Transmittance
Transmittance is the fraction of incident electromagnetic radiation, such as light, that passes through a material without being absorbed or reflected, defined as the ratio of the transmitted intensity I_t to the incident intensity I_0, expressed as T = I_t / I_0, where T ranges from 0 (opaque) to 1 (fully transparent) or equivalently as a percentage.[1] This optical property is wavelength-dependent and characterizes the transparency of materials like glass, liquids, and biological tissues across the ultraviolet, visible, and infrared spectra.[2] In spectroscopy and analytical chemistry, transmittance is central to the Beer-Lambert law, which relates it to absorbance A = -\log_{10}(T) = \epsilon c l, where \epsilon is the molar absorptivity, c is the concentration of the absorbing species, and l is the path length through the sample, enabling quantitative determination of solute concentrations in solutions.[3] Transmittance measurements are typically performed using spectrophotometers, which compare the intensity of light before and after passing through the sample at specific wavelengths to assess material properties or chemical compositions.[4] Transmittance plays a critical role in applications ranging from optical engineering, where it guides the design of lenses, filters, and windows for high transmission in the visible range (400–800 nm), to photovoltaics and biomedical imaging, where it influences energy efficiency and light penetration in encapsulants or tissues.[2][5] Factors affecting transmittance include material thickness, surface reflections, scattering, and absorption bands, with advanced models accounting for these in complex media like the atmosphere or turbid liquids.[6]Basic Principles
Definition
Transmittance is a fundamental concept in optics and radiometry, defined as the fraction of incident radiant flux that is successfully transmitted through a sample, surface, or medium. It quantifies the effectiveness of the material or interface in allowing radiant energy to pass without being reflected or absorbed. This ratio is dimensionless and ranges from 0 (complete opacity) to 1 (perfect transparency).[7] In the context of radiometry, radiant flux \Phi_e represents the total power carried by electromagnetic radiation, measured in watts (W), encompassing the energy emitted, transferred, or received over time. The incident radiant flux \Phi_e^i is the power approaching the sample, while the transmitted radiant flux \Phi_e^t is the power emerging on the other side. The transmittance T is mathematically expressed as T = \frac{\Phi_e^t}{\Phi_e^i}, where both fluxes are evaluated under the same conditions.[8][9] While transmittance specifically denotes the power or intensity ratio in classical optics, the term transmission coefficient is sometimes used interchangeably but more precisely refers to the amplitude transmission factor in wave optics, with the transmittance being its squared magnitude adjusted for refractive indices and angles. In quantum mechanics, the transmission coefficient often describes the probability of transmission for particles or photons. For non-scattering media, transmittance relates to other optical properties through energy conservation, where T + R + A = 1, with R as reflectance and A as absorptance.[10][7]Mathematical Representation
Transmittance is conventionally denoted by the symbol T, representing the fraction of incident radiant energy that is transmitted through a material or optical element. The radiant flux, denoted \Phi_e, quantifies the total power of electromagnetic radiation, measured in watts (W). Subscripts are used to distinguish incident flux (\Phi_{e,i}), transmitted flux (\Phi_{e,t}), and reflected flux (\Phi_{e,r}).[11][12] The general mathematical expression for transmittance is given by the ratio of transmitted radiant flux to incident radiant flux: T = \frac{\Phi_{e,t}}{\Phi_{e,i}} This formulation applies to the total flux crossing a surface or beam. For directional cases, such as collimated beams, the flux may be expressed per unit solid angle in watts per steradian (W/sr). Transmittance is inherently dimensionless, as it is a ratio of like quantities.[11][12] Extensions of this definition account for other radiometric quantities relevant to specific geometries. For planar illumination, transmittance can be defined in terms of irradiance E (radiant flux per unit area, in W/m²), where T = E_t / E_i. Similarly, for imaging or extended sources, it relates to radiance L (radiant flux per unit area per unit solid angle, in W/m²/sr), yielding T = L_t / L_i. These forms assume uniform distribution and conservation along the propagation path.[13][14] The standard formulation assumes negligible scattering within the medium, such that transmitted light consists primarily of directly propagated radiation without diffuse redirection. It typically applies to incoherent light sources, though for coherent illumination (e.g., lasers), interference effects in thin films can modulate transmittance beyond simple absorption. In ideal measurements, the setup isolates transmitted flux without external contributions like ambient light; real measurements may require corrections for scattering or multiple reflections to approximate these conditions.[15][12] Energy conservation in optical systems relates transmittance to complementary quantities: reflectance R = \Phi_{e,r} / \Phi_{e,i} and absorptance A = (\Phi_{e,i} - \Phi_{e,t} - \Phi_{e,r}) / \Phi_{e,i}. For non-scattering, opaque samples where transmission occurs minimally but is considered, the flux balance yields T + R + A = 1, ensuring all incident energy is accounted for via transmission, reflection, or absorption. This relation derives directly from the principle that incident flux equals the sum of outgoing (transmitted and reflected) and absorbed components.[13]Surface Transmittance
Hemispherical Transmittance
Hemispherical transmittance quantifies the fraction of radiant flux that passes through a surface when the incident illumination originates from all directions within a hemisphere, integrating over the projected solid angle to account for the total energy transfer. It is defined as the ratio of the total transmitted radiant flux \Phi_e^t to the total incident radiant flux \Phi_e^i across the hemisphere, assuming uniform or diffuse incidence conditions.[11] In general, for non-uniform radiance distributions, it is expressed as T = \frac{\int_{\text{hemisphere}} L^t(\theta, \phi) \cos \theta \, d\Omega}{\int_{\text{hemisphere}} L^i(\theta, \phi) \cos \theta \, d\Omega}, where L^t and L^i are the transmitted and incident radiances, respectively, \theta is the polar angle from the surface normal, \phi is the azimuthal angle, and d\Omega = \sin \theta \, d\theta \, d\phi is the differential solid angle.[8] This measure is particularly relevant for surface interactions in optics, focusing on the overall energy passage without resolving specific angles. Hemispherical transmittance is commonly applied to opaque surfaces exhibiting partial transmission through interfaces or to thin films where reflection at boundaries dominates the optical behavior, such as in coatings or layered materials. In these scenarios, the value represents the net flux transmission after accounting for interfacial losses, aiding in the design of optical components like mirrors or filters. Key factors influencing hemispherical transmittance include surface roughness, which scatters incident light and alters the reflectance distribution, thereby reducing specular reflection and potentially increasing diffuse transmission in low-absorption materials.[16] Additionally, mismatches in refractive index between the incident medium and the surface lead to reflection governed by the Fresnel equations, where the power transmittance decreases with larger index differences, particularly at oblique angles.[10] For example, a clear soda-lime glass pane under solar illumination exhibits a hemispherical transmittance of approximately 92%, reflecting the high passage of broadband solar flux through the material while minimal absorption and reflection occur at the air-glass interfaces.[17]Directional Transmittance
Directional transmittance quantifies the transmission of radiant energy through an interface or medium for light incident and transmitted along specific directions, emphasizing the angular dependence that arises due to refraction and reflection at boundaries. It is particularly relevant in scenarios involving ray tracing or directional light propagation, where the overall energy transfer cannot be captured by angle-integrated measures. Unlike broader transmittance definitions, directional transmittance focuses on the ratio of radiance or irradiance in precisely defined incident and transmitted directions, governed by the geometry of the interface.[12] Mathematically, the directional transmittance is expressed asT(\theta_i, \phi_i; \theta_t, \phi_t) = \frac{L^t(\theta_t, \phi_t)}{L^i(\theta_i, \phi_i)},
where L^i(\theta_i, \phi_i) is the incident radiance from polar angle \theta_i and azimuthal angle \phi_i, and L^t(\theta_t, \phi_t) is the transmitted radiance in the corresponding transmitted directions \theta_t and \phi_t. The transmitted angles are related to the incident angles via 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 incident and transmitting media, respectively; the azimuthal angle typically remains unchanged (\phi_t = \phi_i) for planar interfaces. This formulation assumes conservation of energy and applies to non-scattering, non-absorbing conditions at the boundary.[18][19] The angular dependence of directional transmittance is primarily determined by the Fresnel coefficients, which describe the amplitude reflection and transmission at the interface for s- (perpendicular) and p- (parallel) polarized light. For oblique incidence, the power transmittance T incorporates these coefficients along with the cosine factors from the Poynting vector: T = \frac{n_2 \cos \theta_t}{n_1 \cos \theta_i} |t|^2, where t is the amplitude transmission coefficient. At normal incidence (\theta_i = 0), the polarization distinction vanishes, and the transmittance simplifies to T = \frac{4 n_1 n_2}{(n_1 + n_2)^2} for non-absorbing media, representing the maximum transmission efficiency without angular complications.[19][20] For oblique angles, transmittance varies significantly with polarization and incidence angle, peaking at the Brewster angle where p-polarized light experiences zero reflection (R_p = 0), yielding T_p = 1 in ideal non-absorbing dielectrics; the Brewster angle is given by \theta_B = \tan^{-1}(n_2 / n_1). In contrast, s-polarized light shows decreasing transmittance with increasing angle, approaching total internal reflection beyond the critical angle if n_2 < n_1. This angular sensitivity is crucial for understanding phenomena like glare reduction in polarized sunglasses.[19] A representative example is the transmission of polarized light through a dielectric interface, such as unpolarized light from air (n_1 = 1) incident on glass (n_2 = 1.5) at the Brewster angle of approximately 56.3°. Here, the p-component transmits fully without reflection, while the s-component partially reflects, resulting in transmitted light that is partially p-polarized and directionally refracted according to Snell's law. For normal incidence on the same interface, T \approx 0.96, illustrating the reduced losses compared to oblique cases.[19]
Internal Transmittance
Optical Depth
Optical depth, denoted as τ, serves as a logarithmic measure of the attenuation experienced by light propagating through a medium, specifically defined as the negative natural logarithm of the internal transmittance T_i, where T_i represents the fraction of incident radiant power transmitted through the bulk material excluding losses due to surface reflection or refraction: \tau = -\ln(T_i). This quantity quantifies the cumulative effect of interactions that remove photons from the beam along the propagation path.[21][22] In its general formulation for inhomogeneous media, the optical depth is expressed as the line integral of the local attenuation coefficient along the path s: \tau = \int \alpha(s) \, ds, where \alpha(s) is the attenuation coefficient at position s, which encompasses both absorption (photon energy conversion to heat or chemical energy) and scattering (photon redirection out of the beam). The distinction between these contributions arises in the Napierian scale (natural logarithm base, typical for optical depth) versus the decadic scale (common logarithm base, used in spectrophotometry), with the former emphasizing exponential decay directly tied to transmittance.[22][23] Optical depth is dimensionless, often expressed in nepers (Np), reflecting its origin in the natural logarithm; equivalently, it can be converted to decibels for certain applications as \tau_{db} = 10 \log_{10}(1/T_i), yielding approximately 4.343 times the value in nepers. Absorption dominates in scenarios with high material opacity, while scattering prevails in turbid media like aerosols, altering the effective \alpha based on particle size and wavelength relative to the scattering regime. A practical example occurs in Earth's atmosphere, where the optical depth for sunlight varies from about 0.1 in clear conditions (dominated by molecular Rayleigh scattering) to 10 or more in heavily polluted or dusty environments, depending on wavelength and aerosol loading; for visible light around 500 nm, typical values range from 0.1 to 1 under varying weather.[24][25]Beer-Lambert Law
The Beer-Lambert law serves as the foundational model for calculating internal transmittance in dilute, homogeneous media, assuming no scattering or other losses beyond absorption. It expresses the internal transmittance T_i as T_i = e^{-\alpha \ell}, where \alpha is the linear absorption coefficient (in units of inverse length) and \ell is the path length through the medium. An equivalent form, commonly used in chemical analysis, is T_i = 10^{-\epsilon c \ell}, where \epsilon is the molar absorptivity (in liters per mole per centimeter), c is the molar concentration of the absorbing species, and the base-10 logarithm relates to the definition of absorbance A = -\log_{10} T_i = \epsilon c \ell. This law quantifies how absorbed radiant power leads to exponential decay of transmitted intensity along the optical path.[26][27] The law's development traces back to Johann Heinrich Lambert's 1760 work Photometria, which established that light intensity diminishes exponentially with distance in an absorbing medium, independent of the incident angle for diffuse sources. August Beer extended this in 1852 by incorporating the effect of solute concentration in solutions, demonstrating that absorption scales linearly with both path length and concentration for dilute systems. Beer's contribution, published in Annalen der Physik und Chemie, merged Lambert's geometric optics insight with chemical principles, forming the modern Beer-Lambert law. This optical depth \tau = \alpha \ell directly corresponds to the exponent in the transmittance formula for uniform media.[26][28] The derivation starts from the infinitesimal attenuation of radiant flux \Phi_e due to absorption along a differential path ds: \frac{d\Phi_e}{ds} = -\alpha \Phi_e. Separating variables and integrating from s = 0 (incident flux \Phi_e(0)) to s = \ell (transmitted flux \Phi_e(\ell)) gives \ln \left( \frac{\Phi_e(\ell)}{\Phi_e(0)} \right) = -\alpha \ell, yielding the exponential form T_i = \frac{\Phi_e(\ell)}{\Phi_e(0)} = e^{-\alpha \ell}. For the molar version, \alpha = \epsilon c \ln 10, leading to the decadic expression after substitution. This assumes monochromatic light and a homogeneous medium where absorption is the sole interaction.[26][27] While powerful for quantitative analysis, the Beer-Lambert law has limitations and applies strictly to dilute, non-scattering media. It breaks down at high concentrations (typically above 0.01 M) due to non-linear effects like intermolecular interactions that alter the effective absorptivity \epsilon, causing deviations from the predicted straight-line calibration. Scattering, such as in turbid solutions, redirects light paths and violates the assumption of unidirectional propagation, while fluorescence re-emits absorbed energy, reducing net attenuation. Validity requires thin samples where \alpha \ell \ll 1 (absorbance A < 1) to minimize higher-order effects like stray light or polychromatic deviations in spectrometers.[29][27] A practical example is spectrophotometric analysis of dye solutions, where the concentration of a colored analyte like Congo red is determined by measuring absorbance at its peak wavelength (around 500 nm) in a 1 cm cuvette; with known \epsilon \approx 4 \times 10^4 L mol⁻¹ cm⁻¹, the law allows solving for c = A / (\epsilon \ell) from observed A, enabling quality control in textile or pharmaceutical industries.[26]Spectral and Luminous Transmittance
Spectral Transmittance
Spectral transmittance describes the fraction of incident radiant flux that is transmitted through a material or interface as a function of wavelength \lambda, denoted as T(\lambda). It is formally defined as the ratio of the differential spectral transmitted radiant flux d\Phi_e^t(\lambda) to the differential spectral incident radiant flux d\Phi_e^i(\lambda), expressed as T(\lambda) = \frac{d\Phi_e^t(\lambda)}{d\Phi_e^i(\lambda)}. [30] This extends the general transmittance concept by differentiating the radiant fluxes with respect to wavelength, allowing analysis of wavelength-dependent transmission behavior. An equivalent formulation exists in terms of frequency \nu, where T(\nu) = d\Phi_e^t(\nu) / d\Phi_e^i(\nu), though the wavelength form is more common in optical spectroscopy.[31] Spectral transmittance applies to both surface and internal transmission, capturing variations due to reflection, scattering, or absorption that differ across the electromagnetic spectrum. For internal spectral transmittance within a bulk material, the formulation arises from the wavelength-dependent absorption along the path, given by T(\lambda) = e^{-\int \alpha(\lambda) \, ds}, [32] where \alpha(\lambda) is the spectral absorption coefficient and the integral is over the path length s. This exponential decay reflects how materials selectively attenuate light at wavelengths corresponding to molecular or atomic transitions. Key features of spectral transmittance include absorption bands, regions of low T(\lambda) where energy is absorbed by electronic excitations (common in ultraviolet) or vibrational modes (prominent in infrared), and transparency windows, intervals of high T(\lambda) approaching unity where minimal interaction occurs. For instance, soda-lime glass exhibits near-complete transmittance (>90%) in the visible range (400–700 nm) due to a transparency window matching human vision, but drops sharply to opacity in the ultraviolet (below ~300 nm) from electronic absorption and in the near-infrared (beyond ~2.5 \mum) from vibrational overtones.[33] Spectral transmittance is measured using a spectrophotometer incorporating a monochromator to isolate narrow wavelength bands from a broadband source, such as a deuterium or tungsten-halogen lamp, and detect the transmitted intensity relative to a reference beam.[34] The monochromator disperses light via a diffraction grating or prism, enabling sequential scanning across the spectrum (typically 200–2500 nm for UV-Vis-NIR instruments), with detectors like photodiodes or photomultipliers quantifying the flux ratio at each \lambda. This yields a transmittance spectrum directly convertible to an absorption spectrum via A(\lambda) = -\log_{10} T(\lambda), facilitating identification of material composition through characteristic peaks.[7] A representative example is water, which displays high spectral transmittance (>99% for thin layers) throughout the visible range (400–700 nm) owing to low absorption in this window, enabling its clarity for optical applications. However, transmittance falls dramatically in the near-infrared due to absorption bands from O-H vibrations, for example, dropping below 50% around 1200 nm for a 1 cm path length from combination modes, and approaching zero beyond 1400 nm from the first overtone of O-H stretching.[35] This wavelength selectivity underscores water's role in biological and environmental optics, where visible transmission supports photosynthesis while IR opacity limits thermal imaging through aqueous media.Luminous Transmittance
Luminous transmittance, denoted as \tau_v, quantifies the fraction of incident luminous flux that passes through a material, weighted according to the human eye's sensitivity in the visible spectrum under photopic conditions. It is defined by the formula \tau_v = \frac{\int V(\lambda) T(\lambda) S(\lambda) \, d\lambda}{\int V(\lambda) S(\lambda) \, d\lambda}, where V(\lambda) is the CIE photopic luminosity function, T(\lambda) is the spectral transmittance of the material, and S(\lambda) is the spectral power distribution of the light source; the constant K_m = 683 lm/W represents the maximum luminous efficacy at 555 nm but cancels out in the ratio, yielding a dimensionless value tied to luminous flux in lumens.[36][37][11] Unlike radiometric transmittance, which treats all wavelengths equally, luminous transmittance accounts for the eye's peak sensitivity at 555 nm via V(\lambda), emphasizing green light while reducing the contribution from red and blue regions; in low-light scotopic conditions, a separate function V'(\lambda) with peak sensitivity at 507 nm applies, though photopic weighting dominates for typical daylight applications.[38][39] This perceptual weighting ensures \tau_v reflects visible brightness rather than total energy, making it essential in photometry for assessing transparency in human vision contexts. In practical applications, luminous transmittance is critical for protective eyewear and vehicle glazing, where standards mandate minimum values to balance visibility and safety. For sunglasses, ANSI Z80.3 specifies categories based on \tau_v ranges, such as 8–18% for general outdoor use to reduce glare while maintaining color perception.[40] Automotive windshields must achieve \tau_v > 70\% per Federal Motor Vehicle Safety Standard No. 205 to ensure driver visibility under normal conditions.[41] A representative example is neutral density filters, which exhibit nearly flat T(\lambda) across the visible spectrum, allowing \tau_v to approximate the uniform attenuation factor (e.g., an optical density of 0.3 corresponds to \tau_v \approx 50\%) without spectral bias, ideal for photography and machine vision where consistent light reduction is needed.[42]Related Quantities and Applications
Other Radiometric Coefficients
In radiometry, transmittance is complemented by reflectance and absorptance, which together describe the interaction of radiant flux with a material. Reflectance R is defined as the ratio of the reflected radiant flux \Phi_e^r to the incident radiant flux \Phi_e^i, expressed as R = \frac{\Phi_e^r}{\Phi_e^i}. Absorptance a is similarly the ratio of the absorbed radiant flux \Phi_e^a to the incident radiant flux, given by a = \frac{\Phi_e^a}{\Phi_e^i}. For non-scattering media, conservation of energy requires that the sum of transmittance T, reflectance R, and absorptance a equals unity: T + R + a = 1. These coefficients exhibit various forms depending on geometry and wavelength. Hemispherical reflectance accounts for reflection integrated over the hemisphere, while directional reflectance specifies the incident and reflected directions. Spectral variants, such as R_\lambda or T_\lambda, describe wavelength-dependent behavior. The table below summarizes key radiometric coefficients related to transmittance.| Coefficient | Symbol | Definition | Common Variants |
|---|---|---|---|
| Transmittance | T | \frac{\Phi_e^t}{\Phi_e^i} | Hemispherical T, directional T, spectral T_\lambda |
| Reflectance | R | \frac{\Phi_e^r}{\Phi_e^i} | Hemispherical R, directional-hemispherical R, spectral R_\lambda |
| Absorptance | a | \frac{\Phi_e^a}{\Phi_e^i} | Spectral a_\lambda |
| Emissivity | \varepsilon | Ratio of emitted radiance to blackbody radiance | Spectral \varepsilon_\lambda = a_\lambda by Kirchhoff's law |