## Abstract

In this paper, we propose a new type of rotational Doppler shift measurement based on the OAM of light which is capable of measuring the rotation of a point source in the plane orthogonal to the observer line of sight. By analysing the correlations between OAM states of light emitted by rotating sources, the rotational Doppler shift, and hence the rate of rotation, can be measured. We demonstrate that an OAM interferometer capable of extracting the rotational Doppler shift from OAM correlations can be constructed from a standard OAM modesorter combined with a phase filter.

© 2017 Optical Society of America

## 1. Introduction

Interferometry is a method of measuring the complex amplitude of an optical signal by decomposing the incoming light from a source into a spectrum of plane waves. Pairs of these plane waves are interfered and by looking at the resulting interference pattern, the relative phase of the plane waves and their amplitudes can be measured. Most often what we measure is the mutual coherence function of a source Γ(*x̄*):

The spatial size of astronomical interferometers are determined by the maximum “baselines” (or pairs of Fourier components (*k*, *k′*)) which the instrument measures. Thus, the mutual coherence function that these interferometers measure is the correlation function in Fourier space and for very long “baseline” separations, extremely high resolution images can be measured.

In addition to a decomposition in plane waves (i.e. *k*-space), a source can be described as a decomposition into different helical OAM modes. OAM interferometry is performed by interfering the different OAM components and extracting the complex amplitudes. From these, the mutual coherence function of a source can be measured in OAM modes. While in the field of quantum optics, the measurement of the OAM intensity correlation function of entangled photons is a routine procedure [2–4], the development of the astronomical OAM interferometer has received much less attention and was only first theoretically analyzed by Elias in 2008 [5].

One problem which limits the development of astronomical OAM interferometry is the lack of practical applications for such a device and unfortunately this problem also applies to the field of OAM detection in astronomy as a whole. While there have been various proposals of interesting astronomical targets, such as the modulation of the OAM spectrum of light due to Kerr black holes [6], as well as the measurement of density inhomogeneities in the ISM [7], these proposals are at the same time extremely technically challenging and the theory behind the measurements is not well developed.

In this paper, we address this concern and propose a new application for the measurement of astronomical optical OAM which is technically feasible and provides information on the rotation rates of sources. By measuring the rotational Doppler shift of light, rotation rates of astronomical objects in the plane orthogonal to the observer’s line of sight can be extracted, complementary to the tangential rotation rates currently available.

The first part of this paper outlines the formalism used to describe the OAM correlation function of an incoherent point source such as the type common to astronomy and derives the OAM correlation functions for *N* such independently incoherent point sources located in a ring. The rotational Doppler shift is obtained by allowing these sources to rotate and as an example, the OAM correlation function of a rotating resolved binary star system is derived. Finally a procedure by which the phase shift can be measured is proposed and an instrument for this purpose is outlined. We now begin this paper with a brief introduction to the rotational Doppler shift.

## 2. Rotational Doppler shift

The traditional linear Doppler shift manifests as a change in the frequency of light emitted by an object travelling at some velocity with respect to an observer. This phenomena is exploited routinely in astronomy to directly measure the rates of rotation of astronomical objects rotating in a plane parallel to the observer line of sight. The caveat is of course that the motion of an object which is rotating in the plane orthogonal to the line of sight has zero tangential velocity and hence no linear Doppler shift. There exists however a rotational analogue to the linear Doppler shift, the rotational Doppler shift. This manifests as a phase shift associated with the OAM of light emitted by sources rotating in the plane orthogonal to the line of sight [8,9].

The light emitted by a rotating source will be observed to experience a total frequency shift depending on both the linear and rotational Doppler shifts:

*f*is the total phase shift due to the translational and rotational Doppler shifts, Δ

*f*

_{‖}and Δ

*f*

_{⊥}are the phase shifts measured by an observer due to a relative velocity parallel (

*v*

_{‖}) and transverse (

*v*

_{⊥}) to the observer line of sight, Φ is the rotational motion of the object.

*k*= 2

*π/λ*is the wavevector,

*ℓ*and

*σ*are the orbital and spin components of the angular momentum, Ω is the rate of rotation of the target.

The first term in Eq. (2) describes the linear Doppler shift and is associated with the wavevector *k*. The second term arises from the rotational Doppler shift and describes a phase shift depending on both Ω as well as the OAM component *ℓ*. In general, any electric field *E*(*r*, *ϕ*) may be expressed as an infinite series of scalar helical modes (ignoring the vector nature of light) with complex amplitudes *A _{ℓ}* (

*r*):

*r*and

*ϕ*are the radial and azimuthal coordinates defined with respect to the axis of decomposition. The rotational Doppler shift thus introduces a phase shift relative to different helical modes

*ℓ*. By measuring the correlations between the different OAM components (

*ℓ*) of a source it is possible to extract the rate of rotation of the source in the plane orthogonal to the line of sight of the observer by measuring the complex amplitudes of the correlation function.

Laboratory experiments have been conducted demonstrating the measurement of the rotational motion of an object using the rotational Doppler shift of light [10, 11]. This is performed by reflecting two laser beams prepared in OAM states +*ℓ* and −*ℓ* respectively from a rotating target. The reflected OAM beams are then interfered with one another, and the phase of the field and the rotational Doppler shift is measured. The studies were performed with the intention of remote sensing of rotational motion, and hence the target was a scattering disc rather than a rotating source, however the applications to the field of observational astronomy are clear. For very slowly rotating objects, astronomers are able to determine rates of rotation using direct imaging and tracking the orbits of the bodies such as the exoplanet system HR8799 however this can take years [12]. By measuring the rotational Doppler shift, the rate of rotation (for circularly symmetric systems) can be immediately measured.

The same method of determining an object’s linear and rotational transverse velocity has since been applied to micro-scale particles under laboratory conditions [13,14], combining into one the fields of linear and rotational Doppler velocimetry. In order to obtain information about the full three-dimensional velocity of a particle, the original method for Doppler rotation detection is extended to use both a beam with high OAM and a beam with zero OAM. The high OAM beam is imprinted with the rotational Doppler shift which depends on *ℓ* (second term in Eq. (2)), while the zero OAM beam extracts out the linear Doppler shift (*ℓ* = 0 beams are insensitive to the rotational Doppler shift). By combining measurements using beams of different total OAM, the full three-dimensional velocities of the particles are acquired.

In all of these studies however, the rotational Doppler shift of coherent light has been reflected from a rotating surface and analyzed. This type of scenario is not particularly applicable to observational astronomy however, and for our purposes we now extend the treatment of the rotational Doppler shift to incoherent point sources.

## 3. OAM correlation function of incoherent sources

The electric field of any source can be written as an infinite Fourier series composed of helical modes which individually have a well defined OAM as in Eq. (3). In this paper we focus on the OAM correlation functions of point-like and extended incoherent sources which would account for the majority of sources in astronomy.

#### 3.1. OAM correlation function of a single incoherent point source

Let us begin with a single resolved star as given in Fig. 1.

We define the coordinates of our problem as follows:

*x*,

*y*) and (

*r*,

*ϕ*) are the real space Cartesian and polar coordinates respectively and (

*k*,

_{x}*k*) and (

_{y}*k*,

_{r}*ϕ*) are the respective Fourier space Cartesian and polar coordinates. Let us model the electric field

_{k}*u*

_{x0}(

*x̄*) arriving at our detector on earth from a distant resolved star located at position

*x̄*

_{0}= (

*r*

_{0}cos

*ϕ*

_{0},

*r*

_{0}sin

*ϕ*

_{0}) relative to the measurement axis, as a point source:

*E*

_{0}is a normalization constant. The OAM spectrum of such a source is given by the OAM decomposition of a Dirac delta function:

*a*(

_{ℓ}*r*,

*r*

_{0}) is given as follows: Where

*δ*(

*r*−

*r*

_{0}) is the Dirac delta and

*𝒫*is a normalization constant. We use the following definition of the correlation function for a fully coherent source:

*r*

_{0}cos

*ϕ*

_{0},

*r*

_{0}sin

*ϕ*

_{0}) from the measurement axis:

*𝒫*. The energy of the source is related to the intensity of the source, measured by taking the diagonal elements of the correlation function $\u3008{u}_{{x}_{0}}(\overline{x}){u}_{{x}_{0}}^{*}({\overline{x}}^{\prime})\u3009$. Setting

*x̄*=

*x̄′*:

*E*

_{0}is the square root of the power of the star (|

*E*|

^{2}). Thus, our final expression for

*a*(

_{ℓ}*r*,

*r*

_{0}) is:

#### 3.2. Correlation function of N incoherent sources

We now consider the correlation function of *N* mutually incoherent sources located at *x̄ _{n}* = (

*r*cos

_{n}*ϕ*,

_{n}*r*sin

_{n}*ϕ*). All sources are spatially incoherent with respect to one another and hence we can write the coherence function as the sum of the individual coherence functions:

_{n}#### 3.3. Rotating sources

By allowing each source to rotate at a rate of Ω* _{n}* rad per unit time, the effect of the rotational Doppler shift is clearly observable by making the substitution

*ϕ*= Ω

_{n}*in Eq. (11), where*

_{n}t*t*is time. The coherence function for one and

*N*sources are given below:

Let us take a moment here to appreciate the significance of this result. We predict that we should be able to measure the rotational Doppler shift, for a system of spatially incoherent point sources. One concern regarding the measurement of OAM from astronomical objects is that Laguerre-Gauss beams which possess well defined OAM diverge as they propagate to infinity [15]. Thus the light from a resolved point source which reaches Earth will have only light associated with the fundamental Gaussian LG(0,0) mode and all information contained within the higher order *ℓ* modes will be lost. This statement is true if the point source is located on-axis with the measurement instrument, in this case the Fourier spectrum is a flat plane wave. However when a single point source is located off-axis from the measurement axis, the Fourier spectrum is a tilted plane wave, and what our derivation of Eqs. (12) and (13) tells us, is that this has a non-zero decomposition in OAM modes and that for a rotating source, the correlations between these modes contains interesting information, in this case the rotational Doppler shift. We briefly mention that these results are still valid when considering the effect of an instrument PSF, since the original input is spatially coherent, the PSF of each source will be spatially coherent also.

#### 3.4. Example problem: rotating binary star pair

Now let us apply this formalism to a problem in astronomy, determining the in plane rate of rotation of a resolved binary star system. From Earth, we can model the binary star system as a pair of incoherent point sources located at (+*x*_{0}, 0) and (−*x*_{0}, 0) (or in polar coordinates (*r*_{0}, *ϕ*) and (*r*_{0}, *ϕ* + *π*)). Let us assume the binary stars are of equal brightness and rotate around the origin at a rate of Ω revolutions per unit time *t*. From Eq. (13) we write down the corresponding coherence function of the binary star system:

*e*

^{−iπ(ℓ−ℓ′)}) arises from the even rotational symmetry of the source and forces all odd values of

*ℓ*to zero. From Eq. (14), one can calculate the on-axis rate of rotation for the binary star system in which both stars are equal brightness. In order to measure the rate of rotation Ω, we need access to the phase of the correlation function. The most direct way to do this is by using interferometry of the OAM components of the electric field. Imagine that we had a way to filter out all but two OAM components (

*ℓ*

_{1},

*ℓ*

_{2}) of the light emitted by the binary star. Interfering the two components, from Eq. (14) the resulting correlation function is:

*a*(

_{ℓ}*r*,

*r*

_{0}) in Section 3.1, Eq. (7), ${a}_{\ell}(r,{x}_{0})={a}_{\ell}^{*}(r,{x}_{0})$. We assume that the light from the binary star system is completely spatially incoherent and hence set

*x̄*=

*x̄′*yielding a simplified correlation function:

*ℓ*of the filtered OAM components Δ

*ℓ*=

*ℓ*

_{1}−

*ℓ*

_{2}, the visibility of the interference fringes as

*A*= (2

*π*)

^{2}2

*a*

_{ℓ1}(

*r*,

*x*

_{0})

*a*

_{ℓ2}(

*r*,

*x*

_{0})[1 + (−1)

^{Δ}

*], and $B={\left(2\pi \right)}^{2}\left[{a}_{{\ell}_{1}}^{2}(r,{x}_{0})+{a}_{{\ell}_{1}+\mathrm{\Delta}\ell}^{2}(r,{x}_{0})\right]$. In addition, the amplitudes of the OAM spectrum*

^{ℓ}*a*

_{ℓ1}and

*a*

_{ℓ2}can be determined from the coefficients

*A*and

*B*:

*r*=

*r′*=

*r*

_{0}. From the definition of

*a*(

_{ℓ}*r*,

*r*

_{0}) in Eq. (7), any other choice of

*r*and

*r′*would result in a correlation function populated by zeros. We note that the amplitude of the correlation function in Fig. 2(a) has only every second element and in addition, the amplitude of the correlation function is also constant in

*ℓ*as predicted by Eq. (10).

The feature of primary interest in Fig. 2 is that phase of the OAM correlation function Fig. 2 contains a variation in phase arising from the rotation of the binary star system. The phase gradient in Fig. 2 is greater for a more rapidly rotating system, and by interfering even OAM modes separated by a specific Δ*ℓ*, the phase gradient can be estimated. For a very slowly rotating source, the phase gradient is much smaller, and hence the Δ*ℓ* and the number of simultaneous measurements must be increased. Every second element of the correlation function in Fig. 2(b) has been removed in order to more accurately represent the information contained in the phase of the correlation function.

## 4. OAM interferometry

The rotational Doppler shift is a phase shift encoding the rotation of an object (in the plane transverse to the observer line of sight), into correlations in the OAM spectrum. This phase shift is impossible to detect using conventional interferometers, or OAM modesorter instruments such as the type described in [16,17], thus a new instrument is required. From the previous section, we know that in order to measure the complex amplitudes of the OAM correlation function we require the following functions:

- Decomposition of light into helical OAM modes.
- Selection of two OAM modes
*ℓ*_{1}and*ℓ*_{2} - Interference between
*ℓ*_{1}and*ℓ*_{2}

Figure 3 describes a simple device with which the angular correlation function of light could be measured at optical wavelengths. Elements MS 1 and MS 2 are optical OAM mode sorting elements which perform an afocal Cartesian (*x*, *y*) to Log-polar (*r*, *ϕ*) coordinate transformation as in [16]. Given a circularly symmetric input to MS 1, the elements MS 1 and MS 2 would transform the circle of light into a rectangle (see Fig. 3). This optical coordinate transformation has the effect of converting an angularly varying phase shift exp(*iℓϕ*) characteristic of OAM modes, to a phase shift linear in a Cartesian coordinate (*x*) of the form exp(*ikx*) which can be recognized as the phase shift of a plane wave carrying transverse momentum *k*.

A lens then disperses the light into separate OAM modes. A filter then selects two different values of *ℓ*_{1} and *ℓ*_{2} to interfere. A SLM could be implemented as a programmable filter similar to the setup in [18], in which phase slits centered on *ℓ*_{1} and *ℓ*_{2} are generated by using diffraction gratings (see inset Fig. 3). This approach has the advantage that the offset Δ*ℓ* can be automatically varied, however in addition to to the loss of light due to the aperture, there will be an additional loss arising from the efficiency of the liquid crystal on silicon (LCOS) display. LCOS display are inherently polarization sensitive and will require the use of additional polarization sensitive optics to split the incoming light into orthogonal linear polarizations and process each polarization. Instead of a LCOS display, a segmented deformable mirror could be used to generate a polarization insensitive OAM filter and at a higher efficiency.

The two OAM modes *ℓ*_{1} and *ℓ*_{2} are then interfered either by free space propagation to the far-field or focusing with a cylindrical lens (focusing along *ϕ*) and the resulting intensity pattern is then recorded. By varying the offset Δ*ℓ* between the two interfered modes and measuring the resulting intensity pattern, the OAM correlation function should vary according to Eq. (15). The phase of the optical field can then be extracted from the interference between the two modes and the amplitudes of the components can be found from Eq. (16). Optical interferometry between OAM modes places specific constraints on the design of modesorter elements. For the measurement of the rotational Doppler shift of a source, interference between higher *ℓ* modes amplifies the measured phase shift. The maximum *ℓ* that the modesorter can detect is dependent on the diameter of the telescope aperture and also the aperture of the modesorter elements. The major constraint on the diameter is the maximum surface relief of the refractive element that needs to be manufactured.

One factor that needs to be addressed is the cross-talk between OAM modes during the OAM decomposition step in Fig. 3 due to the overlap of OAM modes. It can be theoretically shown that approximately 10% of the power in adjacent OAM modes *ℓ* + 1 and *ℓ* − 1 will leak into the mode *ℓ* [19]. This cross-talk is inherent to the modesorter and can be removed by using an additional pair of spatial light modulators placed in-between MS 2 and the final lens to increase the phase ramp associated with individual OAM modes in the transformed plane [20,21], however this approach has the downside of adding significant complexity to the design of the experiment as well as reducing the efficiency. Recently, these phase profiles associated with the additional SLM compensators have been combined with the phase profile of the OAM mode sorting elements [22], resulting in a single pair of phase profiles containing both the OAM sorting function as well as the additional compensation. These represent a very good candidate for future implementations of the OAM interferometer described above.

Another option would be to carefully choose the focal length of the focusing lens after MS 2, and then simply select the slit widths in the OAM filter to be small enough that the phase slit centered on an OAM bin minimizes the amount of cross-talk from adjacent modes, although again this would necessarily result in a large loss of light and lower SNR.

In addition, it might be necessary to incorporate several beam splitter stages, or simultaneously use multiple pairs of phase slits in the OAM filter to simultaneously measure the correlation functions $\u3008{u}_{{\ell}_{1}}{u}_{{\ell}_{1}+\mathrm{\Delta}\ell}^{*}\u3009$ as the source is not temporally coherent. Due to the phase sensitive nature of the measurement, OAM interferometry clearly needs strong AO correction to correct for atmospheric turbulence. As interferometry is a measurement of the phase of the OAM correlation function, any phase noise due to the atmosphere will destroy the measurement.

## 5. Spectrograph coupling

While this concept is very far in the future and significant work is required to demonstrate a practical OAM interferometer, we include this section to demonstrate the complementary nature of OAM “polar” interferometry to current methods in observational astronomy. In this section we describe a combined OAM interferometer coupled to a spectrograph capable of measuring the 3D motion of an object.

A schematic is provided in Fig. 4 for a possible spectrograph to which the output from our proposed OAM interferometer could be coupled. The first lens in the schematic is the telescope lens. This is the effective lens of the collecting telescope, however for the purposes of our design we have selected a lens with a diameter *D*_{tel} = 1 m and focal length *f*_{tel} = 10 m (yielding a *F*/# = 10). A beam splitter is then placed to split the light into two paths, one path is sent to the OAM interferometer to analyze the rotational Doppler shift, the second path leads to a spectrograph that measures the translational Doppler shift. A 100 *μ*m slit spatially filters the input and allows through a 2″ slice of the focal plane.

A lens with focal length *f*_{coll} = 250 mm and diameter 25 mm then collimates the output from the slit. The focal ratio (*F*/#) of the collimating lens has been chosen to match the F/# of the telescope lens. The collimated light is diffracted using a volume phase hologram (VPH) “immersed” between a pair of prisms. The VPH has a grating density of 1800 cycles/mm and the output from the first diffraction order (m=1) is used. The two prisms are used to increase the angle at which the incident beam hits the VPH grating [23].

The required incoming angle of incidence on the VPH grating at 550 nm is *α* = 45°, with the resulting output angle *β* = 16°. A final lens with diameter *D*_{cam} = 25 mm and focal length *f*_{cam} = 50 mm focuses the output from the VPH grating and focuses the dispersed light onto the detector. The final plate scale at the detector is *P _{DP}* = 100″/mm, with the 100

*μ*m, 2″ slit mapping to 20

*μ*m at the detector. This limits the maximum pixel size of the detector to 10

*μ*m pixel size for adequate Nyquist sampling. The resulting resolution of the spectrograph is

*ℛ*= 35000. Considering the focal lengths and the angles involved in this design, this spectrograph is very compact and can be constructed along with the OAM interferometer on an optical breadboard of around 1 × 1 m.

It should be noted that in initial designs of the above instrument, the spectrograph was fed directly by the *ℓ* = 0 mode output from the interferometer, however this design suffers from a major flaw. The light in the final focal plane (*α*, *β*) of the OAM interferometer actually describes the distribution of the source as a function of only *ℓ* and *r*. Thus while the input slit of the spectrograph could select what radial coordinate to measure the transverse Doppler shift, the spectrograph cannot specify what angle, thus the spectrograph would measure the linear Doppler shift of light originating in a ring at the focal plane and for a rotating object would include light that is both red and blue shifted.

One important note to make is that throughout this paper we have discussed the operation of an OAM interferometer with the implicit assumption that the light analyzed is monochromatic and it is unlikely that this assumption would be valid for many astronomical objects. Due to wavelength dispersion by the refractive optical elements which perform the coordinate transformation (MS 1 and MS 2 in Fig. 3) as well as any other optical elements, multiple input wavelengths would clearly be detrimental to the performance of the proposed OAM interferometer. This problem is not unique to OAM interferometry, and indeed the analysis of light using narrow-band filters has already been implemented in conventional astronomical interferometers, reducing the spectral range to the order of Angstroms [24]. Doing so would limit observations to bright objects only, however an analysis of the effect of a non-monochromatic input to the OAM interferometer is required to develop an understanding of the change in measurable interferometer visibility with input bandwidth, and hence the minimum brightness would be observable.

## 6. Conclusion

At the moment, the OAM interferometer envisaged in this paper is constructed from free-space optics as this is the fastest pathway to practical implementation of the device. For our concluding remarks, we briefly discuss how future iterations of this instrument may evolve. The implementation of the OAM interferometer as a photonic device is attractive, the main advantage being that a photonic architecture removes much of the precision alignment required, while improving phase stability [25]. While a lot of work is obviously required, multiple astrophotonic technologies are already capable of implementing the key components of the device from this paper: DRAGONFLY boasts a Cartesian to Polar pupil remapper [26], and the free propagation zones developed for photonic spectrographs [27,28] can be combined to implement the OAM modesorter. In fact, there has already been an laboratory demonstration of an OAM modesorter in a silicon photonic chip [29–31]. To interfere pairs of OAM modes, there are many implementations of photonic beam splitters [32–35], however the real challenge will be that faced by astrophotonics at large, how to couple the light from a pair of binary stars into a single mode fibre [25].

Alternatively, we could also imagine the application of Very Long Baseline Interferometry (VLBI) to the measurement of the OAM correlation function. At the beginning of this paper it was mentioned that interferometry is used to measure the complex amplitudes of very high Fourier coefficients, however we recognize that the Fourier spectrum of the source is the transverse momentum correlation function which should in principle contain the same information as the OAM correlation function. Thus the transverse momentum correlation function obtained using VLBI can be transformed into a discrete approximation of the OAM correlation function. The rotational Doppler shift can then be extracted from the phase and the rotation rate of an unresolved binary star can be measured. The measurement of OAM correlations using VLBI was initially proposed by Elias and we refer interested readers to the original paper in [5] which discusses the various problems which must be overcome.

In this paper we motivate the next generation of instruments with which to measure the OAM of light. By measuring the angular correlation function, new information can be extracted from the OAM of light: the rotational Doppler shift.

The rotational Doppler shift cannot be measured using existing interferometeric and spectroscopic techniques, however it is clear that helical OAM modes are a particularly appropriate set of basis functions with which to describe rotations [36]. To highlight this, we provide a derivation of the OAM correlation function of incoherent point sources, and after expanding the correlation function of a source into OAM modes, upon including a rotating source the rotational Doppler shift naturally appears in the equations.

Finally, this paper provides a schematic of an OAM interferometer capable of measuring the OAM correlation function. In addition, by pairing it with a high resolution spectrograph the combined instrument would be capable of measuring both the translational and rotational Doppler shift of astronomical sources. Importantly, the components of the spectrograph-coupled OAM interferometer are all commercially available with the exception of the custom optics of the OAM mode sorter.

## Funding

Australian Research Council (ARC) Future Fellowship (FT1110924) scheme.

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