Hubbry Logo
Optical pathOptical pathMain
Open search
Optical path
Community hub
Optical path
logo
8 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Contribute something
Optical path
Optical path
Optical path
View on Wikipedia
from Wikipedia
The figure shows the optical path of a Mirau-objective respectively a Mirau-interferometer. Reference beam (4 to 6) and object beam (7) have identical optical path length and can thus cause white light interference.

Optical path (OP) is the trajectory that a light ray follows as it propagates through an optical medium. The geometrical optical-path length or simply geometrical path length (GPD) is the length of a segment in a given OP, i.e., the Euclidean distance integrated along a ray between any two points.[1] The mechanical length of an optical device can be reduced to less than the GPD by using folded optics. The optical path length in a homogeneous medium is the GPD multiplied by the refractive index of the medium.

Factors affecting optical path

[edit]

Path of light in medium, or between two media is affected by the following:

Simple materials used

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Optical path

View on Grokipedia
from Grokipedia
The optical path, also known as the optical path length (OPL), is a fundamental quantity in optics that describes the effective propagation distance of light through a medium, defined as the line integral of the refractive index nn along the geometric path dsds traversed by a light ray: OPL=∫n ds\mathrm{OPL} = \int n \, ds. This measure is physically equivalent to the distance light would travel in vacuum during the same time, given by c×tc \times t, where cc is the speed of light in vacuum and tt is the travel time. In a homogeneous medium with constant refractive index, the OPL simplifies to the product of nn and the physical path length LL, i.e., OPL=nL\mathrm{OPL} = nL. The concept of optical path underpins Fermat's principle, which states that light rays propagate along paths where the OPL is stationary—typically a minimum or maximum with respect to small variations in the trajectory—ensuring the principle of least time for light travel between two points. This principle, originally formulated by in 1662, derives key laws of geometric optics, such as reflection and refraction (), by minimizing the OPL for ray paths at interfaces between media. In inhomogeneous media, where nn varies spatially (e.g., in graded-index materials or atmospheric ), the OPL accounts for bending or distortion of rays, making it essential for analyzing phenomena like mirages or aero-optical effects in high-speed flows. Beyond geometric optics, the optical path plays a critical role in wave optics and , where differences in OPL determine phase shifts and interference patterns; for instance, in a , inserting a sample in one arm alters the OPL, shifting fringes whose count quantifies the change with high precision. Applications extend to optical imaging systems, where equalizing OPLs across rays ensures aberration-free focus, as in lens design and ray tracing algorithms. In for telescopes, real-time adjustments compensate for atmospheric OPL variations to sharpen stellar images. Additionally, OPL measurements enable techniques like low-coherence for in random media, revealing path-length distributions in environments such as biological tissues. In interferometric arrays like the CHARA telescope, OPL equalizers maintain phase coherence across baselines for high-resolution astronomical imaging.

Fundamentals

Definition

The optical path represents the effective distance travels through a medium, equivalent to the distance it would traverse in to accumulate the same phase. This measure accounts for the medium's influence on 's propagation speed, providing a way to quantify the cumulative effect on the wavefront's advance as if were unimpeded in free space. The originated in 17th-century , rooted in Pierre de Fermat's 1657 principle of least time, which posits that follows the path minimizing travel time and, by extension, the . provided early recognition of this idea within wave theory in his 1678 Traité de la Lumière, where he modeled propagation as secondary wavelets advancing at speeds inversely proportional to the , implicitly incorporating optical path considerations for construction. In contrast to the geometric path, which denotes the purely physical along the ray's trajectory regardless of the medium, the optical path integrates the to reflect the actual slowing of , yielding a longer effective distance in denser materials. For instance, in where the n=1n = 1, the optical path coincides exactly with the geometric path, while in air with n≈1n \approx 1, the two are virtually indistinguishable for typical distances.

Optical Path Length

The optical path length (OPL), often denoted as Λ\Lambda, quantifies the effective distance traveled by light through a medium by accounting for the medium's refractive index, providing a vacuum-equivalent measure that determines the accumulated phase of the light wave. It is defined mathematically as the line integral along the ray path: Λ=∫n ds,\Lambda = \int n \, ds, where nn is the refractive index at each point along the infinitesimal path element dsds. This formulation extends the geometric path length to incorporate the slowing of light in denser media, making OPL a fundamental quantity in wave and ray optics. Physically, the OPL corresponds to the time τ\tau light takes to traverse the path, since the speed of light in the medium is c/nc/n (with cc the vacuum speed), yielding τ=∫(n ds)/c=Λ/c\tau = \int (n \, ds)/c = \Lambda / c. This equivalence arises because the phase advance ϕ=(2π/λ)Λ\phi = (2\pi / \lambda) \Lambda (where λ\lambda is the vacuum wavelength) directly governs interference and diffraction phenomena, emphasizing OPL's role in phase accumulation without altering the underlying propagation time. In applications like , OPL ensures that phase shifts are predicted accurately even in varying environments. The units of OPL are meters, identical to the geometric path length, facilitating direct comparison with vacuum propagation distances. A key property in ray optics is that OPL remains invariant under coordinate transformations along the ray trajectory, preserving its scalar nature and utility in geometric optics formulations. This invariance underpins its connection to , where stationary OPL paths correspond to actual light rays.

Mathematical Formulation

Homogeneous Media

In homogeneous media, where the nn remains constant along the path of propagation, the (OPL) simplifies to a straightforward product of the refractive index and the geometric path length LL. This is expressed mathematically as OPL=n×L,\text{OPL} = n \times L, where LL is the physical distance traveled by the ray in the medium. This formulation assumes a uniform medium without spatial variations in nn, allowing to propagate in straight lines according to the principles of . The concept of optical path length originates from the time-of-flight perspective in . In such a medium, the speed of light is v=c/nv = c / n, where cc is the in . Thus, the time tt for to traverse the LL is t=Lv=Lc/n=nLc.t = \frac{L}{v} = \frac{L}{c / n} = \frac{n L}{c}. This travel time is equivalent to the time it would take for to cover a distance nLn L in , establishing the OPL as an effective vacuum-equivalent path that accounts for the medium's slowing effect on . This equivalence underpins for path minimization in uniform media. This simplified expression applies under key assumptions: the medium must be isotropic (with nn independent of light direction), non-absorbing (no energy loss that alters the effective path), and free of refractive index gradients that would curve the light ray. These conditions hold for many common materials like air (n≈1n \approx 1) or crown glass in basic optical setups. For instance, consider a ray passing through a slab of glass with thickness L=1L = 1 cm and refractive index n=1.5n = 1.5. The OPL is then 1.5×11.5 \times 1 cm = 1.5 cm, representing the effective path length as if the light had traveled 1.5 cm in vacuum. This calculation is fundamental in designing simple lenses or windows where uniformity is assured.

Inhomogeneous Media

In inhomogeneous media, where the refractive index nn varies spatially, the (OPL) is computed by integrating along the actual ray path, contrasting with the simple product used in uniform media. This variation in n(r)n(\mathbf{r}) arises from gradients due to factors like or composition, requiring a more complex evaluation to capture the true propagation delay. The general formula for the between points A and B is given by the OPL=∫ABn(r) ds,\text{OPL} = \int_A^B n(\mathbf{r}) \, ds, where dsds is the differential arc length along the ray path. This integral accounts for the local refractive index at each point, providing the effective distance light travels as if in vacuum. For practical computation in media with complex index gradients, such as graded-index (GRIN) fibers, numerical methods like ray tracing software approximate the integral by discretizing the path into segments and summing contributions. These tools solve differential equations governing ray propagation, enabling simulations of light guiding in multimode fibers where the index decreases parabolically from the core center. Within the eikonal approximation, valid for high-frequency light where wavefronts are nearly planar locally, light rays follow curved paths that minimize the OPL according to . The ∣∇S∣=n(r)|\nabla S| = n(\mathbf{r}), where SS is the optical path function, governs this bending, ensuring rays take stationary paths through varying media. This formulation is essential in , where temperature-induced index gradients cause mirages by bending rays toward denser air layers, creating illusory images of distant objects.

Optical Path Difference

The optical path difference (OPD), denoted as Δ, is defined as the difference between the optical path lengths (OPLs) of two light rays or beams traveling along different routes from a common source to a common observation point. This quantity accounts for both the physical distances traversed and the refractive indices of the media involved, providing a measure of the effective path disparity that influences wave superposition. For two paths in potentially different media, the OPD is given by the formula Δ=n1L1−n2L2,\Delta = n_1 L_1 - n_2 L_2, where n1n_1 and n2n_2 are the refractive indices, and L1L_1 and L2L_2 are the physical path lengths along each route, respectively. In vacuum or air (where n≈1n \approx 1), this simplifies to the geometric path difference. The OPD directly determines the phase difference δ\delta between the two waves, expressed as δ=2πλΔ\delta = \frac{2\pi}{\lambda} \Delta, where λ\lambda is the wavelength of the light in vacuum. This phase shift governs interference phenomena: constructive interference occurs when Δ=mλ\Delta = m \lambda for integer mm, resulting in maximum intensity as the waves reinforce each other; destructive interference arises when Δ=(m+12)λ\Delta = (m + \frac{1}{2}) \lambda, leading to minimum intensity due to wave cancellation. These conditions assume coherent, monochromatic sources and equal amplitudes for the beams. A prominent example of OPD's role is in the , where a is split into two perpendicular paths by a partially reflecting mirror and reflected back by movable mirrors before recombination. The OPD here is Δ=2(d1−d2)\Delta = 2(d_1 - d_2), accounting for the round-trip travel in each arm (assuming air, n=1n=1), with d1d_1 and d2d_2 as the distances to the mirrors. Displacing one mirror by a distance xx changes the OPD by 2x2x, producing interference fringes that shift across the observation plane; a displacement of λ/2\lambda/2 corresponds to a full fringe shift, enabling precise measurements of length or variations.

Fermat's Principle

Fermat's principle asserts that a ray of light traveling between two fixed points follows the path for which the optical path length is stationary, corresponding to an extremum—typically a minimum—with respect to small variations in the path. This variational principle underpins ray optics by selecting the trajectory that light actually takes among all possible paths. The principle was first proposed by the French mathematician in a 1662 letter to Cureau de la Chambre, as a means to explain the of . Although formulated prior to the advent of wave optics in the late 17th and 19th centuries, it remains fully consistent with wave-based descriptions of light propagation, such as Huygens' principle. Mathematically, is expressed through the , requiring that the —the of the nn along the path—be stationary: δ∫n ds=0,\delta \int n \, ds = 0, where dsds is the differential along the path and the is taken between the two points. This condition, applied to paths involving interfaces between media, yields the law of reflection (equal angles of incidence and reflection) and of (n1sin⁡θ1=n2sin⁡θ2n_1 \sin \theta_1 = n_2 \sin \theta_2) as consequences of minimization. The referenced here is elaborated in the section on inhomogeneous media.

Factors Affecting Optical Path

Refractive Index

The nn of a medium is defined as the ratio of the in cc to its speed in the medium vv, expressed as n=cvn = \frac{c}{v}. This quantifies how much slower propagates through the material compared to . For most materials, n>1n > 1, reflecting the reduced velocity due to the medium's properties. Physically, the originates from interactions between the 's and the electrons in the medium's atoms or molecules. The oscillating field induces polarization, creating oscillating dipoles that radiate secondary waves; the interference of these with the incident wave effectively delays propagation. Higher material density increases the by providing more electrons per unit volume for these interactions. Variations in environmental conditions also affect the . The temperature coefficient dndT\frac{dn}{dT} measures its sensitivity to temperature changes, which can induce thermal lensing in optical systems where nonuniform heating creates refractive index gradients acting like a lens. Pressure influences arise similarly through changes; for instance, standard atmospheric air has a refractive index of approximately 1.0003 at 15°C and 101.325 kPa. In media that absorb light, the refractive index becomes complex, n=nr+ikn = n_r + i k, where the real part nrn_r governs the phase shift and thus the optical path in non-absorptive scenarios, while the imaginary part kk describes . This refractive index fundamentally scales the optical path length in homogeneous media.

Wavelength Dependence

The (OPL) of propagating through a medium is inherently dependent on the due to material dispersion, where the nn varies as a function of λ\lambda, denoted as n(λ)n(\lambda). This dependence arises from the interaction of with the material's electronic structure, leading to different phase velocities for various colors of and complicating the design of optical systems that must handle or polychromatic . In normal dispersion, which characterizes most transparent media like in the , the increases with decreasing , meaning shorter wavelengths (e.g., blue light) experience a higher nn than longer wavelengths (e.g., red light). As a result, for a fixed physical path length through such a dispersive medium, the OPL is effectively longer for blue light than for red light, since OPL is the product of the and the geometric path. This differential OPL contributes to phenomena like chromatic dispersion in optical fibers and lenses, where signals spread temporally or spatially. The provides an empirical model for describing n(λ)n(\lambda) in dispersive materials, derived from a classical oscillator representation of the bound electrons responding to the of light. Developed by Wolfgang Sellmeier in 1871, it fits experimental data by summing contributions from multiple terms associated with the material's and absorption oscillators, offering accurate predictions of dispersion across a wide spectral range without invoking . However, near regions of strong absorption—such as electronic transition bands in the or vibrational bands in the —materials exhibit anomalous dispersion, where dn/dλ>0dn/d\lambda > 0, causing the to decrease with decreasing . This counterintuitive behavior, first noted in the , stems from the rapid variation in the real part of the complex adjacent to absorption lines, and it can lead to unusual OPL effects in or resonant optical applications.

Applications

Interferometry

In interferometry, the optical path difference (OPD) plays a central role in generating interference fringes by introducing controlled phase shifts between waves traveling along distinct paths. These fringes arise when the OPD between interfering beams results in constructive or destructive interference, enabling high-precision measurements of displacements, refractive indices, and other . In the Fabry-Pérot interferometer, consisting of two parallel partially reflecting mirrors forming an , multiple reflections create successive beams with incremental OPDs of 2ndcos⁡θ2nd\cos\theta, where nn is the , dd is the mirror separation, and θ\theta is the incidence angle. Constructive interference occurs when this OPD equals an multiple of the , producing transmission peaks or reflection minima that form sharp fringes, allowing resolution of lines down to picometer scales. Similarly, the Mach-Zehnder interferometer splits a beam into two arms with adjustable path lengths, recombining them to produce fringes sensitive to OPD variations as small as a fraction of a ; the resulting phase shift manifests as fringe shifts or modulations, making it ideal for dynamic sensing applications like vibration analysis. A key application of OPD in is wavelength measurement, where the fringe spacing or order provides direct . For the mm-th order fringe, the condition for maximum intensity is given by Δ=mλ\Delta = m\lambda, allowing λ\lambda to be determined from the measured OPD Δ\Delta (via arm displacement or cavity tuning) and observed fringe count mm, with accuracies exceeding 1 part in 10810^8 in stabilized setups. Historically, the Michelson-Morley experiment of 1887 employed an interferometer to detect the luminiferous ether by measuring expected fringe shifts from Earth's motion through it, which would alter the OPD in arms due to velocity-dependent light propagation. No such shift was observed, nullifying the ether hypothesis and supporting relativistic principles, with the apparatus achieving OPD sensitivities of about 0.01 fringes. In modern holography, OPD governs the recording and reconstruction of three-dimensional images by capturing the wavefront phase variations from an object, encoded in the interference pattern between object and reference beams. During reconstruction, the hologram diffracts light to replicate these OPDs, restoring the original wavefront and enabling parallax-based 3D visualization without computational aids in classical setups.

Optical Design

In optical design, the optimization of optical path length (OPL) is fundamental to achieving aberration-free imaging in lens systems, telescopes, and microscopes. Ray tracing techniques simulate the propagation of light rays through optical elements, calculating the OPL as the integral of the refractive index along each ray's path to evaluate and minimize aberrations. For instance, spherical aberration is corrected by ensuring that the OPL for paraxial and marginal rays from an object point to the image plane is equal, preventing wavefront distortion. Similarly, chromatic aberration is addressed by balancing OPL variations across wavelengths using materials with appropriate dispersion properties. These simulations enable designers to iteratively refine surface curvatures, thicknesses, and separations for high-performance systems. A key principle in optical design is the conservation of , which quantifies the throughput of an optical system as n2n^2 times the product of the beam's cross-sectional area and the it subtends (for homogeneous media). Along the optical path in lossless, reversible systems, etendue remains invariant, imposing fundamental limits on light collection and concentration. This conservation guides the design of and illumination , ensuring that beam parameters do not degrade due to mismatches in area or angle, as seen in objectives where maximizing etendue enhances light-gathering efficiency without introducing path-induced losses. Telecentric designs exemplify OPL optimization by positioning the aperture stop at the focal plane, making chief rays parallel to the and thus ensuring equal OPL from the object to the across the field of view. This equality minimizes and perspective errors, ideal for precision metrology in microscopes. In fiber optics, OPL mismatches among propagating modes in multimode fibers cause , where rays following longer paths arrive later, broadening pulses and limiting data rates to below 1 Gbit/s over kilometer distances.

Materials and Examples

Common Materials

In optical path calculations, the nn of a is a fundamental parameter that determines the effective path length through the medium. Common materials in are selected for their transparency and well-characterized , with refractive indices typically measured at the sodium D-line of 589 nm unless otherwise specified. These values serve as baselines for designing optical systems where the (OPL) is given by n×dn \times d, with dd as the physical thickness. Air and are standard reference media, with n≈1.000n \approx 1.000 for and nair≈1.000277n_{\text{air}} \approx 1.000277 at (STP), making the difference negligible for most laboratory and engineering applications (approximately 2.7×10−42.7 \times 10^{-4}). This small deviation arises primarily from air's density and composition but does not significantly affect OPL in typical setups. are ubiquitous in optical elements like lenses and prisms. glass, often made from soda-lime compositions, has a refractive index of about 1.52 at 589 nm and a high (typically 50–60), indicating low dispersion suitable for achromatic designs. , with higher lead content, exhibits n≈1.62n \approx 1.62 at the same wavelength and a lower (around 30–40), which introduces greater chromatic dispersion but is valuable for correcting aberrations in compound lenses. Other everyday materials include , with n=1.33n = 1.33 at 589 nm, commonly encountered in aqueous optics or biological . Diamond, prized for its hardness and clarity, has a high n=2.42n = 2.42 at the same wavelength, leading to significant in gem applications. Polymers such as acrylic (polymethyl methacrylate) offer n≈1.49n \approx 1.49 and are favored for lightweight, cost-effective components in displays and protective covers.
MaterialRefractive Index (nn at 589 nm)Abbe Number (if applicable)Typical Use
Vacuum1.000N/AReference medium
Air (STP)1.000277N/A
1.33N/ALiquid lenses,
Acrylic (PMMA)1.49~57Lenses, windows
Crown Glass1.5250–60Achromatic doublets
1.6230–40Aberration correction
2.42~55High-index prisms, tools
These indices highlight how material choice influences OPL, with higher nn values extending the effective path and enabling compact designs, as detailed in the refractive index section.

Calculation Examples

Consider a straightforward calculation of the optical path length (OPL) through a homogeneous medium, such as a crown glass slab with refractive index n=1.5n = 1.5 and physical thickness d=10d = 10 cm. The OPL, denoted as Λ\Lambda, is computed as Λ=nd=1.5×10\Lambda = n d = 1.5 \times 10 cm = 15 cm, effectively equivalent to the distance light would travel in to match the phase delay experienced in the glass. In the context of Young's double-slit experiment conducted in air, where the refractive index is approximately 1, an optical path difference (OPD) of Δ=2\Delta = 2 μm between the light paths from the two slits results in a phase shift that shifts the interference pattern. This OPD corresponds to constructive or destructive interference depending on the wavelength λ\lambda, with adjacent fringes separated by a path difference change of λ\lambda, or λ/2\lambda/2 for transitions between bright and dark fringes. For media with a graded refractive index, such as a linear profile where nn varies from n1n_1 to n2n_2 over path length LL, the OPL Λ=∫0Ln(s) ds\Lambda = \int_0^L n(s) \, ds requires numerical approximation. The simple trapezoidal rule divides the path into NN segments of width Δs=L/N\Delta s = L/N, yielding Λ≈∑i=1Nni+ni+12Δs\Lambda \approx \sum_{i=1}^N \frac{n_i + n_{i+1}}{2} \Delta s, which provides a reliable estimate for smooth gradients by treating each segment as a trapezoid. In homogeneous cases, error analysis reveals that uncertainties propagate directly: a 1% relative uncertainty in the nn induces a 1% relative error in the OPL, assuming the physical path length is precisely known, as derived from standard uncertainty propagation for the product Λ=nd\Lambda = n d.

References

  1. https://ocw.mit.edu/courses/2-71-optics-spring-2009/de5506d16a5bfb92d2ce78486b9d7472_X6cea7dAhBc.pdf
  2. http://courses.washington.edu/me557/sensor/day1.pdf
  3. https://osa.magnet.fsu.edu/terms/general.html
  4. http://scipp.ucsc.edu/~haber/ph5B/fermat09.pdf
  5. https://pages.uoregon.edu/raghu/TeachingFiles/Winter08Phys352/Notes_optics_3Refraction.pdf
  6. https://www3.nd.edu/~sgordeye/AnnRevFlMech.pdf
  7. https://www.physics.purdue.edu/~jones105/phys42200_Spring2016/notes/Phys42200_Lecture41.pdf
  8. https://www.cis.rit.edu/class/simg303/Notes/Ch8-Optical-Imaging.pdf
  9. https://wp.optics.arizona.edu/jcwyant/wp-content/uploads/sites/13/2016/08/Adaptive-Optics.pdf
  10. http://light.ece.illinois.edu/wp-content/uploads/2011/07/1999_OL_24_442_1999_OPS.pdf
  11. http://www.chara.gsu.edu/files/preliminary_engineering_design_reports/appendixg.pdf
  12. https://www.rp-photonics.com/optical_path_length.html
  13. https://farside.ph.utexas.edu/teaching/316/lectures/node125.html
  14. https://ocw.mit.edu/courses/2-71-optics-spring-2009/c5c982b1fe835b982df0090dbd528d39_MIT2_71S09_lec15.pdf
  15. https://kestrel.nmt.edu/~rsonnenf/phys3034/Supp_Text/Ch02_GeoOptics_Pedrotti.pdf
  16. http://mx.nthu.edu.tw/~rklee/files/EE3130-ray
  17. https://application.wiley-vch.de/books/sample/3527413316_c01.pdf
  18. http://depts.washington.edu/me557/reading/geometric_optics_1.pdf
  19. https://opg.optica.org/abstract.cfm?uri=ao-21-6-984
  20. https://www.mdpi.com/2076-3417/11/3/912
  21. http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1207.1.K.pdf
  22. https://aty.sdsu.edu/mirages/mirintro.html
  23. https://www.cis.rit.edu/class/simg303/Notes/Ch10-Waves-in-Imaging.pdf
  24. https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/03%3A_Interference/3.06%3A_The_Michelson_Interferometer
  25. https://www.physics.umd.edu/courses/Phys375/Anlage_Fall09/Lab4MichelsonRevF09.pdf
  26. http://web.mit.edu/8.13/www/JLExperiments/JLExp009.pdf
  27. https://www.feynmanlectures.caltech.edu/I_26.html
  28. http://www2.isb.ac.th/physics_isb/physisb/index/aaaasubs/optics1/fermat1.htm
  29. https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_%28Raymond%29/03%253A_Geometrical_Optics/3.05%253A_Fermats_Principle
  30. https://www.fields.utoronto.ca/programs/scientific/12-13/Marsden/FieldsSS2-FinalSlidesJuly2012.pdf
  31. http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/Fermat.html
  32. https://wp.optics.arizona.edu/optomech/wp-content/uploads/sites/53/2016/10/tie-29_refractive_index_v2_us.pdf
  33. https://optics.byu.edu/docs/opticsbook.pdf
  34. https://www.cis.rit.edu/class/simg455/455-Notes-20073c.pdf
  35. https://www.feynmanlectures.caltech.edu/I_31.html
  36. https://seabass.gsfc.nasa.gov/wiki/NASA_particle_absorption_workshop/Twardowski_0044_Zaneveld.PDF
  37. https://nvlpubs.nist.gov/nistpubs/jres/64A/jresv64An3p191_A1b.pdf
  38. https://courses.washington.edu/phys331/fabry-perot/fabry-perot_rev3-09.pdf
  39. https://pmc.ncbi.nlm.nih.gov/articles/PMC3376560/
  40. https://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/zetie_et_al_mach_zehnder00.pdf
  41. https://www.csub.edu/~adzyubenko/Phys2120/Interferometer_Lab.pdf
  42. http://physics.wm.edu/~inovikova/phys201/historicalpapers/michelson.pdf
  43. https://faculty.etsu.edu/gardnerr/5310/5310pdf/dg2-2.pdf
  44. https://www.spie.org/news/0041-designing-aplanatic-thick-lenses
  45. https://opg.optica.org/jot/abstract.cfm?uri=jot-13-4-704
  46. https://www.eng.yale.edu/caolab/papers/aop23.pdf
  47. http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/indrf.html
  48. https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%253A_Physics_7C_-_General_Physics/8%253A_Waves/8.7%253A_Double-Slit_Interference
  49. https://www.mdpi.com/2077-1312/13/3/473
Add your contribution
Related Hubs
Contribute something
User Avatar
No comments yet.