Physics:Number of independent spatial modes in a spherical volume

Quantum optics 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.

Counting independent spatial modes in a spherical volume: modes are determined in k-space by the number of allowed wavevectors within a sphere of radius k_max, giving N ∝ V k_max^3.

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.

Concept

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. Since the early days of spherical near-field far-field transformations a recommendation for the necessary number of polar modes has been given by $N=k{r}_0+10$ , $k$ being the wavenumber and $r_0$ the radius of the minimum sphere. The almost explosive development in computer speed and storage capacity witnessed during the last two decades has made transformations of fields from antennas exceeding thousands of wavelengths feasible, and a closer investigation of the above expression seems to be appropriate.

An improved expression for the number of modes, $N$, related to the antenna size and the required accuracy will be developed. The impact of truncation of the modal expansion at a given level will be illustrated. This is especially important for measurements where noise is present, or where there is undesirable scattering from objects.
Turchin and Tseytlin have stated that $N=n_o+3/(1-(n_o/(k{r}_o){)}^3)$ but neither a justification nor a parametric expression have been presented.

Keywords: Noise Filtering, Normal Mode, Number of Modes, Spherical Expansion, Spherical Modes, Spherical Near Field, Spherical Waves. Antenna measurement, echo suppression, spherical wave expansion, vector addition theorem, mode filtering.

Modes

Artistic visualization of independent spatial modes (spherical harmonics) inside a spherical volume..

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.
The theorem is central to post-processing techniques for echo and reflection suppression in antenna measurements. By offsetting the antenna under test (AUT) and translating modes back to the origin, direct and scattered signals separate in modal space, allowing filtering of unwanted higher-order modes.

It is particularly valuable in spherical near-field (SNF) measurements, probe compensation, and RCS analysis.

What are the main findings?

• Unlike other perhaps better-known techniques, the generalized vector addition theorem can be successfully utilized to perform antenna position translations in any direction, and for displacements that can be smaller or larger than the maximum radial extent of the antenna.
• Crucially, this article shows how the all-important vector addition coefficients can be obtained from the scalar coefficients for which efficient, accurate, and precise recurrence relations exist, which greatly reduces the computational effort required, which would otherwise likely render the technique impractical.

What is the implication of the main finding?

• The technique presented provides the first-ever algorithm for applying mode-filteringbased reflection suppression without the need to first transform to the asymptotic far-field, yielding a significant generalization of the algorithm;
• The new algorithm represents a notable development as it is rigorous and general, incorporating both reactive and propagating components, thereby making the processing applicable to a wider range of problems than has previously been the case.

Abstract Monochromatic mode-filtering-based scattering suppression techniques have been shown to be applicable to all commonly used forms of far- and near-field antenna and RCS measurement techniques. Traditionally, the frequency-domain mode-filtering technique takes a far-field pattern, either measured directly or obtained using a suitable near-field to far-field transformation, as its starting point. The measurement is required to be conducted such that the antenna under test (AUT) is positioned offset from the origin of the measurement coordinate system. This physical offset introduces a phase taper across the AUT pattern and results in far greater interference occurring between the direct and indirect parasitically coupled spurious scattered signals. The method is very general and can be applied to all forms of near- or far-field measurements. However, for the case of a spherical near-field measurement (SNF) approach, it is somewhat cumbersome and tedious as first we must perform a probe-corrected spherical near-field to far-field transformation, which itself involves the computation of a complete set of spherical mode coefficients, and then after the displacement has been applied to the far-electric-fields, a second spherical wave expansion

History

David A. B. Miller

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.
In 2000, David A. B. Miller used this precise expression for the number of orthogonal spatial channels that can be supported between two spherical volumes in free space, formula N ≈ (kR)³ / (3π²) when both polarisations are included.This result has become the standard reference for the maximum number of “beams in a sphere”.
In 2004 Frank Jensen published "On the number of modes in spherical wave expansions" describing polar modes and improved expression for the number of modes
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.
In 2025 Marc Dirix et al. published Generalized Vector Addition Thorem for displacements that can be smaller or larger than the maximum radial extent of the object

Modern vector addition coefficients

• Unlike other perhaps better-known techniques, the generalized vector addition theorem can be successfully utilized to perform object position translations in any direction, and for displacements that can be smaller or larger than the maximum radial extent of the antenna;
• Crucially, this article shows how the all-important vector addition coefficients can be obtained from the scalar coefficients for which efficient, accurate, and precise recurrence relations exist, which greatly reduces the computational effort required, which would otherwise likely render the technique impractical.

Implications

• The technique provides the first-ever algorithm for applying mode-filtering-based reflection suppression without the need to first transform to the asymptotic far-field, yielding a significant generalization of the algorithm;

• The new algorithm represents a notable development as it is rigorous and general, incorporating both reactive and propagating components, thereby making the processing applicable to a wider range of problems than has previously been the case.

The conventional vector addition theorem is written in a compact notation. Then a new and succinct derivation of the vector addition theorem is as close to the derivation of the scalar addition theorem. Newly derived expressions in this new derivation are used to diagonalize the vector addition theorem. The diagonal form of the vector addition theorem is important in the design of fast algorithms for computational wave physics such as computational electromagnetics and computational acoustics. Monochromatic mode-filtering-based scattering suppression techniques have been shown to be applicable to all commonly used forms of far- and near-field antenna and RCS measurement techniques. Traditionally, the frequency-domain mode-filtering technique takes a far-field pattern, either measured directly or obtained using a suitable near-field to far-field transformation, as its starting point. The measurement is required to be conducted such that the antenna under test (AUT) is positioned offset from the origin of the measurement coordinate system.
While the vector addition theorem required for the spherical near-field to far-field transformation (SNFFFT) algorithm has been described in detail in the open literature, its implementation has been limited to the case of impinging waves and positive z-directed translations where the magnitude of the displacement is necessarily larger than the minimum sphere radius (MRE).

RF background subtraction

Although many scattering suppression methods have been proposed, with examples including time-gating (both hardware and software), spatial filtering, RF background subtraction, parametric repeat measurements, and waveform correlation, it is only during the past two decades that the use of mode orthogonalization and filtering-based postprocessing techniques have become widely deployed in industry and academia to identify and subsequently extract measurement artifacts arising from parasitically coupled spurious scattered fields in antenna pattern and radar cross-section (RCS) measurements . These have been used very successfully to greatly improve the facility-level uncertainty budget and have gradually become available for use with all commonly encountered forms of near-field and far-field antenna range measurements. These single-frequency mode-filtering-based approaches have been extended to admit the possibility of processing data that are sampled on irregular grids and, more recently, have been extended to take advantage of sparse sampling compressive sensing techniques .

Generalized vector addition theorem

The volume element in spherical coordinates is ΔV ≈ ρ² sin φ Δρ Δφ Δθ, leading to dV = ρ² sin φ dρ dφ dθ.

The volume element in spherical coordinates is crucial for deriving the orthogonality relations of spherical harmonics and vector spherical wave functions, as integrals over the sphere involve the Jacobian factor ρ² sin φ.

This Jacobian ensures proper normalization of the modes over spherical surfaces, forming the basis for multipole expansions used in the translation addition theorem.

A generalized form of the theorem, valid for arbitrary translation directions and distances (including those smaller or larger than the antenna's minimum radial extent), was developed by Marc Dirix, Stuart F. Gregson, and Rostyslav F. Dubrovka in 2025.

This generalization enables direct translation of spherical mode coefficients without intermediate far-field transformations, preserving both propagating and evanescent modes.

It supports translations in any direction and for any displacement magnitude, overcoming limitations of classical formulations that required specific conditions (e.g., translation along the z-axis or far-field approximations).

The approach uses efficient recurrence relations to compute translation coefficients from scalar wave expansions, reducing computational complexity.

Key applications include advanced spherical near-field antenna measurements, where it facilitates mode-filtering-based reflection suppression in environments contaminated by parasitic scattering.

By translating the antenna under test back to the measurement origin before filtering, the method maximizes separation between desired radiation modes and unwanted scattering modes, achieving superior echo suppression.

Section 1

The core translation equation (Eq. 1) is:

Translation Addition Theorem for Vector Spherical Wave Functions

$F_{smn}^{(-c)}(r,\theta ,\varphi )=\sum _{g=1}^2\sum _{\nu =1}^{\mathrm{\infty }}\sum _{\mu =-\nu }^{\nu }{C}_{g\mu 0}^{smn(-c)}({\beta }_0,{\varphi }_0,kA)F_{\sigma (g)\mu \nu }^{(-3)}(r^{\prime },{\theta }^{\prime },{\varphi }^{\prime })$ (Eq. 1}

where:

• $F_{smn}^{(-c)}$ are vector spherical wave functions (s=1,2 for TE/TM modes),
• superscript $(-c)$ denotes radial dependence,
• primed coordinates are translated by distance A along the z-axis (general direction via angles ${\beta }_0,{\varphi }_0$),
• $C$ are translation coefficients derived efficiently from scalar recurrence relations.

This form removes previous limitations (e.g., requiring translation distance > minimum radial extent) and affects only the far-field phase when translation is inside the minimum sphere, generalizing the phase shift $e^{jk\hat{r}\cdot \mathbf{𝐀}}$. The translation coefficients are defined as and can further be expressed as

$C_{\sigma \mu \nu }^{smn}({\theta }_0,{\phi }_0,kA)=\alpha [{\delta }_{s\sigma }{A}_{\mu \nu }^{mn}({\theta }_0,{\phi }_0,kA)+{\delta }_{3-s,\sigma }{B}_{\mu \nu }^{mn}({\theta }_0,{\phi }_0,kA)]$(Eq.2}

where α is a normalization correction factor, which is dependent upon the chosen normalization of the spherical wave expansion (SWE), and δ is the Dirac delta function . Several conversions between different normalizations are reported within the open literature, with a summary being presented within (A1) of . For the present work, however, the translation coefficients $A_{\sigma \mu \nu }^{smn}(kA)$ and $B_{\sigma \mu \nu }^{smn}(kA)$ are based on the spherical wave functions , while the SWE utilized to attain the spherical mode coefficients employs the normalization proposed in . Thus, the requisite correction factor for the normalization employed here can be expressed as

$\alpha =\sqrt{\frac{\nu (\nu +1)}{n(n+1)}}$(Eq.3}

with

$A_{\mu \nu }^{mn}({\theta }_0,{\phi }_0,kA)=\sum _{p=|n-\nu |}^{n+\nu }{i}^{-p}[n(n+1)+\nu (\nu +1)-p(p+1)]a(m,n,-\mu ,\nu ,p)z_p^{(c)}(kA)P_p^{m-\mu }(\cos {\theta }_0)e^{i(m-\mu ){\phi }_0}$(Eq.4}

and

$B_{\mu \nu }^{mn}({\theta }_0,{\phi }_0,kA)=\sum _{p=|n-\nu |}^{n+\nu }{i}^{-p}[n(n+1)+\nu (\nu +1)-p(p+1)]b(m,n,-\mu ,\nu ,p-1)z_p^{(c)}(kA)P_p^{m-\mu }(\cos {\theta }_0)e^{i(m-\mu ){\phi }_0}$(Eq.5}

where the linearization coefficient b is
$b(m,n,-\mu ,\nu ,p)=\frac{2p+1}{2p-1}[(\nu -\mu )(\nu +\mu +1)a(m,n,-\mu -1,\nu ,p)$ $-(p-m+\mu )(p-m+\mu +1)a(m,n,-\mu +1,\nu ,p)+2\mu (p-m+\mu )a(m,n,-\mu ,\nu ,p)]$(Eq.6}
Here, a(m, n, −µ, ν, p)’s are the linearization coefficients, which can be expressed in terms of Wigner 3-j symbols , and $z_p^{(c)}$ is the spherical radial function of order p and type (c). Furthermore, $P_p^{m-\mu }(\cos {\theta }_0)$ denotes the associated Legendre function of degree p and order (m −µ). However, given the number of factorials required, these turn out to be enormously computationally expensive to calculate directly. Fortunately, an alternative strategy may be harnessed involving the computation of these coefficients by means of efficient, stable recurrence relations . Recurrence relations greatly reduce the computational complexity of calculating the coefficients, with this becoming increasingly important as the number of coefficients increases. This is especially crucial as the highest-order coefficient depends upon the frequency, minimum sphere radius, and the size of the translation. Once the scalar coefficients have been obtained, these can then be used to derive the vector addition coefficients.

Section 2

When taking this approach, the addition coefficients can be written as follows:

Geometry of two PEC spheres separated by δ along the z-axis, illustrating the translation addition theorem for vector spherical waves.

$A_{\mu \nu }^{mn}({\theta }_0,{\varphi }_0,kA)={\beta }_{\mu \nu }^{mn}({\theta }_0,{\varphi }_0,kA)$ ${\displaystyle }+kA\sin {\theta }_0{e}^{-i{\varphi }_0}\frac{1}{2(\nu +1)}\sqrt{\frac{(\nu -\mu +2)(\nu -\mu +1)}{(2\nu +1)(2\nu +3)}}{\beta }_{\mu -1,\nu +1}^{mn}({\theta }_0,{\varphi }_0,kA)$ ${\displaystyle }-kA\sin {\theta }_0{e}^{-i{\varphi }_0}\frac{1}{2\nu }\sqrt{\frac{(\nu +\mu -1)(\nu +\mu )}{(2\nu -1)(2\nu +1)}}{\beta }_{\mu -1,\nu -1}^{mn}({\theta }_0,{\varphi }_0,kA)$ ${\displaystyle }-kA\sin {\theta }_0{e}^{i{\varphi }_0}\frac{1}{2(\nu +1)}\sqrt{\frac{(\nu +\mu +2)(\nu +\mu +1)}{(2\nu +1)(2\nu +3)}}{\beta }_{\mu +1,\nu +1}^{mn}({\theta }_0,{\varphi }_0,kA)$ ${\displaystyle }+kA\sin {\theta }_0{e}^{i{\varphi }_0}\frac{1}{2\nu }\sqrt{\frac{(\nu -\mu )(\nu -\mu -1)}{(2\nu -1)(2\nu +1)}}{\beta }_{\mu +1,\nu -1}^{mn}({\theta }_0,{\varphi }_0,kA)$ ${\displaystyle }+kA\sin {\theta }_0\frac{1}{\nu +1}\sqrt{\frac{(\nu +\mu +1)(\nu -\mu +1)}{(2\nu +1)(2\nu +3)}}{\beta }_{\mu ,\nu +1}^{mn}({\theta }_0,{\varphi }_0,kA)$ ${\displaystyle }+kA\sin {\theta }_0\frac{1}{\nu }\sqrt{\frac{(\nu +\mu )(\nu -\mu )}{(2\nu -1)(2\nu +1)}}{\beta }_{\mu ,\nu -1}^{mn}({\theta }_0,{\varphi }_0,kA)$ (Eq.7}

and

$B_{\mu \nu }^{mn}({\theta }_0,{\varphi }_0,kA)=kA\cos {\theta }_0\frac{i^{\mu }}{\nu (\nu +1)}{\beta }_{\mu \nu }^{mn}({\theta }_0,{\varphi }_0,kA)$ ${\displaystyle }+ikA\sin {\theta }_0\frac{1}{2\nu (\nu +1)}[\sqrt{(\nu -\mu )(\nu +\mu +1)}{e}^{i{\varphi }_0}{\beta }_{\mu +1,\nu }^{mn}({\theta }_0,{\varphi }_0,kA)$ $+\sqrt{(\nu +\mu )(\nu -\mu +1)}{e}^{-i{\varphi }_0}{\beta }_{\mu -1,\nu }^{mn}({\theta }_0,{\varphi }_0,kA)]$ (Eq.8}

Here, (Eq. 7) and (Eq. 8) refer to the calculation of the scalar translation matrix, which is in the form of ${\beta }_{\mu \nu }^{mn}({\theta }_0,{\varphi }_0,kA)$, for which recurrence relations also exist and are developed in . Here, the starting values for n = 0, m = 0, i.e., ${\beta }_{\nu \mu }^{00}({\theta }_0,{\varphi }_0,kA)$ can be easily derived from the scalar wave equation using

${\beta }_{\nu \mu }^{00}({\theta }_0,{\varphi }_0,kA)=(-1{)}^{\mu +\nu }\sqrt{4\pi }{Y}_{\nu ,-\mu }({\theta }_0,{\varphi }_0)z_n^{(c)}(kA)$ (Eq.9}

Here, \(z^{(c)}_n\) is the spherical radial function of order n and type (c).

$Y_{n,m}({\theta }_0,{\varphi }_0)=(-1{)}^m\sqrt{\frac{(n-m)!(2n+1)}{(n+m)!4\pi }}{P}_n^{|m|}(\cos {\theta }_0)e^{im{\varphi }_0}$ (Eq10}

$P_n^{|m|}(\cos \theta )$ is the associated Legendre function. This calculation also makes use of the standard relation $Y_{n,-m}({\theta }_0,{\varphi }_0)=(-1{)}^m{Y}_{n,m}^{*}({\theta }_0,{\varphi }_0)$ . Then, ${\beta }_{\mu \nu }^{nn}$ may be expanded using the relation where \(P_n^{|m|}(\cos\theta)\) is the associated Legendre function. This calculation also makes use of the standard relation $Y_{n,-m}({\theta }_0,{\varphi }_0)=(-1{)}^m{Y}_{n,m}^{*}({\theta }_0,{\varphi }_0)$ . Then, ${\beta }_{\mu \nu }^{nn}$ may be expanded using the relation

$b_{nn}^{+}{\beta }_{\mu \nu }^{n+1,n+1}=b_{\nu -1,\mu -1}^{+}{\beta }_{\mu -1,\nu -1}^{nn}+b_{\nu +1,\mu -1}^{-}{\beta }_{\mu -1,\nu +1}^{nn}$ (Eq.11}

where

$b_{nm}^{+}=\sqrt{\frac{(n-m)(n-m-1)}{(2n+1)(2n-1)}}$ (Eq.12}

and

$b_{nm}^{-}=\sqrt{\frac{(n+m+2)(n+m+1)}{(2n+1)(2n+3)}}$ (Eq.13}

Section 3

The recurrence relation

Elements of spherical coordinates: constant-r surfaces (spheres), constant-θ surfaces (cones), constant-φ surfaces (half-planes), coordinate lines, and axes.

for ${\beta }_{\mu \nu }^{n+1,m}$ is derived from the ${\beta }_{\mu \nu }^{nn}$ values, where the starting points are m = n so that the requisite recurrence relations are

$a_{nm}^{+}{\beta }_{\mu \nu }^{n+1,m}=-a_{nm}^{-}{\beta }_{\mu \nu }^{n-1,m}+a_{\nu -1,\mu }^{+}{\beta }_{\mu \nu }^{n,\nu -1}+a_{\nu +1,\mu }^{-}{\beta }_{\mu \nu }^{n,\nu +1}$ (Eq.14}

where

$a_{nm}^{+}=\sqrt{\frac{(n+m+1)(n-m+1)}{(2n+1)(2n+3)}}$ (Eq.15}

and

$a_{nm}^{-}=\sqrt{\frac{(n+m)(n-m)}{(2n+1)(2n-1)}}$ (Eq.16}

Lastly, ${\beta }_{\mu \nu }^{n,-m}$ is derived from ${\beta }_{\mu \nu }^{n,m}$ using ${\beta }_{\nu \mu }^{n,-m}=(-1{)}^{\mu +m}{\beta }_{\nu ,-\mu }^{nm}$ (Eq.17}

Thus, using (Eq.1), a translation can be applied in any direction and in either sense, where the displacement is defined as r = θ0 ˆθ + φ0 ˆφ + kAˆr, and where the magnitude of the displacement |r| can be larger or smaller than the minimum sphere radius.

Section 4

Two translated conducting spheres with glowing electromagnetic fields, illustrating memory consumption in multiple scattering computations.

The memory consumption

of the final translation matrix depends on the number of input and output modes considered. If the AUT at the input is displaced, the occupied MRE is expected to be larger than the AUT itself, which, after translation back to the origin, is reduced. The size of the translation matrix can be written as follows, assuming for each point P a double-precision complex number:

$P=J_{in}\cdot J_{out}=2N_{in}(N_{in}+2)\cdot 2N_{out}(N_{out}+2)$ (Eq.18}

with

$N=\frac{2\pi a}{\lambda },$ (Eq.19}

where a is the MRE, and λ is the wavelength.

For the pre-calculation of ${\beta }_{\mu ,\nu }^{mn}$, however, a matrix of the following size is required as well, which is about half the size of the translation matrix:

$P=((N_{in}+N_{out})(N_{in}+N_{out}+2)+1)\cdot (N_{out}(N_{out}+2)+1)$ (Eq.20}

In principle, there is no practical numerical limitation to the maximum number of N-modes that can be used with the recursive formulations to determine the translation matrix. However, the direct calculation of the Legendre polynomials currently utilized has a higher-order limit of approximately N = 85, although when required, this can be easily the generalized translation has no inherent limitation on the magnitude of the displacement A, provided the mode truncation is increased to account for the enlarged minimum radial extent (MRE) after translation.

To verify this, a simulated far-field pattern of a standard gain horn (SGH) centered at the origin is offset by applying a phase taper:

$w_{\text{offset}}(\theta ,\varphi )=w_0(\theta ,\varphi )e^{jk\cos \theta \text{Offset}}$ (Eq.21}

where:

• $w_0(\theta ,\varphi )$: Original far-field pattern centered at the origin,
• $e^{jk\cos \theta \text{Offset}}$: Phase taper introduced by displacing the antenna along the positive z-axis by distance Offset,
• $k=2\pi /\lambda$: Wavenumber.

The offset pattern is then expanded into spherical modes with increased truncation to represent the larger enclosing sphere, translated back using the generalized theorem, filtered if needed, and compared to the original.

Results show excellent agreement (better than numerical precision), confirming no limitation for large offsets when modes are extended accordingly.

Key contributions from Dirix et al. (2025)

• Successful application to translations in any direction and for any displacement magnitude relative to the antenna's radial extent.
• Efficient computation of vector coefficients from scalar coefficients using precise recurrence relations, making the method practical.
• Direct integration into spherical near-field transformation chains, avoiding intermediate far-field transforms and preserving evanescent modes.
• Enables first-ever mode-filtering-based reflection suppression directly on spherical mode coefficients.

The spherical waves

Click for animation.
Concentric spherical shells with rainbow coloring, illustrating nested spherical wave fronts in a 3D rotating view.

Field variations in φ are limited by the order $M$ and field variations in θ are limited by the degree $N$ . For the general case $M=N$ and the total number of modes is $2NN(+2)$. For objects with some rotational symmetry around the -axis we may have $MN$. An empirical value for $N$ is $N=k{r}_o+n_1$ The electric field $\mathbf{𝐄}(\mathbf{𝐫},\theta ,\varphi )=\sum _{n=1}^N\sum _{m=-n}^n{Q}_{smn}{F}_{smn}(\mathbf{𝐫},\theta ,\varphi )$radiated from the primed system ${\overrightarrow{F}}_{smn}^{(c=3)}(r,\theta ,\varphi )=\sum _{\sigma =1}^2\sum _{\nu =1}^{\mathrm{\infty }}\sum _{\mu =-\nu }^{\nu }{C}_{smn\sigma \mu \nu }^{(c=1)}({\theta }_0,{\varphi }_0,kA){\overrightarrow{F}}_{\sigma \mu \nu }^{(c=3)}(r^{\prime },{\theta }^{\prime },{\varphi }^{\prime })$ (Eq.23}
Here, a source of limited extent can be expressed as a weighted sum of spherical waves.

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.

The formula above is for vacuum (refractive index n = 1). In a dielectric medium with refractive index n > 1, the effective wavenumber inside the medium is n k, so the number of modes becomes:

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

In the example calculation (50 µm sphere at 800 nm), an effective refractive index n ≈ 1.5 (typical for fused silica) is implicitly used, giving N ≈ 2.8 × 10⁷.

With finite bandwidth Δf and observation time t, the total number of independent spatio-temporal channels becomes

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

Quantum-mechanical limit

Artistic visualization of independent spatial modes (spherical harmonics) inside a spherical volume, with rainbow-colored symmetric wave patterns and cosmic background.

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

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

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
  

Detailed Mode Number and Time-Bandwidth Product Calculation
kR ≈ 393:
With radius $R$ = 50 µm = 5 × 10⁻⁵ m and wavelength λ = 800 nm = 8 × 10⁻⁷ m,
k = 2π/λ ≈ 7.854 × 10⁶ m⁻¹,
k$R$ = (2π R)/λ = 2π × (R/λ) = 2π × 62.5 ≈ 392.7 (rounded to 393). This matches.
N ≈ 2.8 × 10⁷:This represents the approximate number of spatial electromagnetic modes up to the given wavelength in the sphere's volume, using the formula derived from the photonic density of states (Weyl law) for a large cavity:
N ≈ (8π n³ V)/(3 λ³), where V = (4/3)π $R$³ ≈ 5.24 × 10⁻¹³ m³ is the volume and n accounts for the dielectric.
Without n (vacuum), N ≈ 8.57 × 10⁶. For n ≈ 1.5 (common approximation for silica), n³ = 3.375, so N ≈ 2.89 × 10⁷.
For n ≈ 1.453 (precise value for fused silica at 800 nm), n³ ≈ 3.07, so N ≈ 2.63 × 10⁷.
The given 2.8 × 10⁷ is a reasonable approximation, likely using n ≈ 1.5 and rounding.
Δf · t = 10¹³:With bandwidth Δf = 10 THz = 10¹³ Hz and observation time t = 1 s,
Δf · t = 10¹³ × 1 = 10¹³. This is the time-bandwidth product, representing the number of independent temporal modes. This matches.
N_total ≈ 2.8 × 10²⁰: N_total = N × (Δf · t) ≈ 2.8 × 10⁷ × 10¹³ = 2.8 × 10²⁰. This multiplication is exact and represents the total degrees of freedom (spatial modes × temporal modes) for signals or photons in the system.
The slight discrepancy in N (2.8 × 10⁷ vs. calculated 2.6–2.9 × 10⁷) is typical for approximations in n or constants in such models.

Bekenstein bound limits from quantum gravity

The Bekenstein entropy/information bound
According to the universal entropy bound, the entropy (and hence information capacity) of a complete weakly self-gravitating physical system can be bounded exclusively in terms of its circumscribing radius and total gravitating energy. The bound’s correctness is supported by explicit statistical calculations of entropy, gedanken experiments involving the generalized second law, and Bousso’s covariant holographic bound . On the other hand, it is not always obvious in a particular example how the system avoids having too many states for given energy, and hence violating the bound. We analyze in detail several purported counterexamples of this type (involving systems made of massive particles, systems at low temperature, systems with high degeneracy of the lowest excited states, systems with degenerate ground states, or involving a particle spectrum with proliferation of nearly massless species), and exhibit in each case the mechanism behind the bound’s efficacy.

Information theory and the question of storage capacity

Information theory started as a theory of communication transport of information. The developers of communication channel capacity theorems paid little attention to the akin question of information storage capacity.

In essence such question boils down to a more physically sounding one: what are the limitations on the magnitude of the entropy of a system characterized by general parameters such as size, energy, mass, ...?

In 1981 Bekenstein proposed that the entropy of a complete physical system in asymptotically flat D = 4 spacetime, whose total mass-energy is E, and which fits inside a sphere of radius R, is necessarily bounded from above:

$S\le \frac{2\pi ER}{\hslash c}$

The motivation for this universal entropy bound came from gedanken experiments in which an entropy-bearing object is deposited at a black hole’s horizon with the least possible energy; a violation of the generalized second law seems to occur unless the said bound applies to the object.

The tenor of the argument is that E is to be interpreted as the gravitating energy of the system; this prescription disposes of any ambiguity that would arise if we attempted to redefine the zero of energy.

Quantum wavefunctions in spheres

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

Growth of total channels with bandwidth and time

Real spherical harmonics Y_lm up to l=5

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

Hyperfine structure and Zeeman splitting of the $4^2{P}_{3/2}$ state in ${39}^{}\mathrm{K}$.

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 a spherical volume.

In addition to these spatial degrees of freedom, quantum states may possess internal structure (e.g., spin, fine and hyperfine levels). External fields, as illustrated by the Zeeman effect, lift degeneracies among these internal states, increasing the number of distinguishable quantum states without altering the spatial mode structure.

Index

Core theory

1. Foundations

2. Conceptual and interpretations

3. Mathematical structure and systems

4. Atomic and spectroscopy

5. Wavefunctions and modes

6. Quantum dynamics and evolution

7. Measurement and information

8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments

10. Open quantum systems

11. Quantum field theory

12. Statistical mechanics and kinetic theory

13. Condensed matter and solid-state physics

14. Plasma and fusion physics

15. Timeline

16. Advanced and frontier topics

Book II

• Matter by scale

Book III

• Methods and tools

Book IV

• Data Analysis Techniques

Full contents

1. Foundations (11) ↑ Back to index

1. Physics:Quantum basics

2. Physics:Quantum Postulates

3. Physics:Quantum Hilbert space

4. Physics:Quantum Observables and operators

5. Physics:Quantum mechanics

6. Physics:Quantum mechanics measurements

7. Physics:Quantum state

8. Physics:Quantum system

9. Physics:Quantum superposition

10. Physics:Quantum probability

11. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) ↑ Back to index

12. Physics:Quantum Interpretations of quantum mechanics

13. Physics:Quantum Wave–particle duality

14. Physics:Quantum Complementarity principle

15. Physics:Quantum Uncertainty principle

16. Physics:Quantum Measurement problem

17. Physics:Quantum Bell's theorem

18. Physics:Quantum Hidden variable theory

19. Physics:Quantum nonlocality

20. Physics:Quantum contextuality

21. Physics:Quantum Darwinism

22. Physics:Quantum A Spooky Action at a Distance

23. Physics:Quantum A Walk Through the Universe

24. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

25. Physics:Quantum measurement problem

3. Mathematical structure and systems (13) ↑ Back to index

26. Physics:Quantum Density matrix

27. Physics:Quantum Exactly solvable quantum systems

28. Physics:Quantum Formulas Collection

29. Physics:Quantum A Matter Of Size

30. Physics:Quantum Symmetry in quantum mechanics

31. Physics:Quantum Angular momentum operator

32. Physics:Quantum Runge–Lenz vector

33. Physics:Quantum Approximation Methods

34. Physics:Quantum Matter Elements and Particles

35. Physics:Quantum Dirac equation

36. Physics:Quantum Klein–Gordon equation

37. Physics:Quantum pendulum

38. Physics:Quantum configuration space

4. Atomic and spectroscopy (14) ↑ Back to index

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

39. Physics:Quantum Atomic structure and spectroscopy

40. Physics:Quantum Hydrogen atom

41. Physics:Quantum number

42. Physics:Quantum Multi-electron atoms

43. Physics:Quantum Fine structure

44. Physics:Quantum Hyperfine structure

45. Physics:Quantum Isotopic shift

46. Physics:Quantum defect

47. Physics:Quantum Zeeman effect

48. Physics:Quantum Stark effect

49. Physics:Quantum Spectral lines and series

50. Physics:Quantum Selection rules

51. Physics:Quantum Fermi's golden rule

52. Physics:Quantum beats

5. Wavefunctions and modes (9) ↑ Back to index

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

53. Physics:Quantum Wavefunction

54. Physics:Quantum Superposition principle

55. Physics:Quantum Eigenstates and eigenvalues

56. Physics:Quantum Boundary conditions and quantization

57. Physics:Quantum Standing waves and modes

58. Physics:Quantum Normal modes and field quantization

59. Physics:Number of independent spatial modes in a spherical volume

60. Physics:Quantum Density of states

61. Physics:Quantum carpet

6. Quantum dynamics and evolution (17) ↑ Back to index

62. Physics:Quantum Time evolution

63. Physics:Quantum Schrödinger equation

64. Physics:Quantum Time-dependent Schrödinger equation

65. Physics:Quantum Stationary states

66. Physics:Quantum Perturbation theory

67. Physics:Quantum Time-dependent perturbation theory

68. Physics:Quantum Adiabatic theorem

69. Physics:Quantum Scattering theory

70. Physics:Quantum S-matrix

71. Physics:Quantum tunnelling

72. Physics:Quantum speed limit

73. Physics:Quantum revival

74. Physics:Quantum reflection

75. Physics:Quantum oscillations

76. Physics:Quantum jump

77. Physics:Quantum boomerang effect

78. Physics:Quantum chaos

7. Measurement and information (9) ↑ Back to index

79. Physics:Quantum Measurement theory

80. Physics:Quantum Measurement operators

81. Physics:Quantum Projective measurement

82. Physics:Quantum POVM

83. Physics:Quantum Weak measurement

84. Physics:Quantum Measurement collapse

85. Physics:Quantum entanglement

86. Physics:Quantum Zeno effect

87. Physics:Quantum limit

8. Quantum information and computing (10) ↑ Back to index

88. Physics:Quantum information theory

89. Physics:Quantum Qubit

90. Physics:Quantum Entanglement

91. Physics:Quantum Gates and circuits

92. Physics:Quantum Computing Algorithms in the NISQ Era

93. Physics:Quantum Noisy Qubits

94. Physics:Quantum random access code

95. Physics:Quantum pseudo-telepathy

96. Physics:Quantum network

97. Physics:Quantum money

9. Quantum optics and experiments (5) ↑ Back to index

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

98. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

99. Physics:Quantum optics beam splitter experiments

100. Physics:Quantum Ultra fast lasers

101. Physics:Quantum Experimental quantum physics

102. Physics:Quantum optics

103. Template:Quantum optics operators

10. Open quantum systems (9) ↑ Back to index

104. Physics:Quantum Open systems

105. Physics:Quantum Master equation

106. Physics:Quantum Lindblad equation

107. Physics:Quantum Decoherence

108. Physics:Quantum dissipation

109. Physics:Quantum Markov semigroup

110. Physics:Quantum Markovian dynamics

111. Physics:Quantum Non-Markovian dynamics

112. Physics:Quantum Trajectories

11. Quantum field theory (20) ↑ Back to index

Structural dependency map of quantum field theory.

113. Physics:Quantum field theory (QFT) basics

114. Physics:Quantum field theory (QFT) core

115. Physics:Quantum Fields and Particles

116. Physics:Quantum Second quantization

117. Physics:Quantum Harmonic Oscillator field modes

118. Physics:Quantum Creation and annihilation operators

119. Physics:Quantum vacuum fluctuations

120. Physics:Quantum Propagators in quantum field theory

121. Physics:Quantum Feynman diagrams

122. Physics:Quantum Path integral formulation

123. Physics:Quantum Renormalization in field theory

124. Physics:Quantum Renormalization group

125. Physics:Quantum Field Theory Gauge symmetry

126. Physics:Quantum Non-Abelian gauge theory

127. Physics:Quantum Electrodynamics (QED)

128. Physics:Quantum chromodynamics (QCD)

129. Physics:Quantum Electroweak theory

130. Physics:Quantum Standard Model

131. Physics:Quantum triviality

132. Physics:Quantum confinement problem

12. Statistical mechanics and kinetic theory (9) ↑ Back to index

133. Physics:Quantum Statistical mechanics

134. Physics:Quantum Partition function

135. Physics:Quantum Distribution functions

136. Physics:Quantum Liouville equation

137. Physics:Quantum Kinetic theory

138. Physics:Quantum Boltzmann equation

139. Physics:Quantum BBGKY hierarchy

140. Physics:Quantum Relaxation and thermalization

141. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (13) ↑ Back to index

142. Physics:Quantum Band structure

143. Physics:Quantum Fermi surfaces

144. Physics:Quantum Semiconductor physics

145. Physics:Quantum Phonons

146. Physics:Quantum Electron-phonon interaction

147. Physics:Quantum Superconductivity

148. Physics:Quantum Topological phases of matter

149. Physics:Quantum well

150. Physics:Quantum spin liquid

151. Physics:Quantum spin Hall effect

152. Physics:Quantum phase transition

153. Physics:Quantum critical point

154. Physics:Quantum dot

14. Plasma and fusion physics (8) ↑ Back to index

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

155. Physics:Quantum Fusion reactions and Lawson criterion

156. Physics:Quantum Plasma (fusion context)

157. Physics:Quantum Magnetic confinement fusion

158. Physics:Quantum Inertial confinement fusion

159. Physics:Quantum Plasma instabilities and turbulence

160. Physics:Quantum Tokamak core plasma

161. Physics:Quantum Tokamak edge physics and recycling asymmetries

162. Physics:Quantum Stellarator

15. Timeline (8) ↑ Back to index

163. Physics:Quantum mechanics/Timeline

164. Physics:Quantum mechanics/Timeline/Pre-quantum era

165. Physics:Quantum mechanics/Timeline/Old quantum theory

166. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

167. Physics:Quantum mechanics/Timeline/Quantum field theory era

168. Physics:Quantum mechanics/Timeline/Quantum information era

169. Physics:Quantum mechanics/Timeline/Quantum technology era

170. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (16) ↑ Back to index

171. Physics:Quantum topology

172. Physics:Quantum battery

173. Physics:Quantum Supersymmetry

174. Physics:Quantum Black hole thermodynamics

175. Physics:Quantum Holographic principle

176. Physics:Quantum gravity

177. Physics:Quantum De Sitter invariant special relativity

178. Physics:Quantum Doubly special relativity

179. Physics:Quantum arithmetic geometry

180. Physics:Quantum unsolved problems

181. Physics:Quantum Yang-Mills mass gap

182. Physics:Quantum gravity problem

183. Physics:Quantum black hole information paradox

184. Physics:Quantum dark matter problem

185. Physics:Quantum neutrino mass problem

186. Physics:Quantum matter-antimatter asymmetry problem

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

References

Author: Harold Foppele

Source attribution: Number of independent spatial modes in a spherical volume