Laser beam simulation

• Φ_tmij←ik(τ), the total phase of laserikafter propagation to the photodi-ode of the test-mass interferometer isc_ij.

Any quantities derived from the total phase follow the same notation, with braces around the inner symbols carrying the indices. For example,ν_ij,c^o (τ)is the frequency offset of the carrier virtual local beam at the laser source ij(cf.

section5.2.4), while(ν_refij←ij)^o_c(τ)is the same quantity after it has propagated to the reference interferometer photodiode.

Finally, we list the available interferometric readouts in section5.6.

5.2 laser beam simulation

5.2.1 Laser beam model

We want to model the information content of the EM field of a laser beam.

In all generality, the EM field can be modelled by Or alternatively, it could be described in all generality by the antisymmetric EM field tensorFµν.

two three-dimensional vector fields, the electrical field and the magnetic field. For our simulation modelling, however, we will operate under a number of simplifications.

We work in the plane wave approximation, and assume that any effects due to wavefront imperfections can be modeled as equivalent longitudinal pathlength variations. In addition, we neglect effects related to the fields polarization, consider the waves to propagate in a perfect vacuum and only model the scalar electric field amplitude².

This allows us to use a simple model for the electrical field of our laser beam.

At any fixed point inside a spacecraft, its amplitude can be written as

E(τ) =E0(τ)cos(2πΦ(τ)), (5.1)

using the time coordinates associated with a reference frame at rest for this point.

In practice, it is often more useful to use a complex representation of such a signal,

E(τ) =E₀(τ)e^i2πΦ(τ), (5.2)

withias the unit imaginary number.

The physical electrical field amplitude is then given as the real part of the complex signal,

E(τ) =R[E(τ)]. (5.3)

2 We don’t need to model the magnetic field amplitude, as it is also determined by the electrical field amplitude [103].

54 optical simulation

Figure 5.2:Bessel functions of the first kind.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1

Function argumentm BesselfunctionJn(m)

n =0 n =1 n =2 n =3 n =4

5.2.2 Sideband modulation

As discussed in section3.7, the laser beams in LISA will be modulated using a GHz signal derived from the local clock.

This signal is imprinted on the outgoing laser beam using an EOM, which creates a phase modulation. The electric field then reads

E(τ) =E0e^i2πΦc(τ)e^{imcos(2πΦm(τ))}, (5.4) where m is the modulation depth, Φ_c(τ) is the total phase of the carrier, and Φ_m(τ) is the total phase of the modulating signal, both expressed in cycles.

Following [44], the terme^{imcos(2πΦm(τ))} in the resulting electric field can be expanding using the Bessel functions of the first kindJn. This is known as the Jacobi-Anger expansion [100], given as

e^imcos(Θ) =

∑

∞ n=−∞

iⁿJ_n(m)e^inθ. (5.5)

If the modulation depthmis

We expect that m≈^{0.15 [64].}

not too large, we can further expand the Bessel functions to first order in m. As can be seen in fig. 5.2, only the first two orders have significant contributions form1, and we get

J₀(m)≈^{1, J}1(m)≈ ^m₂ ^{and J}n(m)≈⁰ ∀ⁿ>1 . (5.6) The complex field amplitude of the modulated laser then reads

E(τ)≈^E0

e^i2πΦc(τ)+im

2e^{i2π(Φc(τ)+Φm(τ))}+im

2e^{i2π(Φc(τ)−Φm(τ))}

. (5.7) Let us introduce the upper and lower sideband total phases,

Φ_sb⁺(τ) =Φc(τ) +Φm(τ) and Φ_sb−(τ) =Φc(τ)−^Φm(τ), (5.8)

5.2 laser beam simulation 55 such that the modulated laser beam is written as the superposition of three virtual beams,

E(τ)≈ ^E0

e^i2πΦc(τ)+im

2e^{i2πΦsb+(τ)}+im

2e^{i2πΦsb−(τ)}

(5.9a)

≡^Ec(τ) +E^sb+(τ) +E^sb−(τ) (5.9b) For the purpose of our simulation, we will model E^c(τ), E^sb+(τ)and E^sb−(τ) as independent beams. As before, we are only interested in the phase or frequency of these beams.

We can decompose the phases in eq. (5.8) in terms of large phase drifts and small fluctuations, as described in eq. (4.4), to get

Φ_c(τ) =ν0τ+φ^o_c(τ) +φ^e_c(τ), (5.10a) Φ_m(τ) =φ^o_m(τ) +φ_m^e(τ). (5.10b) Notice that since Φ_m(τ) represents the 2.4 GHz electrical signal used for modulation, it contains no term proportional to the central laser frequencyν0. We define the upper and lower sideband phase drifts and fluctuations,

φ^o_sb⁺(τ) =φ^o_c(τ) +φ_m^o(τ) and φ_sb^e+(τ) =φ^e_c(τ) +φ^e_m(τ), (5.11a) φ^o_sb−(τ) =φ^o_c(τ)−φ_m^o(τ) and φ_sb^e−(τ) =φ^e_c(τ)−φ^e_m(τ), (5.11b) such that by plugging eqs. (5.10a) and (5.10b) in eq. (5.8), we recover the usual two-variable decompositions for the upper and lower sideband total phases,

Φ_sb⁺(τ) =ν0τ+φ^o_sb⁺(τ) +φ^e_sb⁺(τ), (5.12a) Φ_sb−(τ) =ν₀τ+φ^o_sb−(τ) +φ^e_sb−(τ). (5.12b)

Equivalently, we can decompose the carrier and modulating signal phases in terms of frequency offsets and fluctuations,

νc(τ) =ν0+ν_c^o(τ) +ν^e_c(τ), (5.13) νm(τ) =ν_m^o(τ) +ν_m^e(τ), (5.14) such that the upper and lower sideband frequencies can be written as

ν_sb⁺(τ) =ν₀+ν_sb^o +(τ) +ν_sb^e+(τ), (5.15) ν_sb−(τ) =ν0+ν_sb^o −(τ) +ν_sb^e−(τ), (5.16) with the upper and lower sideband frequency offsets and fluctuations,

ν_sb^o +(τ) =ν_c^o(τ) +ν_m^o(τ) and ν_sb^e+(τ) =ν_c^e(τ) +ν^e_m(τ), (5.17) ν_sb^o −(τ) =ν_c^o(τ)−^νm^o(τ) and ν_sb^e−(τ) =ν_c^e(τ)−^νem(τ). (5.18)

56 optical simulation

5.2.3 Pseudo-random noise modulation

As discussed in section3.6.4, the laser beams will carry an additional modula-tion with a PRN code used for absolute ranging and timing synchronizamodula-tion.

This step-wise modulation is performed at a relatively high frequency of around 2 MHz, far outside our simulation bandwidth. We therefore do not model the actual phase modulation. As shown in [34], this modulation also causes a small additional noise in our measurement band, at a level below 1 pm/√

Hz in units of displacement, which we neglect.

Instead, as described in section 6.3.2, we model the PRN measurement by directly propagating the time deviations of each spacecraft timer with respect to their TPSs alongside the laser beams. The model used for the onboard timers themselves is described in section6.1.3.

Note that at the moment, we only model the PRN measurement in the ISC interferometers, and completely ignore their presence in the other interferom-eters.

5.2.4 Model for a modulated beam

From sections 5.2.2 and 5.2.3, we model an actual laser beam using three independent virtual beams for the carrier, the upper and the lower sidebands, as well as an independent variable representing the local timer deviations encoded in the PRN modulation.

In addition, eqs. (5.11a) and (5.11b) show that the information content of the upper and lower sidebands are

One difference is that they lie at a different frequencies, and are thus affected differently by Doppler shifts.

almost identical. We make the assumption that they can be combined in such a way that we can treat them as one signal, with a readout noise that is reduced by a factor of√

2. Therefore, we only simulate the upper sideband. For clarity, we drop the sign in all sideband indices and simply usesbwhen we refer to the upper sideband. Ultimately, each laserijis then implemented by propagating at most5variables,

νij(τ)≡(ν_ij,c^o (τ),ν_ij,c^e (τ),ν_ij,sb^o (τ),ν_ij,sb^e (τ),δτˆi(τ)), (5.19) where

• ν_ij,c^o (τ)andν_ij,c^e (τ)are the carrier frequency offsets and fluctuations,

• ν_ij,sb^o (τ)andν_ij,sb^e (τ)are the sideband frequency offsets and fluctuations, and

• δτˆ_i(τ)describes the local timer deviations of the generating spacecraft as described in section6.1.3, in relevant laser beams only.

We can express all these quantities in units of phase,

Φ_ij(τ)≡ (φ_ij,c^o (τ),φ^e_ij,c(τ),φ_ij,sb^o (τ),φ_ij,sb^e (τ),δτˆi(τ)), (5.20) where

Im Dokument Von der Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz Universität Hannover (Seite 79-83)