10.5 Total internal reflection

Total internal reflection occurs when light travels from a denser medium to a less dense one at an angle greater than the critical angle. This phenomenon is crucial in optics and electromagnetism, governing light behavior in various systems and devices.

Snell's law describes the relationship between angles of incidence and refraction at media interfaces. The critical angle, derived from Snell's law, determines when total internal reflection occurs. Understanding these concepts is essential for analyzing light propagation in different materials.

Conditions for total internal reflection

• Total internal reflection occurs when light traveling from a medium with a higher refractive index to a medium with a lower refractive index is incident at an angle greater than the critical angle
• The phenomenon of total internal reflection is crucial in understanding the behavior of light in various optical systems and devices used in electromagnetism

Snell's law and critical angle

• Snell's law describes the relationship between the angles of incidence and refraction when light passes through the interface between two media with different refractive indices
  • The law is expressed as $n_1 \sin \theta_1 = n_2 \sin \theta_2$, where $n_1$ and $n_2$ are the refractive indices of the two media, and $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, respectively
• The critical angle is the minimum angle of incidence at which total internal reflection occurs
  • It is calculated using the equation $\theta_c = \sin^{-1}(n_2/n_1)$, where $n_1$ and $n_2$ are the refractive indices of the two media, with $n_1 > n_2$
• When the angle of incidence is greater than the critical angle, light is completely reflected back into the original medium (water-air interface, glass-air interface)

Refractive indices of media

• The refractive index is a dimensionless number that describes how light propagates through a medium
  • It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium
• Materials with higher refractive indices (diamond, glass) have a lower speed of light and a greater ability to bend light compared to materials with lower refractive indices (air, water)
• The difference in refractive indices between two media determines the critical angle and the occurrence of total internal reflection

Evanescent waves

• Evanescent waves are non-propagating electromagnetic waves that arise when light undergoes total internal reflection at the interface between two media
• These waves play a crucial role in various optical phenomena and applications, such as near-field microscopy and surface plasmon resonance

Characteristics of evanescent waves

• Evanescent waves have an exponentially decaying amplitude perpendicular to the interface, with the highest intensity at the interface itself
• The wave vector of an evanescent wave has an imaginary component, indicating that the wave does not propagate in the direction perpendicular to the interface
• Evanescent waves exhibit a phase shift relative to the incident light, which depends on the angle of incidence and the refractive indices of the media

Penetration depth

• The penetration depth is the distance from the interface at which the intensity of the evanescent wave decays to 1/e (approximately 37%) of its initial value
  • It is calculated using the equation $d_p = \frac{\lambda}{4\pi\sqrt{n_1^2\sin^2\theta - n_2^2}}$, where $\lambda$ is the wavelength of light, $n_1$ and $n_2$ are the refractive indices of the two media, and $\theta$ is the angle of incidence
• The penetration depth depends on the wavelength of light and the refractive indices of the media
  • Longer wavelengths and smaller differences in refractive indices result in greater penetration depths (red light in glass-air interface, near-infrared light in fiber optics)

Intensity and phase

• The intensity of an evanescent wave decays exponentially with distance from the interface, following the equation $I(z) = I_0e^{-z/d_p}$, where $I_0$ is the initial intensity at the interface, $z$ is the distance from the interface, and $d_p$ is the penetration depth
• Evanescent waves exhibit a phase shift relative to the incident light, which is given by $\phi = \tan^{-1}\left(\frac{2\cos\theta\sqrt{n_1^2\sin^2\theta - n_2^2}}{n_1^2\sin^2\theta - n_2^2 - \cos^2\theta}\right)$, where $n_1$ and $n_2$ are the refractive indices of the two media, and $\theta$ is the angle of incidence
• The phase shift depends on the angle of incidence and the refractive indices of the media (glass-air interface at different angles, evanescent wave coupling in optical waveguides)

Frustrated total internal reflection

• Frustrated total internal reflection occurs when an evanescent wave couples with a nearby medium, allowing some of the light to "tunnel" through the gap and propagate in the second medium
• This phenomenon has important applications in optical devices and sensors, where the coupling of evanescent waves can be used to control and manipulate light

Coupling of evanescent waves

• When two media with similar refractive indices are brought close together (within the range of the penetration depth), the evanescent wave generated by total internal reflection in one medium can couple with the second medium
• The coupling of evanescent waves allows energy to be transferred from one medium to another, even though the light does not propagate in the classical sense (optical waveguides, coupled resonators)

Photon tunneling

• In the context of frustrated total internal reflection, photon tunneling refers to the process by which light can "tunnel" through a gap between two media, even though classical optics would predict total reflection
• The probability of photon tunneling depends on the gap size, the refractive indices of the media, and the angle of incidence
  • Smaller gaps, closer refractive indices, and angles of incidence closer to the critical angle result in higher tunneling probabilities (nanoscale gaps between optical fibers, evanescent wave coupling in photonic crystals)

Applications in optical devices

• Frustrated total internal reflection has numerous applications in optical devices, such as:
  • Optical couplers and splitters, where light can be efficiently transferred between waveguides or split into multiple channels
  • Optical switches and modulators, where the coupling of evanescent waves can be controlled by external factors (electric fields, mechanical deformation) to modulate the light signal
  • Optical sensors, where the presence of a target analyte can alter the evanescent wave coupling and provide a measurable signal (surface plasmon resonance sensors, waveguide-based biosensors)

Goos-Hänchen effect

• The Goos-Hänchen effect is a phenomenon in which a beam of light undergoing total internal reflection experiences a lateral shift along the interface between two media
• This effect has important implications for optical measurements and can be exploited in various applications, such as sensing and imaging

Lateral shift of reflected beam

• When a beam of light undergoes total internal reflection, the reflected beam experiences a lateral shift along the interface, known as the Goos-Hänchen shift
• The magnitude of the shift depends on the angle of incidence, the refractive indices of the media, and the polarization of the light
  • The shift is typically on the order of a fraction of the wavelength of light (nanometer-scale shifts for visible light, micrometer-scale shifts for infrared light)

Dependence on polarization and angle

• The Goos-Hänchen shift is different for s-polarized (transverse electric) and p-polarized (transverse magnetic) light
  • For s-polarized light, the shift is given by $D_s = \frac{\lambda}{2\pi}\frac{\sqrt{n_1^2\sin^2\theta - n_2^2}}{n_1^2\cos\theta}$
  • For p-polarized light, the shift is given by $D_p = \frac{\lambda}{2\pi}\frac{n_1^2\sqrt{n_1^2\sin^2\theta - n_2^2}}{n_2^2\cos\theta}$
• The shift is also dependent on the angle of incidence, with the maximum shift occurring near the critical angle (glass-air interface at different angles, polarization-dependent shifts in optical waveguides)

Implications for optical measurements

• The Goos-Hänchen effect can introduce errors in optical measurements that rely on precise positioning or alignment of light beams
  • In interferometry, the lateral shift can cause a phase difference between the interfering beams, leading to inaccuracies in distance or displacement measurements
  • In optical trapping and manipulation, the shift can affect the position and stability of trapped particles
• Understanding and accounting for the Goos-Hänchen effect is crucial in high-precision optical measurements and applications (gravitational wave detection, optical tweezers)

Attenuated total reflection (ATR)

• Attenuated total reflection is a technique that exploits the evanescent wave generated during total internal reflection to probe the properties of materials in close proximity to the interface
• ATR has widespread applications in spectroscopy, sensing, and material characterization

Principles of ATR spectroscopy

• In ATR spectroscopy, a sample is placed in close contact with a high-refractive-index material (the ATR crystal) that undergoes total internal reflection
• The evanescent wave generated at the crystal-sample interface interacts with the sample, and its properties (intensity, phase, polarization) are modified by the sample's absorption and refractive index
• By measuring the changes in the reflected light, the absorption spectrum and other properties of the sample can be determined (infrared ATR spectroscopy, surface-enhanced Raman spectroscopy)

Evanescent wave absorption

• When the evanescent wave interacts with the sample, it can be absorbed by the sample's molecular vibrations or electronic transitions
• The absorption of the evanescent wave leads to a reduction in the intensity of the reflected light, which can be measured and used to determine the sample's absorption spectrum
• The penetration depth of the evanescent wave determines the sampling volume and the sensitivity of the ATR technique to surface or bulk properties of the sample (thin film characterization, surface adsorption studies)

ATR prisms and waveguides

• ATR spectroscopy can be performed using various optical elements, such as prisms or waveguides, that support total internal reflection and generate evanescent waves
• ATR prisms are commonly used in benchtop spectrometers, where the sample is placed in contact with one of the prism's faces, and the reflected light is collected and analyzed (diamond ATR prisms, germanium ATR prisms)
• ATR waveguides, such as optical fibers or planar waveguides, allow for more compact and integrated ATR systems, suitable for in-situ or remote sensing applications (fiber-optic ATR probes, lab-on-a-chip devices)

Applications of total internal reflection

• Total internal reflection and its associated phenomena have numerous applications in various fields, ranging from telecommunications to biomedical diagnostics
• These applications rely on the unique properties of evanescent waves, the control of light propagation, and the sensitivity to surface or interface properties

Optical fibers and waveguides

• Optical fibers and waveguides are essential components in modern telecommunications and data transmission systems
• They rely on total internal reflection to confine and guide light over long distances with minimal losses
• The evanescent wave generated at the core-cladding interface of an optical fiber can be used for sensing or coupling purposes (fiber-optic sensors, evanescent wave couplers)

Prisms and reflectors

• Prisms and reflectors that exploit total internal reflection are used in various optical systems for beam steering, dispersion control, and polarization management
• Examples include:
  • Dove prisms for image rotation
  • Penta prisms for beam deflection
  • Retroreflectors for precise distance measurements
  • Polarizing beam splitters for separating s- and p-polarized light

Biosensors and surface plasmon resonance

• Total internal reflection and evanescent waves are the basis for several biosensing techniques, such as surface plasmon resonance (SPR) and evanescent wave fluorescence
• In SPR, the evanescent wave generated at a metal-dielectric interface couples with surface plasmons, and the resulting resonance condition is sensitive to the refractive index and thickness of the dielectric layer
• By functionalizing the metal surface with biorecognition elements (antibodies, aptamers), SPR can be used to detect specific analytes (proteins, nucleic acids) with high sensitivity and specificity

Fingerprint recognition and security systems

• Total internal reflection is used in fingerprint recognition systems, where the unique patterns of friction ridges on a finger are captured by imaging the contact area between the finger and a glass prism
• The evanescent wave generated at the glass-air interface scatters off the fingerprint ridges, creating a high-contrast image that can be processed and compared to a database for identification purposes
• Similar principles are employed in other security and authentication systems, such as palm vein recognition or iris scanning, where the evanescent wave is used to probe the subsurface features of the biometric traits