View source for Physics:Number of independent spatial modes in a spherical volume

{{Short description|Maximum number of independent spatial modes inside a spherical volume}}

{{Quantum book backlink|Wavefunctions and modes}}
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<ref name="Miller2000">{{cite journal | last=Miller | first=David A. B. | year=2000 | title=Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths | journal=Applied Optics | volume=39 | issue=11 | pages=1681–1699 | doi=10.1364/AO.39.001681}}</ref>

<ref name="Erhard2020">{{cite journal | last1=Erhard | first1=Manuel | last2=Krenn | first2=Mario | last3=Zeilinger | first3=Anton | year=2020 | title=Advances in high-dimensional quantum entanglement | journal=Nature Reviews Physics | volume=2 | pages=365–381 | doi=10.1038/s42254-020-0193-5}}</ref>

<ref name="Fabre2020">{{cite journal | last1=Fabre | first1=Claude | last2=Treps | first2=Nicolas | year=2020 | title=Modes and states in quantum optics | journal=Reviews of Modern Physics | volume=92 | issue=3 | article=035005 | doi=10.1103/RevModPhys.92.035005}}</ref>

<ref name="Lloyd2000">{{cite journal | last=Lloyd | first=Seth | year=2000 | title=Ultimate physical limits to computation | journal=Nature | volume=406 | pages=1047–1054 | doi=10.1038/35023282}}</ref>


<ref name="01M">{{cite conference |last=Jensen |first=F. |title=The influence of inaccuracies of measurements on antenna near-field far-field correlation |book-title=Proc. Fourth Coll. on Microwave Communication |volume=III |pages=ET-13 |year=1970 |location=Budapest |url=https://backend.orbit.dtu.dk/ws/files/3665733/Hansen.pdf }}</ref>
<ref name="02M">{{cite thesis |last=Jensen |first=F. |title=Electromagnetic near-field far-field correlations |type=Ph.D. |publisher=Lab. of Electromagnetic Theory, Technical University of Denmark |year=1970 |id=LD 15 |url=https://books.google.com/books/about/Electromagnetic_Near_field_Far_field_Cor.html?id=GzklHAAACAAJ }}</ref>
<ref name="03M">{{cite journal |last1=Turchin |first1=V.I. |last2=Tseytlin |first2=N.M. |title=Antenna testing based on near-field measurements (Review) |journal=Radio Engineering and Electronic Physics |volume=24 |issue=12 |pages=1–26 |year=1979 |url=https://publications.rwth-aachen.de/record/815778/files/815778.pdf }}</ref>
<ref name="04M">{{cite book |last=Hansen |first=J.E. |title=Spherical Near-Field Antenna Measurements |publisher=Peter Peregrinus Ltd. |location=London |year=1988 |isbn=978-0-86341-110-6 |doi=10.1049/pbew026e |url=https://digital-library.theiet.org/content/books/ew/pbew026e }}</ref>
<ref name="05M">{{cite conference |last1=Lemanczyk |first1=J. |last2=Jensen |first2=F. |title=The calibration of probes for near-field measurements |book-title=Proc. 13th Annual Meeting and Symposium Antenna Measurements Techniques Association (AMTA) |pages=9-5–9-10 |year=1991 |location=Boulder, CO  |url=https://www.researchgate.net/publication/252157840_ON_THE_NUMBER_OF_MODES_IN_SPHERICAL_WAVE_EXPANSIONS }}</ref>

<ref name="01AP">{{Cite journal
 | last1 = Dirix
 | first1 = Marc
 | last2 = Gregson
 | first2 = Stuart F.
 | last3 = Dubrovka
 | first3 = Rostyslav F.
 | date = 2025-09-05
 | title = Use of the Generalized Vector Addition Theorem for Antenna Position Translation for Spherical Mode-Filtering-Based Reflection Suppression
 | journal = Sensors
 | publisher = MDPI
 | volume = 25
 | issue = 17
 | article-number = 5557
 | doi = 10.3390/s25175557
 | pmid = 40942983
 | pmc = PMC12430979
 | url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12430979/
 | access-date = 2026-01-07
}}</ref>
<ref name="01P">{{Cite journal |last1=Bucci |first1=Ovidio M. |last2=D'Elia |first2=Giuseppe |last3=Migliore |first3=Marco Donald |date=2004 |title=A general and effective clutter filtering strategy in near-field antenna measurements |journal=IEE Proceedings - Microwaves, Antennas and Propagation |volume=151 |issue=3 |pages=227–235 |doi=10.1049/ip-map:20040267}}</ref>

<ref name="02P">{{Cite conference |last1=Hindman |first1=Greg E. |last2=Newell |first2=Allen C. |date=2005-10-30 |title=Reflection Suppression in a Large Spherical Near-Field Range |conference=Proceedings of the 27th Annual Meeting and Symposium of the Antenna Measurement Techniques Association (AMTA) |location=Newport, RI, USA}}</ref>

<ref name="03P">{{Cite conference |last1=Hindman |first1=Greg E. |last2=Newell |first2=Allen C. |date=2006-05-01 |title=Reflection Suppression to Improve Anechoic Chamber Performance |conference=Proceedings of the AMTA Europe |location=Munich, Germany}}</ref>

<ref name="04P">{{Cite conference |last1=Hess |first1=Daniel W. |date=2007-11-11 |title=The IsoFilter™ Technique: Isolating an Individual Radiator from Spherical Near-Field Data Measured in a Contaminated Environment |conference=Proceedings of the EuCAP 2007 |location=Edinburgh, UK}}</ref>

<ref name="05P">{{Cite conference |last1=Gregson |first1=Stuart F. |last2=Newell |first2=Allen C. |last3=Hindman |first3=Greg E. |date=2009-11-01 |title=Reflection Suppression in Cylindrical Near-Field Antenna Measurement Systems—Cylindrical MARS |conference=Proceedings of the AMTA |location=Salt Lake City, UT, USA}}</ref>

<ref name="06P">{{Cite book |last1=Parini |first1=Clive G. |last2=Gregson |first2=Stuart F. |last3=McCormick |first3=John |last4=van Rensburg |first4=Daniël Janse |last5=Eibert |first5=Thomas |date=2021 |title=Theory and Practice of Modern Antenna Range Measurements: Volumes 1 & 2 |edition=2nd |publisher=IET Press |location=Stevenage, UK |isbn=978-1-83953-126-2}}</ref>

<ref name="07P">{{Cite journal |last=Newell |first=Allen C. |date=1988-06 |title=Error Analysis Techniques for Planar Near-Field Measurements |journal=IEEE Transactions on Antennas and Propagation |volume=36 |issue=6 |pages=754–768 |doi=10.1109/8.1177}}</ref>

<ref name="08P">{{Cite conference |last1=Chen |first1=Zhong |last2=Gregson |first2=Stuart F. |last3=Wang |first3=Yihong |date=2023-10-08 |title=Novel Application of Compressed Sensing in Cylindrical Mode Filtering for Far-Field Antenna Measurements |conference=Proceedings of the Antenna Measurement Techniques Association Annual Symposium |location=Seattle, WA, USA}}</ref>

<ref name="09P">{{Cite conference |last1=Dubrovka |first1=Rostyslav F. |last2=Gregson |first2=Stuart F. |last3=Parini |first3=Clive G. |date=2025-03-30 |title=Compressive Sensing Technique in a Complex Electromagnetic Environment |conference=Proceedings of the EuCAP |location=Stockholm, Sweden}}</ref>

<ref name="10P">{{Cite journal |last=Jensen |first=Frank |date=1975 |title=On the probe compensation for near-field measurements on a sphere |journal=Archiv für Elektronik und Übertragungstechnik |volume=29 |pages=305–308}}</ref>

<ref name="11P">{{Cite conference |last=Wacker |first=Peter F. |date=1974 |title=Near-field antenna measurements using a spherical scan: Efficient data reduction with probe correction |conference=Conference on Precision Electromagnetic Measurements |book-title=IEE Conference Publication |volume=113 |pages=286–288}}</ref>

<ref name="12P">{{Cite book |last=Kerns |first=David M. |date=1981 |title=Plane-Wave Scattering-Matrix Theory of Antennas and Antenna-Antenna Interactions |publisher=National Bureau of Standards |location=Boulder, CO, USA |series=NBS Monograph |volume=162}}</ref>

<ref name="13P">{{Cite thesis |last=Larsen |first=Flemming Holm |date=1980 |title=Probe-Corrected Spherical Near-Field Antenna Measurements |type=PhD thesis |publisher=Technical University of Denmark |department=Department of Electromagnetic Systems |location=Lyngby, Denmark}}</ref>

<ref name="14P">{{Cite journal |last1=Chew |first1=Weng Cho |last2=Wang |first2=Yun Ming |date=1993 |title=Efficient Ways to Compute the Vector Addition Theorem |journal=Journal of Electromagnetic Waves and Applications |volume=7 |issue=5 |pages=651–665 |doi=10.1163/156939393X00552}}</ref>

<ref name="15P">{{Cite web |title=§14.10 Zeros |url=https://dlmf.nist.gov/14.10 |website=Digital Library of Mathematical Functions |publisher=National Institute of Standards and Technology |access-date=2025-08-11}}</ref>

<ref name="16P">{{Cite book |editor-last=Hansen |editor-first=Jørgen E. |date=1988 |title=Spherical Near-Field Antenna Measurements |publisher=Peter Peregrinus Ltd. |location=London, UK |isbn=978-0863411106}}</ref>

<ref name="17P">{{Cite book |last=Balanis |first=Constantine A. |date=1989 |title=Advanced Engineering Electromagnetics |publisher=John Wiley & Sons, Inc. |location=Hoboken, NJ, USA |isbn=978-0-471-62194-2}}</ref>

<ref name="18P">{{Cite conference |last1=Jensen |first1=Frank |last2=Frandsen |first2=Aksel |date=2004-10-17 |title=On the Number of Modes in Spherical Wave Expansions |conference=Proceedings of the Antenna Measurement Techniques Association, Annual Symposium |location=Atlanta, GA, USA}}</ref>

<ref name="19P">{{Cite book |last=Dirix |first=Marc |date=2018 |title=Quiet Zone Performance Qualification Using Spherical Near-Field Scanning |publisher=Dr. Hut |location=Munich, Germany |isbn=978-3-8439-3576-0}}</ref>

<ref name="20P">{{Cite web |title=Altair Feko |url=https://altair.com/feko |website=Altair |access-date=2025-05-20}}</ref>

<ref name="21P">{{Cite conference |last1=Dubrovka |first1=Rostyslav F. |last2=Jones |first2=Robert C. |last3=Parini |first3=Clive G. |last4=Gregson |first4=Stuart F. |date=2023-10-08 |title=Predication of Planar Near-Field Measurements Based on Full-Wave Three-Dimensional CEM Measurement Simulation |conference=Proceedings of the AMTA |location=Seattle, WA, USA}}</ref>

<ref name="22P">{{Cite journal |last=Chew |first=W. C. |date=2008 |title=Vector Addition Theorem and Its Diagonalization |journal=Communications in Computational Physics |volume=3 |issue=2 |pages=330–341 |url=https://www.global-sci.com/index.php/cicp/article/download/5517/10971/12201}}</ref>

<ref name="23P">J. D. Bekenstein, "Universal upper bound on the entropy-to-energy ratio for bounded systems", ''Physical Review D'' '''23''', 287 (1981). DOI: [https://doi.org/10.1103/PhysRevD.23.287 10.1103/PhysRevD.23.287]</ref>

<ref name="24P">{{cite journal |last1=Bousso |first1=Raphael |last2=Kaya |first2=Sami |date=April 2025 |title=Holographic entropy cone beyond AdS/CFT |journal=Physical Review D |volume=111 |issue=8 |pages=086014 |doi=10.1103/PhysRevD.111.086014 |arxiv=2502.03516 |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.111.086014}}</ref>

<ref name="Jensen2004">{{cite conference
| last1 = Jensen
| first1 = F.
| last2 = Frandsen
| first2 = A.
| title = On the number of modes in spherical wave expansions
| conference = 26th Annual Meeting of the Antenna Measurement Techniques Association (AMTA)
| pages = 489–494
| year = 2004
| location = Stone Mountain Park, GA, USA
| url = https://www.ticra.com/wp-content/uploads/2018/03/Number-of-spherical-modes.pdf
}}</ref>
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Quantum optics spatial modes inside a spherical volume is based on [[Wikipedia:Quantum optics|'''Quantum optics''']] a branch of [[Wikipedia:atomic, molecular, and optical physics|atomic, molecular, and optical physics]] and [[Wikipedia:quantum chemistry|quantum chemistry]] that studies the behavior of [[Wikipedia:photon|photons]] (individual quanta of light). and refers to the number of simultaneously supportable [[Wikipedia:Propagator|propagable]] [[Wikipedia:Orthogonal frequency-division multiplexing|orthogonal]] [[Wikipedia:Spatial multiplexing|spatial modes]] in the [[Wikipedia:Wave function|wavefield]] confined in a [[Wikipedia:Spherical shell|spherical region]] of [[Wikipedia:Radius|radius]] ''R'' with negligible [[Wikipedia:Crosstalk|crosstalk]]. 

[[File:Independent_spatial_modes_spherical_volume_yellow1.jpg|thumb|400px|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.]]
[[File:Phase-amplitude plots of several Laguerre-Gauss modes.svg|thumb|upright=1.35|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 [[Wikipedia:Quantum channel|quantum channels]] and deals with [[Wikipedia:Microwave|electromagnetic waves]] (light or radio waves), acoustic waves (sound or [[Wikipedia:Ultrasound|ultrasonic waves]]), or [[Wikipedia:Matter wave|de Broglie matter waves]].
Since the early days of spherical near-field far-field transformations <ref name="01M" /><ref name="02M" /> a recommendation for the necessary number of polar modes has been given by <math>N = k r_0 + 10</math> <ref name="Jensen2004" />, <math>k</math> being the wavenumber and <math>r_0</math>  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, <math>N</math>, 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. <br>
Turchin and Tseytlin <ref name="03M" /> have stated that 
<math>N = n_o + 3 / (1 - (n_o / (k r_o))^3 )</math> but neither a justification nor a parametric expression have been presented.<ref name="04M" /><br><br>
'''Keywords''': [[Wikipedia:Generalized_pencil-of-function_method#Noise_filtering|Noise Filtering]], [[Wikipedia:Normal mode|Normal Mode]], [[Wikipedia:Squeezed_coherent_state#Based_on_the_number_of_modes|Number of Modes]], [[Wikipedia:Spherical harmonics|Spherical Expansion]], [[Wikipedia:Bessel function|Spherical Modes]], [[Wikipedia:Antenna_measurement#Spherical_near-field_range|Spherical Near Field]], [[Wikipedia:Spherical wave transformation|Spherical Waves]]. [[Wikipedia:Antenna_measurement|Antenna measurement]], [[Wikipedia:Echo suppression and cancellation|echo suppression]], [[Wikipedia:Bessel function|spherical wave expansion]], vector addition theorem<ref name="22P" />, [[Wikipedia:Waveguide filter|mode filtering]]. 
== Modes==
[[File:Spherical harmonics modes in sphere - colorful waves 2026.jpg|thumb|160px|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, [[Wikipedia:Gaussian beam#Laguerre-Gaussian modes|Laguerre–Gaussian modes]], [[Wikipedia:Gaussian beam#Hermite-Gaussian modes|Hermite–Gaussian modes]], or [[Wikipedia:Spherical harmonics|spherical-harmonic modes]]). The number of such modes in principle depends upon the size of the accessible [[Wikipedia:phase space|phase space]] defined in the sphere and finally upon the [[Wikipedia:Diffraction-limited system|diffraction limit]] contained in the [[Wikipedia:Wavelength|wavelength]] ''λ''.<ref name="Miller2000" /><ref name="Fabre2020" />
This applies in many fields. For quantum optics and [[Wikipedia:Quantum information|quantum information]] research, this determines the [[Wikipedia:Curse of dimensionality|maximum dimensionality]] of [[Wikipedia:Quantum entanglement|quantum entanglement]] in the spatial mode and the [[wikipedia:Quantum capacity|quantum capacity]] of free-space [[Wikipedia:Quantum channel|quantum communication channels]].<ref name="Erhard2020" /> For classical [[Wikipedia:Free-space optical communication|free-space optical communication]] links, the same counting gives the ultimate (hole-burning-free) upper bound on the number of independent [[Wikipedia:Space-time adaptive processing|spatial channels]], the diffraction-limited number of spatial degrees of freedom (a three-dimensional analogue of the familiar A/λ² limit for planar apertures).<ref name="Miller2000" />

==Applications==
[[File:Phasearray.gif|thumb|upright=0.8|Animation showing the [[Wikipedia:radiation pattern|radiation pattern]]]]
For [[Wikipedia:Phased array|phased arrays]] in radar or radio astronomy applications, this defines the number of independently [[Wikipedia:Beam steering|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 [[Wikipedia:5D_optical_data_storage|five-dimensional optical information recording technologies]], this determines the overall data packing density if all possible spatial and polarization freedoms are utilized.<ref name="Miller2000" />
Since the same counting in the [[Wikipedia:Phase space|phase-space]] argument holds equally for classical waves and [[Wikipedia:Quantum state|quantum states]] consisting of one or few photons, this limit turns out to be universal.<ref name="Erhard2020" /><ref name="Miller2000" /><ref name="Fabre2020" />{{efn-ua|''Note: The calculations and limits below assume an ideal sphere, a homogeneous medium, and neglect losses, decoherence, or other non-ideal effects. An “independent mode” is defined here as an orthogonal spatial mode with minimal crosstalk''}}<ref name="Lloyd2000" /><br>
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 [[Wikipedia:Phase-type distribution|phase taper]] across the AUT pattern and results in far greater interference occurring between the direct and indirect [[Wikipedia:Driven and parasitic elements|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 [[Wikipedia:Near and far field|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
<br>

== History ==

[[File:Stanford2010DavidMiller.png|thumb|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.<ref>{{cite journal |last=Toraldo di Francia |first=G. |year=1969 |title=Degrees of freedom of an image |journal=Journal of the Optical Society of America |volume=59 |issue=7 |pages=799–804 |doi=10.1364/JOSA.59.000799}}</ref><ref>{{cite journal |last=Lukosz |first=W. |year=1966 |title=Optical systems with resolving powers exceeding the classical limit |journal=Journal of the Optical Society of America |volume=56 |issue=11 |pages=1463–1472 |doi=10.1364/JOSA.56.001463}}</ref> These works showed that the number of spatial degrees of freedom of a field confined to a region of radius ''R'' scales as (2π''R''/λ)² in two dimensions and (2π''R''/λ)³ in three dimensions.<br>
In 2000, [[Wikipedia:David A. B. Miller|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.<ref name="Miller2000" />This result has become the standard reference for the maximum number of “beams in a sphere”.<br>
In 2004 Frank Jensen published "On the number of modes in spherical wave expansions"<ref name="Jensen2004" /> describing polar modes and improved expression for the number of modes<br>
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.<ref name="Erhard2020" /><ref name="Fabre2020" />
When finite bandwidth Δ''f'' and observation time ''t'' are taken into account, the total number of independent spatio-temporal channels is approximately ''N''<sub>spatial</sub> × Δ''f'' × ''t''. This formula is now routinely used to estimate fundamental limits in quantum communication, radar, and imaging.<ref name="Fabre2020" />
{{efn-ua|'' Note: Recent studies show that practical implementations of high-dimensional spatial modes can be limited by non-ideal media, optical aberrations, and detector limitations.''}}<br>
In 2025 Marc Dirix et al. published Generalized Vector Addition Thorem for displacements<ref name="22P" /> that can be smaller or larger than the maximum radial extent of the object<ref name="01AP" />

==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<ref name="22P" />. 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.<br>
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 [[Wikipedia:Singular value decomposition#Numerical approach|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 <ref name="02P" /><ref name="03P" /><ref name="04P" /><ref name="05P" /><ref name="06P" /><ref name="07P" />. These have been used very successfully to greatly improve the facility-level uncertainty budget <ref name="08P" /> 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 <ref name="07P" /> and, more recently, have been extended to take advantage of sparse sampling compressive sensing techniques <ref name="09P" /><ref name="10P" />.

== Generalized vector addition theorem ==
	
[[File:Triple Integrals in Cylindrical and Spherical Coordinates.png|thumb|275px|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.<ref name="Dirix2025">{{cite journal
 | last1 = Dirix
 | first1 = Marc
 | last2 = Gregson
 | first2 = Stuart F.
 | last3 = Dubrovka
 | first3 = Rostyslav F.
 | title = Use of the Generalized Vector Addition Theorem for Antenna Position Translation for Spherical Mode-Filtering-Based Reflection Suppression
 | journal = Sensors
 | volume = 25
 | issue = 17
 | pages = 5557
 | date = 5 September 2025
 | doi = 10.3390/s25175557
 | url = https://www.mdpi.com/1424-8220/25/17/5557
 | access-date = 8 February 2026
}}</ref>

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 ({{EquationNote|Eq. 1}}) is:====
	
[[File:Translation Addition Theorem for Vector Spherical Wave Functions.png|thumb|600px|center|Translation Addition Theorem for Vector Spherical Wave Functions]]
<math>
F_{smn}^{(-c)}(r,\theta,\phi) = \sum_{g=1}^{2} \sum_{\nu=1}^{\infty} \sum_{\mu=-\nu}^{\nu} C_{g \mu 0}^{s m n (-c)}(\beta_0, \phi_0, k A) \, F_{\sigma(g) \mu \nu}^{(-3)}(r', \theta', \phi')
</math>   ({{EquationRef|1|Eq. 1}}}

where:
* <math>F_{smn}^{(-c)}</math> are vector spherical wave functions (''s''=1,2 for TE/TM modes),
* superscript <math>(-c)</math> denotes radial dependence,
* primed coordinates are translated by distance ''A'' along the ''z''-axis (general direction via angles <math>\beta_0, \phi_0</math>),
* <math>C</math> 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 <math>e^{j k \hat{r} \cdot \mathbf{A}}</math>.
The translation coefficients are defined as <ref name="14P" /> and can further be expressed as

<math>
C_{\sigma\mu\nu}^{s m n}(\theta_0, \varphi_0, kA) = \alpha \left[ \delta_{s\sigma} A_{\mu\nu}^{m n}(\theta_0, \varphi_0, kA) + \delta_{3-s,\sigma} B_{\mu\nu}^{m n}(\theta_0, \varphi_0, kA) \right]
</math>({{EquationRef|2|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 <ref name="06P" />. Several conversions between different normalizations are reported within the open literature, with a summary being presented within (A1) of <ref name="15P" />. For the present work, however, the translation coefficients <math>A_{\sigma\mu\nu}^{s m n}(kA)</math> and <math>B_{\sigma\mu\nu}^{s m n}(kA)</math> are based on the spherical wave functions <ref name="15P" />, while the SWE utilized to attain the spherical mode coefficients employs the normalization proposed in <ref name="13P" /><ref name="16P" />.
Thus, the requisite correction factor for the normalization employed here can be expressed as

<math>
\alpha = \sqrt{\frac{\nu(\nu+1)}{n(n+1)}}
</math>({{EquationRef|3|Eq.3}}}

with

<small><math>
A_{\mu\nu}^{m n}(\theta_0, \varphi_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)\varphi_0}</math></small>({{EquationRef|4|Eq.4}}}<br>

and

<small><math>
B_{\mu\nu}^{m n}(\theta_0, \varphi_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)\varphi_0}</math></small>({{EquationRef|5|Eq.5}}}

where the linearization coefficient ''b'' is<br><small><math>b(m,n,-\mu,\nu,p) = \frac{2p+1}{2p-1} \Big[ (\nu-\mu)(\nu+\mu+1)\, a(m,n,-\mu-1,\nu,p)</math>
<math>\qquad - (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) \Big] \quad </math></small>({{EquationRef|6|Eq.6}}}<br>Here, a(m, n, −µ, ν, p)’s are the linearization coefficients, which can be expressed in terms of Wigner 3-j symbols <ref name="15P" />, and <math>z^{(c)}_p</math> is the spherical radial function of order p and type (c). Furthermore, <math>P_p^{m-\mu}(\cos\theta_0)</math> 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 <ref name="14P" />. 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:====
[[File:Multiple Scattering by Two PEC Spheres Using Translation Addition.png|thumb|650px|center|Geometry of two PEC spheres separated by δ along the z-axis, illustrating the translation addition theorem for vector spherical waves.]]
<math>
A^{mn}_{\mu\nu}(\theta_0, \phi_0, kA) = \beta^{mn}_{\mu\nu}(\theta_0, \phi_0, kA)
</math>
<math>
{} + kA \sin\theta_0 \, e^{-i\phi_0} \, \frac{1}{2(\nu+1)} \sqrt{ \frac{(\nu-\mu+2)(\nu-\mu+1)}{(2\nu+1)(2\nu+3)} } \, \beta^{mn}_{\mu-1,\nu+1}(\theta_0, \phi_0, kA)
</math>
<math>
{} - kA \sin\theta_0 \, e^{-i\phi_0} \, \frac{1}{2\nu} \sqrt{ \frac{(\nu+\mu-1)(\nu+\mu)}{(2\nu-1)(2\nu+1)} } \, \beta^{mn}_{\mu-1,\nu-1}(\theta_0, \phi_0, kA)
</math>
<math>
{} - kA \sin\theta_0 \, e^{i\phi_0} \, \frac{1}{2(\nu+1)} \sqrt{ \frac{(\nu+\mu+2)(\nu+\mu+1)}{(2\nu+1)(2\nu+3)} } \, \beta^{mn}_{\mu+1,\nu+1}(\theta_0, \phi_0, kA)
</math>
<math>
{} + kA \sin\theta_0 \, e^{i\phi_0} \, \frac{1}{2\nu} \sqrt{ \frac{(\nu-\mu)(\nu-\mu-1)}{(2\nu-1)(2\nu+1)} } \, \beta^{mn}_{\mu+1,\nu-1}(\theta_0, \phi_0, kA)
</math>
<math>
{} + kA \sin\theta_0 \, \frac{1}{\nu+1} \sqrt{ \frac{(\nu+\mu+1)(\nu-\mu+1)}{(2\nu+1)(2\nu+3)} } \, \beta^{mn}_{\mu,\nu+1}(\theta_0, \phi_0, kA)
</math>
<math>
{} + kA \sin\theta_0 \, \frac{1}{\nu} \sqrt{ \frac{(\nu+\mu)(\nu-\mu)}{(2\nu-1)(2\nu+1)} } \, \beta^{mn}_{\mu,\nu-1}(\theta_0, \phi_0, kA)
</math>
({{EquationRef|7|Eq.7}}}

and

<math>
B^{mn}_{\mu\nu}(\theta_0, \phi_0, kA) = kA \cos\theta_0 \, \frac{i^{\mu}}{\nu(\nu+1)} \, \beta^{mn}_{\mu\nu}(\theta_0, \phi_0, kA)
</math>
<math>
{} + i kA \sin\theta_0 \, \frac{1}{2\nu(\nu+1)} \left[ \sqrt{ (\nu - \mu)(\nu + \mu + 1) } \, e^{i\phi_0} \, \beta^{mn}_{\mu+1,\nu}(\theta_0, \phi_0, kA) \right.
</math>
<math>
\left.
{} + \sqrt{ (\nu + \mu)(\nu - \mu + 1) } \, e^{-i\phi_0} \, \beta^{mn}_{\mu-1,\nu}(\theta_0, \phi_0, kA) \right]
</math>
({{EquationRef|8|Eq.8}}}

Here, ({{EquationNote|7|Eq. 7}}) and ({{EquationNote|8|Eq. 8}}) refer to the calculation of the scalar translation matrix, which is in the form of <math>\beta^{mn}_{\mu\nu} (\theta_0, \phi_0, kA)</math>, for which recurrence relations also exist and are developed in <ref name="17P" />. Here, the starting values for n = 0, m = 0, i.e., <math>\beta^{00}_{\nu\mu} (\theta_0, \phi_0, kA)</math> can be easily derived from the scalar wave equation using

<math>
\beta^{0 0}_{\nu\mu}(\theta_0, \phi_0, kA) = (-1)^{\mu+\nu} \sqrt{4\pi} \, Y_{\nu, -\mu}(\theta_0, \phi_0) \, z_n^{(c)}(kA)
</math>
({{EquationRef|9|Eq.9}}}

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

<math>
Y_{n,m}(\theta_0, \phi_0) = (-1)^m \sqrt{ \frac{(n - m)! (2n + 1)}{(n + m)! 4\pi} } P_n^{|m|}(\cos\theta_0) e^{i m \phi_0}
</math>
({{EquationRef|10|Eq10}}}

<math>P_n^{|m|}(\cos\theta)</math> is the associated Legendre function. This calculation also makes use of the standard relation <math>Y_{n,-m}(\theta_0, \phi_0) = (-1)^m Y_{n,m}^*(\theta_0, \phi_0)</math> <ref name="17P"/>. Then, <math>\beta^{n n}_{\mu\nu}</math> 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 <math>Y_{n,-m}(\theta_0, \phi_0) = (-1)^m Y_{n,m}^*(\theta_0, \phi_0)</math> <ref name="17P" />. Then, <math>\beta^{n n}_{\mu\nu}</math> may be expanded using the relation

<math>
b_{n n}^{+} \beta_{\mu \nu}^{n+1, n+1} = b_{\nu-1, \mu-1}^{+} \beta_{\mu-1, \nu-1}^{n n} + b_{\nu+1, \mu-1}^{-} \beta_{\mu-1, \nu+1}^{n n}
</math>
({{EquationRef|11|Eq.11}}}

where

<math>
b_{n m}^{+} = \sqrt{ \frac{(n - m)(n - m - 1)}{(2n + 1)(2n - 1)} }
</math>
({{EquationRef|12|Eq.12}}}

and

<math>
b_{n m}^{-} = \sqrt{ \frac{(n + m + 2)(n + m + 1)}{(2n + 1)(2n + 3)} }
</math>
({{EquationRef|13|Eq.13}}}

===Section 3===
====The recurrence relation====
[[File:Spherical coordinate elements.svg|thumb|200px|Elements of spherical coordinates: constant-r surfaces (spheres), constant-θ surfaces (cones), constant-φ surfaces (half-planes), coordinate lines, and axes.]]
for <math>\beta_{\mu \nu}^{n+1, m}</math> is derived from the <math>\beta_{\mu \nu}^{n n}</math> values, where the starting points are ''m'' = ''n'' so that the requisite recurrence relations are

<math>
a_{n m}^{+} \beta_{\mu \nu}^{n+1, m} = -a_{n m}^{-} \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}
</math>
({{EquationRef|14|Eq.14}}}

where

<math>
a_{n m}^{+} = \sqrt{ \frac{(n + m + 1)(n - m + 1)}{(2n + 1)(2n + 3)} }
</math>
({{EquationRef|15|Eq.15}}}

and

<math>
a_{n m}^{-} = \sqrt{ \frac{(n + m)(n - m)}{(2n + 1)(2n - 1)} }
</math>
({{EquationRef|16|Eq.16}}}

Lastly, <math>\beta_{\mu \nu}^{n, -m}</math> is derived from <math>\beta_{\mu \nu}^{n, m}</math> using
<math>
\beta_{\nu \mu}^{n, -m} = (-1)^{\mu + m} \beta_{\nu, -\mu}^{n m}
</math>
({{EquationRef|17|Eq.17}}}

Thus, using ({{EquationNote|1|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===
[[File:Memory consumption illustration electric fields.jpg|thumb|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:

<math>
P = J_{in} \cdot J_{out} = 2 N_{in} (N_{in} + 2) \cdot 2 N_{out} (N_{out} + 2)
</math>
({{EquationRef|18|Eq.18}}}

with

<math>
N = \frac{2 \pi a}{\lambda},
</math>
({{EquationRef|19|Eq.19}}}

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

For the pre-calculation of <math>\beta^{m n}_{\mu, \nu}</math>, however, a matrix of the following size is required as well, which is about half the size of the translation matrix:

<math>
P = ((N_{in} + N_{out})(N_{in} + N_{out} + 2) + 1) \cdot (N_{out}(N_{out} + 2) + 1)
</math>
({{EquationRef|20|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:

<math>
w_{\text{offset}}(\theta, \phi) = w_0(\theta, \phi) \, e^{j k \cos\theta \, \text{Offset}}
</math>
({{EquationRef|21|Eq.21}}}

where:
* <math>w_0(\theta, \phi)</math>: Original far-field pattern centered at the origin,
* <math>e^{j k \cos \theta \, \text{Offset}}</math>: Phase taper introduced by displacing the antenna along the positive z-axis by distance ''Offset'',
* <math>k = 2\pi / \lambda</math>: 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 ==
[[File:Nested spherical wavefronts rainbow rotation.gif|thumb|Click for animation.<br>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 <math>M</math> and field variations in θ are limited by the degree <math>N</math> . For the general case <math>M =N</math> and the total number of modes is <math>2N N( +2)</math>. For objects with some rotational symmetry around the -axis we may have <math>MN</math>. An empirical value for <math>N</math> is <math>N = kr_o + n_1</math>  
The electric field <math>\mathbf{E}(\mathbf{r}, \theta, \phi) = \sum_{n=1}^{N} \sum_{m=-n}^{n} Q_{smn} F_{smn}(\mathbf{r}, \theta, \phi)</math>radiated from the primed system <math>\vec{F}_{smn}^{(c=3)}(r, \theta, \phi) = \sum_{\sigma=1}^{2} \sum_{\nu=1}^{\infty} \sum_{\mu=-\nu}^{\nu} C_{smn\sigma\mu\nu}^{(c=1)}(\theta_0, \phi_0, kA) \, \vec{F}_{\sigma\mu\nu}^{(c=3)}(r', \theta', \phi')</math> ({{EquationRef|1|Eq.23}}}<br>Here, a source of limited extent can be expressed as a weighted sum of spherical waves.<ref name="05M" />

== Diffraction limit ==
The fundamental [[Wikipedia:Diffraction|diffraction limit]] is

:<math>N \approx 2 \times \frac{(kR)^3}{3\pi^2} \approx 0.068 \left( \frac{2\pi R}{\lambda} \right)^3</math>

independent [[Wikipedia:Spatial modulation|spatial modes]] (including both [[Wikipedia:Dielectric|polarisation states]]), where <math>k = \frac{2\pi}{\lambda}</math> is the [[Wikipedia:Wavenumber|wavenumber]].<ref name="Miller2000" /><ref name="Fabre2020" />

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:

:<math>N \approx 2 \times \frac{(n k R)^3}{3\pi^2} \approx 0.068 \left( \frac{2\pi n R}{\lambda} \right)^3</math>

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 × 10<sup>7</sup>.

With [[Wikipedia:Shannon–Hartley theorem#Bandwidth-limited case|finite bandwidth]] Δ''f'' and observation time ''t'', the total number of independent [[Wikipedia:Singular spectrum analysis#Spatio-temporal gap filling|spatio-temporal channels]] becomes

:<math>N_\text{total} \approx N \times \Delta f \times t</math>.<ref name="Fabre2020" />

== Quantum-mechanical limit ==
[[File:Spherical harmonics modes in sphere - rainbow symmetry 2026.jpg|thumb|150px|Artistic visualization of independent spatial modes (spherical harmonics) inside a spherical volume, with rainbow-colored symmetric wave patterns and cosmic background.]]
A [[Wikipedia:Quantum limit|'''quantum limit''']] in physics is a [[Wikipedia:De Broglie–Bohm theory#Operators as observables|limit on measurement accuracy]] at [[Wikipedia:Quantum limit|quantum scales]].<ref >
{{cite book
 |last1=Braginsky |first1=V. B.
 |last2=Khalili |first2=F. Ya.
 |year=1992
 |title=Quantum Measurement
 |publisher=[[Wikipedia:Cambridge University Press|Cambridge University Press]]
 |isbn=978-0-521-48413-8
}}</ref> [[File:Laguerre-Gaussian_modes.png|thumb|450px|center|Intensity profiles of Laguerre–Gaussian modes]]
Per polarisation:
:<math>N_\text{per pol} \approx \frac{(kR)^3}{3\pi^2}</math>
including both polarisations:
:<math>N \approx 2 \times \frac{(kR)^3}{3\pi^2} \approx 0.068 \left( \frac{2\pi R}{\lambda} \right)^3</math>.<ref name="Miller2000" /><ref name="Fabre2020" /><ref name="Erhard2017">{{cite journal |last=Erhard |first=Manuel |last2=Fickler |first2=Robert |last3=Krenn |first3=Mario |last4=Zeilinger |first4=Anton |year=2017 |title=Twisted photons: new quantum perspectives in high dimensions |journal=Light: Science & Applications |volume=6 |issue=10 |pages=e17146 |doi=10.1038/lsa.2017.146 |pmc=5655069 |pmid=29138663}}</ref><ref name="Erhard2020" /> {{efn-ua|''Note: Both polarizations are included in the count. The number N represents a theoretical limit; practically achievable values may be lower due to material imperfections or measurement errors.''}}

== Classical beam packing ==
For [[Wikipedia:Bessel beam|non-diffracting beams]] of diameter ''d'', geometric packing gives a crude (and irrelevant) upper bound <math>N \lesssim (2R/d)^3</math>.<ref>{{cite book |last=Conway |first=John H. |last2=Sloane |first2=Neil J. A. |title=Sphere Packings, Lattices and Groups |edition=3rd |publisher=Springer |year=1999 |isbn=978-0-387-98585-5}}</ref>{{efn-ua| ''Note: This geometric limit is theoretical; in practice, diffraction and interference can reduce the number of effectively usable beams.''}}
=== Including time and bandwidth ===
:<math>N_\text{total} \approx N \times (\Delta f \cdot t)</math><ref>{{cite journal
 |last=Brecht
 |first=Bernd
 |last2=Reddy
 |first2=Divya V.
 |last3=Silberhorn
 |first3=Christoph
 |last4=Raymer
 |first4=Michael G.
 |year=2015
 |title=Photon temporal modes: a complete framework for quantum information science
 |journal=Physical Review X
 |volume=5
 |pages=041017
 |doi=10.1103/PhysRevX.5.041017
}}</ref>{{efn-ua| ''Note: Although the total spatio-temporal modes increase with Δf·t, detector bandwidth, decoherence, and losses can limit the number of effectively measurable modes.''}}

=== Example calculation (50 [[Wikipedia:Micrometre|µm]] sphere, near-infrared light) ===
* ''R'' = 50 µm, ''λ'' = 800 [[Wikipedia:Nanometre|nm]] → ''kR'' ≈ 393
* ''N'' ≈ 2.8 × 10<sup>7</sup>
* Δ''f'' = 10 [[Wikipedia:Terahertz|THz]], ''t'' = 1 s → Δ''f''·''t'' = 10<sup>13</sup>
* ''N''<sub>total</sub> ≈ 2.8 × 10<sup>20</sup><br>
'''Detailed Mode Number and Time-Bandwidth Product Calculation'''<br>kR ≈ 393:<br>
With radius <math>R</math> = 50 µm = 5 × 10⁻⁵ m and wavelength λ = 800 nm = 8 × 10⁻⁷ m,<br>
k = 2π/λ ≈ 7.854 × 10⁶ m⁻¹,<br>
k<math>R</math> = (2π R)/λ = 2π × (R/λ) = 2π × 62.5 ≈ 392.7 (rounded to 393). This matches.<br>
''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:<br>
''N'' ≈ (8π n³ V)/(3 λ³), where V = (4/3)π <math>R</math>³ ≈ 5.24 × 10⁻¹³ m³ is the volume and n accounts for the dielectric.<br>
Without n (vacuum), N ≈ 8.57 × 10⁶.
For n ≈ 1.5 (common approximation for silica), n³ = 3.375, so N ≈ 2.89 × 10⁷.<br>
For n ≈ 1.453 (precise value for fused silica at 800 nm), n³ ≈ 3.07, so N ≈ 2.63 × 10⁷.<br>
The given 2.8 × 10⁷ is a reasonable approximation, likely using n ≈ 1.5 and rounding.<br>
Δf · t = 10¹³:With bandwidth Δf = 10 THz = 10¹³ Hz and observation time t = 1 s,<br>
Δf · t = 10¹³ × 1 = 10¹³. This is the time-bandwidth product, representing the number of independent temporal modes. This matches.<br>
''N''<sub>total</sub> ≈ 2.8 × 10²⁰:
''N''<sub>total</sub> = ''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.<br>
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.{{efn-ua|''Note: This example gives a theoretical estimate (practical detection may yield far fewer modes than the theoretical (''N''<sub>total</sub> ).''}}

== Bekenstein bound limits from quantum gravity ==
'''The Bekenstein [[Wikipedia:Bekenstein bound|entropy/information bound]]'''<br>
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 <ref name="24P" />. 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<ref name="23P"/> 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:

:<math>S \leq \frac{2\pi E R}{\hbar c} \qquad</math>

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.<ref name="23P"/>

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 ==
<div style="column-count:4; column-gap:20px;">

[[File:Orthogonal spatial modes inside a sphere.svg|thumb|Orthogonal spatial modes inside a sphere of radius ''R''.<br /><math>N \approx \frac{(kR)^3}{3\pi^2}</math>]]

[[File:Sphere spatio-temporal-1.svg|thumb|Growth of total channels with bandwidth and time]]

[[File:Spherical Harmonics.png|thumb|Real spherical harmonics ''Y''<sub>lm</sub> up to ''l''=5]]

[[File:Vibrating drum Bessel function.gif|thumb|Bessel functions describe the radial part of [[Wikipedia:vibrations of a circular membrane|vibrations of a circular membrane]].]]

</div>

[[File:Zeeman splitting in 39K.jpg|thumb|Hyperfine structure and Zeeman splitting of the <math>4^2P_{3/2}</math> state in <math>^{39}\mathrm{K}</math>.]]

[[Wikipedia:Spherical harmonics|Spherical harmonics]] and [[Wikipedia:Bessel function|Bessel functions]] are the natural solutions of the [[Wikipedia:Helmholtz equation|Helmholtz]] or [[Wikipedia:Schrödinger equation|Schrödinger equation]]. These functions form a complete [[Wikipedia:Orthonormal basis|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.<ref>{{cite thesis
 |last=Chapurin
 |first=Roman
 |title=Precise Measurements of Few-Body Physics in Ultracold <math>^{39}\mathrm{K}</math> Bose Gas
 |type=PhD thesis
 |institution=University of Colorado Boulder (JILA)
 |year=2019
 |url=https://jila.colorado.edu/sites/default/files/2019-07/Chapurin_PhD_Thesis.pdf
 |pages=~10–30
}}</ref>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
* [[Wikipedia:Phased array|Phased array]]
* [[Wikipedia:Orbital angular momentum of light|Orbital angular momentum of light]]
* [[Wikipedia:5D optical data storage|5D optical data storage]]

== Notes ==
{{notelist-ua}}
== References ==
<div style="column-count:3; break-inside:avoid; column-gap:2em;">
{{Reflist}}
</div>
{{Author|Harold Foppele}}
[[Category:Quantum information science]]

{{Sourceattribution|Number of independent spatial modes in a spherical volume|1}}