Modeling FM lineshapes

Chapter VI: Frequency Modulation Spectroscopy of YbOH

6.3 Modeling FM lineshapes

0.79) I determined that a modulation frequency in the range 0.5 ≲ 𝜔_𝑚/Γ ≲ 0.79 is ideal for our spectroscopic purposes. Specifically, we operate at a modulation frequency of 𝜔_𝑚/Γ ≈ 0.56, indicated by the vertical dashed line in Fig. 6.3 and 6.4. The specific value of 𝜔_𝑚/Γ was determined partially due to the fact that we already had a 2𝜋×50.3 MHz EOM in the laboratory. A 2𝜋×50.3 MHz modulation frequency falls in the optimal range for FM spectroscopy inside our buffer gas cell (the measured in cell linewidth isΓ∼2𝜋×90 MHz).

a function 𝑓(𝑥) is given by

𝐻(𝑓) (𝑦) = 1 𝜋

𝑃

∫ ∞

−∞

𝑓(𝑥) 𝑦−𝑥

𝑑𝑥 . (6.14)

Therefore, given the absorption lineshape,𝛿(𝜔^′) the dispersion line shape is given by the negative Hilbert transform of the absorption lineshape

𝜙(𝜔) =−𝐻(𝛿) (𝜔). (6.15) If we make the substitution 𝑢 = (𝜔^′ − 𝜔_{𝑟 𝑒 𝑠})/(√

2𝜎) where 𝜎 = Γ/2.355 then 𝑑 𝑢 =1/(√

2𝜎)𝑑 𝜔^′,𝛿(𝜔^′)= 𝑓(𝑢) = 𝐴 exp(−𝑢²) and 𝜙(𝜔) = 1

𝜋 𝑃

∫ ∞

−∞

𝑓(𝑢) 𝑢− ^{𝜔−√𝜔𝑟 𝑒𝑠}

2𝜎

𝑑 𝑢

= 1 𝜋

𝑃

∫ ∞

−∞

−𝐴×exp(−𝑢²) 𝑡−𝑢

𝑑 𝑢

=−𝐴 𝐻[exp(−𝑢²)] (𝑡),

(6.16)

where𝑡 = (𝜔−𝜔_{𝑟 𝑒 𝑠}/(

√

2𝜎). The Hilbert transform of 𝑓(𝑢) =exp(−𝑢²) is known and is related to the Dawson function [155, 156]

𝐻[exp(−𝑢²)] (𝑡) = 2

√ 𝜋

𝐹(𝑡), (6.17)

where𝐹(𝑡) is the Dawson function. Therefore, the dispersion lineshape is given by 𝜙(𝜔) =−𝐴

√2 𝜋

𝐹

(𝜔−𝜔_{𝑟 𝑒 𝑠})

√

2(Γ/2.355)

!

. (6.18)

The dispersion lineshape given in Eq. 6.18 is convenient for numerical modeling since the Dawson function is a built in function in several programming languages4. Simulated Gaussian absorption and dispersion lineshapes (Eq. 6.12 and 6.18) are shown in the top left and right panels of Fig. 6.5, respectively.

Ultimately, we want to model a true absorption spectrum which will contain an arbitrary superposition of Gaussian lineshapes

𝛿_{𝑡 𝑜𝑡}(𝜔) =∑︁

𝑖

𝛿_𝑖(𝜔) =∑︁

𝑖

𝐴_𝑖 exp

− (𝜔−𝜔_𝑖)² 2(Γ𝑖/2.355)

, (6.19)

4The Dawson function is a built in special function in python, which is the language used here to model the FM lineshapes.

where 𝑖 denotes the absorption lineshape due to the ith transition, and 𝜔_𝑖 is the resonance frequency of the ith transition. Therefore, the total dispersion lineshape due to the combination of multiple transitions is given by

𝜙_{𝑡 𝑜𝑡}(𝜔)= 1 𝜋

𝑃

∫ ∞

−∞

𝛿_{𝑡 𝑜𝑡}(𝜔^′) 𝜔^′−𝜔

𝑑 𝜔

=∑︁

𝑖

𝐴_𝑖

√2 𝜋

𝐹

(𝜔−𝜔_𝑖)

√

2(Γ𝑖/2.355)

! .

(6.20)

The simulated total Gaussian absorption and dispersion lineshapes for five closely spaced and blended transitions are shown in the top left and right panels of Fig. 6.6, respectively.

With Eq. 6.19 and 6.20, any arbitrary absorption and dispersion line shape can be modeled. Any arbitrary FM lineshape can be modeled by using Eq. 6.19 and 6.20 in Eq. 6.7, 6.8, 6.6, and 6.11. However, Eq. 6.7 and 6.8 account for the attenuation of the carrier due to strong absorption as well as the effects of the sidebands of all orders. While modeling the FM lineshapes using Eq. 6.7 and 6.8 would provide the most accurate models, including the aforementioned effects is excessive and could complicate the fitting by providing too large of an optimization space.

We are specifically utilizing FM spectroscopy to measure weak absorption lines so operating in the weak absorption limit, discussed in Section 6.1, is justified.

Therefore, in our FM linshape model we assume exp[−2𝛿(𝜔)] ≈ 1. Additionally, Eq. 6.9 and 6.10 indicate that at lower modulation depth 𝐴_{𝐹 𝑀}(𝜔)/𝐷_{𝐹 𝑀}(𝜔) ∝ 𝑀 and therefore, operating at higher modulation depths will result in larger FM signals.

Experimentally, we observe this trend with larger signal sizes for larger modulation depths. In order to maximize our FM signal sizes we operate at a modulation depth of 𝑀 = 0.84. The modulation depth is fixed to this value for all modeled FM lineshapes shown in this chapter. At this modulation depth we find that the amplitude of the the first-order sidebands is ≈ 20% of the carrier amplitude and the amplitude of second-order sidebands are ≈ 1% of the carrier amplitude. In order to include the effects of the second order sidebands we model the𝐴_{𝐹 𝑀}(𝜔)and 𝐷_{𝐹 𝑀}(𝜔)lineshapes to second order (the n=0,1 terms in Eq. 6.7 and 6.8). Therefore,

in the small absorption limit the FM absorption lineshape is given by [153]

𝐴^2𝑛𝑑

𝐹 𝑀(𝜔) =𝐽₀(𝑀)𝐽₁(𝑀) [𝛿(𝜔−𝜔_𝑚) −𝛿(𝜔+𝜔_𝑚)]

+𝐽₁(𝑀)𝐽₂(𝑀) [𝛿(𝜔−2𝜔_𝑚) −𝛿(𝜔+2𝜔_𝑚) +𝛿(𝜔−𝜔_𝑚) −𝛿(𝜔+𝜔_𝑚)], (6.21) and the FM dispersion lineshape is given by

𝐷^2𝑛𝑑

𝐹 𝑀(𝜔) =𝐽₀(𝑀)𝐽₁(𝑀) [𝜙(𝜔−𝜔_𝑚) +𝜙(𝜔+𝜔_𝑚) −2𝜙(𝜔)]

+𝐽₁(𝑀)𝐽₂(𝑀) [𝜙(𝜔−2𝜔_𝑚) +𝜙(𝜔+2𝜔_𝑚) −𝜙(𝜔−𝜔_𝑚) −𝜙(𝜔+𝜔_𝑚)]. (6.22) Given a set of transitions (each with transition frequency𝜔_𝑖, widthΓ𝑖, and amplitude 𝐴_𝑖) and a phase angle, 𝜃, the 𝐼_{𝐹 𝑀}(𝜔)and𝑄_{𝐹 𝑀}(𝜔)lineshapes are calculated using Eq. 6.6 and 6.11 respectively, where 𝐴_{𝐹 𝑀}(𝜔) is given by Eq. 6.21, 𝐷_{𝐹 𝑀}(𝜔) is given by Eq. 6.22, 𝛿(𝜔) is given by Eq. 6.19, and 𝜙(𝜔) is given by Eq. 6.20.

The simulated I and Q FM lineshapes at various values of 𝜃 between 𝜃 = 0 and 𝜃 = 𝜋 for a single transition with a Gaussian absorption lineshape are shown in Fig. 6.5. A modulation frequency of𝜔_𝑚/Γ =0.56, the same modulation frequency used in our FM setup, was used when modeling the lineshapes in Fig. 6.5. The simulated I and Q FM lineshapes at various values of 𝜃 between 𝜃 = 0 and 𝜃 = 𝜋 for five closely spaced transitions are shown in Fig. 6.6. Here, the FWHM of all transitions was set to 90 MHz and a modulation frequency of 50.3 MHz was used (𝜔_𝑚/Γ = 0.56). In Fig. 6.6 several of the transitions are blended to illustrate the resulting FM lineshapes when blended lines are present. In all cases, both isolated transitions in Fig. 6.5 and blended transitions in Fig. 6.6, at a phase angle of𝜃 =0 𝐼_{𝐹 𝑀}(𝜔) (𝜃 =0) = 𝐴_{𝐹 𝑀}(𝜔) and𝑄_{𝐹 𝑀}(𝜔) (𝜃 =0) =−𝐷_{𝐹 𝑀}(𝜔). As the phase angle increases the I and Q signals become mixtures of 𝐴_{𝐹 𝑀}(𝜔) and𝐷_{𝐹 𝑀}(𝜔) up to𝜃 = 𝜋/2 where𝐼_{𝐹 𝑀}(𝜔) (𝜃 = 𝜋/2) = 𝐷_{𝐹 𝑀}(𝜔) and𝑄_{𝐹 𝑀}(𝜔) (𝜃 = 𝜋/2) = 𝐴_{𝐹 𝑀}(𝜔). For phase angles between𝜋/2 and𝜋the I and Q signals are again a mixture of𝐴_{𝐹 𝑀}(𝜔) and 𝐷_{𝐹 𝑀}(𝜔). At 𝜃 = 𝜋 𝐼_{𝐹 𝑀}(𝜔) (𝜃 = 𝜋) = −𝐼_{𝐹 𝑀}(𝜔) (𝜃 = 0) = −𝐴_{𝐹 𝑀}(𝜔) and 𝑄_{𝐹 𝑀}(𝜔) (𝜃 = 𝜋) =−𝑄_{𝐹 𝑀}(𝜔) (𝜃 =0) = 𝐷_{𝐹 𝑀}(𝜔). For phase angles of𝜋 < 𝜃 < 2𝜋 𝐼_{𝐹 𝑀}(𝜔) (𝜃) =−𝐼_{𝐹 𝑀}(𝜔) (𝜃^′=𝜃−𝜋) and𝑄_{𝐹 𝑀}(𝜔) (𝜃) =−𝑄_{𝐹 𝑀}(𝜔) (𝜃^′ =𝜃−𝜋). In all cases𝐼_{𝐹 𝑀}(𝜔) (𝜃+2𝜋) =𝐼_{𝐹 𝑀}(𝜔) (𝜃)and𝑄_{𝐹 𝑀}(𝜔) (𝜃+2𝜋) =𝑄_{𝐹 𝑀}(𝜔) (𝜃). Ultimately, we want to perform a simultaneous fit of our modeled I and Q FM linshapes to the measured I and Q FM data. This is accomplished with a non-linear

least squares optimization5which takes the FM data and initial guesses for the phase angle and the parameters of each transition present (the transition frequency, width, and amplitude of each transition) as inputs. The optimization works to minimize the set of residuals provided to it. To accomplish the simultaneous fit the following residual function was used

𝑅(𝜔) =

√︃

[𝐼_{𝑐𝑎𝑙 𝑐}(𝜔) −𝐼_{𝑑𝑎𝑡 𝑎}(𝜔)]²+ [𝑄_{𝑐𝑎𝑙 𝑐}(𝜔) −𝑄_{𝑑𝑎𝑡 𝑎}(𝜔)]², (6.23) where𝐼_{𝑐𝑎𝑙 𝑐}(𝜔)(𝑄_{𝑐𝑎𝑙 𝑐}(𝜔)) is the calculated value of𝐼_{𝐹 𝑀}(𝑄_{𝐹 𝑀}) at the frequency𝜔 and 𝐼_{𝑑𝑎𝑡 𝑎}(𝜔) (𝑄_{𝑑𝑎𝑡 𝑎}(𝜔)) is the measured value of 𝐼_{𝐹 𝑀} (𝑄_{𝐹 𝑀}) at the frequency𝜔. The sum of the squares of the individual I and Q residuals as opposed to just the sum of the I and Q residuals was used to prevent the residual from taking on inaccurately small values due to a cancellation resulting from the I and Q residuals being opposite in sign. In the fit, the phase angle 𝜃 and the lineshape parameters𝜔_𝑖, Γ𝑖, and 𝐴_𝑖 are floated. Any arbitrary number of transitions can be fit by the algorithm. An example of a fit of the model to FM data is shown later in this chapter in Fig. 6.11.

For isolated lines, we find that the linecenters extracted from the fit exactly match our zero crossing measurements and have equivalent or smaller errors.

5The lmfit Python package [157] is used for the non-linear least squares fitting.

-4 -2 0 2 4

( )

-4 -2 0 2 4

( )

-4 -2 0 2 4

-5 0

5 I

FM

( ) = 0

-4 -2 0 2 4

-5 0

5 Q

FM

( ) = 0

-4 -2 0 2 4

-5 0

5 I

FM

( ) = /8

-4 -2 0 2 4

-5 0

5 Q

FM

( ) = /8

-4 -2 0 2 4

-5 0

5 I

FM

( ) = /4

-4 -2 0 2 4

-5 0

5 Q

FM

( ) = /4

-4 -2 0 2 4

-5 0

5 I

FM

( ) = /2

-4 -2 0 2 4

-5 0

5 Q

FM

( ) = /2

-4 -2 0 2 4

-5 0

5 I

FM

( ) = 3 /4

-4 -2 0 2 4

-5 0

5 Q

FM

( ) = 3 /4

-4 -2 0 2 4

-5 0

5 I

FM

( ) =

-4 -2 0 2 4

-5 0

5 Q

FM

( ) =

Frequency ( _res )/

FM signal Amplitude (arb.)

Figure 6.5: Simulated in-phase,𝐼_{𝐹 𝑀}(𝜔), and in-quadrature,𝑄_{𝐹 𝑀}(𝜔), FM signals for an isolated spectral feature with a Gaussian absorption, 𝛿(𝜔), and dispersion, 𝜙(𝜔), lineshape. The in-phase and in-quadrature signals are shown for various phase angles,𝜃, ranging from𝜃 =0 to𝜃 =𝜋. For phase angles> 𝜋 𝐼_{𝐹 𝑀}(𝜔) (𝜃 =𝜋+𝜙)=

−𝐼_{𝐹 𝑀}(𝜔) (𝜃 =𝜙) and𝑄_{𝐹 𝑀}(𝜔) (𝜃 =𝜋+𝜙) =−𝑄_{𝐹 𝑀}(𝜔) (𝜃 =𝜙)for 0 ≤ 𝜙 ≤ 𝜋.

0.00 0.02 0.04 0.06

( )

0.00 0.02 0.04 0.06

( )

0.00 0.02 0.04 0.06 -5 0 5 I

FM

( ) = 0

0.00 0.02 0.04 0.06 -5 0 5 Q

FM

( ) = 0

0.00 0.02 0.04 0.06 -5 0 5 I

FM

( ) = /8

0.00 0.02 0.04 0.06 -5 0 5 Q

FM

( ) = /8

0.00 0.02 0.04 0.06 -5 0 5 I

FM

( ) = /4

0.00 0.02 0.04 0.06 -5 0 5 Q

FM

( ) = /4

0.00 0.02 0.04 0.06 -5 0 5 I

FM

( ) = /2

0.00 0.02 0.04 0.06 -5 0 5 Q

FM

( ) = /2

0.00 0.02 0.04 0.06 -5 0 5 I

FM

( ) = 3 /4

0.00 0.02 0.04 0.06 -5 0 5 Q

FM

( ) = 3 /4

0.00 0.02 0.04 0.06 -5 0 5 I

FM

( ) =

0.00 0.02 0.04 0.06 -5 0 5 Q

FM

( ) =

Frequency (cm ¹ )

FM signal Amplitude (arbitrary)

Figure 6.6: Simulated in-phase,𝐼_{𝐹 𝑀}(𝜔), and in-quadrature,𝑄_{𝐹 𝑀}(𝜔), FM signals for five closely spaced and blended spectral features. All five spectral features were modeled with a Gaussian absorption,𝛿(𝜔), and dispersion, 𝜙(𝜔), lineshape. The in-phase and in-quadrature signals are shown for various phase angles, 𝜃, ranging from𝜃 =0 to𝜃 =𝜋. For phase angles> 𝜋 𝐼_{𝐹 𝑀}(𝜔) (𝜃 =𝜋+𝜙) =−𝐼_{𝐹 𝑀}(𝜔) (𝜃 =𝜙) and 𝑄_{𝐹 𝑀}(𝜔) (𝜃 = 𝜋 + 𝜙) = −𝑄_{𝐹 𝑀}(𝜔) (𝜃 = 𝜙) for 0 ≤ 𝜙 ≤ 𝜋. A modulation frequency of 50.3 MHz and a FWHM for all transitions of 90 MHz were used in the simulations. The absolute value of the frequency of the x-axis is arbitrary.