For unpolarized light: T = 1 2 ( T s + T p ) {\displaystyle T={1 \over 2}(T_{s}+T_{p})} R = 1 2 ( R s + R p ) {\displaystyle R={1 \over 2}(R_{s}+R_{p})} where R + T = 1 {\displaystyle R+T=1} .

Ellipticalpolarization

Nosepiece houses the objectives. The objectives are exposed and are mounted on a rotating turret so that different objectives can be conveniently selected. Standard objectives include 4x, 10x, 40x and 100x although different power objectives are available.

If we do likewise with the formula for rp, the result is easily shown to be equivalent to [12][13] r p = tan ⁡ ( θ i − θ t ) tan ⁡ ( θ i + θ t ) . {\displaystyle r_{\text{p}}={\frac {\tan(\theta _{\text{i}}-\theta _{\text{t}})}{\tan(\theta _{\text{i}}+\theta _{\text{t}})}}.}

Using this convention,[5][6] r s = n 1 cos ⁡ θ i − n 2 cos ⁡ θ t n 1 cos ⁡ θ i + n 2 cos ⁡ θ t , t s = 2 n 1 cos ⁡ θ i n 1 cos ⁡ θ i + n 2 cos ⁡ θ t , r p = n 2 cos ⁡ θ i − n 1 cos ⁡ θ t n 2 cos ⁡ θ i + n 1 cos ⁡ θ t , t p = 2 n 1 cos ⁡ θ i n 2 cos ⁡ θ i + n 1 cos ⁡ θ t . {\displaystyle {\begin{aligned}r_{\text{s}}&={\frac {n_{1}\cos \theta _{\text{i}}-n_{2}\cos \theta _{\text{t}}}{n_{1}\cos \theta _{\text{i}}+n_{2}\cos \theta _{\text{t}}}},\\[3pt]t_{\text{s}}&={\frac {2n_{1}\cos \theta _{\text{i}}}{n_{1}\cos \theta _{\text{i}}+n_{2}\cos \theta _{\text{t}}}},\\[3pt]r_{\text{p}}&={\frac {n_{2}\cos \theta _{\text{i}}-n_{1}\cos \theta _{\text{t}}}{n_{2}\cos \theta _{\text{i}}+n_{1}\cos \theta _{\text{t}}}},\\[3pt]t_{\text{p}}&={\frac {2n_{1}\cos \theta _{\text{i}}}{n_{2}\cos \theta _{\text{i}}+n_{1}\cos \theta _{\text{t}}}}.\end{aligned}}}

Stage is where the specimen to be viewed is placed. A mechanical stage is used when working at higher magnifications where delicate movements of the specimen slide are required.

Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Dividing (4) by (5)) yields Y = n c μ . {\displaystyle Y={\frac {n}{\,c\mu \,}}\,.} For non-magnetic media we can substitute the vacuum permeability μ0 for μ, so that Y 1 = n 1 c μ 0     ;       Y 2 = n 2 c μ 0 ; {\displaystyle Y_{1}={\frac {n_{1}}{\,c\mu _{0}}}~~;~~~Y_{2}={\frac {n_{2}}{\,c\mu _{0}}}\,;} that is, the admittances are simply proportional to the corresponding refractive indices. When we make these substitutions in equations (13) to (16) and equations (21) to (26), the factor cμ0 cancels out. For the amplitude coefficients we obtain:[5][6]

For fields of that form, Faraday's law and the Maxwell-Ampère law respectively reduce to [33] ω B = k × E ω D = − k × H . {\displaystyle {\begin{aligned}\omega \mathbf {B} &=\mathbf {k} \times \mathbf {E} \\\omega \mathbf {D} &=-\mathbf {k} \times \mathbf {H} \,.\end{aligned}}}

If ℓ is the component of r in the direction of k, the field (1) can be written Ek ei(kℓ−ωt).  If the argument of ei(⋯) is to be constant,  ℓ must increase at the velocity ω / k , {\displaystyle \omega /k\,,\,} known as the phase velocity (vp). This in turn is equal to c / n {\displaystyle c/n} . Solving for k gives

As a consequence of conservation of energy, one can find the transmitted power (or more correctly, irradiance: power per unit area) simply as the portion of the incident power that isn't reflected: [4] T s = 1 − R s {\displaystyle T_{\mathrm {s} }=1-R_{\mathrm {s} }} and T p = 1 − R p {\displaystyle T_{\mathrm {p} }=1-R_{\mathrm {p} }}

Details of Fresnel's derivation, including the modern forms of the sine law and tangent law, were given later, in a memoir read to the French Academy of Sciences in January 1823.[24] That derivation combined conservation of energy with continuity of the tangential vibration at the interface, but failed to allow for any condition on the normal component of vibration.[25] The first derivation from electromagnetic principles was given by Hendrik Lorentz in 1875.[26]

So, for the incident, reflected, and transmitted H fields, let the respective components in the −z-direction be Hi, Hr, Ht. Then, since H = YE,

The second form of each equation is derived from the first by eliminating θt using Snell's law and trigonometric identities.

Although these relationships describe the basic physics, in many practical applications one is concerned with "natural light" that can be described as unpolarized. That means that there is an equal amount of power in the s and p polarizations, so that the effective reflectivity of the material is just the average of the two reflectivities: R e f f = 1 2 ( R s + R p ) . {\displaystyle R_{\mathrm {eff} }={\frac {1}{2}}\left(R_{\mathrm {s} }+R_{\mathrm {p} }\right).}

For the case of normal incidence, θi = θt = 0, and there is no distinction between s and p polarization. Thus, the reflectance simplifies to R 0 = | n 1 − n 2 n 1 + n 2 | 2 . {\displaystyle R_{0}=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\,.}

In Cartesian coordinates (x, y, z), let the region y < 0 have refractive index n1, intrinsic admittance Y1, etc., and let the region y > 0 have refractive index n2, intrinsic admittance Y2, etc. Then the xz plane is the interface, and the y axis is normal to the interface (see diagram). Let i and j (in bold roman type) be the unit vectors in the x and y directions, respectively. Let the plane of incidence be the xy plane (the plane of the page), with the angle of incidence θi measured from j towards i. Let the angle of refraction, measured in the same sense, be θt, where the subscript t stands for transmitted (reserving r for reflected).

For the p polarization, the incident, reflected, and transmitted E fields are parallel to the red arrows and may therefore be described by their components in the directions of those arrows. Let those components be Ei, Er, Et  (redefining the symbols for the new context). Let the reflection and transmission coefficients be rp and tp. Then, if the incident E field is taken to have unit amplitude, we have

for the p polarization. Note that when comparing the powers of two such waves in the same medium and with the same cos θ, the impedance and geometric factors mentioned above are identical and cancel out. But in computing the power transmission (below), these factors must be taken into account.

polarization. Polarization refers to the orientation of oscillations in a transverse wave, such as light waves, radio waves, or other electromagnetic waves. In ...

For common glass (n2 ≈ 1.5) surrounded by air (n1 = 1), the power reflectance at normal incidence can be seen to be about 4%, or 8% accounting for both sides of a glass pane.

Note that all such intensities are measured in terms of a wave's irradiance in the direction normal to the interface; this is also what is measured in typical experiments. That number could be obtained from irradiances in the direction of an incident or reflected wave (given by the magnitude of a wave's Poynting vector) multiplied by cos θ for a wave at an angle θ to the normal direction (or equivalently, taking the dot product of the Poynting vector with the unit vector normal to the interface). This complication can be ignored in the case of the reflection coefficient, since cos θi = cos θr, so that the ratio of reflected to incident irradiance in the wave's direction is the same as in the direction normal to the interface.

The above equations relating powers (which could be measured with a photometer for instance) are derived from the Fresnel equations which solve the physical problem in terms of electromagnetic field complex amplitudes, i.e., considering phase shifts in addition to their amplitudes. Those underlying equations supply generally complex-valued ratios of those EM fields and may take several different forms, depending on the formalism used. The complex amplitude coefficients for reflection and transmission are usually represented by lower case r and t (whereas the power coefficients are capitalized). As before, we are assuming the magnetic permeability, µ of both media to be equal to the permeability of free space µ0 as is essentially true of all dielectrics at optical frequencies.

Crosspolarization

This can be used to either verify the consistency of the measurements of Rs and Rp, or to derive one of them when the other is known. This relationship is only valid for the simple case of a single plane interface between two homogeneous materials, not for films on substrates, where a more complex analysis is required.

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In the same memoir of January 1823,[24] Fresnel found that for angles of incidence greater than the critical angle, his formulas for the reflection coefficients (rs and rp) gave complex values with unit magnitudes. Noting that the magnitude, as usual, represented the ratio of peak amplitudes, he guessed that the argument represented the phase shift, and verified the hypothesis experimentally.[27] The verification involved

S-polarized light

In 1808, Étienne-Louis Malus discovered that when a ray of light was reflected off a non-metallic surface at the appropriate angle, it behaved like one of the two rays emerging from a doubly-refractive calcite crystal.[19] He later coined the term polarization to describe this behavior.  In 1815, the dependence of the polarizing angle on the refractive index was determined experimentally by David Brewster.[20] But the reason for that dependence was such a deep mystery that in late 1817, Thomas Young was moved to write:

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When we substitute from equations (8) to (10) and then from (7), the exponential factors cancel out, so that the interface conditions reduce to the simultaneous equations

One can see that ts = rs + 1[7] and ⁠n2/n1⁠tp = rp + 1. One can write very similar equations applying to the ratio of the waves' magnetic fields, but comparison of the electric fields is more conventional.

The s polarization refers to polarization of a wave's electric field normal to the plane of incidence (the z direction in the derivation below); then the magnetic field is in the plane of incidence. The p polarization refers to polarization of the electric field in the plane of incidence (the xy plane in the derivation below); then the magnetic field is normal to the plane of incidence. The names "s" and "p" for the polarization components refer to German "senkrecht" (perpendicular or normal) and "parallel" (parallel to the plane of incidence).

From (4) we obtain the phase velocity c / n = 1 / μ ϵ {\displaystyle c/n=1{\big /}\!{\sqrt {\mu \epsilon \,}}} . For vacuum this reduces to c = 1 / μ 0 ϵ 0 {\displaystyle c=1{\big /}\!{\sqrt {\mu _{0}\epsilon _{0}}}} . Dividing the second result by the first gives n = μ rel ϵ rel . {\displaystyle n={\sqrt {\mu _{\text{rel}}\epsilon _{\text{rel}}}}\,.} For a non-magnetic medium (the usual case), this becomes ⁠ n = ϵ rel {\displaystyle n={\sqrt {\epsilon _{\text{rel}}}}} ⁠. (Taking the reciprocal of (5), we find that the intrinsic impedance is Z = μ / ϵ {\textstyle Z={\sqrt {\mu /\epsilon }}} . In vacuum this takes the value Z 0 = μ 0 / ϵ 0 ≈ 377 Ω , {\textstyle Z_{0}={\sqrt {\mu _{0}/\epsilon _{0}}}\,\approx 377\,\Omega \,,} known as the impedance of free space. By division, Z / Z 0 = μ rel / ϵ rel {\textstyle Z/Z_{0}={\sqrt {\mu _{\text{rel}}/\epsilon _{\text{rel}}}}} . For a non-magnetic medium, this becomes Z = Z 0 / ϵ rel = Z 0 / n . {\displaystyle Z=Z_{0}{\big /}\!{\sqrt {\epsilon _{\text{rel}}}}=Z_{0}/n.} )

In the case of total internal reflection where the power transmission T is zero, t nevertheless describes the electric field (including its phase) just beyond the interface. This is an evanescent field which does not propagate as a wave (thus T = 0) but has nonzero values very close to the interface. The phase shift of the reflected wave on total internal reflection can similarly be obtained from the phase angles of rp and rs (whose magnitudes are unity in this case). These phase shifts are different for s and p waves, which is the well-known principle by which total internal reflection is used to effect polarization transformations.

The equations assume the interface between the media is flat and that the media are homogeneous and isotropic.[1] The incident light is assumed to be a plane wave, which is sufficient to solve any problem since any incident light field can be decomposed into plane waves and polarizations.

Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoir [30] in which he introduced the needed terms linear polarization, circular polarization, and elliptical polarization,[31] and in which he explained optical rotation as a species of birefringence: linearly-polarized light can be resolved into two circularly-polarized components rotating in opposite directions, and if these propagate at different speeds, the phase difference between them — hence the orientation of their linearly-polarized resultant — will vary continuously with distance.[32]

[T]he great difficulty of all, which is to assign a sufficient reason for the reflection or nonreflection of a polarised ray, will probably long remain, to mortify the vanity of an ambitious philosophy, completely unresolved by any theory.[21]

Illuminator is the light source for a microscope, typically located in the base of the microscope. Most light microscopes use low voltage, halogen bulbs with continuous variable lighting control located within the base.

Condenser is used to collect and focus the light from the illuminator on to the specimen. It is located under the stage often in conjunction with an iris diaphragm.

When light travelling in a denser medium strikes the surface of a less dense medium (i.e., n1 > n2), beyond a particular incidence angle known as the critical angle, all light is reflected and Rs = Rp = 1. This phenomenon, known as total internal reflection, occurs at incidence angles for which Snell's law predicts that the sine of the angle of refraction would exceed unity (whereas in fact sin θ ≤ 1 for all real θ). For glass with n = 1.5 surrounded by air, the critical angle is approximately 42°.

In the above formula for rs, if we put n 2 = n 1 sin ⁡ θ i / sin ⁡ θ t {\displaystyle n_{2}=n_{1}\sin \theta _{\text{i}}/\sin \theta _{\text{t}}} (Snell's law) and multiply the numerator and denominator by ⁠1/n1⁠ sin θt, we obtain [10][11] r s = − sin ⁡ ( θ i − θ t ) sin ⁡ ( θ i + θ t ) . {\displaystyle r_{\text{s}}=-{\frac {\sin(\theta _{\text{i}}-\theta _{\text{t}})}{\sin(\theta _{\text{i}}+\theta _{\text{t}})}}.}

For low-precision applications involving unpolarized light, such as computer graphics, rather than rigorously computing the effective reflection coefficient for each angle, Schlick's approximation is often used.

For equal permeabilities (e.g., non-magnetic media), if θi and θt are complementary, we can substitute sin θt for cos θi, and sin θi for cos θt, so that the numerator in equation (31) becomes n2‍sin θt − n1‍sin θi, which is zero (by Snell's law). Hence rp = 0  and only the s-polarized component is reflected. This is what happens at the Brewster angle. Substituting cos θi for sin θt in Snell's law, we readily obtain

At the interface, by the usual interface conditions for electromagnetic fields, the tangential components of the E and H fields must be continuous; that is,

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Iris Diaphragm controls the amount of light reaching the specimen. It is located above the condenser and below the stage. Most high quality microscopes include an Abbe condenser with an iris diaphragm. Combined, they control both the focus and quantity of light applied to the specimen.

The reflectance for s-polarized light is R s = | Z 2 cos ⁡ θ i − Z 1 cos ⁡ θ t Z 2 cos ⁡ θ i + Z 1 cos ⁡ θ t | 2 , {\displaystyle R_{\mathrm {s} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {i} }-Z_{1}\cos \theta _{\mathrm {t} }}{Z_{2}\cos \theta _{\mathrm {i} }+Z_{1}\cos \theta _{\mathrm {t} }}}\right|^{2},} while the reflectance for p-polarized light is R p = | Z 2 cos ⁡ θ t − Z 1 cos ⁡ θ i Z 2 cos ⁡ θ t + Z 1 cos ⁡ θ i | 2 , {\displaystyle R_{\mathrm {p} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {t} }-Z_{1}\cos \theta _{\mathrm {i} }}{Z_{2}\cos \theta _{\mathrm {t} }+Z_{1}\cos \theta _{\mathrm {i} }}}\right|^{2},} where Z1 and Z2 are the wave impedances of media 1 and 2, respectively.

Putting B = μH and D = ϵE, as above, we can eliminate B and D to obtain equations in only E and H: ω μ H = k × E ω ϵ E = − k × H . {\displaystyle {\begin{aligned}\omega \mu \mathbf {H} &=\mathbf {k} \times \mathbf {E} \\\omega \epsilon \mathbf {E} &=-\mathbf {k} \times \mathbf {H} \,.\end{aligned}}} If the material parameters ϵ and μ are real (as in a lossless dielectric), these equations show that k, E, H form a right-handed orthogonal triad, so that the same equations apply to the magnitudes of the respective vectors. Taking the magnitude equations and substituting from (2), we obtain μ c H = n E ϵ c E = n H , {\displaystyle {\begin{aligned}\mu cH&=nE\\\epsilon cE&=nH\,,\end{aligned}}} where H and E are the magnitudes of H and E. Multiplying the last two equations gives

Linearpolarization

When light strikes the interface between a medium with refractive index n1 and a second medium with refractive index n2, both reflection and refraction of the light may occur. The Fresnel equations give the ratio of the reflected wave's electric field to the incident wave's electric field, and the ratio of the transmitted wave's electric field to the incident wave's electric field, for each of two components of polarization. (The magnetic fields can also be related using similar coefficients.) These ratios are generally complex, describing not only the relative amplitudes but also the phase shifts at the interface.

Under the sign convention used in this article, a positive reflection or transmission coefficient is one that preserves the direction of the transverse field, meaning (in this context) the field normal to the plane of incidence. For the s polarization, that means the E field. If the incident, reflected, and transmitted E fields (in the above equations) are in the z-direction ("out of the page"), then the respective H fields are in the directions of the red arrows, since k, E, H form a right-handed orthogonal triad. The H fields may therefore be described by their components in the directions of those arrows, denoted by Hi, Hr, Ht. Then, since H = YE,

If the E fields are in the directions of the red arrows, then, in order for k, E, H to form a right-handed orthogonal triad, the respective H fields must be in the −z-direction ("into the page") and may therefore be described by their components in that direction. This is consistent with the adopted sign convention, namely that a positive reflection or transmission coefficient is one that preserves the direction of the transverse field (the H field in the case of the p polarization). The agreement of the other field with the red arrows reveals an alternative definition of the sign convention: that a positive reflection or transmission coefficient is one for which the field vector in the plane of incidence points towards the same medium before and after reflection or transmission.[34]

Measurements of Rs and Rp at 45° can be used to estimate the reflectivity at normal incidence.[citation needed] The "average of averages" obtained by calculating first the arithmetic as well as the geometric average of Rs and Rp, and then averaging these two averages again arithmetically, gives a value for R0 with an error of less than about 3% for most common optical materials.[citation needed] This is useful because measurements at normal incidence can be difficult to achieve in an experimental setup since the incoming beam and the detector will obstruct each other. However, since the dependence of Rs and Rp on the angle of incidence for angles below 10° is very small, a measurement at about 5° will usually be a good approximation for normal incidence, while allowing for a separation of the incoming and reflected beam.

On the other hand, calculation of the power transmission coefficient T is less straightforward, since the light travels in different directions in the two media. What's more, the wave impedances in the two media differ; power (irradiance) is given by the square of the electric field amplitude divided by the characteristic impedance of the medium (or by the square of the magnetic field multiplied by the characteristic impedance). This results in:[9] T = n 2 cos ⁡ θ t n 1 cos ⁡ θ i | t | 2 {\displaystyle T={\frac {n_{2}\cos \theta _{\text{t}}}{n_{1}\cos \theta _{\text{i}}}}|t|^{2}} using the above definition of t. The introduced factor of ⁠n2/n1⁠ is the reciprocal of the ratio of the media's wave impedances. The cos(θ) factors adjust the waves' powers so they are reckoned in the direction normal to the interface, for both the incident and transmitted waves, so that full power transmission corresponds to T = 1.

So, for a given ω, if we redefine k as the magnitude of the wave vector in the reference medium (for which n = 1), then the wave vector has magnitude n1k in the first medium (region y < 0 in the diagram) and magnitude n2k in the second medium. From the magnitudes and the geometry, we find that the wave vectors are k i = n 1 k ( i sin ⁡ θ i + j cos ⁡ θ i ) k r = n 1 k ( i sin ⁡ θ i − j cos ⁡ θ i ) k t = n 2 k ( i sin ⁡ θ t + j cos ⁡ θ t ) = k ( i n 1 sin ⁡ θ i + j n 2 cos ⁡ θ t ) , {\displaystyle {\begin{aligned}\mathbf {k} _{\text{i}}&=n_{1}k(\mathbf {i} \sin \theta _{\text{i}}+\mathbf {j} \cos \theta _{\text{i}})\\[.5ex]\mathbf {k} _{\text{r}}&=n_{1}k(\mathbf {i} \sin \theta _{\text{i}}-\mathbf {j} \cos \theta _{\text{i}})\\[.5ex]\mathbf {k} _{\text{t}}&=n_{2}k(\mathbf {i} \sin \theta _{\text{t}}+\mathbf {j} \cos \theta _{\text{t}})\\&=k(\mathbf {i} \,n_{1}\sin \theta _{\text{i}}+\mathbf {j} \,n_{2}\cos \theta _{\text{t}})\,,\end{aligned}}} where the last step uses Snell's law. The corresponding dot products in the phasor form (3) are

Thus he finally had a quantitative theory for what we now call the Fresnel rhomb — a device that he had been using in experiments, in one form or another, since 1817 (see Fresnel rhomb § History).

In the absence of Doppler shifts, ω does not change on reflection or refraction. Hence, by (2), the magnitude of the wave vector is proportional to the refractive index.

The simplest way to obtain the power transmission coefficient (transmissivity, the ratio of transmitted power to incident power in the direction normal to the interface, i.e. the y direction) is to use R + T = 1 (conservation of energy). In this way we find

s-polarization vs ppolarization

Put simply, focal length as denoted by the numbers on your lens (35mm, 50mm etc.) is the distance between your sensor or film and your camera lens when it's ...

The Poynting vector for a wave is a vector whose component in any direction is the irradiance (power per unit area) of that wave on a surface perpendicular to that direction. For a plane sinusoidal wave the Poynting vector is ⁠1/2⁠‍Re{E × H∗}, where E and H are due only to the wave in question, and the asterisk denotes complex conjugation. Inside a lossless dielectric (the usual case), E and H are in phase, and at right angles to each other and to the wave vector k; so, for s polarization, using the z and xy components of E and H respectively (or for p polarization, using the xy and −z components of E and H), the irradiance in the direction of k is given simply by EH/2, which is E2/2Z in a medium of intrinsic impedance Z = 1/Y. To compute the irradiance in the direction normal to the interface, as we shall require in the definition of the power transmission coefficient, we could use only the x component (rather than the full xy component) of H or E or, equivalently, simply multiply EH/2 by the proper geometric factor, obtaining (E2/2Z)cos θ.

In the following equations and graphs, we adopt the following conventions. For s polarization, the reflection coefficient r is defined as the ratio of the reflected wave's complex electric field amplitude to that of the incident wave, whereas for p polarization r is the ratio of the waves complex magnetic field amplitudes (or equivalently, the negative of the ratio of their electric field amplitudes). The transmission coefficient t is the ratio of the transmitted wave's complex electric field amplitude to that of the incident wave, for either polarization. The coefficients r and t are generally different between the s and p polarizations, and even at normal incidence (where the designations s and p do not even apply!) the sign of r is reversed depending on whether the wave is considered to be s or p polarized, an artifact of the adopted sign convention (see graph for an air-glass interface at 0° incidence).

This switch of polarizations has an analog in the old mechanical theory of light waves (see § History, above). One could predict reflection coefficients that agreed with observation by supposing (like Fresnel) that different refractive indices were due to different densities and that the vibrations were normal to what was then called the plane of polarization, or by supposing (like MacCullagh and Neumann) that different refractive indices were due to different elasticities and that the vibrations were parallel to that plane.[37] Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest.

From equations (4) and (5), we see that two dissimilar media will have the same refractive index, but different admittances, if the ratio of their permeabilities is the inverse of the ratio of their permittivities. In that unusual situation we have θt = θi (that is, the transmitted ray is undeviated), so that the cosines in equations (13), (14), (21), (22), and (25) to (28) cancel out, and all the reflection and transmission ratios become independent of the angle of incidence; in other words, the ratios for normal incidence become applicable to all angles of incidence.[35] When extended to spherical reflection or scattering, this results in the Kerker effect for Mie scattering.

To advance the phase by the angle ϕ, we replace ωt by ωt + ϕ (that is, we replace −ωt by −ωt − ϕ), with the result that the (complex) field is multiplied by e−iϕ. So a phase advance is equivalent to multiplication by a complex constant with a negative argument. This becomes more obvious when the field (1) is factored as Ek eik⋅re−iωt, where the last factor contains the time-dependence. That factor also implies that differentiation w.r.t. time corresponds to multiplication by −iω. [Note 2]

At a dielectric interface from n1 to n2, there is a particular angle of incidence at which Rp goes to zero and a p-polarised incident wave is purely refracted, thus all reflected light is s-polarised. This angle is known as Brewster's angle, and is around 56° for n1 = 1 and n2 = 1.5 (typical glass).

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Eyepiece or Ocular is what you look through at the top of the microscope. Typically, standard eyepieces have a magnifying power of 10x. Optional eyepieces of varying powers are available, typically from 5x-30x.

An example of interference between reflections is the iridescent colours seen in a soap bubble or in thin oil films on water. Applications include Fabry–Pérot interferometers, antireflection coatings, and optical filters. A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference.

95 MHz LC Band Pass Filter.

In optics, one usually knows the refractive index n of the medium, which is the ratio of the speed of light in vacuum (c) to the speed of light in the medium. In the analysis of partial reflection and transmission, one is also interested in the electromagnetic wave impedance Z, which is the ratio of the amplitude of E to the amplitude of H. It is therefore desirable to express n and Z in terms of ϵ and μ, and thence to relate Z to n. The last-mentioned relation, however, will make it convenient to derive the reflection coefficients in terms of the wave admittance Y, which is the reciprocal of the wave impedance Z.

There are two sets of Fresnel coefficients for two different linear polarization components of the incident wave. Since any polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise, unpolarized (or "randomly polarized") light has an equal amount of power in each of two linear polarizations.

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S polarizationmeaning

Although the reflection and transmission are dependent on polarization, at normal incidence (θ = 0) there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients (and another special case is mentioned below in which that is true).

We assume that the media are non-magnetic (i.e., μ1 = μ2 = μ0), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies).[3] Then the wave impedances are determined solely by the refractive indices n1 and n2: Z i = Z 0 n i , {\displaystyle Z_{i}={\frac {Z_{0}}{n_{i}}}\,,} where Z0 is the impedance of free space and i = 1, 2. Making this substitution, we obtain equations using the refractive indices: R s = | n 1 cos ⁡ θ i − n 2 cos ⁡ θ t n 1 cos ⁡ θ i + n 2 cos ⁡ θ t | 2 = | n 1 cos ⁡ θ i − n 2 1 − ( n 1 n 2 sin ⁡ θ i ) 2 n 1 cos ⁡ θ i + n 2 1 − ( n 1 n 2 sin ⁡ θ i ) 2 | 2 , {\displaystyle R_{\mathrm {s} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}\cos \theta _{\mathrm {t} }}{n_{1}\cos \theta _{\mathrm {i} }+n_{2}\cos \theta _{\mathrm {t} }}}\right|^{2}=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}}{n_{1}\cos \theta _{\mathrm {i} }+n_{2}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}}}\right|^{2}\!,} R p = | n 1 cos ⁡ θ t − n 2 cos ⁡ θ i n 1 cos ⁡ θ t + n 2 cos ⁡ θ i | 2 = | n 1 1 − ( n 1 n 2 sin ⁡ θ i ) 2 − n 2 cos ⁡ θ i n 1 1 − ( n 1 n 2 sin ⁡ θ i ) 2 + n 2 cos ⁡ θ i | 2 . {\displaystyle R_{\mathrm {p} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {t} }-n_{2}\cos \theta _{\mathrm {i} }}{n_{1}\cos \theta _{\mathrm {t} }+n_{2}\cos \theta _{\mathrm {i} }}}\right|^{2}=\left|{\frac {n_{1}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}-n_{2}\cos \theta _{\mathrm {i} }}{n_{1}{\sqrt {1-\left({\frac {n_{1}}{n_{2}}}\sin \theta _{\mathrm {i} }\right)^{2}}}+n_{2}\cos \theta _{\mathrm {i} }}}\right|^{2}\!.}

Thus Fresnel's interpretation of the complex values of his reflection coefficients marked the confluence of several streams of his research and, arguably, the essential completion of his reconstruction of physical optics on the transverse-wave hypothesis (see Augustin-Jean Fresnel).

The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by French engineer and physicist Augustin-Jean Fresnel (/freɪˈnɛl/) who was the first to understand that light is a transverse wave, when no one realized that the waves were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.

Circularpolarization

When light makes multiple reflections between two or more parallel surfaces, the multiple beams of light generally interfere with one another, resulting in net transmission and reflection amplitudes that depend on the light's wavelength. The interference, however, is seen only when the surfaces are at distances comparable to or smaller than the light's coherence length, which for ordinary white light is few micrometers; it can be much larger for light from a laser.

We call the fraction of the incident power that is reflected from the interface the reflectance (or reflectivity, or power reflection coefficient) R, and the fraction that is refracted into the second medium is called the transmittance (or transmissivity, or power transmission coefficient) T. Note that these are what would be measured right at each side of an interface and do not account for attenuation of a wave in an absorbing medium following transmission or reflection.[2]

A high power or compound microscope achieves higher levels of magnification than a stereo or low power microscope. It is used to view smaller specimens such as cell structures which cannot be seen at lower levels of magnification. Essentially, a compound microscope consists of structural and optical components. However, within these two basic systems, there are some essential components that every microscopist should know and understand. These key microscope parts are illustrated and explained below.

The equations consider a plane wave incident on a plane interface at angle of incidence θ i {\displaystyle \theta _{\mathrm {i} }} , a wave reflected at angle θ r = θ i {\displaystyle \theta _{\mathrm {r} }=\theta _{\mathrm {i} }} , and a wave transmitted at angle θ t {\displaystyle \theta _{\mathrm {t} }} . In the case of an interface into an absorbing material (where n is complex) or total internal reflection, the angle of transmission does not generally evaluate to a real number. In that case, however, meaningful results can be obtained using formulations of these relationships in which trigonometric functions and geometric angles are avoided; the inhomogeneous waves launched into the second medium cannot be described using a single propagation angle.

As usual, we drop the time-dependent factor e−iωt, which is understood to multiply every complex field quantity. The electric field for a uniform plane sine wave will then be represented by the location-dependent phasor

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The success of the complex reflection coefficient inspired James MacCullagh and Augustin-Louis Cauchy, beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a complex refractive index.[29]

In order to compute meaningful Fresnel coefficients, we must assume that the medium is (approximately) linear and homogeneous. If the medium is also isotropic, the four field vectors E, B, D, H  are related by D = ϵ E B = μ H , {\displaystyle {\begin{aligned}\mathbf {D} &=\epsilon \mathbf {E} \\\mathbf {B} &=\mu \mathbf {H} \,,\end{aligned}}} where ϵ and μ are scalars, known respectively as the (electric) permittivity and the (magnetic) permeability of the medium. For vacuum, these have the values ϵ0 and μ0, respectively. Hence we define the relative permittivity (or dielectric constant) ϵrel = ϵ/ϵ0, and the relative permeability μrel = μ/μ0.

What ispolarizationin Chemistry

For the s polarization, the E field is parallel to the z axis and may therefore be described by its component in the z direction. Let the reflection and transmission coefficients be rs and ts, respectively. Then, if the incident E field is taken to have unit amplitude, the phasor form (3) of its z-component is

where Ek is the (constant) complex amplitude vector, i is the imaginary unit, k is the wave vector (whose magnitude k is the angular wavenumber), r is the position vector, ω is the angular frequency, t is time, and it is understood that the real part of the expression is the physical field.[Note 1]  The value of the expression is unchanged if the position r varies in a direction normal to k; hence k is normal to the wavefronts.

Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, the power reflection coefficient R is just the squared magnitude of r: [8] R = | r | 2 . {\displaystyle R=|r|^{2}.}

Objective Lenses are the primary optical lenses on a microscope. They range from 4x-100x and typically, include, three, four or five on lens on most microscopes. Objectives can be forward or rear-facing.

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When we substitute from equations (17) and (18) and then from (7), the exponential factors again cancel out, so that the interface conditions reduce to

In the diagram on the right, an incident plane wave in the direction of the ray IO strikes the interface between two media of refractive indices n1 and n2 at point O. Part of the wave is reflected in the direction OR, and part refracted in the direction OT. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as θi, θr and θt, respectively. The relationship between these angles is given by the law of reflection: θ i = θ r , {\displaystyle \theta _{\mathrm {i} }=\theta _{\mathrm {r} },} and Snell's law: n 1 sin ⁡ θ i = n 2 sin ⁡ θ t . {\displaystyle n_{1}\sin \theta _{\mathrm {i} }=n_{2}\sin \theta _{\mathrm {t} }.}

In 1821, however, Augustin-Jean Fresnel derived results equivalent to his sine and tangent laws (above), by modeling light waves as transverse elastic waves with vibrations perpendicular to what had previously been called the plane of polarization. Fresnel promptly confirmed by experiment that the equations correctly predicted the direction of polarization of the reflected beam when the incident beam was polarized at 45° to the plane of incidence, for light incident from air onto glass or water; in particular, the equations gave the correct polarization at Brewster's angle.[22] The experimental confirmation was reported in a "postscript" to the work in which Fresnel first revealed his theory that light waves, including "unpolarized" waves, were purely transverse.[23]

The behavior of light striking the interface is explained by considering the electric and magnetic fields that constitute an electromagnetic wave, and the laws of electromagnetism, as shown below. The ratio of waves' electric field (or magnetic field) amplitudes are obtained, but in practice one is more often interested in formulae which determine power coefficients, since power (or irradiance) is what can be directly measured at optical frequencies. The power of a wave is generally proportional to the square of the electric (or magnetic) field amplitude.

In the case of uniform plane sinusoidal waves, the wave impedance or admittance is known as the intrinsic impedance or admittance of the medium. This case is the one for which the Fresnel coefficients are to be derived.

From equations (13) and (21), taking squared magnitudes, we find that the reflectivity (ratio of reflected power to incident power) is

Eyepiece Tube holds the eyepieces in place above the objective lens. Binocular microscope heads typically incorporate a diopter adjustment ring that allows for the possible inconsistencies of our eyesight in one or both eyes. The monocular (single eye usage) microscope does not need a diopter. Binocular microscopes also swivel (Interpupillary Adjustment) to allow for different distances between the eyes of different individuals.

In optics it is common to assume that the medium is non-magnetic, so that μrel = 1. For ferromagnetic materials at radio/microwave frequencies, larger values of μrel must be taken into account. But, for optically transparent media, and for all other materials at optical frequencies (except possible metamaterials), μrel is indeed very close to 1; that is, μ ≈ μ0.

Comparing (23) and (24) with (15) and (16), we see that at normal incidence, under the adopted sign convention, the transmission coefficients for the two polarizations are equal, whereas the reflection coefficients have equal magnitudes but opposite signs. While this clash of signs is a disadvantage of the convention, the attendant advantage is that the signs agree at grazing incidence.

In the case of an interface between two lossless media (for which ϵ and μ are real and positive), one can obtain these results directly using the squared magnitudes of the amplitude transmission coefficients that we found earlier in equations (14) and (22). But, for given amplitude (as noted above), the component of the Poynting vector in the y direction is proportional to the geometric factor cos θ and inversely proportional to the wave impedance Z. Applying these corrections to each wave, we obtain two ratios multiplying the square of the amplitude transmission coefficient:

Although it is not encountered in practice, the equations can also apply to the case of two media with a common permittivity but different refractive indices due to different permeabilities. From equations (4) and (5), if ϵ is fixed instead of μ, then Y becomes inversely proportional to n, with the result that the subscripts 1 and 2 in equations (29) to (38) are interchanged (due to the additional step of multiplying the numerator and denominator by n1n2). Hence, in (29) and (31), the expressions for rs and rp in terms of refractive indices will be interchanged, so that Brewster's angle (39) will give rs = 0 instead of rp = 0, and any beam reflected at that angle will be p-polarized instead of s-polarized.[36] Similarly, Fresnel's sine law will apply to the p polarization instead of the s polarization, and his tangent law to the s polarization instead of the p polarization.

Coarse and Fine Focus knobs are used to focus the microscope. Increasingly, they are coaxial knobs - that is to say they are built on the same axis with the fine focus knob on the outside. Coaxial focus knobs are more convenient since the viewer does not have to grope for a different knob.

Stage Clips are used when there is no mechanical stage. The viewer is required to move the slide manually to view different sections of the specimen.

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for the p polarization. The last two equations apply only to lossless dielectrics, and only at incidence angles smaller than the critical angle (beyond which, of course, T = 0).

Reflection at 45° incidence is very commonly used for making 90° turns. For the case of light traversing from a less dense medium into a denser one at 45° incidence (θ = 45°), it follows algebraically from the above equations that Rp equals the square of Rs: R p = R s 2 {\displaystyle R_{\text{p}}=R_{\text{s}}^{2}}

These formulas [14][15][16] are known respectively as Fresnel's sine law and Fresnel's tangent law.[17] Although at normal incidence these expressions reduce to 0/0, one can see that they yield the correct results in the limit as θi → 0.