Difference between revisions of "Geometrical optics and ray tracing"
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* Typically in a microscope, the aperture stop will be the objective lens. | * Typically in a microscope, the aperture stop will be the objective lens. | ||
− | [[Image: 20.309 130822 ApertureStop.png|center|thumb|400px|From [http://electron6.phys.utk.edu/optics421/modules/m3/Stops.htm utk.edu]. The aperture stop in system (a) at the top is the left aperture, in (b) in the middle is the first, left lens, and in (c) at the bottom is the second, right lens.]] | + | [[Image: 20.309 130822 ApertureStop.png|center|thumb|400px|From utk.edu <ref>[http://electron6.phys.utk.edu/optics421/modules/m3/Stops.htm utk.edu]</ref>. The aperture stop in system (a) at the top is the left aperture, in (b) in the middle is the first, left lens, and in (c) at the bottom is the second, right lens.]] |
====Numerical aperture==== | ====Numerical aperture==== | ||
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* An oil-immersion objective lens will thus have a greater ''NA'' than an air-immersion objective lens (<math>n_{oil} > n_{air}</math> and <math>\alpha_{oil} > \alpha_{air}</math>; indeed, little refraction occurs at the glass-oil interface where <math>n_{oil} \approx n_{glass}</math>). | * An oil-immersion objective lens will thus have a greater ''NA'' than an air-immersion objective lens (<math>n_{oil} > n_{air}</math> and <math>\alpha_{oil} > \alpha_{air}</math>; indeed, little refraction occurs at the glass-oil interface where <math>n_{oil} \approx n_{glass}</math>). | ||
− | [[Image: 20.309 130822 NumericalAperture.png|center|thumb|400px|From [http://spie.org/Documents/Publications/00%20STEP%20Module%2003.pdf spie.org]. Light-gathering power of oil-immersion and air-immersion lens, showing that <math>\alpha_{oil} > \alpha_{air}</math>.]] | + | [[Image: 20.309 130822 NumericalAperture.png|center|thumb|400px|From spie.org <ref>[http://spie.org/Documents/Publications/00%20STEP%20Module%2003.pdf spie.org]</ref>. Light-gathering power of oil-immersion and air-immersion lens, showing that <math>\alpha_{oil} > \alpha_{air}</math>.]] |
===Field stop and field of view=== | ===Field stop and field of view=== | ||
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* In your 20.309 microscope, the field stop will be the CCD camera. | * In your 20.309 microscope, the field stop will be the CCD camera. | ||
* A finite-size lens can also act as a field stop. Below is an illustration of this vignetting effect: a cone of light rays from an off-axis object is not transmitted in its entirety, but rather partially cut off by the field stop of the system, ''L<sub>2</sub>'''s rim. | * A finite-size lens can also act as a field stop. Below is an illustration of this vignetting effect: a cone of light rays from an off-axis object is not transmitted in its entirety, but rather partially cut off by the field stop of the system, ''L<sub>2</sub>'''s rim. | ||
− | [[Image:20.309 130822 FieldStop.png | + | [[Image:20.309 130822 FieldStop.png|thumb|center|400 px]] |
==Optical aberrations== | ==Optical aberrations== | ||
{{Template:20.309 bottom}} | {{Template:20.309 bottom}} |
Revision as of 19:39, 22 August 2013
Reflection and refraction
When incident light reaches an interface between two optical media, it can be
- reflected, partly or totally,
- scattered in random directions,
- refracted and thus entering the second medium, and/or
- absorbed partly.
We will limit our seminal discussion of basic geometrical optics to cases where interfaces are non-absorbant and smooth (as opposed to uneven) surfaces giving rise to specular (as opposed to diffuse) reflection and refraction.
Reflection at a boundary
- The law of reflection states that the angle θr the reflected ray makes with the normal to the surface is equal to the angle θi the incident ray makes with this normal. Incident and reflected rays are in the same plane.
- $ \theta_i = \theta_r $
- The law of reflection applies to plane and curved (e.g. spherical) interfaces.
Refraction at a plane boundary
Index of refraction
- The index of refraction $ n $ of a transparent optical medium is defined as the ratio of the speed of light in vacuum, $ c $, by the speed of light in the medium, $ v $.
- $ n = c / v $
Snell's law of refraction
- Light incident at an interface between two media of distinct indices of refraction will be partly reflected back into the first medium and partly transmitted to the second medium, or refracted according to the Snell-Descartes law:
- $ n_i\ \sin \theta_i = n_t\ \sin \theta_t $
- with θ the angle measured from the normal of the boundary, $ n $ the refractive index (which is unitless) of the medium, and the subscripts $ i $ and $ t $ referring to the incident and transmitted light, respectively.
- Snell's law implies that light passing from a medium of lower index to a higher index of refraction ($ n_t > n_i $) bends toward the surface normal ($ \theta_t < \theta_i $), whereas light traveling from a higher index to a lower index of refraction bends away from the normal.
Critical angle and total internal reflection
- From Snell's law of refraction ensues the existence of a critical angle $ \theta_c $ for the incident light ray at which it gets bent by 90o at the boundary and thus continues traveling along the interface between the two media. Beyond $ \theta_c $ total internal reflection takes place.
- Recalling that $ \sin (90^o) = 1 $, one can derive
- $ \theta_c = \sin ^{-1} \left ( {n_t \over n_i} \right ) $
- The phenomenon of total internal reflection is the principle of light transmission via optical fibers, whose core medium is cladded by a distinct medium of lower index of refraction.
Refraction at a spherical boundary
Gaussian optics assumptions
The lens formulas and ray tracing techniques we'll be using will rely on the first-order or Gaussian optics approximations:
- Paraxial (or small-angle) approximation:
- $ \theta \approx \sin\ \theta \approx \tan\ \theta $
- $ \cos\ \theta \approx 1 $
- (These relations obtained from the first-order terms of the polynomial expansions of $ \sin \theta = \theta - {\theta^3 \over {3!}} + {\theta^5 \over {5!}} - ... $ and $ \cos \theta = 1 - {\theta^2 \over {2!}} + {\theta^4 \over {4!}} - ... $ are accurate to 1% if $ \theta < 10^o $.)
- Thin-lens approximation:
- $ R << S_o,\ S_i $
- with $ R $ defined as the radius of curvature of the lens, $ S_o $ as the distance between the lens and the object, and $ S_i $ the distance between the lens and the image.
- The paraxial and thin-lens assumptions are engineering approximations that allow you to quickly and intuitively understand most optical systems using a simple set of ray tracing rules.
- Deviations from these assumptions result in optical aberrations.
Object and image positions
- Under Gaussian optics conditions, Snell's law predicts the image position of an object formed by a lens:
- $ n\ \sin \theta_1 = n'\ \sin \theta_2 $
- $ \sin \theta_1 \approx \sin a + \sin b \approx {h \over S_o} + {h \over R} $
- $ \sin \theta_2 \approx \sin b - \sin c \approx {h \over R} - {h \over S_i} $
- $ {n \over S_o} + {n' \over S_i} = {(n'\ - n)\over R} $
Note that
- $ S_i $ does not depend on the angle $ a $.
- Light coming from a point on the filament passes through a point after refraction.
a) $ S_o > {n\ R \over (n'\ - n)}\ \Rightarrow S_i > 0 $ , | b) $ S_o = {n\ R \over (n'\ - n)}\ \Rightarrow S_i \to + \infty $, | c) $ S_o = {n\ R \over (n'\ - n)}\ \Rightarrow S_i < 0 $ |
Lenses
Lens maker formula
A simple lens consists of two spherical interfaces of radii of curvature $ R_1 $ and $ R_2 $. Its focal length $ f $ is given by the lens maker formula:
- $ {1 \over S_o} + {1 \over S_i} = {1 \over f} = {(n'\ - n) \over n} \left ( {1 \over R_1} - {1 \over R_2} \right ) $
Bi-convex lens | Plano-concave lens |
---|---|
- In your ray-tracing studies, you'll use the image from the first refraction as the object for the second.
- Note the sign convention for the second surface: $ R_2 $ < 0 for a convex lens, and $ (n'\ - n) $ has opposite sign.
- Concave lens: the lens maker formula holds for $ R_1 $ < 0.
Types of spherical lenses
To minimize spherical aberrations, the general rule is to position the curved side of the lens toward the collimated/straight light.
Ray Tracing
Principles
Objects and images
Imaging with a lens
Imagine the object is made up of many point sources. After refraction, all the rays from a single point in the object plane reach the same location in the image plane. This forms a real, inverted image.
Magnification
By similar triangles, magnification is equal to the ratio of distances to the lens:
$ M = {h_i \over h_o} = {S_i \over S_o} $
A 4 f or compound microscope
- A 4 f microscope is made of two positive lenses, placed at the sum of their focal lengths apart.
- An object (sample) placed in the focal plane of the first lens gets magnified by the ratio of focal lengths, $ f_2 / f_1 $.
The eye as a lens
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You can observe a virtual image.
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Aperture and field stops
Aperture stop and image brightness
- The aperture stop of an imaging system is the optical element that limits angle of rays passing through the system from a source on the optical axis.
- For an off-axis object, the chief ray is the ray that passes through the center of the aperture stop, and the marginal rays are those that pass through the edge of the aperture stop.
- The size of the aperture stop determines the light-gathering capability of an instrument, and thus the brightness (or irradiance in W/m2) of its images.
- Typically in a microscope, the aperture stop will be the objective lens.
Numerical aperture
- The numerical aperture (NA) is another important measure that characterizes the light-gathering power of a lens.
- $ NA = n\ \sin \alpha $
- where $ n $ is the index of refraction of the medium between the object (sample) and the lens and $ \alpha $ is the half-angle defined by the limiting ray.
- An oil-immersion objective lens will thus have a greater NA than an air-immersion objective lens ($ n_{oil} > n_{air} $ and $ \alpha_{oil} > \alpha_{air} $; indeed, little refraction occurs at the glass-oil interface where $ n_{oil} \approx n_{glass} $).
Field stop and field of view
- The field stop is the optical element that blocks off-axis rays passing through the center of the center of the aperture stop (i.e. the chief rays).
- The field stop determines how much of the object can be viewed, in other words: the field of view of the imaging system.
- In your 20.309 microscope, the field stop will be the CCD camera.
- A finite-size lens can also act as a field stop. Below is an illustration of this vignetting effect: a cone of light rays from an off-axis object is not transmitted in its entirety, but rather partially cut off by the field stop of the system, L2's rim.
Optical aberrations
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