# Influence of Optical Aberrations on the Accuracy of an Atomic Gravimeter, QAFCA 2025.

Louis Pagot, Leonid Sidorenkov, Sébastien Merlet, Franck Pereira dos Santos

► To cite this version:

Louis Pagot, Leonid Sidorenkov, Sébastien Merlet, Franck Pereira dos Santos. Influence of Optical Aberrations on the Accuracy of an Atomic Gravimeter, QAFCA 2025. Quantiques à Atomes Froids: mesure du Champ de pesanteur A toutes les échelles, May 2025, Paris, France. hal-05477754

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Submitted on 26 Jan 2026

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cnrs

LTE

Observatoire
de Paris

PSL

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S

SORBONNE
UNIVERSITÉ

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Université
de Lille

LABORATOIRE
NATIONAL
DE MÉTROLOGIE
ET D'ESSAIS

QAFCA
22<sup>th</sup> May 2025

L. Pagot, L. Sidorenkov, S. Merlet, F. Pereira Dos Santos

QAFCA 22<sup>th</sup> May 2025
Influence of Optical Aberrations on the
Accuracy of an Atomic Gravimeter

**Overview:**

1. Introduction

2. Theoretical Approach

3. Simulations
Gravimeter ($\frac{\pi}{2} - \pi - \frac{\pi}{2}$ Raman pulses):

<figure><img src="image_5.png" /></figure>
Gravimeter ($\frac{\pi}{2} - \pi - \frac{\pi}{2}$ Raman pulses):

<figure><img src="image_6.png" /></figure>

Hamiltonian of stimulated Raman transition:

$$H = \frac{\hbar}{2} \begin{pmatrix} \delta - \delta_{AC} & \Omega_{eff}^* \\ \Omega_{eff} & -(\delta - \delta_{AC}) \end{pmatrix}$$

Differential Light Shift: $\delta_{AC}$

Detuning: $\delta = (\omega_1 - \omega_2) - (\omega_e - \omega_f)$

Effective Rabi Frequency: $\Omega_{eff} = |\Omega_{eff}|e^{i\phi}$

<figure><img src="image_7.png" /></figure>

$$\delta - \delta_{AC} = 0$$
Gravimeter ($\frac{\pi}{2} - \pi - \frac{\pi}{2}$ Raman pulses):

<figure><img src="image_8.png" /></figure>

Hamiltonian of stimulated Raman transition:

$$H = \frac{\hbar}{2} \begin{pmatrix} \delta - \delta_{AC} & \Omega_{eff}^* \\ \Omega_{eff} & -(\delta - \delta_{AC}) \end{pmatrix}$$

Differential Light Shift: $\delta_{AC}$

Detuning: $\delta = (\omega_1 - \omega_2) - (\omega_e - \omega_f)$

Effective Rabi Frequency: $\Omega_{eff} = |\Omega_{eff}|e^{i\phi}$

<figure><img src="image_9.png" /></figure>

$$\delta - \delta_{AC} = 0$$

Phase: $\phi_i = -k_{eff}z_i$

Position: $z_i = z_0 + v_0t_i - \frac{1}{2}gt_i^2$
Gravimeter ($\frac{\pi}{2} - \pi - \frac{\pi}{2}$ Raman pulses):

<figure><img src="image_10.png" /></figure>

Transition probability: $P = P_0 - \frac{c}{2}\cos(\Delta\Phi)$
$\Delta\Phi = \phi_1 - 2\phi_2 + \phi_3 = k_{\text{eff}}gT^2$

Hamiltonian of stimulated Raman transition:

$$H = \frac{\hbar}{2} \begin{pmatrix} \delta - \delta_{\text{AC}} & \Omega_{\text{eff}}^* \\ \Omega_{\text{eff}} & -(\delta - \delta_{\text{AC}}) \end{pmatrix}$$

Differential Light Shift: $\delta_{\text{AC}}$

Detuning: $\delta = (\omega_1 - \omega_2) - (\omega_e - \omega_f)$

Effective Rabi Frequency: $\Omega_{\text{eff}} = |\Omega_{\text{eff}}|e^{i\phi}$

<figure><img src="image_11.png" /></figure>

$$\delta - \delta_{\text{AC}} = 0$$

Phase: $\phi_i = -k_{\text{eff}}z_i$

Position: $z_i = z_0 + v_0 t_i - \frac{1}{2} g t_i^2$
Gravimeter ($\frac{\pi}{2} - \pi - \frac{\pi}{2}$ Raman pulses):

<figure><img src="image_12.png" /></figure>

Transition probability: $P = P_0 - \frac{C}{2}\cos(\Delta\Phi)$
$\Delta\Phi = \phi_1 - 2\phi_2 + \phi_3 = k_{\text{eff}}gT^2$

Gravity: $g_{\text{meas.}} \equiv \frac{\Delta\Phi}{k_{\text{eff}}T^2} = g$

Hamiltonian of stimulated Raman transition:

$$H = \frac{\hbar}{2} \begin{pmatrix} \delta - \delta_{\text{AC}} & \Omega_{\text{eff}}^* \\ \Omega_{\text{eff}} & -(\delta - \delta_{\text{AC}}) \end{pmatrix}$$

Differential Light Shift: $\delta_{\text{AC}}$

Detuning: $\delta = (\omega_1 - \omega_2) - (\omega_e - \omega_f)$

Effective Rabi Frequency: $\Omega_{\text{eff}} = |\Omega_{\text{eff}}|e^{i\phi}$

<figure><img src="image_13.png" /></figure>

$$\delta - \delta_{\text{AC}} = 0$$

Phase: $\phi_i = -k_{\text{eff}}z_i$

Position: $z_i = z_0 + v_0t_i - \frac{1}{2}gt_i^2$
Gravimeter ($\frac{\pi}{2} - \pi - \frac{\pi}{2}$ Raman pulses):

<figure><img src="image_14.png" /></figure>

Transition probability: $P = P_0 - \frac{c}{2}\cos(\Delta\Phi)$
$\Delta\Phi = \phi_1 - 2\phi_2 + \phi_3 = k_{\text{eff}}gT^2$

Gravity: $g_{\text{meas.}} \equiv \frac{\Delta\Phi}{k_{\text{eff}}T^2} = g$

Hamiltonian of stimulated Raman transition:

$$H = \frac{\hbar}{2} \begin{pmatrix} \delta - \delta_{\text{AC}} & \Omega_{\text{eff}}^* \\ \Omega_{\text{eff}} & -(\delta - \delta_{\text{AC}}) \end{pmatrix}$$

Differential Light Shift: $\delta_{\text{AC}}$

Detuning: $\delta = (\omega_1 - \omega_2) - (\omega_e - \omega_f)$

Effective Rabi Frequency: $\Omega_{\text{eff}} = |\Omega_{\text{eff}}|e^{i\phi}$

<figure><img src="image_15.png" /></figure>

$$\delta - \delta_{\text{AC}} = 0$$

Phase: $\phi_i = -k_{\text{eff}}z_i + \delta\phi_i$

Position: $z_i = z_0 + v_0t_i - \frac{1}{2}gt_i^2$
Gravimeter ($\frac{\pi}{2} - \pi - \frac{\pi}{2}$ Raman pulses):

<figure><img src="image_16.png" /></figure>

Transition probability: $P = P_0 - \frac{C}{2}\cos(\Delta\Phi)$
$\Delta\Phi = \phi_1 - 2\phi_2 + \phi_3 = k_{\text{eff}}gT^2 + \Delta\varphi$

Gravity: $g_{\text{meas.}} \equiv \frac{\Delta\Phi}{k_{\text{eff}}T^2} = g + \frac{\Delta\varphi}{k_{\text{eff}}T^2}$

Hamiltonian of stimulated Raman transition:

$$H = \frac{\hbar}{2} \begin{pmatrix} \delta - \delta_{\text{AC}} & \Omega_{\text{eff}}^* \\ \Omega_{\text{eff}} & -(\delta - \delta_{\text{AC}}) \end{pmatrix}$$

Differential Light Shift: $\delta_{\text{AC}}$

Detuning: $\delta = (\omega_1 - \omega_2) - (\omega_e - \omega_f)$

Effective Rabi Frequency: $\Omega_{\text{eff}} = |\Omega_{\text{eff}}|e^{i\phi}$

<figure><img src="image_17.png" /></figure>

$$\delta - \delta_{\text{AC}} = 0$$

Phase: $\phi_i = -k_{\text{eff}}z_i + \delta\phi_i$

Position: $z_i = z_0 + v_0t_i - \frac{1}{2}gt_i^2$
Gravity measurement:

Measurement protocol: A. Louchet-Chauvet et al., 2011, New J. Phys. **13** 065025

Error budget of the Cold Atom Gravimeter: R. Karcher, PhD thesis, 2019

<table><caption>TABLE 5.2 – Nouveau bilan des effets systématiques du CAG et de leurs incertitudes associées. 1 µGal ≡ 10⁻⁸ m s⁻²</caption><thead><tr><th>Effet systématiques du CAG</th><th>Biais (µGal)</th><th>Incertitudes (µGal)</th></tr></thead><tbody><tr><td>Alignements</td><td>1.2</td><td>0.2</td></tr><tr><td>Référence de fréquence</td><td>3.2</td><td>0.1</td></tr><tr><td>Déphasages RF</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Gradient vertical de gravité</td><td>-13</td><td>&lt;0.1</td></tr><tr><td>Auto-attraction du CAG</td><td>-2.1</td><td>0.1</td></tr><tr><td>Coriolis</td><td>-6.3</td><td>0.5</td></tr><tr><td>Aberrations du front d'onde</td><td>-5.6</td><td>1.3</td></tr><tr><td>LS2</td><td>-6.1</td><td>1.3</td></tr><tr><td>Indice du nuage</td><td>0.4</td><td>&lt;0.1</td></tr><tr><td>α LS1</td><td>0.3</td><td>&lt;0.1</td></tr><tr><td>LS1</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Effet Zeeman quadratique</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Vitesse finie de la lumière</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>CPT</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Rampe de fréquence</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Décalage de la détection</td><td>0.0</td><td>0.1</td></tr><tr><td>Collisions froides</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Aire de l'interféromètre</td><td>?</td><td>?</td></tr><tr><td><strong>Total</strong></td><td><strong>-27.9</strong></td><td><strong>2.0</strong></td></tr></tbody></table>
**Gravity measurement:**

Measurement protocol: A. Louchet-Chauvet et al., 2011, New J. Phys. **13** 065025

Error budget of the Cold Atom Gravimeter: R. Karcher, PhD thesis, 2019

<table><caption>TABLE 5.2 – Nouveau bilan des effets systématiques du CAG et de leurs incertitudes associées. 1 µGal ≡ 10⁻⁸ m s⁻²</caption><thead><tr><th>Effet systématiques du CAG</th><th>Biais (µGal)</th><th>Incertitudes (µGal)</th></tr></thead><tbody><tr><td>Alignements</td><td>1.2</td><td>0.2</td></tr><tr><td>Référence de fréquence</td><td>3.2</td><td>0.1</td></tr><tr><td>Déphasages RF</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Gradient vertical de gravité</td><td>-13</td><td>&lt;0.1</td></tr><tr><td>Auto-attraction du CAG</td><td>-2.1</td><td>0.1</td></tr><tr><td>Coriolis</td><td>-6.3</td><td>0.5</td></tr><tr><td>Aberrations du front d'onde</td><td>-5.6</td><td>1.3</td></tr><tr><td>LS2</td><td>-6.1</td><td>1.3</td></tr><tr><td>Indice du nuage</td><td>0.4</td><td>&lt;0.1</td></tr><tr><td>α LS1</td><td>0.3</td><td>&lt;0.1</td></tr><tr><td>LS1</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Effet Zeeman quadratique</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Vitesse finie de la lumière</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>CPT</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Rampe de fréquence</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Décalage de la détection</td><td>0.0</td><td>0.1</td></tr><tr><td>Collisions froides</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Aire de l'interféromètre</td><td>?</td><td>?</td></tr><tr><td><strong>Total</strong></td><td><strong>-27.9</strong></td><td><strong>2.0</strong></td></tr></tbody></table>
**Gravity measurement:**

Measurement protocol: A. Louchet-Chauvet et al., 2011, New J. Phys. **13** 065025

Error budget of the Cold Atom Gravimeter: R. Karcher, PhD thesis, 2019

<table><caption>TABLE 5.2 – Nouveau bilan des effets systématiques du CAG et de leurs incertitudes associées. $1 \mu\text{Gal} \equiv 10^{-8} \text{ m s}^{-2}$</caption><thead><tr><th>Effet systématiques du CAG</th><th>Biais (µGal)</th><th>Incertitudes (µGal)</th></tr></thead><tbody><tr><td>Alignements</td><td>1.2</td><td>0.2</td></tr><tr><td>Référence de fréquence</td><td>3.2</td><td>0.1</td></tr><tr><td>Déphasages RF</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Gradient vertical de gravité</td><td>-13</td><td>&lt;0.1</td></tr><tr><td>Auto-attraction du CAG</td><td>-2.1</td><td>0.1</td></tr><tr><td>Coriolis</td><td>-6.3</td><td>0.5</td></tr><tr><td>Aberrations du front d'onde</td><td>-5.6</td><td>1.3</td></tr><tr><td>LS2</td><td>-6.1</td><td>1.3</td></tr><tr><td>Indice du nuage</td><td>0.4</td><td>&lt;0.1</td></tr><tr><td>α LS1</td><td>0.3</td><td>&lt;0.1</td></tr><tr><td>LS1</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Effet Zeeman quadratique</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Vitesse finie de la lumière</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>CPT</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Rampe de fréquence</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Décalage de la détection</td><td>0.0</td><td>0.1</td></tr><tr><td>Collisions froides</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Aire de l'interféromètre</td><td>?</td><td>?</td></tr><tr><td><strong>Total</strong></td><td><strong>-27.9</strong></td><td><strong>2.0</strong></td></tr></tbody></table>

<figure><img src="image_18.png" /></figure>
**Gravity measurement:**

Measurement protocol: A. Louchet-Chauvet et al., 2011, New J. Phys. **13** 065025

Error budget of the Cold Atom Gravimeter: R. Karcher, PhD thesis, 2019

<table><caption>TABLE 5.2 – Nouveau bilan des effets systématiques du CAG et de leurs incertitudes associées. 1 µGal ≡ 10⁻⁸ m s⁻²</caption><thead><tr><th>Effet systématiques du CAG</th><th>Biais (µGal)</th><th>Incertitudes (µGal)</th></tr></thead><tbody><tr><td>Alignements</td><td>1.2</td><td>0.2</td></tr><tr><td>Référence de fréquence</td><td>3.2</td><td>0.1</td></tr><tr><td>Déphasages RF</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Gradient vertical de gravité</td><td>-13</td><td>&lt;0.1</td></tr><tr><td>Auto-attraction du CAG</td><td>-2.1</td><td>0.1</td></tr><tr><td>Coriolis</td><td>-6.3</td><td>0.5</td></tr><tr><td>Aberrations du front d'onde</td><td>-5.6</td><td>1.3</td></tr><tr><td>LS2</td><td>-6.1</td><td>1.3</td></tr><tr><td>Indice du nuage</td><td>0.4</td><td>&lt;0.1</td></tr><tr><td>α LS1</td><td>0.3</td><td>&lt;0.1</td></tr><tr><td>LS1</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Effet Zeeman quadratique</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Vitesse finie de la lumière</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>CPT</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Rampe de fréquence</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Décalage de la détection</td><td>0.0</td><td>0.1</td></tr><tr><td>Collisions froides</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Aire de l'interféromètre</td><td>?</td><td>?</td></tr><tr><td>Total</td><td>-27.9</td><td>2.0</td></tr></tbody></table>

Wavefront aberrations phase shift:

* not suppressed by the measurement protocol.

<figure><img src="image_19.png" /></figure>
**Gravity measurement:**

Measurement protocol: A. Louchet-Chauvet et al., 2011, New J. Phys. **13** 065025

Error budget of the Cold Atom Gravimeter: R. Karcher, PhD thesis, 2019

<table><caption>TABLE 5.2 – Nouveau bilan des effets systématiques du CAG et de leurs incertitudes associées. 1 µGal ≡ 10⁻⁸ m s⁻²</caption><thead><tr><th>Effet systématiques du CAG</th><th>Biais (µGal)</th><th>Incertitudes (µGal)</th></tr></thead><tbody><tr><td>Alignements</td><td>1.2</td><td>0.2</td></tr><tr><td>Référence de fréquence</td><td>3.2</td><td>0.1</td></tr><tr><td>Déphasages RF</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Gradient vertical de gravité</td><td>-13</td><td>&lt;0.1</td></tr><tr><td>Auto-attraction du CAG</td><td>-2.1</td><td>0.1</td></tr><tr><td>Coriolis</td><td>-6.3</td><td>0.5</td></tr><tr><td>Aberrations du front d'onde</td><td>-5.6</td><td>1.3</td></tr><tr><td>LS2</td><td>-6.1</td><td>1.3</td></tr><tr><td>Indice du nuage</td><td>0.4</td><td>&lt;0.1</td></tr><tr><td>α LS1</td><td>0.3</td><td>&lt;0.1</td></tr><tr><td>LS1</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Effet Zeeman quadratique</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Vitesse finie de la lumière</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>CPT</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Rampe de fréquence</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Décalage de la détection</td><td>0.0</td><td>0.1</td></tr><tr><td>Collisions froides</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Aire de l'interféromètre</td><td>?</td><td>?</td></tr><tr><td><strong>Total</strong></td><td><strong>-27.9</strong></td><td><strong>2.0</strong></td></tr></tbody></table>

Wavefront aberrations phase shift:

* not suppressed by the measurement protocol.

* main uncertainty term.

<figure><img src="image_20.png" /></figure>
**Gravity measurement:**

Measurement protocol: A. Louchet-Chauvet *et al.*, 2011, New J. Phys. **13** 065025

Error budget of the Cold Atom Gravimeter: R. Karcher, PhD thesis, 2019

<table><caption>TABLE 5.2 – Nouveau bilan des effets systématiques du CAG et de leurs incertitudes associées. 1 µGal ≡ 10⁻⁸ m s⁻²</caption><thead><tr><th>Effet systématiques du CAG</th><th>Biais (µGal)</th><th>Incertitudes (µGal)</th></tr></thead><tbody><tr><td>Alignements</td><td>1.2</td><td>0.2</td></tr><tr><td>Référence de fréquence</td><td>3.2</td><td>0.1</td></tr><tr><td>Déphasages RF</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Gradient vertical de gravité</td><td>-13</td><td>&lt;0.1</td></tr><tr><td>Auto-attraction du CAG</td><td>-2.1</td><td>0.1</td></tr><tr><td>Coriolis</td><td>-6.3</td><td>0.5</td></tr><tr><td>Aberrations du front d'onde</td><td>-5.6</td><td>1.3</td></tr><tr><td>LS2</td><td>-6.1</td><td>1.3</td></tr><tr><td>Indice du nuage</td><td>0.4</td><td>&lt;0.1</td></tr><tr><td>α LS1</td><td>0.3</td><td>&lt;0.1</td></tr><tr><td>LS1</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Effet Zeeman quadratique</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Vitesse finie de la lumière</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>CPT</td><td>0.0</td><td>&lt;0.1</td></tr><tr><td>Rampe de fréquence</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Décalage de la détection</td><td>0.0</td><td>0.1</td></tr><tr><td>Collisions froides</td><td>&lt;0.1</td><td>&lt;0.1</td></tr><tr><td>Aire de l'interféromètre</td><td>?</td><td>?</td></tr><tr><td><strong>Total</strong></td><td><strong>-27.9</strong></td><td><strong>2.0</strong></td></tr></tbody></table>

**Wavefront aberrations phase shift:**

* not suppressed by the measurement protocol.

* main uncertainty term.

<figure><img src="image_21.png" /></figure>

**Simulations to estimate:**

* the resulting bias

* the related uncertainty

* to evaluate new mirrors

<figure><img src="image_22.png" /></figure>
**Wavefront aberrations characteristics:** Transverse phase variations

<figure><img src="image_23.png" /></figure>

$$ \Delta\varphi \approx \int_{Atomic\ Dist.} \delta\varphi_1 - 2\delta\varphi_2 + \delta\varphi_3 \\ = \int_{Dist.(t_1)} \delta\varphi_1 - 2\int_{Dist.(t_2)} \delta\varphi_2 + \int_{Dist.(t_3)} \delta\varphi_3 $$

**Hypothesis 1:** No significant phase aberrations: $\gamma \ll 1$ rad

**Hypothesis 2:** No correlation between phase and contrast

<table><thead><tr><td>Characteristic quantity</td><td>Symbol</td><td>Relation</td></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr></tbody></table>
Wavefront aberrations characteristics: Transverse phase variations

<figure><img src="image_24.png" /></figure>

$$ \Delta\varphi \approx \int_{Atomic\ Dist.} \delta\varphi_1 - 2\delta\varphi_2 + \delta\varphi_3 \\ = \int_{Dist.(t_1)} \delta\varphi_1 - 2\int_{Dist.(t_2)} \delta\varphi_2 + \int_{Dist.(t_3)} \delta\varphi_3 $$

Hypothesis 1: No significant phase aberrations: $\gamma \ll 1$ rad

Hypothesis 2: No correlation between phase and contrast

<table><thead><tr><td>Characteristic quantity</td><td>Symbol</td><td>Relation</td></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td></td></tr></tbody></table>
Wavefront aberrations characteristics: Cloud transverse distribution

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

<table><thead><tr><td>Characteristic quantity</td><td>Symbol</td><td>Relation</td></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td></td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td><td></td></tr></tbody></table>

* $l_{xy} \ll \sigma_{\rho}$: dependence on the position of atoms in the beam, averaging of wavefront aberrations
Wavefront aberrations characteristics: Cloud transverse distribution

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

<table><thead><tr><td>Characteristic quantity</td><td>Symbol</td><td>Relation</td></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td></td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td><td></td></tr></tbody></table>

* $l_{xy} \ll \sigma_{\rho}$: dependence on the position of atoms in the beam, averaging of wavefront aberrations

* $l_{xy} \gg \sigma_{\rho}$: all atoms acquire the same phase
Wavefront aberrations characteristics: Cloud transverse distribution

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

<table><thead><tr><td>Characteristic quantity</td><td>Symbol</td><td>Relation</td></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td rowspan="2">Δφ = f(σ<sub>ρ</sub> / l<sub>xy</sub>)</td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td></tr></tbody></table>

* $l_{xy} \ll \sigma_{\rho}$: dependence on the position of atoms in the beam, averaging of wavefront aberrations

* $l_{xy} \gg \sigma_{\rho}$: all atoms acquire the same phase
Wavefront aberrations characteristics: Cloud thermal expansion

<figure><img src="image_31.png" /></figure>

Radial position probability distribution:

$$w(\rho, t)d\rho = \frac{\rho d\rho}{\sigma_{\rho}^2(0) + \frac{k_b \Theta}{m} t^2} \exp\left(-\frac{\rho^2}{2\left[\sigma_{\rho}^2(0) + \frac{k_b \Theta}{m} t^2\right]}\right)$$

Hypothesis 3: The atomic cloud remains within the beam: $\sigma_{\rho} \lesssim$ beam size

<table><thead><tr><th>Characteristic quantity</th><th>Symbol</th><th>Relation</th></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td rowspan="2">Δφ = f(σ<sub>ρ</sub> / l<sub>xy</sub>)</td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td></tr><tr><td>Cloud initial size</td><td>σ<sub>ρ</sub>(0)</td><td>σ<sub>ρ</sub><sup>2</sup>(t)</td></tr><tr><td>Cloud temperature</td><td>θ</td><td>= σ<sub>ρ</sub><sup>2</sup>(0) + (k<sub>b</sub>θ/m) t<sup>2</sup></td></tr></tbody></table>
Wavefront aberrations characteristics: Field propagation

<figure><img src="image_32.png" /></figure>

<table><thead><tr><td>Characteristic quantity</td><td>Symbol</td><td>Relation</td></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td rowspan="2">Δφ = f(σ<sub>ρ</sub> / l<sub>xy</sub>)</td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td></tr><tr><td>Cloud initial size</td><td>σ<sub>ρ</sub>(0)</td><td rowspan="3">σ<sub>ρ</sub><sup>2</sup>(t) = σ<sub>ρ</sub><sup>2</sup>(0) + (k<sub>b</sub>θ/m) t<sup>2</sup></td></tr><tr><td>Cloud temperature</td><td>θ</td></tr><tr><td>Ab. propagation length</td><td>l<sub>z</sub></td></tr></tbody></table>
Wavefront aberrations characteristics: Field propagation

<figure><img src="image_33.png" /></figure>

Maxwell's equations: $\Delta U - \frac{1}{c^2} \partial_t U = 0$

<table><thead><tr><td>Characteristic quantity</td><td>Symbol</td><td>Relation</td></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td rowspan="2">Δφ = f(σ<sub>ρ</sub> / l<sub>xy</sub>)</td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td></tr><tr><td>Cloud initial size</td><td>σ<sub>ρ</sub>(0)</td><td rowspan="3">σ<sub>ρ</sub><sup>2</sup>(t) = σ<sub>ρ</sub><sup>2</sup>(0) + (k<sub>b</sub>θ/m)t<sup>2</sup></td></tr><tr><td>Cloud temperature</td><td>θ</td></tr><tr><td>Ab. propagation length</td><td>l<sub>z</sub></td></tr></tbody></table>
Wavefront aberrations characteristics: Field propagation

<figure><img src="image_34.png" /></figure>

$$ \text{Maxwell's equations: } \Delta U - \frac{1}{c^2} \partial_t U = 0 $$

$$ \text{Paraxial equation } (U = e^{i(\omega t - kz)} A): \Delta_{xy} A - 2ik\partial_z A = 0 $$

<table><thead><tr><th>Characteristic quantity</th><th>Symbol</th><th>Relation</th></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td rowspan="2">Δφ = f(σ<sub>ρ</sub> / l<sub>xy</sub>)</td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td></tr><tr><td>Cloud initial size</td><td>σ<sub>ρ</sub>(0)</td><td rowspan="2">σ<sub>ρ</sub><sup>2</sup>(t) = σ<sub>ρ</sub><sup>2</sup>(0) + (k<sub>b</sub>θ/m)t<sup>2</sup></td></tr><tr><td>Cloud temperature</td><td>θ</td></tr><tr><td>Ab. propagation length</td><td>l<sub>z</sub></td><td></td></tr></tbody></table>
Wavefront aberrations characteristics: Field propagation

<figure><img src="image_35.png" /></figure>

$$ \text{Maxwell's equations: } \Delta U - \frac{1}{c^2} \partial_t U = 0 $$

$$ \text{Paraxial equation } (U = e^{i(\omega t - kz)} A): \begin{aligned} & \Delta_{xy} A - 2ik \partial_z A = 0 \\ & \leftarrow -\frac{A}{l_{xy}^2} + 2k \frac{A}{l_z} = 0 \end{aligned} $$

<table><thead><tr><th>Characteristic quantity</th><th>Symbol</th><th>Relation</th></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td>Δφ = f(σ<sub>ρ</sub>/l<sub>xy</sub>)</td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td><td>Δφ = f(σ<sub>ρ</sub>/l<sub>xy</sub>)</td></tr><tr><td>Cloud initial size</td><td>σ<sub>ρ</sub>(0)</td><td>σ<sub>ρ</sub><sup>2</sup>(t)</td></tr><tr><td>Cloud temperature</td><td>θ</td><td>= σ<sub>ρ</sub><sup>2</sup>(0) + (k<sub>b</sub>θ/m)t<sup>2</sup></td></tr><tr><td>Ab. propagation length</td><td>l<sub>z</sub></td><td>Δφ = f(σ<sub>ρ</sub>/l<sub>xy</sub>)</td></tr></tbody></table>
Wavefront aberrations characteristics: Field propagation

<figure><img src="image_36.png" /></figure>

$$ \text{Maxwell's equations: } \Delta U - \frac{1}{c^2} \partial_t U = 0 $$

$$ \text{Paraxial equation } (U = e^{i(\omega t - kz)} A): \begin{gathered} \Delta_{xy}A - 2ik\partial_z A = 0 \\ \leftrightarrow -\frac{\mathbf{X}}{l_{xy}^2} + 2k\frac{\mathbf{X}}{l_z} = 0 \end{gathered} $$

<table><thead><tr><th>Characteristic quantity</th><th>Symbol</th><th>Relation</th></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td>Δφ = f(σ<sub>ρ</sub>/l<sub>xy</sub>)</td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td><td>Δφ = f(σ<sub>ρ</sub>/l<sub>xy</sub>)</td></tr><tr><td>Cloud initial size</td><td>σ<sub>ρ</sub>(0)</td><td>σ<sub>ρ</sub><sup>2</sup>(t) = σ<sub>ρ</sub><sup>2</sup>(0) + (k<sub>b</sub>θ/m)t<sup>2</sup></td></tr><tr><td>Cloud temperature</td><td>θ</td><td>Δφ = f(σ<sub>ρ</sub>/l<sub>xy</sub>)</td></tr><tr><td>Ab. propagation length</td><td>l<sub>z</sub></td><td>l<sub>z</sub> = 2kl<sub>xy</sub><sup>2</sup></td></tr></tbody></table>
Wavefront aberrations characteristics: Field propagation

<figure><img src="image_37.png" /></figure>

$$ \text{Maxwell's equations: } \Delta U - \frac{1}{c^2} \partial_t U = 0 $$

$$ \text{Paraxial equation } (U = e^{i(\omega t - kz)} A): \begin{gathered} \Delta_{xy}A - 2ik\partial_z A = 0 \\ \leftarrow -\frac{\times}{l_{xy}^2} + 2k\frac{\times}{l_z} = 0 \end{gathered} $$

<table><thead><tr><th>Characteristic quantity</th><th>Symbol</th><th>Relation</th></tr></thead><tbody><tr><td>Aberration amplitude</td><td>γ</td><td>Δφ ∝ γ</td></tr><tr><td>Ab. transverse length</td><td>l<sub>xy</sub></td><td>Δφ = f(σ<sub>ρ</sub> / l<sub>xy</sub>)</td></tr><tr><td>Cloud transverse size</td><td>σ<sub>ρ</sub></td><td>Δφ = f(σ<sub>ρ</sub> / l<sub>xy</sub>)</td></tr><tr><td>Cloud initial size</td><td>σ<sub>ρ</sub>(0)</td><td>σ<sub>ρ</sub><sup>2</sup>(t) = σ<sub>ρ</sub><sup>2</sup>(0) + (k<sub>b</sub>θ/m)t<sup>2</sup></td></tr><tr><td>Cloud temperature</td><td>θ</td><td>Δφ = f(σ<sub>ρ</sub> / l<sub>xy</sub>)</td></tr><tr><td>Ab. propagation length</td><td>l<sub>z</sub></td><td>l<sub>z</sub> = 2kl<sub>xy</sub><sup>2</sup></td></tr><tr><td>Phase Wavefront Aberration</td><td>Δφ</td><td>Δφ = γF(σ<sub>ρ</sub> / l<sub>xy</sub>, l<sub>z</sub>)</td></tr></tbody></table>
Wavefront aberrations due to the mirror surface:

<figure><img src="image_38.png" /></figure>
Wavefront aberrations due to the mirror surface: Zernike polynomials: K. Niu and C. Tian, 2022, J. Opt. 24 123001

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /><figcaption>Figure 7. Pyramid of the non-normalized Zernike circle polynomials up to the sixth degree under the Noll indexing scheme.</figcaption></figure>
Wavefront aberrations due to the mirror surface: Zernike polynomials: K. Niu and C. Tian, 2022, J. Opt. 24 123001

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /><figcaption>Figure 7. Pyramid of the non-normalized Zernike circle polynomials up to the sixth degree under the Noll indexing scheme.</figcaption></figure>
Wavefront aberrations due to the mirror surface: Zernike polynomials: K. Niu and C. Tian, 2022, J. Opt. 24 123001

<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /><figcaption>Figure 7. Pyramid of the non-normalized Zernike circle polynomials up to the sixth degree under the Noll indexing scheme.</figcaption></figure>

$$ \left. \begin{array}{l} \text{Symmetry: Cloud, Incident field} \\ \text{Anti-symmetry: Mirror } Z_n^{m \neq 0} \end{array} \right\} \approx 0 $$
**Wavefront aberrations due to the mirror surface:** Zernike polynomials: K. Niu and C. Tian, 2022, J. Opt. **24** 123001

<figure><img src="image_45.png" /></figure>

**Reflection on the mirror:**

Mirror surface: $S_{mir}^{(n)}(\rho) = \gamma Z_n^0(\frac{\rho}{R})$

Amplitude of the incident field: $U_{inc}(\rho, z_{mir})$

Amplitude of the reflected field: $U_{ref}(\rho, z_{mir}) = U_{inc}(\rho, z_{mir})e^{-i2ks_{mir}^{(n)}}$

<figure><img src="image_46.png" /></figure>
**Wavefront aberrations due to the mirror surface:** Zernike polynomials: K. Niu and C. Tian, 2022, J. Opt. **24** 123001

<figure><img src="image_47.png" /></figure>

**Reflection on the mirror:**

Mirror surface: $S_{mir}^{(n)}(\rho) = \gamma Z_{n}^{0}(\frac{\rho}{R})$

Amplitude of the incident field: $U_{inc}(\rho, z_{mir})$

Amplitude of the reflected field: $U_{ref}(\rho, z_{mir}) = U_{inc}(\rho, z_{mir})e^{-i2ks_{mir}^{(n)}}$

**Approximate analytical solution:**

$$ \Delta\varphi_i(z_i) = \underbrace{\int d\rho \ w(\rho, t_i)}_{\text{Cloud average}} \arg \left[ \underbrace{\int dK J_0(K\rho) e^{i\frac{K^2\Delta z_{mir,i}}{2k}} \int d\rho' J_0(K\rho') U_{ref}(\rho', z_{mir})}_{\text{Propagation with paraxial equation}} \underbrace{\int dK J_0(K\rho) e^{i\frac{K^2\Delta z_{mir,i}}{2k}} \int d\rho' J_0(K\rho') U_{ref}(\rho', z_{mir})}_{\text{Refl. Amp.}} \right] $$

<figure><img src="image_48.png" /></figure>
**Wavefront aberrations due to the mirror surface:** Zernike polynomials: K. Niu and C. Tian, 2022, J. Opt. **24** 123001

<figure><img src="image_49.png" /></figure>

**Reflection on the mirror:**

Mirror surface: $S_{mir}^{(n)}(\rho) = \gamma Z_n^0(\frac{\rho}{R})$

Amplitude of the incident field: $U_{inc}(\rho, z_{mir})$

Amplitude of the reflected field: $U_{ref}(\rho, z_{mir}) = U_{inc}(\rho, z_{mir})e^{-i2ks_{mir}^{(n)}}$

**Approximate analytical solution:**

$$ \Delta\varphi_i(z_i) = \underbrace{\int d\rho \ w(\rho, t_i)}_{\text{Cloud average}} \arg \left[ \underbrace{\int dK J_0(K\rho) e^{i\frac{K^2\Delta z_{mir,i}}{2k}} \int d\rho' J_0(K\rho') U_{ref}(\rho', z_{mir})}_{\text{Propagation with paraxial equation}} \underbrace{\int dK J_0(K\rho) e^{i\frac{K^2\Delta z_{mir,i}}{2k}} \int d\rho' J_0(K\rho') U_{ref}(\rho', z_{mir})}_{\text{Refl. Amp.}} \right] $$

* Small aberrations: $U_{ref}(\rho, z_{mir}) \approx U_{inc}(\rho, z_{mir}) [1 - i2k\gamma Z_n^0(\frac{\rho}{R})]$

<figure><img src="image_50.png" /></figure>
**Wavefront aberrations due to the mirror surface:** Zernike polynomials: K. Niu and C. Tian, 2022, J. Opt. **24** 123001

<figure><img src="image_51.png" /></figure>

**Reflection on the mirror:**

Mirror surface: $S_{mir}^{(n)}(\rho) = \gamma Z_n^0(\frac{\rho}{R})$

Amplitude of the incident field: $U_{inc}(\rho, z_{mir})$

Amplitude of the reflected field: $U_{ref}(\rho, z_{mir}) = U_{inc}(\rho, z_{mir})e^{-i2ks_{mir}^{(n)}}$

**Approximate analytical solution:**

$$ \Delta\varphi_i(z_i) = \underbrace{\int d\rho \ w(\rho, t_i)}_{\text{Cloud average}} \arg \left[ \underbrace{\int dK J_0(K\rho) e^{i\frac{K^2 \Delta z_{mir,i}}{2k}} \int d\rho' J_0(K\rho') U_{ref}(\rho', z_{mir})}_{\text{Propagation with paraxial equation}} \underbrace{\int dK J_0(K\rho) e^{i\frac{K^2 \Delta z_{mir,i}}{2k}} \int d\rho' J_0(K\rho') U_{ref}(\rho', z_{mir})}_{\text{Refl. Amp.}} \right] $$

* Small aberrations: $U_{ref}(\rho, z_{mir}) \approx U_{inc}(\rho, z_{mir})[1 - i2k\gamma Z_n^0(\frac{\rho}{R})]$

* Zernike ↔ Bessel: $Z_n^0(\frac{\rho}{R}) \approx J_0(\frac{\rho(n+1)}{R}) \rightarrow \text{lengths: } \begin{cases} l_{xy}^{(n)} = \frac{R}{n+1} \\ l_z^{(n)} = \frac{2kR^2}{(n+1)^2} \end{cases}$

<figure><img src="image_52.png" /></figure>
**Wavefront aberrations due to the mirror surface:** Zernike polynomials: K. Niu and C. Tian, 2022, J. Opt. **24** 123001

<figure><img src="image_53.png" /></figure>

**Reflection on the mirror:**

Mirror surface: $S_{mir}^{(n)}(\rho) = \gamma Z_{n}^{0}(\frac{\rho}{R})$

Amplitude of the incident field: $U_{inc}(\rho, z_{mir})$

Amplitude of the reflected field: $U_{ref}(\rho, z_{mir}) = U_{inc}(\rho, z_{mir})e^{-i2kS_{mir}^{(n)}}$

**Approximate analytical solution:**

$$ \Delta\varphi_i(z_i) = \underbrace{\int d\rho \ w(\rho, t_i)}_{\text{Cloud average}} \arg \left[ \underbrace{\int dK J_0(K\rho) e^{i\frac{K^2 \Delta z_{mir,i}}{2k}} \int d\rho' J_0(K\rho') U_{ref}(\rho', z_{mir})}_{\text{Propagation with paraxial equation}} \underbrace{\int dK J_0(K\rho) e^{i\frac{K^2 \Delta z_{mir,i}}{2k}} \int d\rho' J_0(K\rho') U_{ref}(\rho', z_{mir})}_{\text{Refl. Amp.}} \right] $$

* Small aberrations: $U_{ref}(\rho, z_{mir}) \approx U_{inc}(\rho, z_{mir})[1 - i2k\gamma Z_{n}^{0}(\frac{\rho}{R})]$

* Zernike ↔ Bessel: $Z_{n}^{0}(\frac{\rho}{R}) \approx J_{0}(\frac{\rho(n+1)}{R}) \rightarrow \text{lengths: } \begin{cases} l_{xy}^{(n)} = \frac{R}{n+1} \\ l_{z}^{(n)} = \frac{2kR^2}{(n+1)^2} \end{cases}$

<figure><img src="image_54.png" /></figure>
Wavefront aberrations due to the mirror surface: Zernike polynomials

Approximate analytical solution:

$$ \Delta\varphi^{(1)} = \gamma Z_{n}^{0}(0) \left[ \cos\left(\frac{\Delta z_{mir,1}}{l_{z}^{(n)}}\right) e^{-\frac{1}{2}\left(\frac{\sigma_{\rho}(t_1)}{l_{xy}^{(n)}}\right)^2} - 2\cos\left(\frac{\Delta z_{mir,2}}{l_{z}^{(n)}}\right) e^{-\frac{1}{2}\left(\frac{\sigma_{\rho}(t_2)}{l_{xy}^{(n)}}\right)^2} + \cos\left(\frac{\Delta z_{mir,3}}{l_{z}^{(n)}}\right) e^{-\frac{1}{2}\left(\frac{\sigma_{\rho}(t_3)}{l_{xy}^{(n)}}\right)^2} \right] $$

* Zero temperature limit ($\sigma_{\rho}(t_i) \to \sigma_{\rho}(0)$): $\Delta\varphi^{(1)} \neq 0$
Wavefront aberrations due to the mirror surface: Zernike polynomials

Approximate analytical solution:

$$ \Delta\varphi^{(1)} = \gamma Z_{n}^{0}(0) \left[ \cos\left(\frac{\Delta z_{mir,1}}{l_{z}^{(n)}}\right) e^{-\frac{1}{2}\left(\frac{\sigma_{\rho}(t_{1})}{l_{xy}^{(n)}}\right)^{2}} - 2\cos\left(\frac{\Delta z_{mir,2}}{l_{z}^{(n)}}\right) e^{-\frac{1}{2}\left(\frac{\sigma_{\rho}(t_{2})}{l_{xy}^{(n)}}\right)^{2}} + \cos\left(\frac{\Delta z_{mir,3}}{l_{z}^{(n)}}\right) e^{-\frac{1}{2}\left(\frac{\sigma_{\rho}(t_{3})}{l_{xy}^{(n)}}\right)^{2}} \right] $$

* Zero temperature limit ($\sigma_{\rho}(t_{i}) \to \sigma_{\rho}(0)$): $\Delta\varphi^{(1)} \neq 0$

Gravimeter:

* Free fall time: $2T = 160$ ms

* Cloud initial size: $\sigma_{\rho}(0) = 10$ µm

* Cloud temperature: $30$ nK $\leftrightarrow$ $2$ µK

Mirror:

* Diameter: $2R = 27$ mm

* Amplitude: $\gamma = \frac{\pi}{100} = k \frac{\lambda}{200}$

<figure><img src="image_55.png" /></figure>
Simulation with real mirror surface:

Mirror surface:

* Radius: R = 11 mm

* Peak-to-Valley: $\Delta S_{mir} = 77.1$ nm

* Surface RMS: $\sigma(S_{mir}) = 7.8$ nm

<figure><img src="image_56.png" /></figure>

<figure><img src="image_57.png" /></figure>
Simulation with real mirror surface:

Mirror surface:

* Radius: $R = 11$ mm

* Peak-to-Valley: $\Delta S_{mir} = 77.1$ nm

* Surface RMS: $\sigma(S_{mir}) = 7.8$ nm

<figure><img src="image_58.png" /></figure>

<figure><img src="image_59.png" /></figure>

* Typical order of Zernike polynomial for transverse effects: $n_{xy}^* = \frac{R}{\sigma_{\rho}(t_1)} \approx 30 \leftrightarrow i_{xy}^* = \left(\frac{n_{xy}^*}{2} + 1\right)^2 \approx 256$

* Typical order of Zernike polynomial for propagation effects ($\Delta z_{mir,1} \approx 40$ cm): $n_z^* = R \sqrt{\frac{2k}{\Delta z_{mir,1}}} \approx 70 \leftrightarrow i_z^* \approx 1250$
Simulation with real mirror surface:

Mirror surface:

* Radius: $R = 11$ mm

* Peak-to-Valley: $\Delta S_{mir} = 77.1$ nm

* Surface RMS: $\sigma(S_{mir}) = 7.8$ nm

<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>

Typical parameters of the atomic gravimeter:

* Free fall time: $2T = 160$ ms

* Cloud initial size: $\sigma_{\rho}(0) = 300$ µm

* Cloud temperature: $2$ µK

<figure><img src="image_62.png" /></figure>

* Typical order of Zernike polynomial for transverse effects: $n_{xy}^* = \frac{R}{\sigma_{\rho}(t_1)} \approx 30 \leftrightarrow i_{xy}^* = \left(\frac{n_{xy}^*}{2} + 1\right)^2 \approx 256$

* Typical order of Zernike polynomial for propagation effects ($\Delta z_{mir,1} \approx 40$ cm): $n_z^* = R \sqrt{\frac{2k}{\Delta z_{mir,1}}} \approx 70 \leftrightarrow i_z^* \approx 1250$
Simulation with real mirror surface:

Mirror surface:

* Radius: R = 11 mm

* Peak-to-Valley: $\Delta S_{mir} = 77.1$ nm

* Surface RMS: $\sigma(S_{mir}) = 7.8$ nm

<figure><img src="image_63.png" /></figure>

Typical parameters of the atomic gravimeter:

* Free fall time: $2T = 160$ ms

* Cloud initial size: $\sigma_{\rho}(0) = 300$ µm

* Cloud temperature: $2$ µK

<figure><img src="image_64.png" /></figure>
Simulation with real mirror surface:

Mirror surface:

* Radius: $R = 11$ mm

* Peak-to-Valley: $\Delta S_{mir} = 77.1$ nm

* Surface RMS: $\sigma(S_{mir}) = 7.8$ nm

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

Typical parameters of the atomic gravimeter:

* Free fall time: $2T = 160$ ms

* Cloud initial size: $\sigma_{\rho}(0) = 300$ µm

* Cloud temperature: $2$ µK

<figure><img src="image_67.png" /></figure>

* $g_{meas.}$ bias vs. initial position displacement $\approx 10$ µGal mm$^{-1}$

* Local extremum with lower gradient
Simulation with real mirror surface:

Mirror surface:

* Radius: $R = 11$ mm

* Peak-to-Valley: $\Delta S_{mir} = 77.1$ nm

* Surface RMS: $\sigma(S_{mir}) = 7.8$ nm

<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /></figure>

Gravimeter + Delta kicked atomic source:

* Free fall time: $2T = 160$ ms

* Cloud initial size: $\sigma_{\rho}(0) = 200$ µm

* Cloud expansion temperature: 5 nK

<figure><img src="image_70.png" /></figure>

* $g_{meas.}$ bias vs. initial position displacement $\lesssim 1$ µGal mm$^{-1}$

* Gaussian beam + perfect mirror: $\Delta g = -0.21(1)$ µGal
**Conclusion:**

* Theoretical approach taking into account the propagation of wavefront aberrations

* Representation of the mirror surface with Zernike polynomials

* Sensitivity to the initial cloud position
**Conclusion:**

* Theoretical approach taking into account the propagation of wavefront aberrations

* Representation of the mirror surface with Zernike polynomials

* Sensitivity to the initial cloud position

**Outlook:**

* Correlations between wavefront aberrations and intensity fluctuations

* Sensitivity to initial cloud transverse velocity

* Gradiometer: common mode rejection
**Conclusion:**

* Theoretical approach taking into account the propagation of wavefront aberrations

* Representation of the mirror surface with Zernike polynomials

* Sensitivity to the initial cloud position

**Outlook:**

* Correlations between wavefront aberrations and intensity fluctuations

* Sensitivity to initial cloud transverse velocity

* Gradiometer: common mode rejection

Thank you for your attention!

**Bibliography:**

* A. Louchet-Chauvet *et al.*, 2011, New J. Phys. **13** 065025

* R. Karcher, PhD thesis, 2019, Sorbonne Université

* K. Niu and C. Tian, 2022, J. Opt. **24** 123001

* L. Pagot, S. Merlet, and F. Pereira Dos Santos, 2025, Opt. Express **33**, 18843-18854
Approximation of Zernike polynomials by Bessel functions:

<figure><img src="image_71.png" /></figure>
Approximate analytical propagation 2nd order:

$$U(\rho, 0) = Ae^{i\alpha Z_n^0(\frac{\rho}{R})} \approx Ae^{i\alpha_n J_0(\frac{\rho}{l_{xy}^{(n)}})}$$

$$\approx A \left\{ 1 + i\alpha_n J_0 \left( \frac{\rho}{l_{xy}^{(n)}} \right) - \frac{\alpha_n^2}{2} J_0^2 \left( \frac{\rho}{l_{xy}^{(n)}} \right) \right\}$$

$$U^{(1)}(\rho, z) = Ae^{-ikz} \left\{ 1 + i\alpha_n e^{\frac{i z}{l_{xy}^{(n)}}} J_0 \left( \frac{\rho}{l_{xy}^{(n)}} \right) \right\}$$

$$U^{(2)}(\rho, z) = -\frac{\alpha^2}{2} Ae^{-ikz} e^{\frac{i 2z}{l_{xy}^{(n)}}} \times \left\{ \begin{aligned} & J_0 \left( \frac{2z}{l_{xy}^{(n)}} \right) J_0^2 \left( \frac{\rho}{l_{xy}^{(n)}} \right) \\ & + 2 \sum_{p=1}^{\infty} (-1)^p J_{2p} \left( \frac{2z}{l_{xy}^{(n)}} \right) J_{2p}^2 \left( \frac{\rho}{l_{xy}^{(n)}} \right) \\ & - 2i \sum_{p=0}^{\infty} (-1)^p J_{2p+1} \left( \frac{2z}{l_{xy}^{(n)}} \right) J_{2p+1}^2 \left( \frac{\rho}{l_{xy}^{(n)}} \right) \end{aligned} \right\}$$
### Comparison simulation with/without propagation:
#### Gravimeter:

* Free fall time: $2T = 160$ ms

* Cloud initial size: $\sigma_{\rho}(0) = 10$ µm

* Cloud temperature: $30$ nK $\leftrightarrow$ $2$ µK

$$ \Delta\varphi^{(1)} = \gamma Z_n^0(0) \left[ \cos\left(\frac{\Delta z_{mir,1}}{l_z^{(n)}}\right) e^{-\frac{1}{2}\left(\frac{\sigma_{\rho}(t_1)}{l_{xy}^{(n)}}\right)^2} - 2\cos\left(\frac{\Delta z_{mir,2}}{l_z^{(n)}}\right) \right. \\ \left. e^{-\frac{1}{2}\left(\frac{\sigma_{\rho}(t_2)}{l_{xy}^{(n)}}\right)^2} + \cos\left(\frac{\Delta z_{mir,3}}{l_z^{(n)}}\right) e^{-\frac{1}{2}\left(\frac{\sigma_{\rho}(t_3)}{l_{xy}^{(n)}}\right)^2} \right] $$

With propagation

<figure><img src="image_72.png" /></figure>

Without propagation ($l_z^{(n)} \to \infty$)

<figure><img src="image_73.png" /></figure>

Zernike polynomials: $2R = 27$ mm, $\gamma = \frac{\pi}{100}$
ISO-14999 classification:

$$ (n_F, m_F, s_F) = \left( \frac{n + |m|}{2}, |m|, \text{sign}(m) \right) $$

$$ i_{ISO} = (n_F + 1)^2 - 2m_F + \left\lfloor \frac{1 - s_F}{2} \right\rfloor $$

$$ i_{ISO}^{m=0} = \left( \frac{n}{2} + 1 \right)^2 $$

<figure><img src="image_74.png" /><figcaption>Figure 7. Pyramid of the non-normalized Zernike circle polynomials up to the sixth degree under the Noll indexing scheme. K. Niu and C. Tian, 2022, J. Opt. 24 123001</figcaption></figure>

<figure><img src="image_75.png" /></figure>