessen, “Class-AB techniques for high-dynamic-range microwave-photonic links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006).
[CrossRef]

V. J. Urick, M. S. Rogge, F. Bucholtz, and K. J. Williams, “The performance of analog photonic links employing highly-compressed erbium-doped fiber amplifiers,” IEEE Trans. Microw. Theory Tech. 54(7), 3141–3145 (2006).
[CrossRef]

2000

1993

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[CrossRef]

1986

B. Moslehi, “Analysis of optical phase noise in fiber-optic systems employing a laser source with arbitrary coherence time,” J. Lightwave Technol. 4(9), 1334–1351 (1986).
[CrossRef]

1980

Bucholtz, F.

V. J. Urick, M. Godinez, P. S. Devgan, J. D. McKinney, and F. Bucholtz, “Analysis of an analog fiber-optic link employing a low-biased mach-zehnder modulator followed by an erbium-doped fiber amplifier,” J. Lightwave Technol. 27(12), 2013–2019 (2009).
[CrossRef]

V. J. Urick, F. Bucholtz, P. S. Devgan, J. D. McKinney, and K. J. Williams, “Phase Modulation With Interferometric Detection as an Alternative to Intensity Modulation With Direct Detection for Analog-Photonic Links,” IEEE Trans. Microw. Theory Tech. 55(9), 1978–1985 (2007).
[CrossRef]

V. J. Urick, M. S. Rogge, F. Bucholtz, and K. J. Williams, “The performance of analog photonic links employing highly-compressed erbium-doped fiber amplifiers,” IEEE Trans. Microw. Theory Tech. 54(7), 3141–3145 (2006).
[CrossRef]

Capmany, J.

Chi, H.

Darcie, T. E.

T. E. Darcie, J. Zhang, P. F. Driessen, and J.-J. Eun, “Class-B Microwave-Photonic Link Using Optical Frequency Modulation and Linear Frequency Discriminators,” J. Lightwave Technol. 25(1), 157–164 (2007).
[CrossRef]

T. E. Darcie and P. F. Driessen, “Class-AB techniques for high-dynamic-range microwave-photonic links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006).
[CrossRef]

Devaux, F.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[CrossRef]

Devgan, P. S.

V. J. Urick, M. Godinez, P. S. Devgan, J. D. McKinney, and F. Bucholtz, “Analysis of an analog fiber-optic link employing a low-biased mach-zehnder modulator followed by an erbium-doped fiber amplifier,” J. Lightwave Technol. 27(12), 2013–2019 (2009).
[CrossRef]

V. J. Urick, F. Bucholtz, P. S. Devgan, J. D. McKinney, and K. J. Williams, “Phase Modulation With Interferometric Detection as an Alternative to Intensity Modulation With Direct Detection for Analog-Photonic Links,” IEEE Trans. Microw. Theory Tech. 55(9), 1978–1985 (2007).
[CrossRef]

Driessen, P. F.

T. E. Darcie, J. Zhang, P. F. Driessen, and J.-J. Eun, “Class-B Microwave-Photonic Link Using Optical Frequency Modulation and Linear Frequency Discriminators,” J. Lightwave Technol. 25(1), 157–164 (2007).
[CrossRef]

T. E. Darcie and P. F. Driessen, “Class-AB techniques for high-dynamic-range microwave-photonic links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006).
[CrossRef]

Eun, J.-J.

Gasulla, I.

Godinez, M.

Hoekman, M.

Kerdiles, J. F.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[CrossRef]

Leinse, A.

Marcuse, D.

Marpaung, D.

McKinney, J. D.

V. J. Urick, M. Godinez, P. S. Devgan, J. D. McKinney, and F. Bucholtz, “Analysis of an analog fiber-optic link employing a low-biased mach-zehnder modulator followed by an erbium-doped fiber amplifier,” J. Lightwave Technol. 27(12), 2013–2019 (2009).
[CrossRef]

J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microw. Theory Tech. 57(8), 2093–2099 (2009).
[CrossRef]

V. J. Urick, F. Bucholtz, P. S. Devgan, J. D. McKinney, and K. J. Williams, “Phase Modulation With Interferometric Detection as an Alternative to Intensity Modulation With Direct Detection for Analog-Photonic Links,” IEEE Trans. Microw. Theory Tech. 55(9), 1978–1985 (2007).
[CrossRef]

Moslehi, B.

B. Moslehi, “Analysis of optical phase noise in fiber-optic systems employing a laser source with arbitrary coherence time,” J. Lightwave Technol. 4(9), 1334–1351 (1986).
[CrossRef]

Novak, D.

J. Capmany and D. Novak, “Microwave Photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[CrossRef]

Roeloffzen, C.

Rogge, M. S.

V. J. Urick, M. S. Rogge, F. Bucholtz, and K. J. Williams, “The performance of analog photonic links employing highly-compressed erbium-doped fiber amplifiers,” IEEE Trans. Microw. Theory Tech. 54(7), 3141–3145 (2006).
[CrossRef]

Seeds, A. J.

Sorel, Y.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[CrossRef]

Urick, V. J.

V. J. Urick, M. Godinez, P. S. Devgan, J. D. McKinney, and F. Bucholtz, “Analysis of an analog fiber-optic link employing a low-biased mach-zehnder modulator followed by an erbium-doped fiber amplifier,” J. Lightwave Technol. 27(12), 2013–2019 (2009).
[CrossRef]

V. J. Urick, F. Bucholtz, P. S. Devgan, J. D. McKinney, and K. J. Williams, “Phase Modulation With Interferometric Detection as an Alternative to Intensity Modulation With Direct Detection for Analog-Photonic Links,” IEEE Trans. Microw. Theory Tech. 55(9), 1978–1985 (2007).
[CrossRef]

V. J. Urick, M. S. Rogge, F. Bucholtz, and K. J. Williams, “The performance of analog photonic links employing highly-compressed erbium-doped fiber amplifiers,” IEEE Trans. Microw. Theory Tech. 54(7), 3141–3145 (2006).
[CrossRef]

Williams, K. J.

J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microw. Theory Tech. 57(8), 2093–2099 (2009).
[CrossRef]

V. J. Urick, F. Bucholtz, P. S. Devgan, J. D. McKinney, and K. J. Williams, “Phase Modulation With Interferometric Detection as an Alternative to Intensity Modulation With Direct Detection for Analog-Photonic Links,” IEEE Trans. Microw. Theory Tech. 55(9), 1978–1985 (2007).
[CrossRef]

A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24(12), 4628–4641 (2006).
[CrossRef]

V. J. Urick, M. S. Rogge, F. Bucholtz, and K. J. Williams, “The performance of analog photonic links employing highly-compressed erbium-doped fiber amplifiers,” IEEE Trans. Microw. Theory Tech. 54(7), 3141–3145 (2006).
[CrossRef]

Wu, M. C.

Wyrwas, J. M.

Yabre, G.

Yao, J.

Zhang, J.

Zou, X.

Appl. Opt.

IEEE Photon. Technol. Lett.

T. E. Darcie and P. F. Driessen, “Class-AB techniques for high-dynamic-range microwave-photonic links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

J. D. McKinney and K. J. Williams, “Sampled analog optical links,” IEEE Trans. Microw. Theory Tech. 57(8), 2093–2099 (2009).
[CrossRef]

V. J. Urick, M. S. Rogge, F. Bucholtz, and K. J. Williams, “The performance of analog photonic links employing highly-compressed erbium-doped fiber amplifiers,” IEEE Trans. Microw. Theory Tech. 54(7), 3141–3145 (2006).
[CrossRef]

V. J. Urick, F. Bucholtz, P. S. Devgan, J. D. McKinney, and K. J. Williams, “Phase Modulation With Interferometric Detection as an Alternative to Intensity Modulation With Direct Detection for Analog-Photonic Links,” IEEE Trans. Microw. Theory Tech. 55(9), 1978–1985 (2007).
[CrossRef]

J. Lightwave Technol.

B. Moslehi, “Analysis of optical phase noise in fiber-optic systems employing a laser source with arbitrary coherence time,” J. Lightwave Technol. 4(9), 1334–1351 (1986).
[CrossRef]

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[CrossRef]

G. Yabre, “Comprehensive Theory of Dispersion in Graded-Index Optical Fibers,” J. Lightwave Technol. 18(2), 166–177 (2000).
[CrossRef]

A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24(12), 4628–4641 (2006).
[CrossRef]

T. E. Darcie, J. Zhang, P. F. Driessen, and J.-J. Eun, “Class-B Microwave-Photonic Link Using Optical Frequency Modulation and Linear Frequency Discriminators,” J. Lightwave Technol. 25(1), 157–164 (2007).
[CrossRef]

H. Chi, X. Zou, and J. Yao, “Analytical Models for Phase-Modulation-Based Microwave Photonic Systems With Phase Modulation to Intensity Modulation Conversion Using a Dispersive Device,” J. Lightwave Technol. 27(5), 511–521 (2009).
[CrossRef]

V. J. Urick, M. Godinez, P. S. Devgan, J. D. McKinney, and F. Bucholtz, “Analysis of an analog fiber-optic link employing a low-biased mach-zehnder modulator followed by an erbium-doped fiber amplifier,” J. Lightwave Technol. 27(12), 2013–2019 (2009).
[CrossRef]

J. M. Wyrwas and M. C. Wu“Dynamic Range of Frequency Modulated Direct-Detection Analog Fiber Optic Links,” J. Lightwave Technol. 27(24), 5552–5562 (2009).
[CrossRef]

Nat. Photonics

J. Capmany and D. Novak, “Microwave Photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[CrossRef]

Opt. Express

Other

C. H. Cox III, Analog Photonic Links: Theory and Practice (Cambridge University Press, Cambridge, U.K., 2004).

I. Gasulla and J. Capmany, “Principal mode coefficients for multimode fibers,” in Proceedings of 34th European Conference on Optical Communication ECOC, (Brussels, Belgium, 2008), pp. 1–2.

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Figures (9)

Fig. 1
Fig. 1

Schematic of a general single-port filtered MWP link.

Fig. 2
Fig. 2

Schematic of a filtered MWP link applying phase modulation with a general balanced detection scheme.

Fig. 3
Fig. 3

Multiport filtered MWP link implementing WDM distribution.

Fig. 4
Fig. 4

Phase modulation with interferometric detection scheme.

Fig. 5
Fig. 5

Schematic of a filtered MWP link composed of a dispersive fiber link and a FBG-based notch filter.

Fig. 6
Fig. 6

Measured FBG transmission magnitude and phase shift frequency responses.

Fig. 7
Fig. 7

GRF response in a dispersive link for intensity (IM) and phase (ΦM) modulation with direct detection. (a) 5 km SMF. (b) 20 km SMF. (c) 5 km MMF and (d) 20 km MMF with central launching. (e) 5 km MMF and (f) 20 km MMF with uniform launching.

Fig. 8
Fig. 8

GRF response in a link comprising a dispersive fiber and a FBG notch filter comparing different sideband suppression levels (FBGA and FBGB), for intensity (IM) and phase (ΦM) modulation with direct detection. (a) 20 km SMF. (b) 20 km MMF with central launching.

Fig. 9
Fig. 9

RF photodetected power for the signal, IMD 2 and IMD 3 terms and output noise level as function of the input RF power for a link comprising a dispersive 20 km SMF and a FBG notch filter (maximum attenuation: FBGA) when employing a low-linewidth laser, Δf = 10 MHz.

Equations (58)

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E o u t | M Z M ( t ) = j α M Z M E i n ( t ) sin [ ϕ d c / 2 + ϕ r f / 2 sin ( Ω t ) ]
E o u t | M Z M ( ω ) = 2 π α M Z M n = B n J n ( ϕ r f / 2 ) E i n ( ω n Ω )
B n = ( 1 ) n j | n | + 1 sin ( ϕ d c / 2 + | n | π / 2 ) ,
E o u t | M Z M ( ω ) E o u t * | M Z M ( ω ' ) = ( 2 π ) 2 α M Z M 2 P 0 n = m = B n B m * J n ( ϕ r f / 2 ) J m ( ϕ r f / 2 ) P s ( ω ω 0 + n Ω ) δ [ ω ' ( ω + ( n m ) Ω ) ]
P s ( ω ) = R ( u ) e i ω u d u
P ( t ) = 1 ( 2 π ) 2 E o u t ( ω ) E o u t * ( ω ' ) e j ( ω ω ' ) t d ω d ω ' = α M Z M 2 P 0 n = m = B n B m * J n ( ϕ r f / 2 ) J m ( ϕ r f / 2 ) e j ( n m ) Ω t P s ( ω ω 0 + n Ω ) H ( ω ) H * [ ω + ( n m ) Ω ] d ω
P R F I | o u t ( Ω ) = 2 I d c 2 R o u t ( ϕ r f / 4 ) 2 sin 2 ( ϕ d c ) | A Ω I | 2
A Ω I = n   =   {   0 , 1   } P s ( ω ω 0 n Ω ) H ( ω ) H * ( ω Ω ) d ω .
G R F I ( Ω ) = ( I d c π / V π ) 2 sin 2 ( ϕ d c ) R i n R o u t | A Ω I / 2 | 2 .
N F I = 4 R I N t o t V π 2 π 2 R i n k B T [ 1 cos ( ϕ d c ) ] 2 sin 2 ( ϕ d c ) | A D C I | 2 | A Ω I | 2 ,
R I N i , t h I = k B T π 2 R i n 4 V π 2 sin 2 ( ϕ d c ) [ 1 cos ( ϕ d c ) ] 2 | A Ω I | 2 | A D C I | 2 ,
R I N o , t h I = k B T I d c 2 R o u t 1 [ 1 cos ( ϕ d c ) ] 2 | A D C I | 2    and
R I N s h o t I = 2 e I d c 1 [ 1 cos ( ϕ d c ) ] | A D C I | .
A D C I = P s ( ω ω 0 ) | H ( ω ) | 2 d ω .
P R F I | o u t ( Ω 1 ± Ω 2 ) = 2 I d c 2 R o u t ( ϕ r f / 4 ) 4 | A Ω 1 ± Ω 2 I | 2
A Ω 1 ± Ω 2 I = ± n   =   {   0 , 1   }   m   =   {   1 , 0   } [ ( 1 ) n + m cos ( ϕ d c ) ] P s ( ω ω 0 n Ω 1 ± m Ω 2 ) H ( ω ) H * ( ω Ω 1 Ω 2 ) d ω .
S F D R 2 I = 2 R I N t o t sin 2 ( ϕ d c ) | 1 cos ( ϕ d c ) | | A Ω 1 I | 2 | A Ω 1 ± Ω 2 I | | A D C I | .
P R F I | o u t ( 2 Ω 1 Ω 2 ) = 2 I d c 2 R o u t ( ϕ r f / 4 ) 6 sin 2 ( ϕ d c ) | A 2 Ω 1 Ω 2 I | 2
A 2 Ω 1 Ω 2 I = n   =   {   0 , 1 , 2   }   m   =   {   1 , 0   } ( 1 / 2 ) | n 1 | P s ( ω ω 0 n Ω 1 m Ω 2 ) H ( ω ) H * ( ω 2 Ω 1 + Ω 2 ) d ω
S F D R 3 I = [ 2 R I N t o t sin 2 ( ϕ d c ) | 1 cos ( ϕ d c ) | | A Ω 1 I | 3 | A 2 Ω 1 Ω 2 I | | A D C I | 2 ] 2 / 3 .
A Ω I | δ = n   =   {   0 , 1   } H ( ω 0 + n Ω ) H * [ ω 0 + ( n 1 ) Ω ] ,
A D C I | δ = | H ( ω 0 ) | 2 ,
A Ω 1 ± Ω 2 I | δ = ± n   =   {   0 , 1   }   m   =   {   1 , 0   } [ ( 1 ) n + m cos ( ϕ d c ) ] H ( ω 0 + n Ω 1 m Ω 2 ) H * [ ω 0 + ( n 1 ) Ω 1 ( m + 1 ) Ω 2 ]
A 2 Ω 1 Ω 2 I | δ = n = {   0 , 1 , 2   }   m = {   1 , 0   } ( 1 / 2 ) | n 1 | H ( ω 0 + n Ω 1 + m Ω 2 ) H * [ ω 0 + ( n 2 ) Ω 1 + ( m + 1 ) Ω 2 ] .
E o u t | Φ M ( t ) = j α Φ M E i n ( t ) e j ϕ r f sin ( Ω t )
P R F Φ | o u t ( Ω ) = 2 I d c 2 R o u t ϕ r f 2 | A Ω Φ | 2
A Ω Φ = n   =   {   0 , 1   } ( 1 ) n P s ( ω ω 0 n Ω ) [ H 11 ( ω ) H 11 * ( ω Ω ) H 21 ( ω ) H 21 * ( ω Ω ) ] d ω
G R F Φ ( Ω ) = 4 ( I d c π / V π ) 2 R i n R o u t | A Ω Φ | 2 .
A D C Φ = P s ( ω ω 0 ) [ | H 11 ( ω ) | 2 + | H 21 ( ω ) | 2 ] d ω
N F Φ = R I N t o t V π 2 π 2 R i n k B T | A D C Φ | 2 | A Ω Φ | 2 ,
R I N o , t h Φ = k B T 4 I d c 2 R o u t 1 | A D C Φ | 2 ,
R I N i , t h Φ = k B T π 2 R i n V π 2 | A Ω Φ | 2 | A D C Φ | 2    and
R I N s h o t Φ = e I d c 1 | A D C Φ | .
P R F Φ | o u t ( Ω 1 ± Ω 2 ) = I d c 2 R o u t 2 ϕ r f 4 | A Ω 1 ± Ω 2 Φ | 2
A Ω 1 ± Ω 2 Φ = ± n = { 0 , 1 }   m = { 1 , 0 } ( 1 ) n + m P s ( ω ω 0 n Ω 1 ± m Ω 2 ) [ H 11 ( ω ) H 11 * ( ω Ω 1 Ω 2 ) H 21 ( ω ) H 21 * ( ω Ω 1 Ω 2 ) ] d ω
S F D R 2 Φ = 2 R I N t o t | A Ω 1 Φ | 2 | A Ω 1 ± Ω 2 Φ | | A D C Φ | .
P R F Φ | o u t ( 2 Ω 1 Ω 2 ) = I d c 2 R o u t 8 ϕ r f 6 | A 2 Ω 1 Ω 2 Φ | 2
A 2 Ω 1 Ω 2 Φ = n  =  {   0 , 1 , 2   } m  =  {  -1,0  } ( 1 / 2 ) | n 1 | ( 1 ) m P s ( ω ω 0 n Ω 1 m Ω 2 ) [ H 11 ( ω ) H 11 * ( ω 2 Ω 1 + Ω 2 ) H 21 ( ω ) H 21 * ( ω 2 Ω 1 + Ω 2 ) ] d ω
S F D R 3 Φ = [ 4 R I N t o t | A Ω 1 Φ | 3 | A 2 Ω 1 Ω 2 Φ | | A D C Φ | 2 ] 2 / 3 .
A Ω Φ | δ = n   =   {   0 , 1   } ( 1 ) n [ H 11 ( ω 0 + n Ω ) H 11 * ( ω 0 + ( n 1 ) Ω ) H 21 ( ω 0 + n Ω ) H 21 * ( ω 0 + ( n 1 ) Ω ) ] ,
A D C Φ | δ = | H 11 ( ω 0 ) | 2 + | H 21 ( ω 0 ) | 2 ,
A Ω 1 ± Ω 2 Φ | δ = ± n = {   0 , 1   }   m = {   1 , 0   } ( 1 ) n + m [ H 11 ( ω 0 + n Ω 1 m Ω 2 ) H 11 * ( ω 0 + ( n 1 ) Ω 1 ( m + 1 ) Ω 2 ) H 21 ( ω 0 + n Ω 1 m Ω 2 ) H 21 * ( ω 0 + ( n 1 ) Ω 1 ( m + 1 ) Ω 2 ) ]
A 2 Ω 1 Ω 2 Φ | δ = n  =  {   0 , 1 , 2   } m  =  {  -1,0  } ( 1 / 2 ) | n 1 | ( 1 ) m { H 11 ( ω 0 + n Ω 1 + m Ω 2 ) [ H 11 * ω 0 + ( n 2 ) Ω 1 + ( m + 1 ) Ω 2 ] H 21 ( ω 0 + n Ω 1 + m Ω 2 ) H 21 * [ ω 0 + ( n 2 ) Ω 1 + ( m + 1 ) Ω 2 ] } .
H 11 ( ω ) | M Z I = j α M Z I e j ω τ / 2 sin ( ω τ / 2 )     and     H 21 ( ω ) | M Z I = j α M Z I e j ω τ / 2 cos ( ω τ / 2 ) ,
G R F Φ ( Ω ) | M Z I = 16 ( I d c α M Z I π / V π ) 2 R i n R o u t sin 2 ( Ω τ / 2 ) ,
N F Φ | M Z I = R I N t o t V π 2 4 π 2 R i n k B T sin 2 ( Ω τ / 2 ) ,
R I N o , t h Φ | M Z I = k B T 4 I d c 2 α M Z I 2 R o u t ,     R I N i , t h Φ | M Z I = 4 k B T π 2 R i n V π 2    and    R I N s h o t Φ | M Z I = e I d c α M Z I .
S F D R 3 Φ | M Z I = [ 4 R I N t o t 1 cos ( Ω 1 τ ) | cos [ ( Ω 1 Ω 2 ) τ / 2 ] cos [ ( Ω 1 + Ω 2 ) τ / 2 ] | ] 2 / 3 .
β ( ω ) β ( ω 0 ) + d β ( ω ) d ω | ω = ω 0 ( ω ω 0 ) + 1 2 d β 2 ( ω ) d ω 2 | ω = ω 0 ( ω ω 0 ) 2 = β 0 + β ' ( ω ω 0 ) + 1 2 β ' ' ( ω ω 0 ) 2 .
P s ( ω ) = P 0 π Δ W e ω 2 ( 2 Δ W ) 2 .
H | S M F ( ω ) = e α 0 L / 2 e j [ β 0 + β ' ( ω ω 0 ) + β ' ' / 2 ( ω ω 0 ) 2 ] L ,
G R F I ( Ω ) | S M F = ( I d c π / V π ) 2 sin 2 ( ϕ d c ) R i n R o u t e 2 ( β ' '   L   Ω   Δ W ) 2 cos 2 ( β ' ' Ω 2 L / 2 ) e 2 α 0 L    and
G R F Φ ( Ω ) | S M F = ( 4 I d c π / V π ) 2 R i n R o u t e 2 ( β ' ' L   Ω   Δ W ) 2 sin 2 ( β ' ' Ω 2 L / 2 ) e 2 α 0 L .
H | M M F ( ω ) = e j β 0 ' ' 2 ( ω ω 0 ) 2 L m = 1 M c m ( L )   ε m b   | b m ( L ) e α m 0 L / 2 e j β m 0 L e j τ m ( ω ω 0 ) ,
A Ω I | M M F = 2 e ( β 0 ' ' L Ω Δ W ) 2 cos ( β 0 ' ' Ω 2 L / 2 ) m = 1 M 2 m ( C m m + G m m )   e α m 0 L e j τ m Ω     and
A Ω Φ | M M F = 2 j e ( β 0 ' ' L Ω Δ W ) 2 sin ( β 0 ' ' Ω 2 L / 2 ) m = 1 M 2 m ( C m m + G m m )   e α m 0 L e j τ m Ω
| c m | 2 | ε m b | 2 = 2 m ( C m m + G m m ) ,
H 11 ( ω 0 + Ω ) | l i n = A ( ω 0 + Ω ) e j ( ω 0 + Ω ) τ    and    H 21 ( ω 0 + Ω ) | l i n = A ( ω 0 Ω ) e j ( ω 0 + Ω ) τ .

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