Microwave Rainbow Phenomena

• Microwave rainbow is a phenomenon that maps microwave frequency content into spatial or color channels, exemplified by SAR imaging and artificial sensing systems.
• It leverages geometry, dielectric properties, and nonreciprocal waveguides to selectively trap, disperse, or transform frequency components.
• Applications range from enhanced radar target analysis and frequency-to-angle beamforming in mmWave communications to novel spectral sensing techniques.

Searching arXiv for recent and foundational papers on “microwave rainbow” and closely related usages of the term. “Microwave rainbow” is not a single standardized term across the arXiv literature. In the cited works, it denotes several distinct microwave-domain phenomena in which frequency, colour-like representation, or spatial dispersion is central: geometry-driven colour signatures in colourised sub-aperture synthetic aperture radar (SAR) imagery; artificial colour perception derived from microwave transmission measurements; spatial trapping of different microwave frequencies at different positions in nonreciprocal waveguides; and engineered frequency-to-angle mapping in millimeter-wave multiple access. Other papers use the language more metaphorically, referring to sharply selective microwave emission or to the joint spectral and morphological study of Galactic microwave structures rather than to a literal rainbow process (Yang, 14 Sep 2025, Choudhury et al., 2013, Liu et al., 2016, Li et al., 2021, Grigoryan et al., 2010, Dobler, 2011).

1. Terminological scope and conceptual distinctions

In the most literal technical sense represented here, a microwave rainbow is a mechanism that maps microwave frequency content into distinct spatial positions, angular directions, or pseudo-colour channels. In the SAR formulation, colourised sub-aperture images map Doppler sub-bands to RGB hues, so target geometry appears as colour because geometry redistributes backscatter across the Doppler spectrum (Yang, 14 Sep 2025). In the nonreciprocal-waveguide formulation, different frequency components of a wave packet are slowed and stopped at different positions where the local group velocity vanishes, which is the operational meaning of a “trapped rainbow” (Liu et al., 2016). In Rainbow-link, the “rainbow beam” is an intentional frequency-to-angle mapping produced by a true-time-delay array, so different OFDM subcarriers or frequency resource blocks illuminate different directions at the same time (Li et al., 2021).

A second usage is explicitly artificial rather than literal. The microwave-colour-perception study argues that natural colour is a percept generated from visible-band spectral information, whereas artificial colour is any human-designed mapping from sensed spectral differences to colour-like categories or displays; in that framing, microwave transmission differences among samples can be rendered as artificial colour channels (Choudhury et al., 2013). This directly rejects the misconception that microwaves sense visible colour itself in the same way that visible light does.

A third usage is metaphorical. The dielectric-sphere radiation paper is only indirectly related to “microwave rainbow”: it does not discuss broadband angular dispersion or rainbow caustics in the usual scattering sense, but it does describe frequency-selective, resonantly enhanced, morphology-dependent microwave emission from a dielectric object (Grigoryan et al., 2010). Similarly, the Galactic haze paper concerns a diffuse, anomalously hard-spectrum microwave excess toward the Galactic center and its multiwavelength relation to the Fermi gamma-ray bubbles; the “rainbow” connection is interpretive, in the sense of spectral and morphological behavior across bands rather than literal rainbow formation (Dobler, 2011).

2. Geometry-driven colour in SAR imaging

The most explicit definition of a microwave rainbow is given in the SAR study “The Microwave Rainbow: How Geometry Paints Colours in Microwave Vision,” where the phenomenon is the systematic appearance of man-made and some natural structures in different colours within colourised sub-aperture images derived from high-resolution spaceborne SAR (Yang, 14 Sep 2025). The central claim is that these colours are not arbitrary display artefacts. CSI products visualize different Doppler sub-bands as RGB hues, and a target’s geometry selects or concentrates scattering into particular Doppler frequencies; hence geometry appears as colour.

The model starts from the standard SAR relation between squint angle and Doppler frequency,

$f_d = \frac{2V}{\lambda}\sin(\theta_{sq}),$

where $f_d$ is Doppler frequency, $V$ is platform velocity, $\lambda$ is radar wavelength, and $\theta_{sq}$ is squint angle. For a continuous linear target in the SAR imaging plane $(x,y_s)$,

$t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$

and a line oriented at azimuth angle $\theta_{az}$ is written as

$$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$

The paper derives the constructive-response condition

$f_d + f_c K \cos\theta_{sq} = 0,$

which yields

$f_d$0

This is the paper’s zero-order diffraction law. A continuous linear reflector responds most strongly when the imaging squint angle cancels its azimuth orientation, so colour becomes a direct proxy for orientation.

For periodic structures with spacing $f_d$1, constructive interference can occur for higher diffraction orders $f_d$2. The paper gives

$f_d$3

leading to

$f_d$4

and

$f_d$5

This is the grating law used to explain repeating colour cycles such as RGB$f_d$6RGB from periodic structures.

The paper distinguishes two regimes. Zero-order diffraction ($f_d$7) describes continuous linear features and curved metallic structures, where local tangent orientation varies continuously and therefore the preferred Doppler shifts continuously, producing a smooth colour gradient. High-order diffraction ($f_d$8) describes periodic discrete structures such as fences, ribbed roofs, and stadium seating, where multiple diffraction orders generate repeating microwave rainbows. Real-world examples are based on very-high-resolution spotlight SAR data from the Umbra Open Data initiative, with a representative calculation using $f_d$9, $V$0, spatial resolution $V$1, and $V$2. The paper states that the model was validated through numerical simulations and that these simulations reproduced both continuous zero-order rainbows and repeating high-order rainbows, but it does not provide explicit numerical error metrics or formal goodness-of-fit tables. The resulting interpretation is mechanistic and visual rather than large-sample statistical.

The paper’s broader implication is that colour in SAR CSI can function as a measurable proxy for physical form. According to the paper, SAR colour may allow inference of local tangent orientation of curves, broadside versus off-broadside alignment, sign of orientation, periodic spacing $V$3 when higher-order diffraction is present, deviations from ideal flatness or straightness, large-scale orientation patterns such as urban street-grid direction, and wave morphology on the ocean surface. The paper also states limitations: complex urban multiple scattering is not fully modeled, inversion methods remain incomplete, material and atmospheric effects require systematic study, dynamic scenes can smear colour signatures, and interpretation depends on the specific CSI Doppler partitioning and display choices.

3. Artificial colour perception under microwave illumination

A different but directly related meaning of microwave rainbow appears in “Artificial color perception using microwaves,” which investigates whether colour-like discrimination can be recreated artificially in the microwave domain (Choudhury et al., 2013). The authors define artificial colour perception as a customized spectral discrimination process not limited to natural human cone responses. In that framing, microwave illumination does not recover visible colour as such; rather, if microwave transmission through objects differs systematically, one can define an artificial colour space from microwave measurements.

The experimental arrangement is deliberately simple. A Gunn diode serves as the microwave source at approximately $V$4, corresponding to a wavelength of about $V$5. The source radiates toward a small liquid-filled object. A horn antenna connected to a microwave diode detector collects the transmitted microwave radiation, and the detector output is sent to a microwave power meter. The measured observable is therefore received transmitted power after propagation through the sample. The authors ideally wanted to scan the detector antenna along a straight line orthogonal to the propagation axis, but because the antenna was fixed they instead moved the object laterally and recorded the detector reading at successive positions.

The test objects are a small polythene cup filled with three liquids: plain water, milk tea, and water coloured blue using blue marker-pen ink. The measured quantity is transmitted microwave power, reported in milliwatts, as a function of lateral object position in arbitrary units. The paper’s interpretation is explicitly dielectric rather than chromatic. Microwaves are “affected by the dielectric properties of matter,” and the transmission variations are attributed to changes in dielectric constant caused by the contents of the liquid. Water is noted to have a high dielectric constant, about 73, and adding milk tea or blue ink changes that value slightly. The authors also mention possible dielectric lensing by the water-filled cup.

The physical basis can be expressed with the complex relative permittivity

$V$6

where $V$7 governs stored electric energy and phase velocity, and $V$8 represents dielectric loss. For a nonmagnetic medium,

$V$9

in the low-loss approximation. A simplified transmission model is

$\lambda$0

These relations were not presented explicitly in the paper, but the data block identifies them as the implicit electromagnetic basis of the authors’ dielectric interpretation.

The key result is qualitative rather than statistically rigorous. The three transmitted-power-versus-position curves are not identical and are described as “distinctly different,” which the authors take as evidence that the samples are discriminable. They further suggest that the measured curves could be transformed into “three orthonormal signals” using either the Caulfield–Maloney filter or Gram–Schmidt orthogonalization, and that an arbitrary mapping between these microwave sensitivity curves and the normal human three colour sensitivity curves would enable display to a human observer. However, the paper does not actually implement this transformation, provides no classifier, and reports no confidence intervals, repeated-trial statistics, or classification accuracy.

The major limitation is central to the interpretation of any microwave rainbow based on this work: discrimination is due to material composition correlated with visible colour, not to visible wavelength selectivity. The authors explicitly caution that “the same color on different objects with different dielectric properties may also result in ambiguities in the sensed colors.” A more precise summary given in the data block is that the experiment shows discrimination among three visually coloured liquid samples because their microwave dielectric properties differ, not because microwaves encode colour per se. The work is therefore a proof-of-feasibility for colour-like discrimination in microwaves, not a demonstration of literal microwave colour.

4. Trapped rainbow in nonreciprocal microwave waveguides

In wave physics, a microwave rainbow can refer to frequency-resolved spatial trapping. “Truly trapped rainbow by utilizing nonreciprocal waveguides” proposes and numerically demonstrates a “truly trapped rainbow” for electromagnetic waves by using a nonreciprocal gyromagnetic waveguide under a spatially tapered static magnetic bias (Liu et al., 2016). The authors define a trapped rainbow as a structure in which different frequency components of an incident wave packet are slowed and stopped at different positions in space, so that the spectrum is spatially separated and stored. A “truly” trapped rainbow must satisfy two conditions simultaneously: no premature reflection before reaching the stopping point, and different frequencies must stop at different positions rather than all at the same sink or absorber.

The platform is a 2D three-layer slab waveguide in the microwave regime. The top layer is a perfect electric conductor, the middle layer is a dielectric slab of thickness $\lambda$1 and permittivity $\lambda$2, and the bottom layer is yttrium-iron-garnet (YIG). The guided wave is a TE mode with field components $\lambda$3, propagating along the $\lambda$4-direction. Representative values are $\lambda$5, $\lambda$6 for YIG, saturation magnetization $\lambda$7, gyromagnetic ratio

$\lambda$8

and two dielectric thicknesses,

$\lambda$9

with

$\theta_{sq}$0

The YIG permeability tensor is

$\theta_{sq}$1

with

$\theta_{sq}$2

and effective permeability

$\theta_{sq}$3

The TE surface-mode dispersion relation is

$\theta_{sq}$4

where

$\theta_{sq}$5

The crucial nonreciprocal feature is the term linear in $\theta_{sq}$6,

$\theta_{sq}$7

which makes the dispersion asymmetric, $\theta_{sq}$8.

The magnetic bias is tapered as

$\theta_{sq}$9

For the thin-dielectric case $(x,y_s)$0, the critical field is

$(x,y_s)$1

and the stopping position obeys

$(x,y_s)$2

Different $(x,y_s)$3 therefore correspond to different $(x,y_s)$4, which is the mathematical basis of the rainbow effect. For the thicker case $(x,y_s)$5, two critical fields $(x,y_s)$6 and $(x,y_s)$7 exist, and the wave becomes effectively caged between two critical positions; for the thinner case, the relevant branch is strictly one-way, so the wave slows continuously and stops at a single critical position without a backward guided mode available for coupling.

The paper’s central argument is that earlier trapped-rainbow proposals failed because reciprocal slow-light systems suffer strong forward–backward coupling near the zero-group-velocity point, producing reflection before true standstill. Broken time-reversal symmetry removes or strongly suppresses the backward escape channel. Frequency-domain and time-domain simulations in COMSOL show field enhancement around the critical positions, hot spots, and relatively long duration time of the trapped wave. The authors further state that the trapping effect is stable even under fabrication disorders.

5. Frequency-to-angle rainbow beams in millimeter-wave communications

In wireless communications, a microwave rainbow is implemented as an engineered frequency-to-angle mapping in the antenna array itself. “Rainbow-link: Beam-Alignment-Free and Grant-Free mmW Multiple Access using True-Time-Delay Array” introduces a multiple-access protocol that exploits wide bandwidth at millimeter-wave frequencies and a true-time-delay array with frequency dependent beamforming capability (Li et al., 2021). The base station is equipped with the TTD array to simultaneously steer different frequency resource blocks toward distinct directions covering the entire cell sector. The paper is explicit that this is a real spectral-to-spatial dispersion effect: frequency resources are mapped to specific spatial directions.

The system model is a TDD mmWave network with center frequency $(x,y_s)$8, total bandwidth $(x,y_s)$9, OFDM waveform with cyclic prefix, and total number of subcarriers $t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$0. The main evaluation uses

$t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$1

The base station uses a reconfigurable TTD array with a single RF chain. The main performance analysis simplifies the user equipment to the case $t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$2, that is, single-antenna narrowband users. Each user can only access a small subset $t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$3 of the total subcarriers, with

$t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$4

and the main evaluation uses

$t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$5

For full-range rainbow beam operation, the uniform inter-element delay spacing is set to

$t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$6

This gives the analog beamforming vector on subcarrier $t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$7,

$t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$8

Comparing this with the ULA response

$t_a = \frac{x}{V}, \qquad t_r = \frac{2y_s}{c},$9

gives the approximate frequency-to-angle relation

$\theta_{az}$0

The paper then defines the anchor subcarrier as

$\theta_{az}$1

where the simplified single-path beam gain is

$\theta_{az}$2

This peaks when

$\theta_{az}$3

which is the formal frequency-to-angle map.

The multiple-access protocol combines this rainbow beam with grant-free contention access. The base station uses one fixed TTD configuration for both downlink synchronization and uplink access, so no beam switching is required on the base-station side. Users detect the downlink beacon, find the subcarrier region where received power is highest, and associate with that local angular/frequency region. After synchronization, a user transmits immediately in an arrive-and-go manner, without requesting a grant, repeating the same packet on $\theta_{az}$4 resource blocks chosen among $\theta_{az}$5 RBs in its narrowband. Packet loss is modeled through collisions among users occupying overlapping narrowbands, and the paper derives an approximate packet loss rate in terms of $\theta_{az}$6, $\theta_{az}$7, $\theta_{az}$8, $\theta_{az}$9, a binomial mass function, and a transition matrix $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$0.

The headline performance claims are stated under the paper’s assumptions: quasi-static users, predominantly LOS, single-antenna narrowband UEs, a semi-circle of radius $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$1, and the numerology used in the simulations. The abstract states that, given less than $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$2 probability of packet loss, a rainbow-link cell over $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$3 bandwidth using a $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$4-element antenna array attains sub-millisecond user-plane latency and Mbps user rates with an approximate $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$5 line-of-sight coverage and a density of up to $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$6 active single antenna users per second per meter square. The paper also reports that, for grouping factor $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$7 and with $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$8 active users transmitting simultaneously, packet loss rate remains below $$t_r = t_a \tan(\theta_{az}) \frac{2V}{c} = t_a K, \qquad K = \tan(\theta_{az})\frac{2V}{c}.$$9 under the simulated settings. At the same time, the paper explicitly treats low spectral efficiency as a drawback and acknowledges sensitivity to blockage, mobility, path switching, and non-line-of-sight channels.

This usage of microwave rainbow differs physically from optical rainbows and from material dispersion. The paper states that the effect is not caused by wavelength-dependent refractive index in a medium but by engineered antenna delays that make the array phase progression proportional to frequency. It is therefore an array-factor dispersion mechanism deliberately used as the operating principle of multiple access.

6. Metaphorical and adjacent uses: resonant emission and Galactic microwave structure

A more limited and explicitly indirect connection appears in “Microwave Radiation from a Particle Revolving Along a Shifted Equatorial Orbit About a Dielectric Ball” (Grigoryan et al., 2010). The physical system is a dielectric ball of radius $f_d + f_c K \cos\theta_{sq} = 0,$0 in vacuum, with a relativistic electron uniformly rotating on a circular orbit of radius $f_d + f_c K \cos\theta_{sq} = 0,$1 in the equatorial plane, and with center-to-center offset $f_d + f_c K \cos\theta_{sq} = 0,$2 between the sphere center and the orbit center in the shifted configuration. The main numerical example uses a loss-free dielectric with $f_d + f_c K \cos\theta_{sq} = 0,$3, corresponding to molten quartz in the $f_d + f_c K \cos\theta_{sq} = 0,$4 range, sphere radius $f_d + f_c K \cos\theta_{sq} = 0,$5, electron energy $f_d + f_c K \cos\theta_{sq} = 0,$6, orbit radius $f_d + f_c K \cos\theta_{sq} = 0,$7, and $f_d + f_c K \cos\theta_{sq} = 0,$8.

The radiation mechanism combines synchrotron radiation in vacuum with Cherenkov radiation generated inside the dielectric sphere by the particle-associated field that partially penetrates the ball. Radiation is emitted at discrete harmonics

$f_d + f_c K \cos\theta_{sq} = 0,$9

and the dimensionless quantity

$f_d$00

is introduced as the number of electromagnetic field quanta emitted during one revolution period at harmonic $f_d$01. The enhancement is not broadband. It is frequency-selective, resonantly enhanced, and boundary-structured. For the $f_d$02 harmonic, the paper reports in the centered case $f_d$03 that

$f_d$04

is possible over

$f_d$05

while the comparison values are

$f_d$06

For the shifted-orbit case, the paper highlights

$f_d$07

Strong enhancement requires accurate tuning to the “resonance” rotation frequency

$f_d$08

with $f_d$09 error. The paper itself is clear that this is not a rainbow in the usual angular-scattering sense; the most that can be said, as the data block states, is that it is a microwave analogue of rainbow-like spectral selectivity in a metaphorical sense.

A different adjacent usage concerns the microwave haze or microwave bubbles analyzed by Dobler in WMAP data (Dobler, 2011). Here the subject is diffuse emission toward the center of the Galaxy, extracted by linear template regression from seven years of WMAP observations and compared with the Fermi gamma-ray haze/bubbles. The observational basis emphasizes the K, Ka, and Q bands at $f_d$10, $f_d$11, and $f_d$12. The haze brightness-temperature spectrum is given as

$f_d$13

harder than ordinary soft synchrotron and distinct from free-free, for which $f_d$14. The favored interpretation is synchrotron from a hard electron population with

$f_d$15

Morphologically, the microwave haze is elongated in latitude with respect to longitude by a factor of roughly two, with longitude extent about

$f_d$16

and microwave emission reaching to about

$f_d$17

before cutting off sharply, in contrast to the gamma-ray bubbles extending to about

$f_d$18

Dobler argues that the microwave and gamma-ray structures correspond at low and intermediate latitude but are not identical in extent, and interprets the high-latitude microwave fading as likely evidence for a drop in magnetic field strength above about $f_d$19, with implied magnetic-field scale height about $f_d$20 kpc if the structure is at the Galactic center. The paper also emphasizes that there is no evidence for significant haze polarization in the 7-year WMAP data, but that such a signal would be unlikely to be detectable above the noise. This is not a microwave rainbow in the strict sense; the connection is that the paper studies a spectrally hard, morphologically distinctive microwave component across frequency, sky position, and multiwavelength correspondence.

Taken together, these adjacent usages show that “microwave rainbow” can function as a strict physical term, a display-space interpretation, an engineered systems concept, or a metaphor for frequency-selective microwave structure. The precise meaning depends on whether the governing variable is Doppler colour in SAR, artificial microwave sensing channels, nonreciprocal slow-light localization, true-time-delay beam dispersion, resonant harmonic enhancement, or multi-frequency Galactic morphology.