Abstract

The authors describe a general method to extract the frequency chirping of Mach-Zehnder modulators based on direct measurements of the output spectrum. This method is independent of the modulator extinction ratio and allows measurement of the intrinsic chirp parameter to an accuracy of 5%.

©2004 Optical Society of America

1. Introduction

Chromatic dispersion associated with large frequency chirping of transmitters is one of the limiting factors of high speed and long haul telecommunication systems. Therefore, external Mach-Zehnder modulators, which can provide a low or adjustable chirp parameter, are a key component of high capacity optical telecommunications systems. Knowledge of the chirp parameter α of Mach-Zehnder modulators is thus essential to understand the propagation of short pulses in fibres. If the RF electric fields applied to the optical channels of the Mach-Zehnder structure are symmetric, a zero-chirp modulation can be obtain. This is the case for X-cut LiNbO3 modulators. However, many commercial modulators, such as Z-cut LiNbO3 ones, exhibit an assymetric structure in order to reduce the half-wave voltage. The electric fields of such devices are not equal in both arms of the Mach-Zehnder interferometer, which causes a residual chirp. Chirp may also appear when the optical branches are unbalanced, i.e., when the extinction ratio is finite [1]. Consequently, there is thus much interest in developing convenient experimental techniques for the measurement of chirp, and these techniques should allow for a finite extinction ratio. Some direct spectral-based methods have been recently reported [2,3,4]. A significant limitation of these techniques, however, is the fact that they are restricted to modulators with large extinction ratios, typically exceeding 25dB. In this paper, we describe a modified spectral technique, which allows for extinction-ratio independent measurements of the modulator chirp parameter. We apply it to characterising the chirp of both X-cut and Z-cut LiNbO3 modulators, obtaining results over the 1–9 GHz frequency range with an accuracy of 5%.

2. Experimental setup and theory

The experimental setup for measuring the α chirp parameter of Mach-Zehnder modulators is shown in Fig. 1 Polarized 1.55µm light is launched into the external modulator. An adjustable DC bias imposes a phase shift ΔφDC between the two optical branches of the Mach-Zehnder structure. A high frequency sinusoidal voltage: V(t)=V0 cos(Ωt) is applied to the transmission lines. V0 is the amplitude of the applied voltage, and Ω is the modulation frequency. The spectral density of the output optical field is measured with a scanning Fabry Perot. The output optical field can be written [5]:

E¯(t)=E0×exp(jωot)×[exp(jA1×cos(Ωt))+γ×exp(jA2×cos(Ωt)+jΔφDC)]

where ω0 and E0 are respectively the frequency and amplitude of the optical wave, and the scaling factor γ (0≤γ≤1) accounts for a finite extinction ratio. A1 and A2 denote the magnitude of the optical phase induced in each optical path.

 

Fig. 1. Schematic Experimental Setup.

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The chirp parameter of an external modulator is defined as [6] :

α=dφdt12I(t)×dIdt

where φ(t) and I(t) are the instantaneous phase and intensity of the output optical wave. They can be derived from Eq. (1) as :

φ(t)=tan1(sin(A1×cos(Ωt))+γ×sin(A2×cos(Ωt)+ΔφDC)cos(A1×cos(Ωt))+γ×cos(A2×cos(Ωt)+ΔφDC))

and

I(t)=E02(1+γ2+2×γ×cos((A1A2)×cos(Ωt)ΔφDC))

In the small signal regime and with quadrature bias (i.e., ΔφDC=π/2), Eqs. (2), (3) and (4) can be simplified to obtain :

απ2=A1+γ2×A2γ×(A1A2)

This equation is more general than the one usually used for analysis [2,3,4], because it can account for a finite extinction ratio if the scaling factor γ is different from 1. It should be noted that A1 and A2 are directly proportional to the Γ1 and Γ2 overlap coefficient between optical and electrical field in each arm of the Mach-Zehnder modulator. So απ/2 is an intrinsic parameter of the modulator. It stands for the chirp parameter induced at the output of the modulator in case of a linear modulation.

The proposed method consists in evaluating A1, A2, and γ from the spectral density. The intrinsic chirp parameter απ/2 is then determined with Eq. (5). We can infer from Eq. (1) that the spectral density of the optical field may enable the determination of A1, A2 and γ provided that the bias is correctly chosen. In fact, if the bias is adjusted to zero, the output spectrum at small signal regime can be written as :

I0(ω)=E02×(1+γ)2×δ(ωω0)+E024×(A1+γA2)2×δ[ω(ω0±Ω)]

If the experimental conditions remain the same, except for a bias adjusted to ΔφDC=π, the output spectrum becomes:

Iπ(ω)=E02×(1γ)2×δ(ωω0)+E024×(A1γA2)2×δ[ω(ω0±Ω)]

To determine the intrinsic chirp parameter απ/2, we first set the 0-bias by monitoring the power supply so that the central peak of density spectrum is maximal. The measurements of the central peak I00) and the first sideband magnitude I00+Ω) are then followed with the same measurements at π-bias, which is characterised by the minimum of the central peak. The amplitude of the central peak and the first sideband are now noted as Iπ0) and Iπ0+Ω). γ, A1, A2, and απ/2 parameters can then be determined from the ratio of the central peaks :

γ=I012(ω0)Iπ12(ω0)Iπ12(ω0)+I012(ω0)
A1=2×I012(ω0+Ω)+Iπ12(ω0+Ω)Iπ12(ω0)+I012(ω0)
A2=2×I012(ω0+Ω)Iπ12(ω0+Ω)Iπ12(ω0)I012(ω0)
απ2=I012(ω0)×I0(ω0+Ω)+Iπ12(ω0)·Iπ12(ω0+Ω)I012(ω0)×Iπ12(ω0+Ω)Iπ12(ω0)×I012(ω0+Ω)

An expansion of the density spectrum in term of Bessel functions shows that Eq. (8), and Eq. (11) are accurate at either large or small signal (cf appendix), removing drive-voltage restrictions during measurements.

3. Experimental results

This method was used to characterise the intrinsic chirp of both an X-cut and a Z-cut LiNbO3 Mach-Zehnder modulator. As shown in Fig. 2(a), an X-cut LiNbO3 modulator provides a symmetric structure, which can lead to a zero-chirp parameter when the optical channels are equally balanced (i.e. : γ=1). Conversely, a Z-cut modulator is bound to exhibit a non zero-chirp parameter. This situation is due to the push-pull configuration of the coplanar waveguide (CPW) electrodes (Fig. 2(b)), which introduces an asymmetry between the two arms of the Mach Zehnder modulator, the waveguide aligned under the hot line of the CPW electrode being submitted to a higher electrooptic efficiency than that placed under one of the lateral ground plane electrodes. For the presented Z-cut modulator, the central line width is 10µm, the gap between electrodes is 17µm, and the silica buffer layer is 1.2µm thick. The theoretical prediction evaluated with a finite element method is a απ/2=0.65 in case of an infinite extinction ratio (i.e. : γ=1). The following experimental results will show that a finite extinction ratio (γ<1) introduces a shift in this απ/2 value.

 

Fig. 2. Transversal structure of LiNbO3 modulators.

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The chirp measurements described above were carried out with 9 dBm microwave power (V0=0.89 V) and a 10 GHz free spectral range (FSR) scanning Fabry Perot, permitting measurements over the range 1–9 GHz. For the X-cut and Z-cut modulators respectively, Figs. 3(a) and 3(b) show the 0-bias line spectra for a 3 GHz modulation frequency, whilst the corresponding results for a π-bias are shown in Figs. 3(c) and 3(d). The optical power and wavelength were the same for all spectra (P=0.2mW and λ=1.55µm).

First, the γ coefficients of the two modulators were measured to be 0.89±0.02and 0.85±0.02 respectively. Contrary to γ, A1 and A2 are dependent on frequency, has shown in Fig. 4. Figure 5 shows the experimental results as a function of frequency for the X-cut (squares) and Z-cut (circles) modulators. These are compared with finite difference-based theoretical calculations which are shown as the dashed and dotted lines respectively, showing very good agreement. The validity of the measurements was also checked independently using an alternative technique measuring the frequency-domain small-signal frequency response (with a network analyzer) through 50 km of dispersive fiber [7].

 

Fig. 3. Measured optical spectra.

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Fig. 4. Frequency dependance of A1 and A2 for both modulators. Squares and circles: measured results for the X-cut and the Z-cut modulator respectively.

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Fig. 5. Frequency behavior of the απ/2 chirp parameter for X-cut modulator (experiment: squares, theory: dashed) and a Z-cut modulator (experiment: circles, theory: dotted).

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The mean value of the chirp extracted from our spectral measurements is 0.1 for the X-cut modulator and 0.82 for the Z-cut modulator. The measurement accuracy (limited primarily by noise in the spectral measurements) was estimated at 5%. Importantly, we note that for these modulators, analysis of the spectra assuming an infinite extinction ratio [2, 3, 4] would significantly underestimate the intrinsic chirp parameters, yielding values 0 and 0.65 respectively.

4. Conclusion

Two simple measurements with an optical spectrum analyser readily yield the determination of the chirp parameter of Mach Zehnder modulators, without any restriction on the drive voltage. This spectral technique yields accurate results independent of a finite modulator extinction ratio, and can be conveniently used to measure the chirp parameter’s frequency dependence. Moreover, there is no fundamental restriction on the technique other than the FSR of the Fabry Perot, and applications to the characterisation of 40GHz modulators should be straightforward.

Appendix:

An expansion of Eqs. (3) and (4) in term of Bessel function leads to γ and απ/2 parameters that can be expressed as:

γ=J0(A2)J0(A1)
απ2=J0(A1)×J1(A1)+γ2×J0(A2)×J1(A2)γ×(J0(A2)×J1(A1)J0(A1)×J1(A2))

Now, if we consider the density spectrum, its developpement in term of Bessel functions can be expressed as:

I0(ω)=E02×[J0(A1)+γ×J0(A2)]2×δ(ωω0)+
E024×(J1(A1)+γ×J1(A2))2×δ[ω(ω0±Ω)]

if the DC bias is equal to zero, and as

Iπ(ω)=E02×[J0(A1)γ×J0(A2)]2×δ(ωω0)+
E024×(J1(A1)γ×J1(A2))2×δ[ω(ω0±Ω)]

Taking I00), Iπ0), Iπ0+Ω), Iπ0+Ω) as the magnitudes of the central peak and first order peak for the cases of a 0 bias or π bias, we can show that Eqs. (8) and (11) are equivalent to (3’) and (4’) respectively. This point is essential, because it shows that a Bessel expansion is not necessary to accurately determine γ and απ/2 parameters. Moreover, Eqs. (8) and (9) can lead to an accurate evaluation of the chirp parameter without any restriction on the electrical power: a small signal regime is not absolutely required.

References and Links

1. S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).

2. L.S. Yan, Q. Yu, A.E. Willner, and Y. Shi, “Measurement of the chirp parameter of electro-optic modulators by comparison of the phase between the two sidebands,” Opt. Lett. 28, 1114–1116 (2003). [CrossRef]   [PubMed]  

3. S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15, 682–684 (2003). [CrossRef]  

4. N. Courjal and H. Porte, “Method for measuring frequency chirping in external Mach-Zehnder modulators,” presented at European Conference on Integrated Optics, ECIO’03, Prague, Czek Republic, 2–4 April 2003

5. H. Kim and A.H. Gnauck, “Chirp characteristics of Dual-drive Mach-Zehnder modulator with a finite DC extinction ratio,” IEEE Photonics Technol. Lett. 14, 298–301 (2002). [CrossRef]  

6. F. Koyama and K. Iga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6, 87–93 (1988). [CrossRef]  

7. 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, 1937–1940 (1993). [CrossRef]  

References

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  1. S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).
  2. L.S. Yan, Q. Yu, A.E. Willner, and Y. Shi, “Measurement of the chirp parameter of electro-optic modulators by comparison of the phase between the two sidebands,” Opt. Lett. 28, 1114–1116 (2003).
    [Crossref] [PubMed]
  3. S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15, 682–684 (2003).
    [Crossref]
  4. N. Courjal and H. Porte, “Method for measuring frequency chirping in external Mach-Zehnder modulators,” presented at European Conference on Integrated Optics, ECIO’03, Prague, Czek Republic, 2–4 April 2003
  5. H. Kim and A.H. Gnauck, “Chirp characteristics of Dual-drive Mach-Zehnder modulator with a finite DC extinction ratio,” IEEE Photonics Technol. Lett. 14, 298–301 (2002).
    [Crossref]
  6. F. Koyama and K. Iga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6, 87–93 (1988).
    [Crossref]
  7. 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, 1937–1940 (1993).
    [Crossref]

2003 (2)

L.S. Yan, Q. Yu, A.E. Willner, and Y. Shi, “Measurement of the chirp parameter of electro-optic modulators by comparison of the phase between the two sidebands,” Opt. Lett. 28, 1114–1116 (2003).
[Crossref] [PubMed]

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15, 682–684 (2003).
[Crossref]

2002 (1)

H. Kim and A.H. Gnauck, “Chirp characteristics of Dual-drive Mach-Zehnder modulator with a finite DC extinction ratio,” IEEE Photonics Technol. Lett. 14, 298–301 (2002).
[Crossref]

1999 (1)

S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).

1993 (1)

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, 1937–1940 (1993).
[Crossref]

1988 (1)

F. Koyama and K. Iga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6, 87–93 (1988).
[Crossref]

Courjal, N.

N. Courjal and H. Porte, “Method for measuring frequency chirping in external Mach-Zehnder modulators,” presented at European Conference on Integrated Optics, ECIO’03, Prague, Czek Republic, 2–4 April 2003

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, 1937–1940 (1993).
[Crossref]

Gnauck, A.H.

H. Kim and A.H. Gnauck, “Chirp characteristics of Dual-drive Mach-Zehnder modulator with a finite DC extinction ratio,” IEEE Photonics Technol. Lett. 14, 298–301 (2002).
[Crossref]

Iga, K.

F. Koyama and K. Iga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6, 87–93 (1988).
[Crossref]

Izutsu, M.

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15, 682–684 (2003).
[Crossref]

Jeong, J.

S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).

Kawanishi, T.

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15, 682–684 (2003).
[Crossref]

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, 1937–1940 (1993).
[Crossref]

Kim, H.

H. Kim and A.H. Gnauck, “Chirp characteristics of Dual-drive Mach-Zehnder modulator with a finite DC extinction ratio,” IEEE Photonics Technol. Lett. 14, 298–301 (2002).
[Crossref]

Kim, S.K

S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).

Kim, Y.S.

S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).

Koyama, F.

F. Koyama and K. Iga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6, 87–93 (1988).
[Crossref]

Mizuhara, O.

S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).

Oikawa, S.

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15, 682–684 (2003).
[Crossref]

Park, Y.K.

S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).

Porte, H.

N. Courjal and H. Porte, “Method for measuring frequency chirping in external Mach-Zehnder modulators,” presented at European Conference on Integrated Optics, ECIO’03, Prague, Czek Republic, 2–4 April 2003

Shi, Y.

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, 1937–1940 (1993).
[Crossref]

Tzeng, L.D.

S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).

Willner, A.E.

Yan, L.S.

Yu, Q.

IEEE Photonics Technol. Lett. (2)

S. Oikawa, T. Kawanishi, and M. Izutsu, “Measurement of chirp parameters and halfwave voltages of Mach-Zehnder-type optical modulators by using a small signal operation,” IEEE Photonics Technol. Lett. 15, 682–684 (2003).
[Crossref]

H. Kim and A.H. Gnauck, “Chirp characteristics of Dual-drive Mach-Zehnder modulator with a finite DC extinction ratio,” IEEE Photonics Technol. Lett. 14, 298–301 (2002).
[Crossref]

J. Lightwave Technol. (3)

F. Koyama and K. Iga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6, 87–93 (1988).
[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, 1937–1940 (1993).
[Crossref]

S.K Kim, O. Mizuhara, Y.K. Park, L.D. Tzeng, Y.S. Kim, and J. Jeong, “Theoritical and experimental study of 10 Gb/s transmission performance using 1.55 µm LiNbO3-Based transmitters with adjustable extinction ration and chirp,” J. Lightwave Technol. 7, 1320–1325 (1999).

Opt. Lett. (1)

Other (1)

N. Courjal and H. Porte, “Method for measuring frequency chirping in external Mach-Zehnder modulators,” presented at European Conference on Integrated Optics, ECIO’03, Prague, Czek Republic, 2–4 April 2003

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Figures (5)

Fig. 1.
Fig. 1. Schematic Experimental Setup.
Fig. 2.
Fig. 2. Transversal structure of LiNbO3 modulators.
Fig. 3.
Fig. 3. Measured optical spectra.
Fig. 4.
Fig. 4. Frequency dependance of A1 and A2 for both modulators. Squares and circles: measured results for the X-cut and the Z-cut modulator respectively.
Fig. 5.
Fig. 5. Frequency behavior of the απ/2 chirp parameter for X-cut modulator (experiment: squares, theory: dashed) and a Z-cut modulator (experiment: circles, theory: dotted).

Equations (17)

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E ¯ ( t ) = E 0 × exp ( j ω o t ) × [ exp ( j A 1 × cos ( Ω t ) ) + γ × exp ( j A 2 × cos ( Ω t ) + j Δ φ DC ) ]
α = d φ dt 1 2 I ( t ) × dI dt
φ ( t ) = tan 1 ( sin ( A 1 × cos ( Ω t ) ) + γ × sin ( A 2 × cos ( Ω t ) + Δ φ DC ) cos ( A 1 × cos ( Ω t ) ) + γ × cos ( A 2 × cos ( Ω t ) + Δ φ DC ) )
I ( t ) = E 0 2 ( 1 + γ 2 + 2 × γ × cos ( ( A 1 A 2 ) × cos ( Ω t ) Δ φ DC ) )
α π 2 = A 1 + γ 2 × A 2 γ × ( A 1 A 2 )
I 0 ( ω ) = E 0 2 × ( 1 + γ ) 2 × δ ( ω ω 0 ) + E 0 2 4 × ( A 1 + γ A 2 ) 2 × δ [ ω ( ω 0 ± Ω ) ]
I π ( ω ) = E 0 2 × ( 1 γ ) 2 × δ ( ω ω 0 ) + E 0 2 4 × ( A 1 γ A 2 ) 2 × δ [ ω ( ω 0 ± Ω ) ]
γ = I 0 1 2 ( ω 0 ) I π 1 2 ( ω 0 ) I π 1 2 ( ω 0 ) + I 0 1 2 ( ω 0 )
A 1 = 2 × I 0 1 2 ( ω 0 + Ω ) + I π 1 2 ( ω 0 + Ω ) I π 1 2 ( ω 0 ) + I 0 1 2 ( ω 0 )
A 2 = 2 × I 0 1 2 ( ω 0 + Ω ) I π 1 2 ( ω 0 + Ω ) I π 1 2 ( ω 0 ) I 0 1 2 ( ω 0 )
α π 2 = I 0 1 2 ( ω 0 ) × I 0 ( ω 0 + Ω ) + I π 1 2 ( ω 0 ) · I π 1 2 ( ω 0 + Ω ) I 0 1 2 ( ω 0 ) × I π 1 2 ( ω 0 + Ω ) I π 1 2 ( ω 0 ) × I 0 1 2 ( ω 0 + Ω )
γ = J 0 ( A 2 ) J 0 ( A 1 )
α π 2 = J 0 ( A 1 ) × J 1 ( A 1 ) + γ 2 × J 0 ( A 2 ) × J 1 ( A 2 ) γ × ( J 0 ( A 2 ) × J 1 ( A 1 ) J 0 ( A 1 ) × J 1 ( A 2 ) )
I 0 ( ω ) = E 0 2 × [ J 0 ( A 1 ) + γ × J 0 ( A 2 ) ] 2 × δ ( ω ω 0 ) +
E 0 2 4 × ( J 1 ( A 1 ) + γ × J 1 ( A 2 ) ) 2 × δ [ ω ( ω 0 ± Ω ) ]
I π ( ω ) = E 0 2 × [ J 0 ( A 1 ) γ × J 0 ( A 2 ) ] 2 × δ ( ω ω 0 ) +
E 0 2 4 × ( J 1 ( A 1 ) γ × J 1 ( A 2 ) ) 2 × δ [ ω ( ω 0 ± Ω ) ]

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