Physics
Total Internal Reflection
Total internal reflection occurs when a light ray traveling through a denser medium strikes the boundary of a less dense medium at an angle greater than the critical angle. Instead of refracting into the less dense medium, the light is completely reflected back into the denser medium. This phenomenon is used in various optical devices such as fiber optics and prisms.
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12 Key excerpts on "Total Internal Reflection"
eBook - PDFColor Imaging
Fundamentals and Applications
- Erik Reinhard, Erum Arif Khan, Ahmet Oguz Akyuz, Garrett Johnson(Authors)
- 2008(Publication Date)
- A K Peters/CRC Press(Publisher)
For angles of incidence larger than θ c , the wave will not refract but will only reflect. This phenomenon is called total reflection , or sometimes Total Internal Reflection , and is illustrated in Figure 2.18. It can be demonstrated by punching a hole in a plastic bottle and draining water through it. By aligning a laser with the opening in the bottle and adding a drop of milk (to make the laser beam visible), Total Internal Reflection such as that seen in Figure 2.19 can be ob-tained. The light stays within the water stream through multiple reflections. The transmittance of light through strands of fiber, known as fiberoptics, works on the same principle (Figure 2.20) . While Snell’s law predicts the angles of reflection and transmission, it does not give insight into the amplitude of the reflected and refracted fields, E and B , respectively. To evaluate these amplitudes, we first need to look at the boundary conditions related to the interface between the two media, which are most easily evaluated by means of Maxwell’s equations in integral form. The components of D and B normal to the boundary are assessed first, followed by the tangential components of these fields. 6 n -n θ i θ t s (t) s (r) s (i) Medium 1 Medium 2 x z z=0 θ c θ r θ r θ i s (i) s (r) θ < θ i c θ > θ i c : Reflection + Refraction : Total Reflection Figure 2.18. Geometry associated with total reflection. 6 As D = ε E , the analysis is equivalent. 2.5. Reflection and Refraction 53 Figure 2.19. A stream of water exits a plastic bottle. Coming from the right, a laser beam enters the bottle and exits through the same hole as the water. Multiple reflections inside the stream of water are visible, essentially keeping the light trapped inside the stream. Figure 2.20. Fiberoptic strands transmitting light with very little loss of energy due to Total Internal Reflection inside the strands.
eBook - PDF- Jerry D. Gibson(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
Beyond the critical angle, there is Total Internal Reflection. cladding n 2 n 1 core 2a 50 µ m reflected ray incident ray reflected incident n 1 n 2 n 1 n 2 refracted ray decaying θ 1 θ 1 θ 2 n 2 n 1 θ 1 θ c θ 2 90 ° n 2 n 1 n 1 sin θ 1 n 2 sin θ 2 = Optical Fibers and Lightwave Propagation 45 -3 reflection occurs (internal to the denser medium in which the light originated), and the escape of the light into the less dense medium has been thwarted. There is more to the process of Total Internal Reflection at a planar boundary than merely reflection from the interface. Although the light does not leak into the less dense medium and thereby get trans-mitted outward, it does seep across the interface and propagate, but only parallel to that interface surface [Diament, 1990]. In the direction normal to the interface, there is no propagation on the less dense side of the surface but the strength of the light that appears there decays exponentially. The spatial rate of decay increases with the angle of incidence, beyond the critical angle at which Total Internal Reflection begins. If the decay is sufficiently strong, the light that seeps across is effectively constrained to a thin layer beyond the interface and propagates along it, while the light left behind is reflected back into the denser region with its original strength. The reflection coefficient is 100%. There is undiminished optical power flow along the reflected ray in the denser medium, but only storage of optical energy in the light that seeps across the interface into the less dense region. Much the same process occurs along the curved, cylindrical interface between the core and cladding of the optical fiber. When the light strikes this interface at a sufficiently high grazing angle, Total Internal Reflection occurs, keeping the optical power propagating at an angle within the core and along the axial direction in the cladding.
eBook - ePub- Galen C. Duree, Jr.(Authors)
- 2011(Publication Date)
- For Dummies(Publisher)
Frustrated Total Internal Reflection: Dealing with the evanescent wave(contain real and imaginary parts). When this increase happens, the magnitude is still 1, indicating that all light incident on a surface undergoing internal reflection is totally internally reflected — no light goes through to the other material. This major phenomenon is used with fiber optics, which I talk about in more detail in Chapter 16.In the preceding section, I mention that no light enters the second material as long as the incident light has an angle of incidence greater than the critical angle. However, if you look very close to the surface where the light hits, you find a small, exponentially decaying wave in the second material. If you place another material close to, but not touching, the first boundary, the second material can pick up and propagate the wave away. This wave that passes into the second material is called anevanescent wave.If this evanescent wave is intercepted with a high-index material before its amplitude is zero, the wave continues on in a process called frustrated Total Internal Reflection.(No, that’s not what you do during a midlife crisis.) The energy from this wave comes from the incident wave, so when Total Internal Reflection is frustrated, the energy in the reflected beam decreases by the amount carried away in the evanescent wave. If the process isn’t frustrated, the energy returns to the reflected beam so that the reflected beam experiences no energy loss.The fiber-optics industry uses the phenomenon of frustrated Total Internal Reflection quite a bit to control the amount of light in different parts of the fiber network (head to Chapter 16 for more information about this application). This phenomenon is also used in devices called
eBook - PDF- Sam Beddar, Luc Beaulieu(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
Basic principles and theory 38 Scintillation dosimetry 3.2 BASIC PRINCIPLES OF OPTICAL FIBER 3.2.1 PRINCIPLE OF LIGHT TRANSMISSION IN OPTICAL FIBER As a dielectric waveguide, an optical fiber can transmit light signals from a light source to a light-measuring device at a long distance using Total Internal Reflection. Under the condition of Total Internal Reflection, light can be reflected at a dielectric interface region between two transparent media that have different refractive indices, without any reflectors. Here, the ray of light must be incident on a boundary from the medium with a high refractive index to another with a low refractive index. If a ray of light transmits with a certain incidence angle ( θ 1 ) that is less than the critical angle ( θ c ) from medium 1 with a high refractive index ( n 1 ) to medium 2 with a low refractive index ( n 2 ), a portion of light will be reflected back to medium 1 with a reflection angle ( θ 1 ), and another portion of light will be refracted into medium 2 with a refraction angle ( θ 2 ), as shown in Figure 3.1a. The relationship between the refractive index and the angle of light is expressed by Snell’s law [1,2]: n n 1 2 sin sin θ θ 1 2 nobreakspace = (3.1) Here, the refractive index ( n ) is the ratio of the velocity of light in a vacuum ( c ) to its velocity in a specific medium ( v ) and is given by the following relation: n c v = (3.2) From Equation 3.2, the velocity of light slows down according to the increase of the refractive index of the medium that the light is traveling in, and Equation 3.1 can be changed using Equation 3.2 as follows: sin sin nobreakspacenobreakspacenobreakspace θ θ 2 1 = = n n V V 1 2 2 1 (3.3) When the incidence angle ( θ 1 ) is increased to the critical angle ( θ c ), the refractive angle ( θ 2 ) reaches 90 ° , as shown in Figure 3.1b and thereby, a ray of light travels along the interface region between two contacted media.
eBook - PDF- Ray Tricker(Author)
- 2002(Publication Date)
- Newnes(Publisher)
Theory 47 2 If light emerging from a glass with a high refractive index is incident on a material with a lower refractive index, it changes direction provided it strikes the interface at a steep angle. If the angle of incidence is shallower than the critical angle, TIR occurs. 3 When a light beam is passed from air to glass, because of the different refractive indices at the point of penetration, the light is propagated from within the glass core at various angles. A variable refraction is therefore present at the end face- which is sometimes referred to as the aperture. The critical angle is defined as 'the angle of incidence, which provides an angle of refraction of 90 degrees'. Note: The actual value of the critical angle is dependent upon the combination of materialspresenton each side of the boundary.For the water/air boundary, the critical angle is 48.6 degrees. For the crown glass/waterboundary,the critical angle is 61.0 degrees. 2.6 Acceptance cone Since fiber is circular, a cone (termed the acceptance cone, Figure 2.11) defines the angles of incident light that are totally, internally, reflected by the core. A large acceptance cone, although allowing a fiber to accept and propagate light from a larger field, coincides with greater dispersion. Conversely, a narrow acceptance cone accompanies low dispersion but requires a narrow, more precise source of light. CC o , dc ,, ,, Cladding i fiiiii C!ilol r! e!iiiiiiiiiil Figure 2.11 Acceptance cone 48 Optoelectronics and Fiber Optic Technology _ .... =, Acceptance i an~e i __ ~.,,. ..... nl (cladding) / / n~ (cladding) / Figure 2.12 Acceptance angle of fight incident on the end face of an optical fiber 2.7 Acceptance angle The acceptance angle is a function of the refractive indices of the core and cladding materials and is defined as 'the half angle of vertex of a cone within which rays are incident on the end face of the fiber and will be captured and propagated by that fiber' (see Figure 2.12).
eBook - PDF- Y K Lim(Author)
- 1986(Publication Date)
- WSPC(Publisher)
Let ty be the angle between the electric vector E of the incident wave and the plane of incidence. The components of E corresponding to cases (a) and (b) after total reflection are E'l = E sinijjcos (k • r - cot - <|> x ) , and Ej! = - E 0 costpcos (k • r - ut - $..) . (5.39) In general, x and cb,, will not be equal and the terminus of the resultant electric vector E at a fixed point on the path of the reflected wave will trace out an ellipse in the plane normal to the direction of propagation [Prob. (5.7)]. Thus the totally reflected wave is elliptically polarized. In the special case of !f»=ir/4 and 4^ -<)>,. = ±TT/2, the totally reflected wave is circularly polarized. An example of the application of this effect is Fresnel's rhomb, which is a parallelepiped of crown glass. A beam of light incident normally on one face of the rhomb undergoes internal total reflection twice and emerges normally from the opposite face. If the incident light is linearly polarized with the plane of polarization making an angle ir/4 with the plane of internal reflection, and if the material of the rhomb and the angle of reflection are such that the relative phase change cb i -t}i, j is ±TT/4 on each reflection, the light emerging from the rhomb will be circularly polarized. Another interesting aspect of the theory is the prediction that some penetration of the electromagnetic field into the second medium will occur even under the conditions of total reflection as can be seen from Eq. (5.36). The tacit assumption made in choosing the sign of cose 1 which prevents the amplitude in the second medium from becoming infinitely large is that the thickness of the second medium is large compared with the wavelength. If this condition is not satisfied, both 200 signs are to be used; the wave can penetrate through the second medium and total reflection will not occur.
eBook - ePub- Jerry Marion(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
degree of polarization may be defined asFIG. 6-5(6.34)so that P (00 ) = P (900 ) = 0 and P (θB ) = 1; P (θ0 ) < 0 except for normal and for grazing incidence.6.4 Total Internal Reflection
An interesting application of the Fresnel equations is to consider the electromagnetic wave to be incident on the boundary surface from the more dense of the two media; i.e., we have n 1 < n 2 . If we start with θ0 = 0 and allow the angle of incidence to increase, all goes well until we reach an angle such that θ2 = π/2. From Snell’s law this occurs when sin θ0 = n 2 /n 1 , and the angle is called the critical angle :(6.35)We may write the cosine of the angle of refraction as(6.36)so that cos θ2 = 0 for θ0 = θc , but as θ0 is increased beyond θC , cos θ2 becomes a pure imaginary number. Therefore we write(6.37a)where Q is defined as the positive square root(6.37b)In order to determine the effect on the field in Medium 2 when the angle of incidence exceeds the critical angle, we write E 2 as [see Eq. (6.16) ](6.38)If we take the coordinate axes as in Fig. 6-2 , this becomes(6.39)But(6.40)where W is a real number, greater than unity. Thus, using Eqs. (6.37a) for cos θ2 , we have(6.41)We therefore see that for θ0 < θc , the wave is propagated along the surface and is attenuated in the direction into the medium. The phase velocity of the surface wave is(6.42)which is a function of the angle of incidence θ0 . We see that in the event Medium 2 is air (n2 = 1), the phase velocity is justV1 = c/n1 for grazing incidence.The amplitudes of the reflected electric vectors may be calculated from Fresnel’s equations [Eqs. (6.25a) and (6.28a) ]. For the case of E
eBook - PDF- Charles A. Bennett(Author)
- 2015(Publication Date)
- Wiley(Publisher)
In the next section, we consider materials where the absorption is not zero. If the transmitted medium is modified before the evanescent wave damps to zero, energy can be transmitted across the interface. Figure 3.22 illustrates frustrated total reflection produced by two prisms separated by a gap that is a fraction of a wavelength wide. The 94 REFLECTION AND REFRACTION incident beam encounters the internal surface of the first prism at an angle greater than the critical angle. Nevertheless, a beam transmits into the second prism if it can be positioned close enough to the first prism such that the evanescent wave can couple into it before damping to zero. The width of the gap determines the amount of energy transmitted across "forbidden region." This is the classical optical analog of the quantum mechanical phenomenon of tunneling. By adjusting the width of the gap, the arrangement in Figure 3.22 can be used as a variable attenuator. Figure 3.22. Frustrated total reflection. The gap between the two prisms is a fraction of a wavelength wide. Problem 3.20 A slab of glass (n = 1.52) is immersed in water (n = 1.33). a) Find the critical angle for Total Internal Reflection. b) Find Brewster’s angle for both external and internal incidence. c) Find the reflectivity R for both internal and external incidence at normal incidence. Problem 3.21 As a gemstone, diamond is valued not only for its hardness and dispersion, but also for the brilliance that results from its high reflectivity. Find the reflectivity of an air-diamond interface at normal incidence. Problem 3.22 A beam of light is incident in air normally on a pane of glass with index of refraction equal to 1.52. Neglecting interference and absorption effects, what fraction of the incident beam irradiance is transmitted by the slab? Repeat for a slab of diamond with index of refraction equal to 2.42. Problem 3.23 Show that R + T = 1 for arbitrary angle of incidence.
eBook - PDF- Charles A. Bennett(Author)
- 2022(Publication Date)
- Wiley(Publisher)
If the transmitted medium is modified before the evanescent wave damps to zero, energy can be transmitted across the interface. Figure 3.22 illustrates frustrated total reflection pro- duced by two prisms separated by a gap that is a fraction of a wavelength wide. The incident beam encounters the internal surface of the first prism at an angle greater than the criti- cal angle. Nevertheless, a beam transmits into the second prism if it can be positioned close enough to the first prism such that the evanescent wave can couple into it before damping to zero. The width of the gap determines the amount of energy transmitted across “forbid- den region.” This is the classical optical analog of the quantum mechanical phenomenon of tunneling. By adjusting the width of the gap, the arrangement in Figure 3.22 can be used as a variable attenuator. 3.7 Scattering Scattering occurs when electromagnetic waves interact with particles. If the particles have diameters much larger than the wavelength, the scattering can be understood mostly on a basis of reflection and refraction. Examples of large-particle scattering include rainbows and scattering from clouds. Scattering from all sizes of spherical particles can be calculated rigor- ously using Mie scattering theory, 9 which takes Maxwell’s equations as a starting point and thus includes effects of diffraction. In Rayleigh scattering, 10 the particle size is much smaller 9 Gustav Mie (1869–1957). 10 John Strutt, Lord Rayleigh (1842–1919). 82 3 Reflection and Refraction than the scattering wavelength. Both Rayleigh scattering and large-particle scattering can utilize simplifying assumptions, but when the particle size is on the order of the scattering wavelength, Mie scattering must be used. It just so happens that Earth’s atmosphere has just the right buoyancy to suspend particles that are about equal to the wavelength of visible light.
- Constantine A. Balanis(Author)
- 2023(Publication Date)
- Wiley(Publisher)
This angle is known as the critical angle, and it is independent of wave polarization; however, in order for this angle to occur, the incident wave must exist in the denser medium. The critical angle concept plays a crucial role in the design of transmission lines (such as optical fiber, slab wave-guides, and coated conductors; the microstrip is one example). 5.2 NORMAL INCIDENCE—LOSSLESS MEDIA We begin the discussion of reflection and transmission from planar boundaries of lossless media by assuming the wave travels perpendicular (normal incidence) to the planar interface formed by two semi-infinite lossless media, as shown in Figure 5-1, each characterized by the constitutive CHAPTER 5 Reflection and Transmission 180 REFLECTION AND TRANSMISSION parameters of ε µ 1 1 , and ε µ 2 2 , . When the incident wave encounters the interface, a fraction of the wave intensity will be reflected into medium 1 and part will be transmitted into medium 2. Assuming the incident electric field of amplitude E 0 is polarized in the x direction, we can write expressions for its incident, reflected, and transmitted electric field components, respec- tively, as E a i x j z E e = - ˆ 0 1 β (5-1a) E a r x b j z E e = + ˆ Γ 0 1 β (5-1b) E a t x b j z T E e = - ˆ 0 2 β (5-1c) where Γ b and T b are used here to represent, respectively, the reflection and transmission coeffi- cients at the interface. Presently these coefficients are unknowns and will be determined by applying boundary conditions on the fields along the interface. Since the incident fields are linearly polarized and the reflecting surface is planar, the reflected and transmitted fields will also be linearly polarized. Because we do not know the direction of polarization (positive or negative) of the reflected and transmitted electric fields, they are assumed here to be in the same direction (positive) as the incident electric fields.
eBook - PDF- Pradeep Fulay, Jung-Kun Lee(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
When a transverse wave is fixed at an end, a reflected wave experiences a phase change of 180°. However, if the boundary is flexible, the phase change decreases. For an end where the rope moves freely, there is no phase change between the incident wave and reflected wave. This implies that the level of refractive index difference at the boundary controls the phase shift of reflected light. If n 1 > n 2 , the reflection coefficient and phase change are very different from the data in Figure 8.19. In this case, the phase of reflected light does not change for an incident angle smaller than the Brewster’s angle, which is analogous to the case of free end motion in Figure 8.20. One last factor to be considered about light reflectance (shown in Figure 8.18) is how E t (the elec-tric field component of transmitted light) would change if the incident light is completely reflected (for a case of total reflectance). If the incidence angle exceeds the critical angle (see Equation 8.43), the transmitted light intensity must be zero. However, this does not mean that the electric field and magnetic field in Medium 2 are zero, too. If E t and H t are zero, there is no analytic solution for Equations 8.47, 8.48, 8.58, and 8.59, which are driven from Maxwell’s equations. This contradiction 1.0 –1.0 0 Reflection coefficient, r Incidence angle, θ i (a) (b) Incidence angle, θ i Phase change in degree π π 0 0 0° 30° 60° 90° 0° 30° 60° 90° 0° 30° 60° 90° Brewster’s angle r ⊥ r ⊥ r || r || r || =0 FIGURE 8.19 Dependence of (a) the reflection coefficient and (b) phase change on incident light angle, θ i ; r ⊥ = for perpendicular polarization, parallelto r = for parallel polarization. 328 Electronic, Magnetic, and Optical Materials between no transmitted light and non-zero electric/magnetic field is resolved by introducing a non-vanishing electromagnetic surface wave that is bound to the surface region of Medium 2.
eBook - PDF- Raimo Silvennoinen, Kai-Erik Peiponen, Kari Myller(Authors)
- 2010(Publication Date)
- Elsevier Science(Publisher)
In microscopic world, the refractive index is closely related to the light Ch. 2: Light Reflection from Ideal Surface 21 field and electron interaction as it will be shown later. Substitution of Eq. 2.43 in Eq. 2.42 yields v g = c n 1 + n d n d (2.44) The formula of Eq. 2.44 has much importance in cases where information along an optical path is transferred from one place to another, such as in the case of tele-phone communication through optical fibres. In the case of measurement of the specular gloss, the dependence of the refractive index of the medium has a role because the irradiance of specularly reflected light depends on the dispersion of the medium, both for smooth and rough surfaces. We wish to emphasize once again that refractive index of medium that depends on the wavelength of the incident light, and also on the thermodynamical conditions of the medium, is a crucial intrinsic optical property that all kind of media, irrespective of their shape, texture or surface roughness, have to obey. 2.6. Normal reflection of light The simplest case of light reflection is the one where the plane wave is normally incident on planar surface. In other words, the light beam is in the direction of the normal of the plane that constitutes the interface between two media, i.e. air-transparent medium in the present case. Next, we derive a law that allows one to calculate the strength of electric field after reflection from an ideal surface. We assume that light is incident from air to a denser medium. The key points needed are results from the theory of electromagnetism, namely 22 Specular Gloss that at the interface the component of electric field that is parallel to the interface, and the component of magnetic induction that is normal to the interface are continuous functions in the interface (Born et al., 1960).
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