Total internal reflection

Total internal reflection (TIR) is the phenomenon that makes the water-to-air surface in a fish-tank look like a perfectly silvered mirror when viewed from below the water level (Fig. 1). Technically, TIR is the total reflection of a wave incident at a sufficiently oblique angle on the interface between two media, of which the second ('external') medium is transparent to such waves but has a higher wave velocity than the first ('internal') medium. TIR occurs not only with electromagnetic waves such as light waves and microwaves, but also with other types of waves, including sound and water waves. In the case of a narrow train of waves, such as a laser beam, we tend to speak of the total internal reflection of a 'ray' (Fig. 2). sin ⁡ θ 1 v 1 = sin ⁡ θ 2 v 2 {displaystyle {frac {sin heta _{1}}{v_{1}}}={frac {sin heta _{2}}{v_{2}}},} .    (1) θ c = arcsin ⁡ ( v 1 / v 2 ) {displaystyle heta _{ ext{c}}=arcsin(v_{1}/v_{2}),} .    (2) n 1 sin ⁡ θ 1 = n 2 sin ⁡ θ 2 {displaystyle n_{1}sin heta _{1}=n_{2}sin heta _{2}}     (3) θ c = arcsin ⁡ ( n 2 / n 1 ) {displaystyle heta _{ ext{c}}=arcsin(n_{2}/n_{1}),} .    (4) E k e i ( k ⋅ r − ω t ) , {displaystyle mathbf {E_{k}} e^{i(mathbf {kcdot r} -omega t)},}     (5) k = n k 0 , {displaystyle k=nk_{0},,}     (6) E t = E k t e i ( k t ⋅ r − ω t ) , {displaystyle mathbf {E} _{ ext{t}}=mathbf {E} _{mathbf {k} { ext{t}}}e^{i(mathbf {k_{ ext{t}}cdot r} -omega t)},,}     (7) E t = E k t e i ( n 1 k 0 x sin ⁡ θ i + n 2 k 0 y cos ⁡ θ t − ω t ) . {displaystyle mathbf {E} _{ ext{t}}=mathbf {E} _{mathbf {k} { ext{t}},}e^{i(n_{1}k_{0}xsin heta _{ ext{i}}+n_{2}k_{0}ycos heta _{ ext{t}}-omega t)}.}     (8) cos ⁡ θ t = ± i ( n 1 / n 2 ) 2 sin 2 ⁡ θ i − 1 {displaystyle cos heta _{ ext{t}}=pm i,{sqrt {(n_{1}/n_{2})^{2}sin ^{2} heta _{ ext{i}}-1}},} .    (9) E t = E k t e ∓ n 1 2 sin 2 ⁡ θ i − n 2 2 k 0 y e i ( ( n 1 k 0 sin ⁡ θ i ) x − ω t ) , {displaystyle mathbf {E} _{ ext{t}}=mathbf {E} _{mathbf {k} { ext{t}},}e^{mp {sqrt {n_{1}^{2}sin ^{2} heta _{ ext{i}},-,n_{2}^{2}}};k_{0}y};e^{i{ig (}(n_{1}k_{0}sin heta _{ ext{i}})x-omega t{ig )}},,}     (10) E t ∝ e − κ y e i ( k x x − ω t ) , {displaystyle mathbf {E} _{ ext{t}}propto ,e^{-kappa y,}e^{i(k_{x}x-omega t)},,}     (11) κ = k 0 n 1 2 sin 2 ⁡ θ i − n 2 2 k x = n 1 k 0 sin ⁡ θ i   , {displaystyle {egin{aligned}kappa &=k_{0},{sqrt {n_{1}^{2}sin ^{2} heta _{ ext{i}},-,n_{2}^{2}}}\k_{x!}&=n_{1}k_{0}sin heta _{ ext{i}}~,end{aligned}}}     (12) r s = n 1 cos ⁡ θ i − n 2 cos ⁡ θ t n 1 cos ⁡ θ i + n 2 cos ⁡ θ t {displaystyle r_{s}={frac {n_{1}cos heta _{ ext{i}}-n_{2}cos heta _{ ext{t}}}{n_{1}cos heta _{ ext{i}}+n_{2}cos heta _{ ext{t}}}}}     (13) t s = 2 n 1 cos ⁡ θ i n 1 cos ⁡ θ i + n 2 cos ⁡ θ t {displaystyle t_{s}={frac {2n_{1}cos heta _{ ext{i}}}{n_{1}cos heta _{ ext{i}}+n_{2}cos heta _{ ext{t}}}}}     (14) r p = n 2 cos ⁡ θ i − n 1 cos ⁡ θ t n 2 cos ⁡ θ i + n 1 cos ⁡ θ t {displaystyle r_{p}={frac {n_{2}cos heta _{ ext{i}}-n_{1}cos heta _{ ext{t}}}{n_{2}cos heta _{ ext{i}}+n_{1}cos heta _{ ext{t}}}}}     (15) t p = 2 n 1 cos ⁡ θ i n 2 cos ⁡ θ i + n 1 cos ⁡ θ t {displaystyle t_{p}={frac {2n_{1}cos heta _{ ext{i}}}{n_{2}cos heta _{ ext{i}}+n_{1}cos heta _{ ext{t}}}},} .    (16) δ s = 2 arctan ⁡ n 2 sin 2 ⁡ θ i − 1 n cos ⁡ θ i {displaystyle delta _{s}=,2arctan {frac {sqrt {n^{2}sin ^{2} heta _{ ext{i}}-1}}{ncos heta _{ ext{i}}}},} .    (17) δ p = 2 arctan ⁡ n n 2 sin 2 ⁡ θ i − 1 cos ⁡ θ i {displaystyle delta _{p}=,2arctan {frac {,n{sqrt {n^{2}sin ^{2} heta _{ ext{i}}-1}}}{cos heta _{ ext{i}}}},} .    (18) r s = − sin ⁡ ( θ i − θ t ) sin ⁡ ( θ i + θ t ) , {displaystyle r_{s}=-{frac {sin( heta _{ ext{i}}- heta _{ ext{t}})}{sin( heta _{ ext{i}}+ heta _{ ext{t}})}},,}     (19) r p = tan ⁡ ( θ i − θ t ) tan ⁡ ( θ i + θ t ) , {displaystyle r_{p}={frac { an( heta _{ ext{i}}- heta _{ ext{t}})}{ an( heta _{ ext{i}}+ heta _{ ext{t}})}},,}     (20) Total internal reflection (TIR) is the phenomenon that makes the water-to-air surface in a fish-tank look like a perfectly silvered mirror when viewed from below the water level (Fig. 1). Technically, TIR is the total reflection of a wave incident at a sufficiently oblique angle on the interface between two media, of which the second ('external') medium is transparent to such waves but has a higher wave velocity than the first ('internal') medium. TIR occurs not only with electromagnetic waves such as light waves and microwaves, but also with other types of waves, including sound and water waves. In the case of a narrow train of waves, such as a laser beam, we tend to speak of the total internal reflection of a 'ray' (Fig. 2). Refraction is generally accompanied by partial reflection. When a wavetrain is refracted from a medium of lower propagation speed (higher refractive index) to a medium of higher propagation speed (lower refractive index), the angle of refraction (between the refracted ray and the normal to the refracting interface) is greater than the angle of incidence (between the incident ray and the normal to the interface). Hence, as the angle of incidence approaches a certain limit, called the critical angle, the angle of refraction approaches 90°, at which the refracted ray becomes tangential to the interface. As the angle of incidence increases beyond the critical angle, the conditions of refraction can no longer be satisfied; so we have no refracted ray, and the partial reflection becomes total. In an isotropic medium such as air, water, or glass, the ray direction is simply the direction normal to the wavefront. If the internal and external media are isotropic with refractive indices n1 and n2 respectively, the critical angle is given by‍ θ c = arcsin ⁡ ( n 2 / n 1 ) {displaystyle heta _{{ ext{c}}!}=arcsin(n_{2}/n_{1})} , and is defined if‍ n2 ≤ n1.  For example, for visible light, the critical angle is about 49° for incidence from water to air, and about 42° for incidence from common glass to air. Details of the mechanism of TIR give rise to more subtle phenomena. While total reflection, by definition, involves no continuing transfer of power across the interface, the external medium carries a so-called evanescent wave, which travels along the interface with an amplitude that falls off exponentially with distance from the interface. The 'total' reflection is indeed total if the external medium is lossless (perfectly transparent), continuous, and of infinite extent, but can be conspicuously less than total if the evanescent wave is absorbed by a lossy external medium ('attenuated total reflectance'), or diverted by the outer boundary of the external medium or by objects embedded in that medium ('frustrated' TIR). Unlike partial reflection between transparent media, total internal reflection is accompanied by a non-trivial phase shift (not just zero or 180°) for each component of polarization (normal or parallel to the plane of incidence), and the shifts vary with the angle of incidence. The explanation of this effect by Augustin-Jean Fresnel, in 1823, added to the evidence in favor of the wave theory of light. The phase shifts are utilized by Fresnel's invention, the Fresnel rhomb, to modify polarization. The efficiency of the reflection is exploited by optical fibers (used in telecommunications cables and in image-forming fiberscopes), and by reflective prisms, such as erecting prisms for binoculars. Although total internal reflection can occur with any kind of wave that can be said to have oblique incidence, including (e.g.) microwaves and sound waves,  it is most familiar in the case of light waves. Total internal reflection of light can be demonstrated using a semicircular-cylindrical block of common glass or acrylic glass. In Fig. 3, a 'ray box' projects a narrow beam of light (a 'ray') radially inward. The semicircular cross-section of the glass allows the incoming ray to remain perpendicular to the curved portion of the air/glass surface, and thence to continue in a straight line towards the flat part of the surface, although its angle with the flat part varies. Where the ray meets the flat glass-to-air interface, the angle between the ray and the normal to the interface is called the angle of incidence. If this angle is sufficiently small, the ray is partly reflected but mostly transmitted, and the transmitted portion is refracted away from the normal, so that the angle of refraction (between the refracted ray and the normal to the interface) is greater than the angle of incidence. For the moment, let us call the angle of incidence θi and the angle of refraction θt (where t is for transmitted, reserving r for reflected). As θi increases and approaches a certain 'critical angle', denoted by θc (or sometimes θcr), the angle of refraction approaches 90° (that is, the refracted ray approaches a tangent to the interface), and the refracted ray becomes fainter while the reflected ray becomes brighter. As θi increases beyond θc, the refracted ray disappears and only the reflected ray remains, so that all of the energy of the incident ray is reflected; this is total internal reflection (TIR). In brief: The critical angle is the smallest angle of incidence that yields total reflection. For light waves and other electromagnetic waves in isotropic media, there is a well-known formula for the critical angle in terms of the refractive indices. For some other types of waves, it is more convenient to think in terms of propagation velocities rather than refractive indices. The latter approach is more direct and more general, and will therefore be discussed first.