occurs when light travels from a denser medium to a less dense one at an angle greater than the . This phenomenon is crucial in optics and electromagnetism, governing light behavior in various systems and devices.
describes the relationship between angles of incidence and at media interfaces. The critical angle, derived from Snell's law, determines when total internal reflection occurs. Understanding these concepts is essential for analyzing in different materials.
Conditions for total internal reflection
Total internal reflection occurs when light traveling from a medium with a higher 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
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4.3 The Law of Refraction – Snell’s Law – Douglas College Physics 1207 View original
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 n1sinθ1=n2sinθ2, where n1 and n2 are the refractive indices of the two media, and θ1 and θ2 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 θc=sin−1(n2/n1), where n1 and n2 are the refractive indices of the two media, with n1>n2
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 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 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 dp=4πn12sin2θ−n22λ, where λ is the wavelength of light, n1 and n2 are the refractive indices of the two media, and θ 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 )
Intensity and phase
The intensity of an evanescent wave decays exponentially with distance from the interface, following the equation I(z)=I0e−z/dp, where I0 is the initial intensity at the interface, z is the distance from the interface, and dp is the penetration depth
Evanescent waves exhibit a phase shift relative to the incident light, which is given by ϕ=tan−1(n12sin2θ−n22−cos2θ2cosθn12sin2θ−n22), where n1 and n2 are the refractive indices of the two media, and θ 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
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, 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 , 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 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 Ds=2πλn12cosθn12sin2θ−n22
For p-polarized light, the shift is given by Dp=2πλn22cosθn12n12sin2θ−n22
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)
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 '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