The extraordinary ray violates both Snell’s Law and the Law of Reflection.  It is not necessarily confined to the plane of incidence.  Its speed changes with direction.  The index of refraction for the extraordinary ray is a continuous function of direction.  The index of refraction for the ordinary ray is independent of direction.  When the ordinary index of refraction is plotted against wavelength, the dispersion curve for the ordinary ray is a single unique curve.  The dispersion curve for the extraordinary ray is a family of curves with different curves for different directions.  A ray normally incident on a birefringent crystalline surface will be divided into two rays at the boundary, unless it is in a special polarization state or unless the crystalline surface is perpendicular to an optic axis.  The extraordinary ray will deviate from the incident direction while the ordinary ray will not.  The ordinary ray index n0 and the most extreme extraordinary ray index ne are together known as the principal indices of refraction of the material.  The direction of the lesser index is called the fast axis because light polarized in that direction has the higher speed.

Polarization

Resolution in a scope is a combination of how magnification and objective lens diameter allow the eye to see small details better than using the unaided eye. Larger lenses have the potential of providing smaller details but depending on magnification there is an optimum objective lens diameter to see small details better than the unaided eye. To put it simply, a scope needs to be designed in a way that it AIDS your eye in seeing resolution at magnification, but there is a point where resolution can exceed the capabilities of the human eye. By designing a system that meets, but does not excessively exceed the capabilities of the human eye, we can offer an optic with the perfect balance of optical quality and cost efficiency.

Electric polarization

The fundamental definition of resolution is the ability of an optical system to see fine detail in an object being imaged. However, in reality, resolution is a somewhat broad term as it includes image sharpness and contrast. The resolving power of a scope is the sum total of several components such as optical design, glare, color correction, and focus.

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Linear polarization example

polarization中文

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The units commonly used to measure resolution are seconds of angle or 1/3600th of a degree. For example, a rifle scope having a resolution of 1 arc seconds is equivalent to resolving a feature 0.017 inch at 100 yards. But this is only theoretical, since in application, a good quality rifle scope with a large 50 to 60mm objective at high power can realistically resolve down to about 2.5 arc seconds or slightly less than ½ inch at 100 yards.

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Circularlypolarized light

A: Theoretical resolution based on what the eye can see in seconds of arcB: Resolution without optical aid and the eye pupil at 7mm in seconds of arcC: Potential resolution using a 25mm objective lens in seconds of arcD: Potential resolution using a 40mm lens in seconds of arcE: Potential resolution using a 50mm lens in seconds of arcF: Potential resolution using a 60mm lens in seconds of arcG: Potential resolution using a 80mm lens in seconds of arc

When the sun is at a low angle in the sky, the sunlight reflecting off the surface of water is nearly 100% horizontally polarized because the angle of incidence is close to the Brewster angle.  Glare-reducing sunglasses are coated with a polarizer with a vertical transmission axis and therefore block the reflected light.

This chart illustrates how larger objectives have potential of providing sharper images. However in the upper powers, the lens design plays a much larger role in providing better resolution. The numbers above are only valid for diffraction limited systems. At 60 power and using an 80mm objective the detail comes very close to that of the human eye but at 2 power a scope with an 80mm aperture is producing an image with detail much smaller than can be appreciated by the human eye.

Circular polarization

The clarity or sharpness of the smallest detail seen is never totally resolved, but if the smallest detail can just be distinguished it is considered resolved. How sharp the feature is resolved compared to how much it blurs out in the field in is called MTF (modulation transfer function), which describes how well one can see small details with respect to black and white lines placed closely together. As the lines get closer together it becomes harder to see them as line pairs, and as you approach the resolving limit they are just barely distinguishable as a pair. We can say the limit is if they are 10% resolved. At 10% resolved this can be stated as having an MTF for that pair of 10%. For the human eye with no optical aid, a person with 20/20 vision can distinguish a one minute angle, which would be 1 inch black and white bars at 100 yards. With an optical system with a magnification of 2 power we can see line pairs of ½ inch spacing at 100 yards.

Linearly polarized light is a special case of elliptically polarized light.  If the light is linearly polarized, then the two components oscillate in phase,  for example Ex = E0xexp(i(kz - ωt)), Ey = E0yexp(i(kz - ωt)), φ = 0.  The direction of E and the direction of propagation define a plane.  The electric vector traces out a straight line.  For example, E = Ei = E0xexp(i(kz - ωt))i.

The two beams within the birefringent crystal are referred to as the ordinary and extraordinary ray, respectively.  The polarization of the extraordinary ray lies in the plane containing the direction of propagation and the optic axis, and the polarization of the ordinary ray is perpendicular to this plane.

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Polarization oflight

In other devices the changes in direction of propagation between the two rays is used to separate the incoming beam into two orthogonally polarized beams as in the Wollaston and Thompson beam-splitting prisms.

The figure below shows the trace of the field vector Ex = E0exp(i(kz - ωt)), Ey = E0exp(i(kz - ωt + φ)) in a plane perpendicular to the z-axis when looking towards the source.  (E0x = E0y = E0)

The electric field vector E can always be resolved into two perpendicular components.  The light is elliptically polarized, then the two components have a constant phase difference, and the tip of the electric field vector traces out an ellipse in the plane perpendicular to the direction of propagation.

In the Glan-Taylor polarizing prism shown on the right the rejected (ordinary) ray is absorbed by black mounting material in the prism housing.

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Elliptical polarization

A quarter-wave plate δ = π/2 can be used to convert linearly polarized light to circularly polarized light.  The incident linearly polarized light must be oriented at 45o to the wave plate's axes.  A half-wave plate δ = π can be used to rotate the plane of linearly polarized light.  The angle of rotation is 2θ, where θ is the angle between the angle of polarization and the wave plate's fast axis.

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In a rifle scope at low power, the image is so well resolved that it is beyond the human eye to see the potential smallest detail because of the limits of the human eye. As we use scopes of higher magnification the image details become closer to the human eye limit and as we get up to 30 to 60 power the scopes optical system needs to be designed such that it maintains the capability of the eye.

A half-wave plate δ = π can be used to rotate the plane of linearly polarized light.  The angle of rotation is 2θ, where θ is the angle between the angle of polarization and the wave plate's fast axis.

NOTE: Letters B-G are all the same number because they represent resolution for the respective aperture listed regardless of magnification.

If a beam of linearly polarized monochromatic light enters a birefringent crystal along a direction not parallel to the optical axis of the crystal, the beam will may be divided into two separate beams.  Each will be polarized at right angles to the other, and they will travel in different directions.  The intensity of the original beam will be divided between the two new beams in a manner which depends on the original orientation of the electric field vector with respect to the crystal.  The ratio or the intensities of the two orthogonally polarized beams can have any value.

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