Angle of incidence (optics)

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Angle of incidence (optics)

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The angle of incidence, in geometric optics, is the angle between a ray incident on a surface and the line perpendicular (at 90 degree angle) to the surface at the point of incidence, called the normal. The ray can be formed by any waves, such as optical, acoustic, microwave, and X-ray. In the figure below, the line representing a ray makes an angle θ with the normal (dotted line). The angle of incidence at which light is first totally internally reflected is known as the critical angle. The angle of reflection and angle of refraction are other angles related to beams.

In computer graphics and geography, the angle of incidence is also known as the illumination angle of a surface with a light source, such as the Earth's surface and the Sun.^[1] It can also be equivalently described as the angle between the tangent plane of the surface and another plane at right angles to the light rays.^[2] This means that the illumination angle of a certain point on Earth's surface is 0° if the Sun is precisely overhead and that it is 90° at sunset or sunrise.

Determining the angle of reflection with respect to a planar surface is trivial, but the computation for almost any other surface is significantly more difficult.

Refraction of light at the interface between two media

Grazing angle or glancing angle

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When dealing with a beam that is nearly parallel to a surface, it is sometimes more useful to refer to the angle between the beam and the surface tangent, rather than that between the beam and the surface normal. The 90-degree complement to the angle of incidence is called the grazing angle or glancing angle. Incidence at small grazing angles is called "grazing incidence."^{[citation needed]}

Grazing incidence diffraction is used in X-ray spectroscopy and atom optics, where significant reflection can be achieved only at small values of the grazing angle. Similarly, a Wolter telescope used for X-ray astronomy is based on the principle of total external reflection at small graze angles. Moreover, ridged mirrors are designed to reflect atoms coming at a small grazing angle, usually measured in milliradians. In optics, there is Lloyd's mirror.

References

[edit]

^ Godse, A. P. (2008). Computer Graphics. Technical Publications. p. 292. ISBN 9788189411008.
^ Hengl, Tomislav; Reuter, Hannes I. (2022). Geomorphometryoncepts, Software, Applications. Developments in soil science. Vol. 33. Farha. p. 202. ISBN 9780123743459.

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Angle of incidence (optics)

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In optics, the angle of incidence is defined as the angle between an incident ray of light and the normal (a line perpendicular to the surface) at the point where the ray strikes the surface.^[1] This angle, typically denoted as

\theta_i

, is a fundamental parameter in the interaction of light with interfaces between media, such as air-glass boundaries or mirror surfaces.^[2] The angle of incidence plays a central role in the law of reflection, which states that the angle of reflection

\theta_r

equals the angle of incidence, ensuring that light rays bounce off smooth surfaces predictably.^[3] This equality holds for specular reflection on polished surfaces like mirrors, where the incident, reflected, and normal rays all lie in the same plane, enabling applications in optical devices such as telescopes and periscopes.^[1] On rough surfaces, however, diffuse reflection scatters light in multiple directions, reducing the prominence of a distinct angle of reflection.^[3] In refraction, the angle of incidence determines how light bends when passing from one medium to another due to a change in speed, governed by Snell's law:

n_1 \sin \theta_i = n_2 \sin \theta_t

, where

n_1

and

n_2

are the refractive indices of the respective media, and

\theta_t

is the angle of transmission (or refraction).^[2] This law explains phenomena like the apparent bending of objects viewed through water or lenses, and it defines critical angles leading to total internal reflection when

\theta_i

exceeds a threshold value, essential for fiber optics and prisms.^[3] The behavior at interfaces also depends on polarization and the magnitude of

\theta_i

, influencing reflection coefficients for s- and p-polarized light.^[2]

Fundamentals

Definition

In optics, the angle of incidence is defined as the angle between an incident ray of light and the normal to the interface surface at the point of incidence.^[4] This angle, commonly denoted as θ_i, quantifies the orientation of the incoming light ray relative to the surface and is typically measured in degrees or radians.^[5] The normal line serves as the reference for this measurement, defined as the straight line perpendicular to the surface at the exact point where the ray strikes.^[6] This geometric concept applies specifically to the interaction of light rays with boundaries or interfaces between different media, such as air and glass, where the ray's direction changes upon encounter.^[7] In geometric optics, which treats light as propagating in straight-line rays, the angle of incidence provides a fundamental parameter for analyzing how light behaves at these junctions, independent of wavelength effects.^[8] The term and its geometric basis were formalized in 17th-century optics, notably by René Descartes in his 1637 treatise La Dioptrique, where he used it to describe ray paths in refraction, and by Pierre de Fermat in his 1662 principle of least time, which incorporated incidence angles to derive optical laws.^[9]^[10] A typical ray diagram illustrates this by depicting an incident ray approaching a flat interface from one medium, with the normal drawn as a dashed vertical line at the incidence point; the angle θ_i (between 0° and 90°) is then the angle formed between the ray and this normal, often highlighted for clarity.^[11]

Relation to Other Angles

In optics, the angle of reflection, denoted as

\theta_r

, is defined as the angle formed between the reflected ray and the normal to the interface at the point of incidence. This angle exhibits a key symmetry with the angle of incidence

\theta_i

, such that

\theta_r = \theta_i

, which underpins the behavior of light upon reflection from a surface.^[2] For light transmission across an interface, the angle of refraction, denoted as

\theta_t

, is the angle between the refracted ray and the normal to the surface. This angle describes the deviation of the transmitted ray relative to the incident direction and normal.^[12] Standard notation in geometric optics employs subscripts to distinguish these angles:

i

for incidence,

r

for reflection, and

t

for transmission (or refraction). This convention facilitates clear communication of ray paths in diagrams and equations.^[12]^[2] In vector-based representations, the angle of incidence is computed via the dot product of the incident ray's unit direction vector

\vec{d_i}

and the unit normal vector $\vec{n}$

to the surface, yielding $\cos \theta_i = |\vec{d_i} \cdot \vec{n}|$

⋅n

∣. Similar calculations apply to the angles of reflection and refraction using their respective direction vectors.^[12]^[13]

Reflection

Law of Reflection

The law of reflection states that, for a light ray incident on a reflecting surface, the angle of incidence, defined as the angle between the incident ray and the normal to the surface at the point of incidence, is equal to the angle of reflection, the angle between the reflected ray and the same normal.^[10] This equality, denoted as

\theta_i = \theta_r

, holds in the plane containing the incident ray, the reflected ray, and the surface normal.^[10] This law can be derived from Fermat's principle, which posits that light travels between two points along the path that minimizes the travel time compared to nearby paths.^[14] Consider a light ray traveling from point A above a reflecting surface to point B above the surface, reflecting at point P on the surface; the time

t

for the path is

t = \frac{\sqrt{x^2 + h_1^2} + \sqrt{(l - x)^2 + h_2^2}}{c}

t=cx2+h12

+(l−x)2+h22

, where $x$ is the horizontal position of P, $l$ is the horizontal separation between A and B, $h_1$ and $h_2$ are the vertical distances from A and B to the surface, and $c$ is the speed of light.^[14] Minimizing $t$ with respect to $x$ by setting $\frac{dt}{dx} = 0$ yields $\sin \theta_i = \sin \theta_r$ , implying $\theta_i = \theta_r$ for the geometry involved.^[14] An equivalent geometric construction reflects point B over the surface to B', making the minimal path a straight line from A to B', which enforces the angle equality due to symmetry.^[10] In vector notation, the law is expressed as $\vec{r} = \vec{i} - 2 (\vec{i} \cdot \hat{n}) \hat{n}$

−2(i

⋅n^)n^, where $\vec{i}$

is the unit incident direction vector, $\vec{r}$

is the unit reflected direction vector, and $\hat{n}$ is the unit normal vector to the surface pointing toward the incident side; this form conserves the tangential component of the wave vector while reversing the normal component.^[15] The law was experimentally verified in the 11th century by Ibn al-Haytham (Alhazen), who used a purpose-built apparatus involving pins and a mirror to demonstrate that incident and reflected rays lie in the same plane and that $\theta_i = \theta_r$ , providing empirical proof alongside his mathematical derivation in Kitab al-Manazir.^[16] Modern verifications employ optical elements such as a glass plate mounted on a rotatable base to confirm $\theta_i = \theta_r$ , as in economic, compact, safe, and portable setups using fiberboard enclosures.^[17] The law applies specifically to specular reflection, which occurs on smooth surfaces where the roughness is much smaller than the light wavelength (typically <1 μm for visible light), ensuring the reflected rays are parallel and follow the angle equality; on rougher surfaces, diffuse scattering predominates instead.^[18]

Incidence in Mirrors and Surfaces

In specular reflection, light rays incident on a smooth mirror surface reflect such that the angle of incidence equals the angle of reflection, producing a clear, undistorted image through ray tracing where the incident angle

\theta_i

determines the direction of the reflected ray.^[10] This behavior assumes an ideal, perfectly polished surface, enabling applications like periscopes or rearview mirrors where the precise path governed by

\theta_i

maintains image fidelity.^[19] On rough or matte surfaces, diffuse reflection occurs as incident rays scatter in multiple directions due to microscopic irregularities, with the angle of incidence

\theta_i

influencing the overall brightness distribution according to Lambert's cosine law. This law states that the observed intensity

I

from a diffusely reflecting surface is proportional to the cosine of the angle between the incident ray and the surface normal:

I \propto \cos \theta_i

.^[20] As a result, surfaces appear brighter when viewed near normal incidence and dimmer at grazing angles, a principle fundamental to rendering realistic shading in computer graphics and photometry.^[21] For image formation in plane mirrors, the virtual image position is determined via ray tracing, where rays from the object reflect off the mirror with

\theta_i = \theta_r

, and backward extensions of these reflected rays intersect at the image point located symmetrically behind the mirror at a distance equal to the object's perpendicular distance in front.^[19] Although the image distance itself is independent of

\theta_i

for a given object position, the specific

\theta_i

values for individual rays dictate the apparent location and orientation as perceived by the observer, ensuring the image appears upright and unreversed laterally.^[10] Surface characteristics significantly modulate reflection based on

\theta_i

; highly polished surfaces promote specular reflection for low

\theta_i

, while roughening induces diffusion, scattering light regardless of exact

\theta_i

but still modulated by the cosine law.^[22] Curved surfaces, such as parabolic mirrors, leverage varying

\theta_i

across the reflector to focus parallel incident rays to a single focal point, with the local angle of incidence at each point on the parabola ensuring convergence without spherical aberration, as derived from the parabolic equation

y = \frac{x^2}{4f}

where

f

is the focal length.^[23] Polarization effects arise during reflection, where the angle of incidence

\theta_i

differentially impacts the s- (perpendicular) and p- (parallel) components of the electric field, as described by the Fresnel equations; for instance, the reflection coefficient for p-polarization decreases with increasing

\theta_i

, while s-polarization increases, leading to partial polarization of the reflected beam.^[24] This dependence is key in optical isolators and polarizers, though full details emerge at specific angles like Brewster's.^[25]

Refraction

Snell's Law

Snell's law, also known as the Snell–Descartes law, describes the relationship between the angle of incidence and the angle of refraction when light passes from one medium to another./25:_Geometric_Optics/25.03:_The_Law_of_Refraction) The law is mathematically expressed as

n_1 \sin \theta_i = n_2 \sin \theta_t,

where

n_1

and

n_2

are the refractive indices of the first and second media, respectively,

\theta_i

is the angle of incidence measured from the normal to the interface, and

\theta_t

is the angle of refraction also measured from the normal./25:_Geometric_Optics/25.03:_The_Law_of_Refraction) This equation ensures that the component of the wave vector parallel to the interface remains constant across the boundary, maintaining phase continuity.^[26] The refractive index

n

of a medium is defined as the ratio of the speed of light in vacuum

c

to the speed of light in the medium

v

, given by

n = c / v

.- /25:_Geometric_Optics/25.03:_The_Law_of_Refraction) For air,

n \approx 1.000

, while for typical crown glass,

n \approx 1.5

.^[27] These values indicate that light travels slower in glass than in air, influencing the bending behavior at the interface. When light enters a denser medium (higher

n

) from a rarer one (lower

n

), such as from air to glass, the angle of refraction

\theta_t

is smaller than the angle of incidence

\theta_i

, causing the ray to bend toward the normal.^[28] Conversely, entering a rarer medium results in bending away from the normal, with

\theta_t > \theta_i

.^[29] The law of refraction was first described in 984 AD by the Persian mathematician and physicist Ibn Sahl in his treatise on burning mirrors and lenses.^[30] It was independently rediscovered in Europe by Thomas Harriot before 1602 (unpublished), Willebrord Snell in 1621 (also unpublished),^[31] and René Descartes, who published it in 1637 in his work La Dioptrique.^[32] A common derivation of Snell's law uses Huygens' principle, which posits that every point on a wavefront acts as a source of secondary spherical wavelets, and the new wavefront is the tangent envelope of these wavelets.^[26] Consider a plane wave incident from medium 1 (speed

v_1 = c/n_1

) to medium 2 (speed

v_2 = c/n_2

) at an interface. At time

t = 0

, the wavefront reaches the boundary at point A. The portion of the wavefront at A begins propagating in medium 2 at speed

v_2

, while the rest continues in medium 1 at

v_1

. After time

t

, the secondary wavelet from A has radius

v_2 t

in medium 2, and the wavelet from a point B (distance

l

along the interface) has radius

v_1 t

in medium 1. The new wavefront in medium 2 is the line tangent to these wavelets, forming the refracted ray.^[33] From the geometry, the distances along the interface yield

\sin \theta_i / v_1 = \sin \theta_t / v_2

, which rearranges to

(c / v_1) \sin \theta_i = (c / v_2) \sin \theta_t

, or equivalently

n_1 \sin \theta_i = n_2 \sin \theta_t

.^[26] This derivation highlights the wave nature of light and the boundary conditions at the interface. An alternative derivation based on Fermat's principle of least time leads to the same result by minimizing the optical path length.^[10]

Critical Angle and Total Internal Reflection

In optics, the critical angle is defined as the angle of incidence in a denser medium (refractive index

n_1 > n_2

) at which the angle of refraction in the rarer medium is exactly 90 degrees, resulting in the refracted ray grazing along the interface.^[34] This condition arises from Snell's law, where

n_1 \sin \theta_i = n_2 \sin \theta_t

; setting

\theta_t = 90^\circ

(so

\sin 90^\circ = 1

) yields

\sin \theta_c = \frac{n_2}{n_1}

, and thus

\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)

.^[35] When the angle of incidence exceeds the critical angle (

\theta_i > \theta_c

), total internal reflection (TIR) occurs: no light refracts into the second medium, and all incident light is reflected back into the first medium with the angle of reflection equal to the angle of incidence.^[2] This phenomenon is a direct consequence of the wave nature of light, where the transmitted wave becomes an evanescent wave that decays exponentially without propagating energy across the interface.^[36] TIR finds practical use in devices like optical fibers, where light signals are confined within the core by repeated internal reflections at the core-cladding boundary, enabling efficient long-distance transmission.^[7] In cases of frustrated TIR, if a third medium with a refractive index higher than the gap medium is placed very close to the interface (typically within the evanescent wave's decay length), some energy can tunnel through via the evanescent field, partially coupling light across the boundary.^[37]^[38]

Special Cases

Grazing Angle

The grazing angle, denoted as

\alpha

, is defined as the complement of the angle of incidence

\theta_i

, such that

\alpha = 90^\circ - \theta_i

. It measures the acute angle between the incident ray and the reflecting or refracting surface, providing an alternative perspective to the conventional angle of incidence, which is measured from the surface normal.^[39]^[40] Also referred to as the glancing angle, this parameter is particularly prevalent in X-ray optics, where it facilitates the design of reflective elements operating at shallow incidence angles, and in atmospheric science, including studies of low-angle scattering from ocean surfaces and mirage phenomena.^[40]^[41]^[42] One key advantage of using the grazing angle arises in scenarios of near-parallel incidence to the surface, where

\alpha

becomes small, enhancing the interaction with surface layers and proving especially useful for thin-film analysis in techniques like grazing-incidence X-ray diffraction (GIXRD). This approach increases sensitivity to ultra-thin films by promoting shallower penetration depths and larger illuminated areas on the sample.^[43]^[44] The trigonometric relation between the angles is expressed as

\sin \theta_i = \cos \alpha

, which follows directly from their complementary nature.^[39] Historically, the grazing angle gained prominence in radar applications and low-angle scattering investigations following World War II, as researchers analyzed sea clutter and surface reflections to improve detection systems amid post-war advancements in electromagnetic wave propagation studies.^[45]^[46]

Brewster's Angle

Brewster's angle, denoted as

\theta_B

, is defined as the angle of incidence at which light polarized parallel to the plane of incidence (p-polarized light) experiences zero reflection from a dielectric interface, resulting in the reflected beam being completely s-polarized (perpendicular to the plane of incidence).^[47] This phenomenon occurs because the reflected light consists solely of the s-component, achieving full linear polarization in the reflected ray.^[48] The mathematical expression for Brewster's angle is given by

\theta_B = \arctan\left(\frac{n_2}{n_1}\right)

, where

n_1

is the refractive index of the incident medium and

n_2

is that of the transmitting medium.^[47] This formula arises from the condition that the reflected and refracted rays are orthogonal, such that

\theta_i + \theta_t = 90^\circ

, leading to

\tan \theta_i = n_2 / n_1

.^[48] As a specific value of the angle of incidence,

\theta_B

depends on the relative refractive indices of the two media involved; for example, air-to-glass interfaces yield

\theta_B \approx 56^\circ

when

n_2 = 1.5

.^[47] This polarization effect is explained by the Fresnel equations, which describe the reflection coefficients for light at an interface. At Brewster's angle, the p-polarized reflection coefficient

r_p = 0

, meaning no p-polarized light is reflected, while the s-polarized coefficient

r_s

remains non-zero. The Fresnel coefficient for p-polarization is

r_p = \frac{n_2 \cos \theta_i - n_1 \cos \theta_t}{n_2 \cos \theta_i + n_1 \cos \theta_t}

, and setting

r_p = 0

directly yields the Brewster condition

\theta_B = \arctan(n_2 / n_1)

. The phenomenon was discovered by Scottish physicist David Brewster through a series of experiments on the polarization of light by reflection from transparent bodies, conducted between 1811 and 1815. Brewster published his findings in 1815, establishing the empirical law that the tangent of the polarizing angle equals the refractive index of the medium.^[49]

Applications

In Optical Devices

In optical devices, the angle of incidence plays a crucial role in prisms, where it determines the dispersion of light through refraction at multiple surfaces. In a triangular prism, light enters at an angle of incidence θ_i, refracts according to Snell's law, and undergoes further refraction at the exit face, resulting in angular separation of wavelengths due to the wavelength-dependent refractive index. This dispersion is maximized or minimized based on θ_i; for instance, the condition of minimum deviation occurs when the ray passes symmetrically through the prism, with θ_i equal to the angle of emergence, allowing precise control in spectrometers for spectral analysis.^[50]^[51]^[52] Lenses rely on the paraxial approximation, which assumes small angles of incidence relative to the optical axis to simplify ray tracing and predict image formation accurately. Under this approximation, rays incident at shallow θ_i follow paths where sin(θ_i) ≈ θ_i (in radians), enabling the use of linear equations for focal length and magnification without higher-order aberrations dominating. This approach is fundamental in designing thin lenses for cameras and microscopes, where deviations from small θ_i introduce spherical aberrations that degrade image quality.^[53]^[54] In fiber optics, the angle of incidence is essential for maintaining total internal reflection (TIR) within the core, ensuring efficient signal propagation over long distances. For TIR to occur, θ_i at the core-cladding interface must exceed the critical angle θ_c, defined by the refractive indices of the core and cladding materials, typically around 42° for glass-air interfaces but adjusted to about 80°-90° in fibers by designing a small refractive index difference between the core and cladding. This condition confines light within the core, minimizing losses, and the maximum acceptance angle for incoming light is derived from θ_c to define the fiber's numerical aperture, which quantifies light-gathering capability.^[55]^[56]^[57] Anti-reflective (AR) coatings on optical surfaces are engineered to minimize reflection by exploiting interference, with design optimized for a specific angle of incidence to achieve destructive interference between reflected waves. These multilayer dielectric films, often with quarter-wave thicknesses at the design wavelength, reduce reflectance to below 0.5% for normal incidence (θ_i ≈ 0°), but performance shifts at oblique angles due to path length changes in the film. For example, in high-precision lenses, AR coatings tailored for θ_i up to 30° enhance transmission across visible wavelengths, critical for reducing ghost images in photographic objectives.^[58]^[59]^[60] In modern laser diodes, optimizing the angle of incidence during fiber coupling significantly boosts efficiency, particularly in post-1980s developments for telecommunications. Light from the diode's output facet, which has an elliptical beam profile, is directed at a controlled θ_i to a coupling lens or tapered fiber, maximizing overlap with the fiber mode and achieving efficiencies up to 90% with graded-index microlenses. This optimization mitigates losses from misalignment or astigmatism inherent in diode structures, enabling high-power delivery in systems like wavelength-division multiplexing.^[61]^[62]^[63]

In Natural Phenomena

The angle of incidence plays a crucial role in the formation of rainbows, where sunlight enters spherical water droplets in the atmosphere at specific angles, leading to refraction, internal reflection, and dispersion of wavelengths that produce the characteristic colored arcs. For the primary rainbow, light rays incident on the droplet surface at angles around 59.5° for red light undergo total internal reflection after refraction, exiting the droplet such that the observer sees the bow at an angular radius of approximately 42° from the antisolar point, with shorter wavelengths like violet deviating slightly more due to greater dispersion. This fixed angular separation arises from the geometry of ray paths within uniform spherical droplets, creating the vivid spectrum observed in the sky.^[64] Mirages, such as inferior mirages over hot roads or superior mirages over cold surfaces, result from the bending of light rays due to refractive index gradients in the air caused by temperature variations. For inferior mirages, temperature decreases with height near the ground, increasing air density and refractive index with height; rays from distant objects enter these layers at shallow angles of incidence, curving concave upward toward the denser air above and creating inverted, displaced images that appear as shimmering illusions like "wet" roads. For superior mirages, a temperature inversion increases temperature with height, decreasing density and refractive index with height, causing rays to curve concave downward toward the denser air below and producing elevated or hovering images. These effects are most pronounced for rays near grazing incidence, where the gradient amplifies the curvature, altering the apparent position of objects like the horizon or sky.^[65] Glare on bodies of water, particularly when the sun is low in the sky, intensifies due to high angles of incidence approaching grazing, where the Fresnel reflection coefficients for light increase dramatically, reflecting a large portion of incident sunlight specularly off the surface. This strong reflection, often over 90% for unpolarized light near 90° incidence, overwhelms transmitted light and scatters into the observer's view, reducing underwater visibility by masking details beneath the surface with bright, blinding highlights. Such glare is a common issue for boaters or anglers during dawn or dusk, as the shallow entry angle minimizes absorption and maximizes the mirrored sky or sun image on the water.^[66] The vibrant colors of sunsets emerge from the interaction of sunlight with the atmosphere at low solar elevations, corresponding to high angles of incidence for rays grazing through extensive air layers. At these shallow angles, the path length through the atmosphere can exceed 20 times the vertical thickness at noon, preferentially scattering shorter blue and violet wavelengths via Rayleigh scattering while allowing longer red and orange hues to transmit directly to the observer. This selective attenuation intensifies as the sun dips lower, with dust and aerosols further enhancing the reddish tones by absorbing blues more effectively along the elongated trajectory.^[67] Atmospheric refraction shifts the apparent position of celestial bodies like the sun due to light rays bending as they pass through layers of air with varying refractive indices, influenced by density gradients from pressure and temperature. Rays from the sun enter the atmosphere at low angles of incidence near the horizon, refracting progressively more in denser lower layers, which elevates the sun's image by up to 0.5° at the horizon, allowing it to remain visible briefly after geometric sunset. This cumulative effect, strongest for grazing rays, ensures the sun appears higher and more circular than its true distorted shape, a phenomenon observable in clear skies worldwide.

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