Multi-pass cell, atomic magnetometer, and methods of forming the same

Abstract

Various embodiments may relate to a multi-pass cell. The multi-pass cell may include a concave mirror. The multi-pass cell may also include a rotatable mirror including a first mirror portion and a second mirror portion such that one or both of the first mirror portion and the second mirror portion are rotatable about a rotation axis passing through the first mirror portion and the second mirror portion.

Description

MULTI-PASS CELL, ATOMIC MAGNETOMETER, AND METHODS OF FORMING THE SAMECROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of priority of Singapore application No. 10202402269Y filed July 30, 2024, the contents of it being hereby incorporated by reference in its entirety for all purposes.TECHNICAL FIELD

[0002] Various embodiments of this disclosure may relate to a multi-pass cell. Various embodiments of this disclosure may relate to an atomic magnetometer. Various embodiments of this disclosure may relate to a method of forming a multi-pass cell. Various embodiments of this disclosure may relate to a method of forming an atomic magnetometer.BACKGROUND

[0003] FIG. 1 A shows the growing market size of the global quantum sensor market. Quantum sensors are expected to surpass and replace classical sensors. The global magnetometer mater is expected to reach USD 4.2 billion by 2032.

[0004] Operation of ultrahigh sensitivity magnetometers requires improvements in optical depth (OD) and active volume, thereby improving the photon-shot noise and spin-projection noise, improving the fundamental sensitivity of the atomic magnetometer. This is particularly beneficial for applications requiring sufficient dynamic range to work in Earth’s field where spin-exchange relaxation is not fully suppressed, it leads to a lower relaxation time and worsens the fundamental sensitivity of the magnetometer. Thus, a higher OD can improve photon shot noise and hence the fundamental sensitivity of the magnetometer. FIG. IB shows a schematicillustrating an atomic magnetometer including a multi-pass alkali cell. One strategy involves directing the probe beam through a multi-pass alkali cell to allow the beam to travel several times through the alkali cell, increasing both the OD and active volume in the alkali cell This amplifies paramagnetic Faraday rotation while suppressing quantum noise, specifically spinprojection noise (dBspn) and photon shot noise (5Bpsn).

[0005] Scalar atomic magnetometers, which measure the Larmor precession frequency, can operate effectively in finite magnetic fields, offering metrological advantages such as higher fractional resolution and inherent self-calibration. These magnetometers are employed in a wide range of applications, including biomedical sensing in ambient environments, magnetic anomaly detection, magnetic resonance imaging (MRT), space exploration, and magnetic navigation. However, their performance in finite fields is fundamentally limited by spin-exchange relaxation, which shortens spin coherence times. Although this effect can be partially suppressed using light narrowing techniques, complete elimination is not possible.

[0006] Multi-pass cells offer a promising solution by enabling quantum non-demolition (QND) measurements in scalar magnetometers under a pulsed pump-probe regime, enhancing sensitivity without increasing the measurement volume. A key factor enabling QND measurements is high optical depth, which multi-pass cells significantly improve. For example, sub-femtotesla sensitivity has been demonstrated using two cylindrical multi-pass alkali cells in scalar magnetometers. Moreover, a vector magnetometer incorporating a rapidly rotating bias field and a multi-pass scalar magnetometer was developed, achieving a fractional resolution of 0.7 parts per billion and angular sensitivities of 6 nrad / Hz for two polar angles in Earth’s field. This configuration retains the metrological advantages of scalar systems while enabling directional sensitivity, improving the localization of magnetic sources.

[0007] Under ideal technical conditions, measurement sensitivity is fundamentally limited Under ideal technical conditions, measurement sensitivity is fundamentally limited by quantum noise sources such as spin projection noise (SBspn) and photon shot noise (riB,,.,,) Additionally, noise arising from atomic spin diffusion plays a significant role in multi-pass alkali cells. Atomic spin diffusion (6Basd), in particular, is driven by the thermal motion of atoms. As atoms move, their spatial distribution and magnetic moments fluctuate, leading to time-dependent variations in the detected spin signal. This effect is further influenced by atomic diffusion in and out of the detection region, which can be analyzed through the spin noise correlation function. These noise sources are generally considered independent and combine additively with other contributions. The quantum noise contributions are given by:1 1,’8Bvsn- y T2^ ,nVrrvrOD0’ where y ' is the gyromagnetic ratio,’ Tz is the transverse relaxation time, n is the vapor density of alkali atoms, and I ' is the active volume. T^r represents the probe rate for far-detuned light, and ODo = nol is the optical depth on resonance, where o is the absorption cross-section at resonance and I denotes the probe laser’s path length through the cell.

[0008] ODo is a key parameter in QND measurements of atomic spins. A multi-pass alkali vapor cell enhances the sensitivity of scalar optical magnetometers by increasing the effective interaction length, thereby boosting ODo.

[0009] However, multi-pass cells conventionally made using spherical or cylindrical mirrors with the Herriott and White configurations result in a probe beam that does not adequately utilize the full volume of the alkali cell as the beam spot patterns on the mirror does not fully cover the area of the mirror, resulting in a low active -to-cell volume. Specifically, the dense- beam multi-pass cell often employed in atomic magnetometry is constructed using two cylindrical mirrors arranged with orthogonal axes of curvature, with a hole in the center of one mirror for beam entrance and exit, and one of the mirrors is tilted. FIG. 1C shows a schematic of a conventional multi-pass cell. Although this configuration leads to a high beam density onthe mirrors due to laser reflections that create a two-dimensional Lissajous pattern, this pattern fails to sufficiently cover the mirror surface. Specifically, the pattern can cover at best 2 / it of a circular mirror surface.

[0010] This results in a low beam area to mirror area ratio, which fundamentally limits the number of passes, therefore decreasing the active-to-cell volume ratio. Furthermore, an intrinsic feature of the Lissajous pattern is the tightly focused beams at the edges of the pattern as shown in FIG. ID, which results in a worsened spin correlation decay. FIG. ID illustrates simulation of Lissajous pattern showing large beam spots near the center and tightly focused beam spots near the edges for (a) front and (b) rear cylindrical mirror. This may be detrimental as the decay of spin correlation is primarily influenced by a small number of tightly concentrated beam spots within the multi-pass cell, thereby restricting sensitivity improvement.

[0011] Finally, the cylindrical cell lacks versatility, as it is restricted to generating Lissajous patterns exclusively. Therefore, its applicability is constrained to scenarios where this pattern suffices, and it may not be suitable for applications requiring more diverse or customizable patterns.

[0012] Accordingly, the above problems present a major bottleneck in the miniaturization of compact and ultrahigh sensitivity atomic magnetometers. It is more desirable to make the atomic magnetometers more compact (e g., in applications such as bioimaging) and sensitive (e.g., in applications such as magnetic anomaly detection).SUMMARY

[0013] Various embodiments may relate to a multi-pass cell. The multi-pass cell may include a concave mirror. The multi-pass cell may also include a rotatable mirror including a first mirror portion and a second mirror portion such that one or both of the first mirror portionand the second mirror portion are rotatable about a rotation axis passing through the first mirror portion and the second mirror portion.

[0014] Various embodiments may relate to an atomic magnetometer. The atomic magnetometer may include a multi-pass cell as described herein. The atomic magnetometer may also include a pump beam optical system configured to provide circularly polarized light beam to pump atoms contained in an alkali cell contained within or at least partially defined by the multi-pass cell. The atomic magnetometer may further include a probe beam optical system configured to provide linearly polarized light beam such that the linearly polarized light beam is reflected between the concave mirror and the rotatable mirror. The atomic magnetometer may additionally include a detector optical system configured to detect an optical rotation of the linearly polarized light beam exiting the multi-pass cell.

[0015] Various embodiments may relate to a method of forming a multi-pass cell. The method may include providing or forming a concave mirror. The method may also include providing or forming a rotatable mirror including a first mirror portion and a second mirror portion such that one or both of the first mirror portion and the second mirror portion are rotatable about a rotation axis passing through the first mirror portion and the second mirror portion.

[0016] Various embodiments may relate to a method of fonning an atomic magnetometer. The method may include forming or providing a multi-pass cell as described herein. The method may also include providing or forming a pump beam optical system configured to provide circularly polarized light beam to pump atoms contained in an alkali cell contained within or at least partially defined by the multi-pass cell. The method may further include providing or forming a probe beam optical system configured to provide linearly polarized light beam such that the linearly polarized light beam is reflected between the concave mirror and the rotatable mirror. The method may additionally include providing or forming a detectoroptical system configured to detect an optical rotation of the linearly polarized light beam exiting the multi-pass cell.BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Tn the drawings, like reference characters generally refer to the same parts throughout the different views. The drawings are not necessarily drawn to scale, emphasis instead generally being placed upon illustrating the principles of various embodiments. In the following description, various embodiments of the invention are described with reference to the following drawings.FIG. 1 A shows the growing market size of the global quantum sensor market. Quantum sensors are expected to surpass and replace classical sensors.FIG. IB shows a schematic illustrating an atomic magnetometer including a multi-pass alkali cell.FIG. 1C shows a schematic of a conventional multi-pass cell.FIG. ID illustrates simulation of Lissajous pattern showing large beam spots near the center and tightly focused beam spots near the edges for (a) front and (b) rear cylindrical mirror.FIG. 2 shows a general illustration of a multi-pass cell according to various embodiments.FIG. 3A shows a general illustration of an atomic magnetometer according to various embodiments.FIG. 3B shows a general illustration of another atomic magnetometer according to various embodiments.FIG. 4 shows a general illustration of a method of forming a multi-pass cell according to various embodiments.FIG. 5 shows a general illustration of a method of forming an atomic magnetometer according to various embodiments.FIG. 6A shows the elliptical beam spot pattern in a typical Herriott cavity with 60 reflections.FIG. 6B is a schematic showing a cross-sectional side view of a multi-pass cell according to various embodiments.FIG. 6C is a schematic showing a perspective view of the multi-pass cell according to various embodiments.FIG. 7A shows (a) a schematic illustrating an alkali cell between the concave mirror and the rotatable mirror according to various embodiments; and (b) a schematic illustrating the concave mirror and the rotatable mirror being used to form reflecting walls of an alkali cell according to various embodiments.FIG. 7B shows a setup including the multi-pass cell shown in (a) of FIG. 7A according to various embodiments.FIG. 8 shows schematics of two different configurations for mirror machining of a multi-pass cell according to various embodiments.FIG. 9 A shows (left) a fluorescence image of the concave mirror according to various embodiments; and (right) a fluorescence image of an alkali cell according to various embodiments.FIG. 9B shows another fluorescence image of the alkali cell according to various embodiments.FIG. 9C shows a schematic of an atomic magnetometer according to various embodiments.FIG. 10A shows atable illustrating the simulation parameters used, including cavity parameters and laser parameters according to various embodiments.FIG. 10B shows (a) simulation results of the recirculating beam spot reflections on the concave mirror M2 according to various embodiments; and (b) a three-dimensional (3D) ray tracingdiagram showing optical path of the beam within the multi-pass cell according to various embodiments.FIG. IOC shows the different beam spots patterns in the recirculating multi-pass cell according to various embodiments at different rotations of the second mirror portion Ml’ .FIG. 11A shows (above) a multi-pass cell including the concave mirror (M2), bottom mirror portion (Ml) and top mirror portion (MF), similar to that shown in FIG. 6C according to various embodiments; and (below) the beam spot pattern observed on the bottom mirror portion (Ml) and top mirror portion (Ml’) according to various embodiments showing 78 reflections.FIG. 1 IB shows the beam spots positions on the top mirror portion (Ml’) , with hollow circles marking their positions before the mirror portion’s rotation and filled circles indicating their positions after the rotation according to various embodimentsFIG. 11C shows beam spot distributions on mirror portions Ml and Ml’, along with shift of optical centers, vary with the relative rotation between the two mirror portions according to various embodiments.FIG. 1 ID shows (a) a plot of the initial input angle in the y-direction, y ’o (in radians or rad) as a function of the distance (in millimeters or mm) between the concave mirror and the rotatable mirror illustrating the calculated total number of possible reflections in a recirculating cell as a function of the distance between mirrors and input angle in the y-direction at 2 = 780 nm, mo = 1 mm, 0x= 0.02°, x’o = 0°, and / i = 1 m according to various embodiments; and (b) a three- dimensional (3D) plot of number of reflections as a function of the distance (in millimeters or mm) between the concave mirror and the rotatable mirror and the initial input angle in the y- direction, y ’o (in radians or rad) of the recirculating cell according to various embodiments.FIG. 1 IE shows a plot of beam radius (in millimeters or mm) as a function of number of round trips for various mirror separations d according to various embodiments.FIG. HF shows the analytical beam positions and waist (hollow circles) and simulated light rays (scatter points) according to various embodiments.FIG. 11G shows a plot of spin noise diffusion correlation G / ftime r ) as a function of time r (in seconds or s) calculated illustrating normalized from Cd of a recirculating cell with 78 reflections and d = 30 mm according to various embodiments; and (inset) a plot of normalized power spectral density PSD as a function of f- ftarmor (in Hertz or Hz) of the recirculating cell according to various embodiments.FIG. 11H shows a plot of spin noise diffusion correlation Q(time r ) as a function of time T (in seconds or s) illustrating a slower decay of the spin correlation function by inserting a barrier to prevent atoms from entering the tightly focused region according to various embodiments; and (inset) normalized power spectral density PSD as a function of f (in Hertz or Hz)illustrating the reduction of spin diffusion noise due to insertion of the barrier according to various embodiments.FIG. 1 II shows a conventional cylindrical cell and the resultant beam spot distribution, similar to FIG. 1C.FIG. 1 1 J shows a plot of spin noise diffusion correlation Crf(time T ) as a function of time T (in seconds or s) illustrating the beam width dependence in a cylindrical cell (d = 30.15 mm, / = 100 mm); and (inset) a plot of normalized power spectral density PSD as a function of f- ftannor (in Hertz or Hz) of the cylindrical cell.FIG. 1 IK shows a plot of spin noise diffusion correlation G / (time T) as a function of time r (in milliseconds or ms) comparing the recirculating cell according to various embodiments and a conventional cylindrical cell; and (inset) a plot of normalized power spectral density PSD as a function of f- ftarmor (in Hertz or Hz) of the cylindrical cell according to various embodiments and the conventional cylindrical cell.FIG. 1 IL shows a schematic depicting a screenshot of developed python software and interface used for the simulation according to various embodiments.DESCRIPTION

[0018] The following detailed description refers to the accompanying drawings that show, by way of illustration, specific details and embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. Other embodiments may be utilized, and structural, logical, and electrical changes may be made without departing from the scope of the invention. The various embodiments are not necessarily mutually exclusive, as some embodiments can be combined with one or more other embodiments to form new embodiments

[0019] Features that are described in the context of an embodiment may correspondingly be applicable to the same or similar features in the other embodiments. Features that are described in the context of an embodiment may correspondingly be applicable to the other embodiments, even if not explicitly described in these other embodiments. Furthermore, additions and / or combinations and / or alternatives as described for a feature in the context of an embodiment may correspondingly be applicable to the same or similar feature in the other embodiments.

[0020] In the context of various embodiments, the articles “a”, “an” and “the” as used with regard to a feature or element include a reference to one or more of the features or elements.

[0021] In the context of various embodiments, the terms “about” or “approximately” as applied to a numeric value encompasses the exact value and a reasonable variance, e g. within 10% of the specified value.

[0022] As used herein, the term “and / or” includes any and all combinations of one or more of the associated listed items.

[0023] By “comprising” it is meant including, but not limited to, whatever follows the word “comprising”. Thus, use of the term “comprising” indicates that the listed elements are required or mandatory, but that other elements are optional and may or may not be present.

[0024] By “consisting of’ is meant including, and limited to, whatever follows the phrase “consisting of’. Thus, the phrase “consisting of’ indicates that the listed elements are required or mandatory, and that no other elements may be present.

[0025] Various embodiments may increase or maximize both beam area to mirror area ratio and active-to-cell volume ratio in atomic magnetometers, while maintaining a high OD.

[0026] FIG. 2 shows a general illustration of a multi-pass cell according to various embodiments. The multi-pass cell may include a concave mirror 202 (i.e., a mirror with a concave reflecting surface). The multi-pass cell may also include a rotatable mirror 204 including a first mirror portion 204a and a second mirror portion 204b such that one or both of the first mirror portion 204a and the second mirror portion 204b are rotatable about a rotation axis passing through the first mirror portion 204a and the second mirror portion 204b.

[0027] In other words, various embodiments may relate to a multi-pass cell including a concave mirror 202 and a rotatable mirror 204, the rotatable mirror having two reflecting parts 204a, 204b, in which one or both parts 204a, 204b may be tilted about a common rotational axis passing through both parts 204a, 204b.

[0028] For avoidance of doubt, FIG. 2 seeks to illustrate certain features of a multi-pass cell according to various embodiments, and is not intended to limit, for instance, the shape, size, dimensions etc. of these features. For instance, in various embodiments, and as shown in FIG.2, the first mirror portion 204a may have a planar reflecting surface, while in various other embodiments, the first mirror portion 204a may have a concave reflecting surface. Likewise, and as shown in FIG. 2, the second mirror portion 204b may have a planar reflecting surface, while in various other embodiments, the second mirror portion 204b may have a concavereflecting surface. In various embodiments, both the first mirror portion 204a and the second mirror portion 204b may each have a planar reflecting surface or a concave reflecting surface, while in various other embodiments, one of the mirror portions 204a, 204b may have a concave reflecting surface, and the other one of the mirror portions 204a, 204b have a planar reflecting surface.

[0029] In various embodiments, the concave mirror 202 may have a reflecting surface (i.e., concave reflecting surface) facing the rotatable mirror 204. The reflecting surface of the first mirror portion 204a, and the reflecting surface of the second mirror portion 204b may be facing the concave mirror 202.

[0030] In various embodiments, the first mirror portion 204a may have a straight edge substantially perpendicular to the rotation axis. The second mirror portion 204b may also have a straight edge substantially perpendicular to the rotation axis. The straight edge of the first mirror portion and the straight edge of the second mirror portion may face each other. In various embodiments, the straight edge of the first mirror portion and the straight edge of the second mirror portion may be in contact with each other.

[0031] The first mirror portion 204a may have any suitable shape In various embodiments, the first mirror portion 204a may have a shape selected from a group consisting of a semi-circle, a rectangle, a half-ellipse and a triangle. The second mirror portion 204b may have any suitable shape. In various embodiments, the second mirror portion 204b may have a shape selected from a group consisting of a semi-circle, a rectangle, a half-ellipse and a triangle. In various embodiments, the shape of the first mirror portion 204a and the shape of the second mirror portion 204b may be substantially same, while in various other embodiments, the shape of the first mirror portion 204a and the shape of the second mirror portion 204b may be different.

[0032] During operation, one or both of the first mirror portion 204a and the second mirror portion 204b may be rotated about the rotation axis (e.g., an axis in y-direction) such that thereflecting surface of the first mirror portion 204a and the reflecting surface of the second mirror portion 204b may face different directions. However, the reflecting surface of the first mirror portion 204a and the reflecting surface of the second mirror portion 204b may still be oriented in such a manner to face different parts of the concave mirror 202, so that light (i.e., linearly polarized light) may be reflected between the first mirror portion 204a / the reflecting surface of the second mirror portion 204b and the reflecting surface of the concave mirror 202.

[0033] In various embodiments, the multi-pass cell may further include an alkali cell between the concave mirror 202 and the rotatable mirror 204. In various other embodiments, the concave mirror 202 and the rotatable mirror 204 may form reflecting walls of an alkali cell. The alkali cell may be configured to hold or contain an alkali metal vapor, such as sodium (Na), cesium (Cs), potassium (K) or rubidium (Rb). The alkali cell may also include a buffer gas, such as nitrogen (N2), helium (He) or neon (Ne).

[0034] In various embodiments, the multi-pass cell may be positioned within a magnetic shield, while in various other embodiments, the multi-pass cell may be positioned outside a magnetic shield. In various embodiments, the multi-pass cell may include non-metallic kinematic mounts for mounting the concave mirror 202 and / or the rotatable mirror 204 In various embodiments, the multi-pass cell may include actuator(s) coupled to the first mirror portion 202a and / or the second mirror portion 202b, the actuator(s) configured to control tilting of the first mirror portion 202a and / or the second mirror portion 202b.

[0035] In various embodiments, the multi-pass cell may include a barrier in the alkali cell to prevent alkali metal vapor atoms from reaching a tightly focused region of the alkali cell, i.e., a region within the alkali cell including a focus point of the concave mirror 202. In other words, the barrier may isolate the region of tight beam focusing from the alkali metal vapor atoms. As such, the barrier may help to reduce spin diffusion noise. For avoidance of doubt, thefocus point may occur anywhere within the alkali cell due to the concave mirror 202, and may not necessarily be at the focal length of the concave mirror 202.

[0036] FIG. 3 A shows a general illustration of an atomic magnetometer according to various embodiments. The atomic magnetometer may include a multi-pass cell 302 as described herein. The atomic magnetometer may also include a pump beam optical system 304 configured to provide circularly polarized light beam to pump atoms contained in an alkali cell contained within or at least partially defined by the multi-pass cell 302. The atomic magnetometer may further include a probe beam optical system 306 configured to provide linearly polarized light beam such that the linearly polarized light beam is reflected between the concave mirror and the rotatable mirror. The atomic magnetometer may additionally include a detector optical system 308 configured to detect an optical rotation of the linearly polarized light beam exiting or leaving the multi-pass cell 302.

[0037] In other words, the atomic magnetometer may include a multi-pass cell 302 as described herein, a pump beam optical system 304, a probe beam optical system 306 and a detector optical system 308

[0038] For avoidance of doubt, FIG 3A seeks to illustrate certain features of an atomic magnetometer according to various embodiments, and is not intended to limit, for instance, the shape, size, dimensions, arrangement etc of these features. For instance, while FIG 3A shows the incoming linearly polarized light beam (i.e., provided by the probe beam optical system 306 ) and the exiting linearly polarized light beam at opposite sides of the multi-pass cell 302, in various embodiments, the incoming linearly polarized light beam and the exiting linearly polarized light beam may be at a same side of the multi-pass cell 302, as shown in FIG. 3B. Further, while FIG. 3 A shows the circularly polarized light being substantially perpendicular to the incoming linearly polarized light, in various embodiments, the circularly polarized light and the incoming linearly polarized light may not be substantially perpendicular to each other.

[0039] In various embodiments, the atomic magnetometer may include one or more shields around the multi-pass cell.

[0040] In various embodiments, the detector optical system may include a polarizing prism, e.g., a Wollaston prism (WP), configured to separate or split the linearly polarized light beam exiting the multi-pass cell into separate beams. The detector optical system may also include a pair of photodetectors configured to detect the separate beams.

[0041] FIG. 4 shows a general illustration of a method of forming a multi-pass cell according to various embodiments. The method may include, in 402, providing or forming a concave mirror. The method may also include, in 404, providing or forming a rotatable mirror including a first mirror portion and a second mirror portion such that one or both of the first mirror portion and the second mirror portion are rotatable about a rotation axis passing through the first mirror portion and the second mirror portion.

[0042] In other words, various embodiments may relate to forming a multi-pass cell which includes a concave mirror and a rotatable mirror including a first mirror portion and a second mirror portion.

[0043] For avoidance of doubt, FIG. 4 seeks to illustrate certain steps of forming a multipass cell according to various embodiments, and is not intended to limit the sequence of the various steps. Step 402 may occur before, after or at the same time as step 404.

[0044] In various embodiments, the first mirror portion may have a straight edge substantially perpendicular to the rotation axis. The second mirror portion may also have a straight edge substantially perpendicular to the rotation axis

[0045] In various embodiments, the straight edge of the first mirror portion and the straight edge of the second mirror portion may face each other.

[0046] In various embodiments, the first mirror portion may have a shape selected from a group consisting of a semi-circle, a rectangle, a half-ellipse and a triangle. The second mirrorportion may have a shape selected from a group consisting of a semi-circle, a rectangle, a halfellipse and a triangle.

[0047] In various embodiments, the first mirror portion may have a planar reflecting surface, while in various other embodiments, the first mirror portion may have a concave reflecting surface. In various embodiments, the second mirror portion may have a planar reflecting surface, while in various other embodiments, the second mirror portion may have a concave reflecting surface.

[0048] In various embodiments, the method may further include providing or fonning an alkali cell between the concave mirror and the rotatable mirror. In various other embodiments, the concave mirror and the rotatable mirror may form reflecting walls of an alkali cell.

[0049] FIG. 5 shows a general illustration of a method of forming an atomic magnetometer according to various embodiments. The method may include, in 502, forming or providing a multi-pass cell as described herein. The method may also include, in 504, providing or forming a pump beam optical system configured to provide circularly polarized light beam to pump atoms contained in an alkali cell contained within or at least partially defined by the multi-pass cell. The method may further include, in 506, providing or forming a probe beam optical system configured to provide linearly polarized light beam such that the linearly polarized light beam is reflected between the concave mirror and the rotatable mirror. The method may additionally include, in 508, providing or forming a detector optical system configured to detect an optical rotation of the linearly polarized light beam exiting or leaving the multi-pass cell.

[0050] In otherwords, various embodiments may relate to forming an atomic magnetometer including a multi-pass cell as described herein, a pump beam optical system, a probe beam optical system and a detector optical system.

[0051] For avoidance of doubt, FIG. 5 seeks to illustrate certain steps of forming an atomic magnetometer according to various embodiments, and is not intended to limit the sequence of the various steps. For instance, step 504 may occur before, after or at the same time as step 506.

[0052] In various embodiments, the method may also include providing or forming one or more shields around the multi-pass cell.

[0053] In various embodiments, the detector optical system may include a polarizing prism configured to separate or split the linearly polarized light beam exiting the multi-pass cell into separate beams. The polarizing prism may be a Wollaston prism (WP).

[0054] Various embodiments may relate to or include a recirculating multi-pass alkali cell for advancing the performance of atomic magnetometers, resulting in an enhanced beam-to- mirror area ratio These characteristics may represent an advancement over existing multi-pass cells for atomic magnetometry. Specifically, various embodiments may increase the number of passes in a compact volume, which is fundamentally limited by the ratio of the mirror area to the beam area. Various embodiments may be an extension of the standard Herriott cavity, which may include two concave mirrors facing each other, with the light beam entering through a hole in one mirror and reflecting multiple times between the two mirrors. This process generates beam spots that form an elliptical pattern on both concave mirrors, as shown in FIG. 6A. FIG. 6A shows the elliptical beam spot pattern in a typical Herriott cavity with 60 reflections.

[0055] FIG. 6B is a schematic showing a cross-sectional side view of a multi-pass cell according to various embodiments. FIG. 6C is a schematic showing a perspective view of the multi-pass cell according to various embodiments To increase the number of beams passes while maintaining the same mirror size, a crucial aspect may involve halving one of the mirrors 604 to create two semi-circular mirror portions 604a, 604b, bottom mirror portion 604a (Ml) and top mirror portion 604b (MF), positioned opposite spherical concave mirror 602 (M2) at a distance d apart, as shown in FIGS. 6B - 6C. By tilting bottom mirror portion 604a (Ml) andtop mirror portion 604b (Ml’) in opposite directions about the y-axis (i.e., the rotation axis passing through mirror portions 604a, 604b), the optical center of the beam spots may shift parallel to the x-axis, effectively generating two Herriott cavities within the same volume. The beam spot patterns on the top mirror portion 604b (Ml’) may now feature a shifted optical center, denoted as Cl’, while the bottom mirror portion 604a (Ml) may exhibit a similar shift in the opposite direction, denoted as Cl This configuration may cause the probe beam spots to "recirculate" between mirrors bottom mirror portion 604a (Ml) and top mirror portion 604b (Ml’), forming half ellipses with centers at Cl and Cl’ respectively. As the circulation progresses between Cl and Cl’, the direction of circulation may reverse until exiting the cavity at an output position directly opposite to the input position. While a typical Herriott cavity may achieve 50 - 60 reflections, the recirculating cell design may significantly increase the number of reflections to 180 within the same mirror area.

[0056] The recirculating beam pattern may allow nearly the entire circular mirror surface of mirror 604 to be covered by beam spots, resulting in a nearly unity beam-to-mirror area ratio, whereas the cylindrical cell can cover at best 2 / TT of the same mirror surface. This may allow the atomic magnetometers to achieve an ultrahigh sensitivity despite miniaturization as an improved beam-to-mirror area ratio results in an improved active-to-cell volume ratio given that there are no overlapping beams. This means that the probe beam can fully utilize the entire alkali cell volume. Furthermore, the higher number of beam passes may increase optical depth (OD).

[0057] The fundamental sensitivity of atomic magnetometers may be determined by the quantum fluctuations of the ensemble of atoms and photons. Specifically, spin-projection noise may arise from the uncertainty in measuring the transverse spin projection because the transverse spin operators do not commute so they adhere to the uncertainty principle. The photon shot noise may arise from the uncertainty in measuring the light properties due to thequantum fluctuation in the number of detected photons. The quantum-noise-limited sensitivity of the magnetic field measurement is given by 8B = yj8B^PK+ 8B^SN, where 8BSPNand 8BPSNare respectively contributions from the spin-projection noise and photon shot noise. When spinexchange relaxation is not suppressed, it may lower transverse relaxation time T?.

[0058] The photon shot noise 8BPSNis described by BPSN= — — - A] — - — , where Pois theequilibrium spin polarization, yeis the gyromagnetic ratio of electron, n is atomic density V is active volume, Rpris absorption rate of photons from the probe beam, and rj is the quantum efficiency of the photodiodes used to detect the probe beam.

[0059] The spin projection noise 8BSPNis described by 8BSPN• A higher ODis required to improve photon-shot noise, while a higher active volume V improves both photon-shot noise and spin -projection noise. OD, is described by ODn is the atomic density and o(v) is the frequency-dependent photon absorption cross-section. OD can be increased by passing the probe beam through the alkali cell multiple times using a multipass cell, increasing interaction length I between the probe beam and the alkali cell.

[0060] As the multi-pass cell increases V, 8BPSNand 8BSPNmay reduce, and by increasing OD, 8BPSNcan be further reduced, improving the fundamental sensitivity of the atomic magnetometer. Furthermore, the multi-pass cell may amplify the optical rotation signal. As the probe beam propagates multiple times within the alkali cell such that interaction length / of the probe beam in the alkali cell increases, the optical rotation of probe beam 0 increases accordingthe radius of electron, Pxis the initial transverse spin polarization of the alkali atom, V is the Voigt profile, v is frequency of probe beam, vD1( / DI ) and vD2( / DZ )arethe resonance frequencies (oscillation strengths) of the DI and D2 transitions, respectively. Therefore, the improvedbeam-to-mirror area ratio allows the atomic magnetometers to achieve an ultrahigh sensitivity despite miniaturization.

[0061] Further, the recirculating multi-pass cell according to various embodiments may also form nearly collimated beams whereas the conventional cylindrical cells always result in tightly focused beams as they are an intrinsic feature of the cylindrical cell. The tightly focused beams may be detrimental as the decay of spin correlation due to atomic diffusion is primarily influenced by a small number of tightly concentrated beam spots within the cylindrical cell. Tightly focused beams can be avoided in recirculating multi-pass cells according to various embodiments by maintaining a distance between the rotatable mirror 604 (with the two mirror portions 604a, 604b) and the alkali cell. Furthermore, if the radius of curvature of mirrors 602, 604 used is far longer than the distance between mirrors 602, 604, the beams within the cell can become nearly collimated if total optical path length is much lower than the focal length of the concave mirror 602. This can decrease the spin correlation decay, resulting in a lower T2 time which further reduces SBPSNand 6BSPN.

[0062] The use of recirculating multi-pass cell for atomic magnetometry may be highly versatile as it has several implementation options. For example, instead of utilizing plane mirrors for Ml 604a and Ml ’ 604b, concave mirrors could be used instead, allowing for further increases in the number of passes, although at the expense of increased alignment complexity. Additionally, the multi-pass cell can be positioned either within or outside the magnetic shield, as the recirculating beam spot patterns can be generated over a broad range of distances. When located inside the magnetic shield, employing non-metallic kinematic mounts for mirror mounting can help to reduce or minimize Johnson noise. Conversely, when positioned outside the magnetic shield, mirror adjustment can be managed using actuators, opening up the potential for system automation and the application of machine learning techniques.

[0063] FIG. 7A shows (a) a schematic illustrating an alkali cell 706 between the concave minor 702 and the rotatable mirror 704 according to various embodiments; and (b) a schematic illustrating the concave minor 702 and the rotatable mirror 704 being used to form reflecting walls of an alkali cell 706 according to various embodiments. When the minors 702, 704 are positioned outside the alkali cell 706 as shown in FIG. 7A(a), the mirrors 702, 704 may be easily adjusted to obtain various beam patterns. However, the alkali cell 706 may require an anti -reflection (AR) coating 708, e.g., a double anti-reflection (AR) coating, to reduce the beam power loss, which increases with more beam passes. In order to circumvent this problem, in various embodiments, the mirrors 702, 704 may be placed within the alkali cell 706 (i.e., form walls of the alkali cell 706) using anodic bonding to reduce power loss, particularly when the cell has a high number of passes. The alkali cell 706 may further include intervening walls 710a, 710b connecting the concave mirror 702 and the rotatable mirror 704. An anti -refl ection (AR) coating 708, e.g., a double anti-reflection (AR) coating, may be applied on the intervening walls 710a, 710b. FIG. 7B shows a setup including the multi-pass cell shown in (a) of FIG. 7A according to various embodiments. The alkali cell 706 may include naturally occurring rubidium (Rb) including radioactive rubidium (Rb 87). The AR coating 708 may for instance seek to reflect less than 1% of the light incident onto the coating 708 at 780 nm. The concave mirror 702 may include a high reflectivity coating (99.8% at 780 nm) coated onto its reflecting surface. For the setup shown in FIG. 7A(b), the AR coating 708 may be optional. For the setups shown in FIGS. 7A(a) and FIG. 7B, where laser beams pass through the surfaces, a high-quality AR coating 708 optimized for the probe beam wavelength corresponding to the alkali species may be required.

[0064] Furthermore, by simply adjusting the multi-pass cell parameters without changing the design of the multi-pass cell, various beam spot patterns can be obtained, thereby enhancing the versatility of the magnetometer for different applications. For example, by introducing twolaser inputs at opposite positions on Ml to probe two atomic interaction areas within the same alkali cell, a gradiometer capable of measuring magnetic field gradients may be constructed. Since the reflections on Ml’, Ml and M2 are similar, different configurations of the recirculating cell may be possible, depending on parts available and ease of fabrication.

[0065] FIG. 8 shows schematics of two different configurations for mirror machining of a multi-pass cell according to various embodiments

[0066] FIG. 9A shows (left) a fluorescence image of the concave mirror according to various embodiments; and (right) a fluorescence image of an alkali cell according to various embodiments. FIG. 9B shows another fluorescence image of the alkali cell according to various embodiments.

[0067] FIG. 9C shows a schematic of an atomic magnetometer according to various embodiments. The atomic magnetometer may include a multi-pass cell 902 (which includes the rotatable mirror, the concave mirror and the alkali cell). The atomic magnetometer may also include a pump beam optical system 904 configured to provide circularly polarized light beam to pump atoms contained in the alkali cell contained within or at least partially defined by the multi-pass cell 902 The atomic magnetometer may further include a probe beam optical system 906 configured to provide linearly polarized light beam such that the linearly polarized light beam is reflected between the concave mirror and the rotatable mirror. The atomic magnetometer may additionally include a detector optical system 908 configured to detect an optical rotation of the linearly polarized light beam exiting the multi-pass cell 902. The linearly polarized light beam from the probe beam optical system 906 may undergo the optical rotation due to interaction between the linearly polarized light and the pumped atoms contained in the alkali cell.

[0068] The atomic magnetometer may include a vacuum chamber 910 in which the multipass cell 902 is arranged. The atomic magnetometer may also include one or more shields (e.g., 1Mu-metal shields 912a and ferrite shield 912b) around the multi-pass cell 902 and the vacuum chamber 910. The one or more shields 912a, 912b may allow circularly polarized light beam from the pump beam optical system 904 and linearly polarized light beam from the probe beam optical system 906 to enter the multi-pass cell 902, and may also allow linearly polarized light exiting the multi-pass cell 902 to pass through to the detector optical system 908, e.g., by including openings, gaps or holes.

[0069] The pump beam optical system 904 may include a pump source 914 configured to generate a light (e g., a nearly linearly polarized light), and a polarizer 916 and quarter wave plate 918 (QWP) to generate the circularly polarized light based on the light from the pump source 914. The probe beam optical system 906 may include a probe source 920, a half wave plate 922 (HWP) and a polarizer 924 to generate linearly polarized light The probe beam optical system 906 may also include mirrors 926a, 926b to direct the linearly polarized light from the probe beam optical system 906 to the multi-pass cell 902.

[0070] The detector optical system 908 may include a polarizing prism 928, e.g., a Wollaston prism (WP), configured to separate or split the linearly polarized light beam exiting or leaving the multi-pass cell into separate beams of s- and p- polarized components, and a pair of photodetectors 930 configured to detect the separate beams. The detector optical system may also include mirrors 932a, 932b to direct the linearly polarized light exiting or leaving the multipass cell 902 to the polarizing prism 928.

[0071] Theory of Spin Noise in Atomic Vapor Cells

[0072] The sensitivity of atomic magnetometers is fundamentally constrained by spin projection noise, which sets a quantum limit on measurement precision. Additionally, spin noise due to diffusion introduces an extra layer of classical noise, further degrading sensitivity. This makes a thorough understanding of spin noise essential. Given the rich correlations present in atomic vapors, correlation functions are crucial for capturing the temporal and spatial dynamicsof atomic spin noise. While spin projection noise sets a fundamental quantum limit on measurement sensitivity, spin noise due to diffusion adds an additional layer of classical noise, further degrading sensitivity. By the Wiener-Khinchin theorem, the spin noise power spectral density S(f) of the optical rotation signal can be expressed as the Fourier transform of its time autocorrelation. This relationship simplifies towhere S(f) is expressed in terms of the spin noise time-correlation function C(T), which is given by the normalized time autocorrelation of the Faraday rotation <|> of the probe beamThis correlation function forms the basis for understanding how spin noise is influenced by spin projection-induced Faraday rotation, which we explore in the following section.

[0073] Spin Noise Time-Correlation Function

[0074] The spin noise time-correlation function resulting from atomic diffusion may be derived to quantify the decay of atomic spin correlations over time. Given that the probe beam propagates along the z direction, and thereby measures atomic spins proj ected onto the z-axis, the spin expectation value in z, denoted as ation of szat positions ri and r? is given bywhere F is the total atomic spin, nvis the atomic density and d (n - n) is the Dirac delta function which represents spatial coincidence at ri = m The spin expectation values szfor randomly polarized atoms may be correlated only at the same position. This correlation1 1 • r» i • • i i ? > i / 7\ 2F + 1 1 F(F+1) equals the variance or the spin expectation value given by }F=2(-2J+1^ (2 / +1j2 — 3 — •Given a time lag T, the evolution of atomic spins in vapor cells with buffer gas is may be described bywhere the spin dynamics depend on the spin precession at Larmor frequency COL, the transverse spin relaxation time Ti, and G(ri-r2,r), the Green’s function describing the spatial distribution of diffusing spins over the time lag, which is given bywhere D is the diffusion constant.

[0075] From Equation 4, the correlation of spin expectation values at n and at n may be derived given a time lag T, which can be simplified to{sz(r1, t')sz{r2, t + T^p = cos(mLT) e“T / L f C(r3- r2,z)(sz(r1, t)sz(r3, t))Fd3r3, (5) where r3is introduced as a dummy variable for the integration to distinguish it from n and n. As the spin expectation values 5z(n, / ) and 5z(r3, / ) are the variables that need to be averaged, the constants cos(« / r),Gr'2, and G(r3-r2, r) are independent of these spin variables and can therefore be factored out of the averaging operation {d)F.Substituting Equation 3 into Equation 5 giveswhich can be simplified intoSince the 8^ - r3) is only non-zero at ri = r3. This expression represents a generalized form of Equation 3 as it includes the effects given a finite time lag T. Intuitively, this reflects how atoms initially at ri diffuse to n over time r, causing the spin at ri to become correlated with the spin at r? at the time lag T.

[0076] Optical Rotation Correlation

[0077] Since spin projection fluctuations of diffusion-driven atomic spins along the probe beam affect the polarization rotation of an off-resonant, linearly polarized probe beam, how this manifests as optical rotation may be exploited. Specifically, when there is a spinprojection along the probe beam, it induces birefringence, leading to a measurable rotation of the probe beam’s polarization. The paramagnetic Faraday rotation of the probe beam is given bywhere reis the radius of electron, fOSc is the oscillator strength and / (r ) is intensity distribution of the probe beam. The term D(y — vF) = represents the imaginary part of theLorentzian profde. The Lorentzian lineshape is as an approximation of the Voigt profile, which characterizes the magnetometer response when the cell contains a sufficient amount of buffer gas. Here, v denotes the probe beam frequency, VF represents the resonance frequencies of the DI and D2 transitions and T is the pressure broadening linewidth with full width at half maximum (FWHM).

[0078] From Equation 8, the time autocorrelation of rotation may be given by:

[0079] Similarly, from Equation 8, the variance of the optical rotation can be derived as

[0080] Substituting Equations 9 and 10 into Equation 2, the time correlation function may be obtained:

[0081] The diffusion component Cd(f) is given bywhich can be understood as the likelihood that atoms initially in the probe beam remain within it after time T, even if they diffuse out and later return. It depends solely on the intensity distribution of the probe beam and the Green’s function. Therefore, to determine Cd(i) for multi-pass cells, the intensity distribution of the probe beam within such cells would need to be first characterized. The recirculating multi-pass cell according to various embodiments, such as that shown in FIG. 6C, may feature a high number of passes to reduce photon shot noise and a large active volume that enhances both photon shot noise and spin projection noise performance.

[0082] Modeling

[0083] Both computational and analytical modeling may be utilized to understand complex optical behavior and predict the performance of recirculating multi-pass cells, facilitating the design and optimization of the number of passes. Furthermore, modeling may enable sensitivity analysis, allowing researchers to assess the system's tolerance to various parameters, providing insights that aid experimental efforts.

[0084] Computational Modeling

[0085] To visualize and optimize key design parameters of the multi-pass cell, simulations were conducted using Zemax OpticStudio, which is useful for handling multi -parameter problems. Using this software, a virtual model is constructed using mirrors and detectors, enabling precise adjustment of cavity and laser parameters to achieve optimal performance. The software facilitates analyses by calculating beam paths, thereby providing both visual representations of beam propagation and reflection patterns. FIG. 10A shows atable illustrating the simulation parameters used, including cavity parameters and laser parameters according to various embodiments. In particular, a laser with wavelength of 780 nm, corresponding with the D2 transition of Rubidium atoms essential for spin probing, is used. Moreover, the spatial constraints of the vacuum chamber housing dictate the distance between mirrors In the simulation, Ml was configured as a plane mirror with an infinite radius of curvature (Rl), while R2 (radius of curvature of M2) was set at 2000 mm. Key variables such as input angles, input positions, Rl, and R2 were systematically varied to optimize system performance. The simulation results confirm that the recirculating multi-pass cell according to variousembodiments may be able to achieve a high number of passes with a high beam area to mirror area ratio. FIG. 10B shows (a) simulation results of the recirculating beam spot reflections on the concave mirror M2 according to various embodiments; and (b) a three-dimensional (3D) ray tracing diagram showing optical path of the beam within the multi-pass cell according to various embodiments. The parameters used are: Ml rotation = 0°, Ml’ rotation = -0.04°, Incident angle [1.2°, 0], d = 86.455 mm, R2 = 2000 mm, number of reflections = 120, and number of passes = 240. An analysis of beam spot distribution is conducted by adjusting the rotation of Ml’ to determine the system's tolerance, as illustrated in FIG. 10C.

[0086] FIG. 10B(a) shows the intended recirculation of beam spot patterns observed on M2, showing 120 reflections on a mirror of 1 inch in diameter, corresponding to 240 beam passes as the number of passes is twice of the number of reflections on one mirror. This may enable a high active-to-cell volume in an alkali cell placed within the cavity. FIG. 10C shows the different beam spots patterns in the recirculating multi-pass cell according to various embodiments at different rotations of the second mirror portion Ml’. Parameters used are: Ml rotation = 0°, Laser input angle [1.2°,0.0°], Input position [8.0,0], and Output position [-8.2,0], Specifically, at a Ml ’ rotation of -0.04°, 90 beam spots and 4 circulations are observed. However, with further rotation of Ml’, both the number of beam spots and circulations decrease. For instance, at a Ml’ rotation of -0.12°, the number of beam spots and circulations decreases to 30 and 2, respectively.

[0087] Analytical Modelline

[0088] The optical path within the recirculating cell is first determined to accurately calculate the spin noise.

[0089] FIG. 11 A shows (above) a multi-pass cell including the concave mirror (M2), bottom mirror portion (Ml) and top mirror portion (ML), similar to that shown in FIG. 6C according to various embodiments; and (below) the beam spot pattern observed on the bottom mirrorportion (Ml) and top mirror portion (Ml’) according to various embodiments showing 78 reflections. FIG. HA shows an incident beam entering from one side of Ml and exiting from the opposite side The internal beam paths are simplified for clarity and do not represent the actual number of passes. Various embodiments may allow for nearly collimated beams. Various embodiments may achieve at least 57% more coverage as compared to a conventional multipass cell

[0090] As mentioned above, the recirculating multi-pass cell, based on the Herriott cell design, may be understood as multiple Herriott cells within a single, compact cavity. It may include Ml and Ml’, each with focal length / i, and M2, with a focal length / i. Mirrors Mi and Ml’ may be positioned at distance d from M2, and they may form the lower and upper halves, respectively. They are tilted in opposite directions by small angles ftc (for Ml) and Ox (for Ml ’) about the central axis (i.e., rotation axis) of the mirror pair, aligned along they- axis, forming a recirculating optical pattern with substantially more passes than the Herriott cell. To examine the spread of beam spots in the recirculating cell, the calculation of their distribution as observed on the mirrors Ml and Ml’ may be focused on.

[0091] Beam Spot Distribution of Recirculating Multi-pass Cell

[0092] The distribution of beam spots within the recirculating multi-pass cell may be analyzed using ray transfer matrix analysis. As the rays travel between the mirrors repeatedly, the transfer matrix of the ray within the multi-pass cell can be simply described using the matrix for a single round trip:

[0093] Such that the ray enters from Ml, travels a distance d to M2, reflects off M2, then returns to Ml and reflects off it. Given that Ml and Ml’ are plane mirrors (fi co), the y- position y„ and y’nslope of the nth reflection off Ml can be described using the transfer matrix as:where yo and ) ’u are the initial y-position and y slope. yncan be rewritten in the form:where 9 — arccos is the angle 9 between the beam spots on themirrors. 9 depends solely on the focal length j2' of the mirror and the distance d between the mirrors, and does not change with the position and angle of the incident light The x-position x„ and x slope x can be described in a similar way when Mi and M l ’ are not tilted, and xncan be written in the form:

[0094] The quantities X and Y are the maximum amplitude of the beam spot in the x and y direction Equations 15 and 16 describe the beam spots on the mirrors of a typical Herriot cell, which traces an elliptical path such the amplitude of the beam spot distribution (X and F) increases with increasing the position and angle of the incident light.

[0095] However, x„ and x ’„ differ in the recirculating multi-pass cell due to the tilt of Ml and Ml’, which introduces an offset of the optical center. This offset is precisely what enables the recirculating multi-pass cell to achieve a higher number of passes compared to Herriot cells. Given a small rotation of 0Xof a mirror, x and x' changes such thatwhere 5 is a specific identifier indicating the incident beam on the rotated mirror. Since the angular components of the parameters are relative to the normal of the mirror surface, and in a rotated mirror system, the normal also rotates by the same angle it is necessary to include an additional term to ensure that the ray transfer matrix parameters are referenced back to the original coordinate system. This can be expressed by the following formula:

[0096] Thus, a single round trip may be given bywhich describes a recursive relationship where a perturbation term is added for each round trip due to reflection from a tilted mirror. Consequently, the parameters after n reflections from the rotated mirror can be expressed as:where the first term represents the ray characteristics of a non-rotated mirror, while the second term sums up the perturbations arising from the rotated mirror. This equation relies on the specific case that simplifies Equation 18 - mirror Ml is flat, with a focal length that is effectively infinite.

[0097] The y-offset, A„, where the subscript n indicates the number of times the rotated mirror reflects, corresponds to the first term of the second component in Equation 20. The following expression may be derived:Hn= B ^=1Ui^ x 2Qx, (21) whereand0 = arccos

[0098] By substituting specific parameters, Equation 21 can be simplified to:An- 20Yx 2df - c / 2Xiti sin i 0, (24) indicating the offset for each beam spot reflected off the mirror. This offset accumulates with increasing reflection number, as shown in FIG. 11B. FIG. 11B shows the beam spots positions on the top mirror portion (Ml ’) , with hollow circles marking their positions beforethe mirror portion’s rotation and filled circles indicating their positions after the rotation according to various embodiments. Notably, the offset of the last reflected spot (after a 180° turn about the optical center) is approximately twice the offset of the optical center Axsuch that Ax= 2A„. Substituting this into Equation 24, the offset in the optical center may be expressed as

[0099] To calculate the x-position of the beam spots on the recirculating multi-pass cell, the offset can simply be added to the initial position to account for the tilt of the mirrors. Assuming the beam spot rotates counterclockwise upon laser incidence into the cavity, the first influence on the position offset is from mirror Ml . At this point, the position of the beam spot can be expressed as shown in Equation 20, where the summation term mo is the difference between the parameter n for the desired position calculation and the parameter no for the first reflection on Ml .After a 180° anti-clockwise circulation about the optical center, the beam now enters Ml’ and the position of this beam is given by Equation 26. As there is a rotation in Ml ’ relative to Ml at angle of 0] — 0Xnow the summation term is instead m\ which is the difference between the parameter n for the desired calculation and the parameter m for the first reflection on the second affected mirror Ml ’. It subsequently undergoes further circulations, making this a recirculating multi-pass cell. As the beam spots recirculate, they alternate between Mi and Ml’. The integer k may be defined to describe the circulation number of a beam spot and whether it is located on Ml or Ml ’ such that k iswhere the floor function ensures that k takes integer values.

[0100] k may be derived by taking the floor function of the reflection angle parameter divided by TT. This computation allows k to quantify the number of recirculation cycles encountered by the light within the optical system. Additionally, the parity of k — whether it is odd or even — indicates the positional context of the reflection. Considering that the beam spots alternate between Mi and Ml’, Equation 26 may be modified to arrive at the x-position of the / / -th beam spot given bywhere each change of the reflection mirror adds another summation term to the position description, where each term’s summation index m, is the difference between the parameter n for the desired calculation and the parameter rij for the first reflection on the corresponding mirror. Here, 6j represents the relative rotation angle of the mirror, taking values of 0x, ~20x, 20x, ~20x, 20x, and so on Therefore, beam spot positions may be obtained that indicate spatially distributed beams through the alkali cell, which increase active volume of measurement.

[0101] FIG. 11C shows beam spot distributions on mirror portions Ml and Ml’, along with shift of optical centers, vary with the relative rotation between the two mirror portions according to various embodiments. As shown in FIG. 11C, mirror portions Ml and Ml’ rotate in opposite directions by the same angle, denoted as Az= — .x' = A. This causes the optical center offset of Ml to be in the positive x-direction, and Ml’ in the negative direction. The incident light enters at the position (xo,O) with an angle of (0, y ’o ), where y ’o is set to a negative angle, enabling the light spots to rotate clockwise. The schematic diagram below specifically illustrates the configuration of Ml and Ml’ within the cavity design. This configuration extends the optical path length of the probe beam through the alkali cell, while spatially distributing the beam across the cell. Furthermore, since the optical path length of the recirculating cell affects the spin noise dynamics of the atomic vapor, the total number ofallowed reflections in this type of cell may also be determined, as it is proportional to the path length.

[0102] Total Number of Reflections

[0103] Next, the total number of possible reflections on the mirror may be determined, which depends on the parameters of the multi-pass cell. The initial beam spot on Ml originates from (x«, 0) and first passes through the Ml section, where the beam center is at (A, 0). This trajectory intersects the x-axis again at (2A - xo, 0). Subsequently, the light passes through the Ml’ section, where the beam center is at (-A, 0). Thus, after one complete cycle, the light returns to (xo - 4\, O)'. Following this pattern, after N cycles, the position can be represented as (xo ~ 4NA, O'). Given the symmetrical nature of this design, the criterion for the exit condition is that the position must be less than xo. Therefore, the total number of recirculation required to meet this condition can be calculated based on the inequality: xo - 4NA < - xo- (29)

[0104] Since the number of recirculation, N, must be an integer, it can be calculated using the ceiling function. With N recirculation, the total angular displacement of the light spots amounts to ITTN. Considering Equation 23, where the angle between two consecutive reflection points is 0, the total number of reflectionson MI can be expressed as follows:

[0105] Furthermore, the total number of reflections depends on both the initial input angle in the y-direction, y ’o, and the distance between the mirrors, as shown in FIG 1 ID FIG. 1 ID shows (a) a plot of the initial input angle in the y-direction, y ’o (in radians or rad) as a function of the distance (in millimeters or mm) between the concave mirror and the rotatable mirror illustrating the calculated total number of possible reflections in a recirculating cell as a function of the distance between mirrors and input angle in the y-direction at z = 780 nm, CJC, = 1 mm, 0:, = 0.02°, x’o = 0°, and fi = 1 m according to various embodiments; and (b) a three-dimensional (3D) plot of number of reflections as a function of the distance (in millimeters or mm) between the concave mirror and the rotatable mirror and the initial input angle in the y-direction, y ’o (in radians or rad) of the recirculating cell according to various embodiments. Each data point represents a specific combination of these parameters, with the shading and size indicating the number of reflections. For each mirror separation, there may exist a specific range of allowable}' Rvalues. Tn general, y ’u must increase as the mirror separation decreases to ensure that the beam remains within the mirror’s radius. The number of reflections increases quicky with distance at shorter distances, which, while beneficial for increasing interaction length, may also suggest greater sensitivity to slight variations in mirror separation, making precise alignment more challenging. As the mirror distance increases, the required y ’o to achieve a high number of reflections stabilizes, saturating at around 1.25 mm.

[0106] Beam Width

[0107] In addition to the total number of allowed reflections, the focusing of the probe beam (beam width) may also influence the spin noise of the atoms in the multi-pass cell. To account for this effect, the beam width throughout the cell may be derived, allowing the calculation of the beam’s intensity distribution. This, in turn, enables us to determine Cd in Equation 12. Assuming that the probe beam propagates as a Gaussian beam, the complex beam parameter at Ml and Ml ’ changes with reflection number n such thatwhere the ABCD matrix is given by Equation 13. The initial beam parameter is q0= z + A with position z, beam waist coo and wavelength A. In general, the complex beamof the beam’s wavefront, Cis the refractive index of the medium, and w is the beam radius,which is defined as the distance from the beam center at which the intensity falls to I t'2(13.5%) of its maximum value on the axis. It is given by w(z) = w01 + ( — ) where ZR is the Rayleigh range, defined as zR— —In the case that that the first reflection is positioned at the beam waist of the initial beam, the initial beam has z = 0 set at mo = 1 mm. The beam radius of the / / -th successive reflection on Ml or Ml’ is given by:FIG. HE shows a plot of beam radius (in millimeters or mm) as a function of number of round trips for various mirror separations d according to various embodiments. FIG. HE shows the effect of varying mirror separations on the beam radius for the first 20 reflections, calculated from Equation 32 for plane mirrors M1 and Ml ’ and a concave mirror M2 with fa = 1000 mm, coo = 1 mm. The beam radius varies periodically with successive reflections, with a larger mirror separation resulting in a longer period of oscillation.

[0108] Validating Analytical Model Against Ray-tracing Simulation Results

[0109] To validate the analytical calculations, the predicted beam positions and widths are compared with those obtained from commercial ray-tracing simulations using Zemax.The simulation parameters are set as follows: the focal length of mirror M2 is / 2 = 1000 mm, the inter-mirror distance is d= 86.455 mm, the rotation angles A. of mirrors Ml and Ml’ are 0.02° and -0.02°, respectively. The incident beam has positions xo = 11 andyo = 0, and angles x ’o= 0, and y’o = 1.2°. A Gaussian beam is simulated by using 100 light rays as represented by the black scatter points on the mirror surface in FIG. 1 IF. FIG. 1 IF shows the analytical beam positions and waist (hollow circles) and simulated light rays (scatter points) according to various embodiments. The black hollow circles indicate the beam positions and width calculated analytically using the same parameters as the simulation. These parameters also give a 0 = 0.41887°. This implies that in a single recirculation, there are 2it / d ~ 15 reflections,with 7 reflections occurring on a single rotated mirror. The offset of the optical center A, calculated using Equation 24, is 0.6754 mm. Substituting this into Equation 30, the total number of reflections is found to be 120, which is consistent with the simulation results.

[0110] Furthermore, to validate the analytical beam positions, the average position of the rays may be calculated to determine the spot location. This approach was chosen because it reduces random errors associated with the position of individual rays, thereby providing a more accurate representation of the actual spot location. The average error distance for the 120 reflection positions is found to be 0.0122 mm.

[0111] The beam width evolution is observed to follow the expected periodic variations in FIG. HE. The calculation is further validated by computing the proportion of the light rays within the beam width at each beam spot. For a Gaussian beam, the spots within the beam width should contain 87% of the total number of spots. The proportion of light spots within the beam width converges towards 87% with increasing number of rays simulated, demonstrating the validity of the analytical model, which may now be applied to the calculation of the spin-noise diffusion correlation function in multi-pass cells. Moreover, the model can provide the distribution of the beam waist size as the probe beam multiply propagates through the cell.

[0112] Spin Noise in Multi-pass Cells

[0113] A model is proposed to derive the spin-noise diffusion correlation function for a general multi-pass cell by solving Equation 12 using the intensity distribution of the multipass beams Specifically, it may be required to account for beam ellipticity caused by astigmatism, particularly from reflections off cylindrical mirrors, as they introduce asymmetry in the beam profile. The intensity distribution of an elliptical Gaussian beam can be expressed as / (^?7,z) = \E^,z)E,,(t ,z)\, (33)where E is the relative field strength of an elliptical Gaussian beam with principal axes c:and q. For a circular Gaussian beam, E{= E,hhence its intensity distribution can be simplified to1= |E(r,z)|2. (34)

[0114] In general, E(f) is expressed aswhere q(z) is the complex beam parameter across the probe beam’s travel in z. By propagating the beam in z after each reflection on Ml, the complex beam parameter for the 11thpass is obtained, which is given by q(z) = qn+ z such that qnis the beam parameter on Ml and z is the probe beam’s travel within that round trip q„ is given by q„ = R"#o (36) for the round trip matrix R of any multi-pass cell. For ease of derivation, Equation 35 is , giving E(r, z) =quation 34, the intensity distribution can be expressed in a more concise form:

[0115] Equation 12 is now solved by evaluating the numerator and denominator separately. Using Equation 37, the numerator of Equation 12, J / (r1) / (r2) G (r1— r2, t) d3r1d3r2„ for the nlhround trip can be expressed as

[0116] Assuming that the cell size is much larger than beam width, we can subsequently integrate with respect to x and y from -co to co to obtain

[0117] To further simplify the evaluation, it = z\~Z2 and v = zi+z2are substituted intoEquation 39 to obtain:

[0118] Since the spin noise at zi is only weakly correlated with that at Z2 when the distance |zi Z2I is much greater than the cell length, it can be assumed u « d. Taking u = 0 everywhere except at the Gaussian function e “' / 4Dt, which is even, the numerator becomesUsing the definition of the error function, it is found that fgdsu2 / 4Dxdu — \ / TT7JT X erf , which can also be written in terms of the complementary error function: erf ^^L) — 1 — erfc (-4=y For large d, the asymptotic expansion may be applied giving crfc (- large and D is small, this expression tends to 0, therebyarriving at the simplified numerator:

[0119] The denominator of Equation 12, J / (r)2d3r, for the nthround trip can be expressed as:

[0120] Finally, the normalized spin noise diffusion correlation Equation 12 for multi-pass cells can be evaluated by taking the sum of Equation 42 over all passes divided by the sum of Equation 43 over all passes.

[0121] Equation 12 can be solved in a similar manner for astigmatic Gaussian beams, but using Equation 33 instead. Since the focal points of the two principal axes do not coincide in astigmatic beams, two distinct ^-parameters, i / andare propagated, / t and £,?may then becomputed using Equation 35, which allows us to obtain Equation 33. The numerator of Equationand the denominator, f I (r)2d3r, is given bySimilarly, the normalized spin noise diffusion correlation Equation 12 for astigmaticGaussian beams in multi-pass cells can be evaluated by taking the sum of Equation 44 over all round trips divided by the sum of Equation 45 over all round trips. With R defined for any multi-pass cell, the spin-noise diffusion correlation function for the atoms within the cell can be derived.

[0122] Spin Noise of Recirculating Multi-pass Alkali Cells

[0123] To study the spin noise in the recirculating multi-pass cell in Equation 12, round trip matrix R from Equation 13 is taken. For a beam reflecting off a concave mirror, the focal length in the tangential and sagittal direction is fan = f cos(? and / ]ag= / zcos<^ respectively, where is the angle between the beam and the normal of incidence. For small <j>, kf=fian~ fsaS~ f(p / 2 ~ 0. Therefore, the beams in the recirculating multi-pass cell can be considered approximately stigmatic. FIG. 11G shows a plot of spin noise diffusion correlation Qftime r ) as a function of time T (in seconds or s) calculated illustrating normalized from Cd of a recirculating cell with 78 reflections and d = 30 mm according to various embodiments; and (inset) a plot of normalized power spectral density PSD (in per Hertz or 1 / Hz) as a function of f- ftarmor (in Hertz or Hz) of the recirculating cell according to various embodiments. The normalized from Cd is calculated from Equation 42 and Equation 43 which are applicable to stigmatic beams. The spin noise power spectral densities are also calculated using Fouriertransform of the correlation, with the spectra normalized to their peak values. The calculation is performed for wo = 1 mm and coo = 10 mm, and focal lengths f= 1 mm and / = 10 mm. In these calculations, the total number of reflections nx = 78, cell length 1 = 30 mm, y = 780 nm, cell temperature T= 120°C and buffer gas pressure = 70 Torr. In particular, for alkali vapor cells containing Rb atoms and N2 buffer gas, the diffusion constant Do = 0.159 cm2 / s at temperature 7o = 60 °C and pressure po = 760 Torr is used. The diffusion constant O(pRb-N2) is given by D which depends on the temperature and pressure ofthe rubidium (Rb) and nitrogen (N2 ) mixture in the alkali cell. If different alkali metal vapors, such as sodium (Na), cesium (Cs), or potassium (K), or buffer gases, such as helium (He) or neon (Ne), are used, the diffusion constant in the calculations may need to be adjusted accordingly.

[0124] Interestingly, a larger beam waist may result in a more rapid decay of the spin correlation coefficient Cd, which is somewhat counterintuitive. Analysis also reveals that regions of high optical intensity — typically those with tight focusing — may contribute disproportionately to spin diffusion noise. These regions, despite their strong weighting in the correlation function (as described by Equation 12), may offer limited spin coherence due to their confined spatial scale and the quick transit of atoms through them.

[0125] To mitigate this effect, the inclusion of a barrier that restricts atomic motion in these high-intensity zones for a single pass may be simulated. FIG. 11H shows a plot of spin noise diffusion correlation Q(time r ) as a function of time r (in seconds or s) illustrating a slower decay of the spin correlation function by inserting a barrier to prevent atoms from entering the tightly focused region according to various embodiments; and (inset) normalized power spectral density PSD as a function of f- fLu mor (in Hertz or Hz) illustrating the reduction of spin diffusion noise due to insertion of the barrier according to various embodiments. This modification may lead to a notable improvement in spin diffusion noise performance. Sucha strategy may prove especially effective when using concave mirrors with shorter focal lengths, where tight focusing would otherwise degrade noise characteristics.

[0126] Comparison With Conventional Cylindrical Multi-pass Alkali CellsCylindrical multi-pass cells have been used in scalar and vector atomic magnetometers based on fast rotating fields, demonstrating unprecedented accuracies. FIG. HI shows a conventional cylindrical cell and the resultant beam spot distribution, similar to FIG 1 C As shown in FIG. HI, an incident beam enters a hole in the center of Ml and exits from the same hole. The internal beam paths are simplified for clarity and do not represent the actual number of passes. Arrowed spheres indicate atomic spins. The Lissajous pattern observed on M2 (bottom) shows unavoidable large unfilled regions and tightly focused regions.

[0127] Furthermore, the spin noise in the cylindrical multi-pass cell is calculated to compare it with that in the recirculating multi-pass cell. Since the beams in the cylindrical cells are astigmatic, Equations 44 and 45 may be used to calculate Cd. FIG. 11 J shows a plot of spin noise diffusion correlation (time T ) as a function of time T (in seconds or s) illustrating the beam width dependence in a cylindrical cell d = 30.15 mm, / = 100 mm); and (inset) a plot of normalized power spectral density PSD as a function of f- f aimor (in Hertz or Hz) of the cylindrical cell.

[0128] When comparing the two types of multi-pass cells, the results for the cylindrical configuration are reproduced from a prior reference, while the recirculating multi-pass cell was simulated under identical conditions: 42 passes, a mirror separation of 3 cm, and an initial beam waist diameter of 1 .9 mm. The cylindrical cell may include one planar mirror and one concave mirror with a focal length of 5 m. FIG. 1 IK shows a plot of spin noise diffusion correlation Cd (time r) as a function of time T (in milliseconds or ms) comparing the recirculating cell according to various embodiments and a conventional cylindrical cell; and (inset) a plot of normalized power spectral density PSD as a function of f- f armor (in Hertz or Hz) of thecylindrical cell according to various embodiments and the conventional cylindrical cell. As shown in FIG. 1 IK, the recirculating cell exhibits a slower decay in the correlation function, indicating reduced spin diffusion noise. It is also important to note that the original cylindrical cell calculation did not account for beam spot overlap. Since the recirculating design features significantly less overlap, the previously calculated spin correlation coefficient Cd for the cylindrical cell is likely an overestimate. FIG 1 1L shows a schematic depicting a screenshot of developed python software and interface used for the simulation according to various embodiments.

[0129] Various embodiments may relate to a recirculating multi-pass alkali cell designed to enhance the active-to-cell volume ratio, thereby addressing key spin noise limitations in traditional cylindrical Herriott cavity multi-pass cells Through an analytical model based on the ABCD matrix approach, the laser beam distribution within the cell may be predicted, with results closely matching Zemax simulations. The model may also account for astigmatism, which plays a significant role in determining power intensity distribution and its subsequent impact on spin diffusion noise.

[0130] A detailed analysis of shot noise characteristics using correlation function methods has been conducted. Unlike conventional cylindrical cells — which exhibit large unfilled regions due to Lissajous-pattern beam spot distributions — the proposed recirculating cell may achieve more uniform spatial coverage. This results in a substantially larger active volume V , defined as the overlap between the laser beam and polarized atomic spins, which directly improves spin projection noise and photon shot noise performance, both of which scale as V1 / 2.

[0131] By utilizing concave mirrors with longer focal lengths, the recirculating cell may further demonstrate superior spin correlation and reduced spin diffusion noise. Importantly, prior diffusion noise analyses often neglected beam spot overlap (as shown in FIG. I ll),which can lead to worsening spin noise calculated in cylindrical multi-pass cell designs. In contrast, various embodiments may reduce or minimize this overlap (FIG. 11 A), resulting in lower diffusion-induced noise

[0132] It has also been found that tightly focused regions, due to their high optical intensity, may degrade overall spin correlation due to their small spatial extent and rapid atomic transit. According to Equation 12, such regions may be heavily weighted yet weakly correlated, leading to degraded noise performance. To address this, a barrier may be introduced to prevent atoms from entering these regions, which significantly reduced spin diffusion noise. This approach may be particularly useful when employing short focal length concave mirrors while maintaining low diffusion noise levels.

[0133] Overall, various embodiments may offer a practical advancement in multi-pass cell design, enhancing both theoretical understanding and experimental performance for high-sensitivity applications such as atomic magnetometry. The developed analytical model may not only aid in predicting beam behavior and astigmatism but may also hold potential for broader applications, including optical quantum memory. Furthermore, the approach to spin diffusion noise modeling as described herein may contribute to improving the fundamental noise limits of optical magnetometers and supports techniques like quantum non-demolition (QND) detection by leveraging the high optical depth achieved through multi-pass geometries.

[0134] Various embodiments may have applications in quantum sensing, biomedical research, geomagnetic exploration, defense and security, navigation and positioning. And / or non-destructive testing. For instance, the recirculating multi-pass cell may be employed in quantum sensing technology that can benefit from measurements requiring high OD, including technologies such as the atomic magnetometer, quantum memory based on warm atomic cell and radio frequency (RF) sensing based on Rydberg atoms. In biomedical research, atomicmagnetometers may be employed in neuroscience research for studying brain activity, biomagnetic fields, and neuronal signaling. For instance, various embodiments may be used in magnetocardiography (MCG) or magnetoencephalography (MEG). In particular, the recirculating multi-pass cell may enable precise bioimaging in the ambient earth field without a bulky and expensive magnetic shielding room. In addition, the atomic magnetometer may be used for geological exploration, such as mineral prospecting, oil and gas exploration, and groundwater detection. In defense / security, the atomic magnetometer may be deployed for detection of magnetic anomalies, surveillance, and detection of unexploded ordnance in military and security applications. Atomic magnetometers may also be integrated into navigation systems for precise positioning in Global Positioning System (GPS)-denied environments, underwater navigation, and autonomous vehicle guidance. Further, atomic magnetometers may be implemented for flaw detection, material characterization, and / or quality control in manufacturing processes.

Claims

Claims1. A multi-pass cell comprising: a concave mirror; and a rotatable mirror comprising a first mirror portion and a second mirror portion such that one or both of the first mirror portion and the second mirror portion are rotatable about a rotation axis passing through the first mirror portion and the second mirror portion.

2. The multi-pass cell according to claim 1, wherein the first mirror portion has a straight edge substantially perpendicular to the rotation axis; and wherein the second mirror portion also has a straight edge substantially perpendicular to the rotation axis.

3. The multi-pass cell according to claim 2, wherein the straight edge of the first mirror portion and the straight edge of the second mirror portion face each other.

4. The multi-pass cell according to any one of claims 1 to 3, wherein the first mirror portion has a shape selected from a group consisting of a semi-circle, a rectangle, a half-ellipse and a triangle; and wherein the second mirror portion has a shape selected from a group consisting of a semi-circle, a rectangle, a half-ellipse and a triangle.

5. The multi-pass cell according to any one of claims 1 to 4, wherein the first mirror portion has a planar reflecting surface.

6. The multi-pass cell according to any one of claims 1 to 4, wherein the first mirror portion has a concave reflecting surface7. The multi-pass cell according to any one of claims 1 to 6, wherein the second mirror portion has a planar reflecting surface.

8. The multi-pass cell according to any one of claims 1 to 6, wherein the second mirror portion has a concave reflecting surface.

9. The multi-pass cell according to any one of claims 1 to 8, further comprising: an alkali cell between the concave mirror and the rotatable mirror.

10. The multi-pass cell according to any one of claims 1 to 8, wherein the concave mirror and the rotatable mirror form reflecting walls of an alkali cell.

11. An atomic magnetometer comprising: a multi-pass cell comprising: a concave mirror; and a rotatable mirror comprising a first mirror portion and a second mirror portion such that one or both of the first mirror portion and the second mirror portion are rotatable about a rotation axis passing through the first mirror portion and the second mirror portion; a pump beam optical system configured to provide circularly polarized light beam to pump atoms contained in an alkali cell contained within or at least partially defined by the multi-pass cell;a probe beam optical system configured to provide linearly polarized light beam such that the linearly polarized light beam is reflected between the concave mirror and the rotatable mirror; and a detector optical system configured to detect an optical rotation of the linearly polarized light beam exiting the multi-pass cell.

12. The atomic magnetometer according to claim 11, further comprising: one or more shields around the multi-pass cell.

13. The atomic magnetometer according to claim 11 or claim 12, wherein the detector optical system comprises a polarizing prism configured to split the linearly polarized light beam exiting the multi-pass cell into separate beams; and a pair of photodetectors configured to detect the separate beams.

14. The atomic magnetometer according to claim 13, wherein the polarizing prism is a Wollaston prism (WP).

15. A method of forming a multi-pass cell, the method comprising: providing a concave mirror; and providing a rotatable mirror comprising a first mirror portion and a second mirror portion such that one or both of the first mirror portion and the second mirror portion are rotatable about a rotation axis passing through the first mirror portion and the second mirror portion.

16. The method according to claim 15,wherein the first mirror portion has a straight edge substantially perpendicular to the rotation axis, and wherein the second mirror portion also has a straight edge substantially perpendicular to the rotation axis.

17. The method according to claim 16, wherein the straight edge of the first mirror portion and the straight edge of the second mirror portion face each other.

18. The method according to any one of claims 15 to 17, wherein the first mirror portion has a shape selected from a group consisting of a semi-circle, a rectangle, a half-ellipse and a triangle; and wherein the second mirror portion has a shape selected from a group consisting of a semi-circle, a rectangle, a half-ellipse and a triangle.

19. The method according to any one of claims 15 to 18, wherein the first mirror portion has a planar reflecting surface.

20. The method according to any one of claims 15 to 18, wherein the first mirror portion has a concave reflecting surface21. The method according to any one of claims 15 to 20, wherein the second mirror portion has a planar reflecting surface.

22. The method according to any one of claims 15 to 20, wherein the second mirror portion has a concave reflecting surface.

23. The method according to any one of claims 15 to 22, further comprising: providing an alkali cell between the concave mirror and the rotatable mirror.

24. The method according to any one of claims 15 to 22, wherein the concave mirror and the rotatable mirror form reflecting walls of an alkali cell.

25. A method of forming an atomic magnetometer, the method comprising: forming a multi-pass cell comprising: a concave mirror; and a rotatable mirror comprising a first mirror portion and a second mirror portion such that one or both of the first mirror portion and the second mirror portion are rotatable about a rotation axis passing through the first mirror portion and the second mirror portion; providing a pump beam optical system configured to provide circularly polarized light beam to pump atoms contained in an alkali cell contained within or at least partially defined by the multi-pass cell; providing a probe beam optical system configured to provide linearly polarized light beam such that the linearly polarized light beam is reflected between the concave mirror and the rotatable mirror; and providing a detector optical system configured to detect an optical rotation of the linearly polarized light beam exiting the multi-pass cell.

26. The method according to claim 25, further comprising: providing one or more shields around the multi-pass cell.

27. The method according to claim 25 or claim 26, wherein the detector optical system comprises a polarizing prism configured to split the linearly polarized light beam exiting the multi-pass cell into separate beams; and a pair of photodetectors configured to detect the separate beams.

28. The method according to claim 27, wherein the polarizing prism is a Wollaston prism (WP).

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