Resolving power of telescope, microscope and grating

May 25 • General • 27333 Views • 30 Comments on Resolving power of telescope, microscope and grating

Resolving power of various devices are very necessary from experimental point of view so lets begin with some background.

Resolving power of telescope:

A simple telescope consists of a large aperture objective lens with high focal length and an eye piece  with a lower focal length and smaller aperture. An eye piece consists of two lenses separated a distance. The lens towards the objective is called the field lens and other which is near to the eye is known as eye lens.

The resolving power of an optical instrument, say a telescope or microscope, is its ability to produce  separate images of two closely spaced objects/ sources. The plane waves from each source after passing through an aperture from diffraction pattern characteristics of the aperture. It is the overlapping of diffraction patterns formed by two sources sets a theoretical upper limit to the resolving power.

Consider two narrow slit sources, d distance apart, kept at a distance D away from the aperture, i.e. objective, of a telescope. In the following we will stick to rectangular aperture. Then the angular separation α of the slits at the aperture is α = d/D. Each slit will produce its own single slit diffraction pattern, for which the intensity distribution is given by


I=I0Sin^2b/b^2 where b= 3.14asin@/lemda

And  is the slit width, θ is the angle of diffraction and λ is the wave length of light from the sources. The principal maximum of each slit corresponds to θ = 0   0 and the position of the minima, which are points of zero intensity, corresponds to, 2 etc. The angular separation of the two principal maxima is equal to the angular separation of the sources, i.e. α. 

The Rayleigh’s criterion for resolution of two diffraction patterns states that two sources or their diffraction patterns are resolved when the principal maximum of one falls exactly on the first minimum of the other. Since the first minimum is formed at , then angular separation α of the maxima is equal to the corresponding θ1,The angle is known as minimum angle of resolution while 1” is sometimes called the resolving power of the aperture . For circular aperture, Rayleigh’s criterion is modified to   1.22 $” . To find the intensity at the centre of the resultant minimum for the overlapping diffraction fringes separated by , we note the curves of principal maxima cross at 2” for either pattern. Therefore intensity at the centre, relative to the maximum, is sum of the intensity of either at 2

Questions
1. How the minimum angle of resolution changes with the wavelength of the light $?
2. Which one of the two telescopes, optical and radio has more resolving power for a given aperture?
3. What is the for human eye for red (700nm), yellow (600nm), and blue (400nm) light, assuming
dark-adapted average pupil size is 5mm? (Use the Rayleigh’s criterion for circular apert.

Resolving power of microscope:

The imaging system’s resolution can be limited either by aberration or by diffraction causing blurring of the image. These two phenomena have different origins and are unrelated. Aberrations can be explained by geometrical optics and can in principle be solved by increasing the optical quality — and cost — of the system. On the other hand, diffraction comes from the wave nature of light and is determined by the finite aperture of the optical elements. The lens’ circular aperture is analogous to a two-dimensional version of the single-slit experiment. Light passing through the lens interferes with itself creating a ring-shape diffraction pattern, known as the Airy pattern, if the wavefront of the transmitted light is taken to be spherical or plane over the exit aperture.

The interplay between diffraction and aberration can be characterised by the point spread function (PSF). The narrower the aperture of a lens the more likely the PSF is dominated by diffraction. In that case, the angular resolution of an optical system can be estimated (from the diameter of the aperture and the wavelength of the light) by the Rayleigh criterion invented by Lord Rayleigh:

Two point sources are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other, If the distance is greater, the two points are well resolved and if it is smaller, they are regarded as not resolved. If one considers diffraction through a circular aperture.

The resolution R (here measured as a distance, not to be confused with the angular resolution of a previous subsection) depends on the angular aperture :

 where .^[4]

Here NA is the numerical aperture,  is half the included angle  of the lens, which depends on the diameter of the lens and its focal length,  is the refractive index of the medium between the lens and the specimen, and  is the wavelength of light illuminating or emanating from (in the case of fluorescence microscopy) the sample.

It follows that the NAs of both the objective and the condenser should be as high as possible for maximum resolution. In the case that both NAs are the same, the equation may be reduced to:

The practical limit for  is about 70°. In an air objective or condenser, this gives a maximum NA of 0.95. In a high-resolution oil immersion lens, the maximum NA is typically 1.45, when using immersion oil with a refractive index of 1.52. Due to these limitations, the resolution limit of a light microscope using visible light is about 200 nm. Given that the shortest wavelength of visible light is violet ( ≈ 400 nm),

which is near 200 nm.

Oil immersion objectives can have practdue to their shallow depth of field and extremely short working distance, which calls for the use of very thin (0.17mm) cover slips, or, in an inverted microscope, thin glass-bottomed Petri dishes.

However, resolution below this theoretical limit can be achieved using optical near-fields (Near-field scanning optical microscope) or a diffraction technique called 4Pi STED microscopy. Objects as small as 30 nm have been resolved with both technique.

Questions:

1.what is ment by resolving power of microscope?

2.how can the formula pf resolving power be computed

Resolving power of grating:

the resolution (resolving power) of a microscope means its ability to distinguish two items at its highest magnification. the same goes for any other optical instrument. its like watching two lines which are extremely close to each other with unaided eye and then watching them with the microscope. with the unaided eye they will appear as one line. with the microscope they will appear distinct.

Questions:

1.what do you mean by grating

2.how could the effeciency be improved for resolving power

resolving power of microscope

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