Number of independent spatial modes inside a spherical volume

Phase and amplitude plots of several Laguerre–Gaussian modes with different radial index p and azimuthal index ℓ. These form a complete orthogonal basis commonly used to count the number of independent spatial modes ("beams") that can fit inside a spherical or cylindrical volume of radius R.

Number of independent spatial modes inside a spherical volume is based on Quantum optics a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light), and refers to the number of simultaneously supportable propagable orthogonal spatial modes in the wavefield confined in a spherical region of radius R with negligible crosstalk. The concept applies equally well in both the classical and quantum channels and deals with electromagnetic waves (light or radio waves), acoustic waves (sound or ultrasonic waves), or de Broglie matter waves.

Modes

Every beam corresponds to a different orthogonal mode in the spatial structure of the electromagnetic field (for example, Laguerre–Gaussian modes, Hermite–Gaussian modes, or spherical-harmonic modes). The number of such modes in principle depends upon the size of the accessible phase space defined in the sphere and finally upon the diffraction limit contained in the wavelength λ. This applies in many fields. For quantum optics and quantum information research, this determines the maximum dimensionality of quantum entanglement in the spatial mode and the quantum capacity of free-space quantum communication channels. For classical free-space optical communication links, the same counting gives the ultimate (hole-burning-free) upper bound on the number of independent spatial channels, the diffraction-limited number of spatial degrees of freedom (a three-dimensional analogue of the familiar A/λ² limit for planar apertures).

Applications

Animation showing the radiation pattern

For phased arrays in radar or radio astronomy applications, this defines the number of independently beam-steerable or spatially resolvable beams or channels. For medical ultrasonic imaging or sonar applications, this closely-related idea defines the number of independently addressable focal areas or image pixels. Theoretically, in the latest volumetric or five-dimensional optical information recording technologies, this determines the overall data packing density if all possible spatial and polarization freedoms are utilized. Since the same counting in the phase-space argument holds equally for classical waves and quantum states consisting of one or few photons, this limit turns out to be universal.

History

The fundamental limit imposed by a finite volume on the number of independent communication channels was first analysed in optics during the 1960s and 1970s. These works showed that the number of spatial degrees of freedom of a field confined to a region of radius R scales as (2πR/λ)² in two dimensions and (2πR/λ)³ in three dimensions.

Since the mid-2010s, the same spatial-mode counting argument has been widely applied in high-dimensional quantum optics, where approximately (kR)³ modes (orbital-angular-momentum, Laguerre–Gaussian, or pixel bases) are used as independent quantum channels. When finite bandwidth Δf and observation time t are taken into account, the total number of independent spatio-temporal channels is approximately N_spatial × Δf × t. This formula is now routinely used to estimate fundamental limits in quantum communication, radar, and imaging.

Diffraction limit

The fundamental diffraction limit is

$N\approx 2\times \frac{(kR{)}^3}{3{\pi }^2}\approx 0.068{\left(\frac{2\pi R}{\lambda }\right)}^3$

independent spatial modes (including both polarisation states), where $k=\frac{2\pi }{\lambda }$ is the wavenumber.
With finite bandwidth Δf and observation time t, the total number of independent spatio-temporal channels becomes

$N_{\text{total}}\approx N\times \Delta f\times t$

Quantum-mechanical limit

A quantum limit in physics is a limit on measurement accuracy at quantum scales.

Intensity profiles of Laguerre–Gaussian modes

Per polarisation:

$N_{\text{per pol}}\approx \frac{(kR{)}^3}{3{\pi }^2}$

including both polarisations:

$N\approx 2\times \frac{(kR{)}^3}{3{\pi }^2}\approx 0.068{\left(\frac{2\pi R}{\lambda }\right)}^3$

Classical beam packing

For non-diffracting beams of diameter d, geometric packing gives a crude (and irrelevant) upper bound $N\lesssim (2R/d{)}^3$

Including time and bandwidth

$N_{\text{total}}\approx N\times (\Delta f\cdot t)$

Example calculation (50 µm sphere, near-infrared light)

• R = 50 µm, λ = 800 nm → kR ≈ 393
• N ≈ 2.8 × 10⁷
• Δf = 10 THz, t = 1 s → Δf·t = 10¹³
• N_total ≈ 2.8 × 10²⁰
  

Quantum wavefunctions in spheres

Spatial modes
inside a sphere

Orthogonal spatial modes inside a sphere of radius R. N≈(kR)3/3π2

Orthogonal spatial modes inside a sphere of radius R.
$N\approx (kR{)}^3/3{\pi }^2$

Angular part:
spherical harmonics

Real spherical harmonics Ylm up to l=5

Real spherical harmonics Y_lm up to l=5

Radial part:
spherical Bessel

File:Vibrating drum Bessel function.gif

Bessel functions describe the radial part of vibrations of a circular membrane.

Spatio-temporal
modes over time

Growth of total channels with bandwidth and time

Growth of total channels with bandwidth and time

Spherical harmonics and Bessel functions are the natural solutions of the Helmholtz or Schrödinger equation (these functions form a complete orthogonal set for the spatial modes in the sphere).

See also

• Phased array
• Orbital angular momentum of light
• 5D optical data storage

References

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