Physics RESOLVING POWER OF MICROSCOPE AND TELESCOPE

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RESOLVING POWER OF MICROSCOPE AND TELESCOPE

Resolving Power of Telescope

Consider a parallel beam of light falling on a convex lens. If the lens is well corrected for aberrations, then geometrical optics tells us that the beam will get focused to a point. However, because of diffraction, the beam instead of getting focused to a point gets focused to a spot of finite area. In this case the effects due to diffraction can be taken into account by considering a plane wave incident on a circular aperture followed by a convex lens (Fig.). The analysis of the corresponding diffraction pattern is quite involved; however, in principle, it is similar to the analysis carried out to obtain the single-slit diffraction pattern. Taking into account the effects due to diffraction, the pattern on the focal plane would consist of a central bright region surrounded by concentric dark and bright rings (Fig.). A detailed analysis shows that the radius of the central bright region is approximately given by

`r_o=(1.22lamdaf)/(2a)=(0.61lamdaf)/a`

where f is the focal length of the lens and 2a is the diameter of the circular aperture or the diameter of the lens, whichever is smaller.

Although the size of the spot is very small, it plays an important role in determining the limit of resolution of optical instruments like a telescope or a microscope. For the two stars to be just resolved

`fDeltathetaapprox(r_o)=(0.61lamdaf)/a`

`=>` `Deltathetaapprox(0.61lamda)/a......(1)`

Thus Δθ will be small if the diameter of the objective is large. This implies that the telescope will have better resolving power if a is large. It is for this reason that for better resolution, a telescope must have a large diameter objective.

Consider a parallel beam of light falling on a convex lens. If the lens is well corrected for aberrations, then geometrical optics tells us that the beam will get focused to a point. However, because of diffraction, the beam instead of getting focused to a point gets focused to a spot of finite area. In this case the effects due to diffraction can be taken into account by considering a plane wave incident on a circular aperture followed by a convex lens (Fig.). The analysis of the corresponding diffraction pattern is quite involved; however, in principle, it is similar to the analysis carried out to obtain the single-slit diffraction pattern. Taking into account the effects due to diffraction, the pattern on the focal plane would consist of a central bright region surrounded by concentric dark and bright rings (Fig.). A detailed analysis shows that the radius of the central bright region is approximately given by

`r_o=(1.22lamdaf)/(2a)=(0.61lamdaf)/a`

where f is the focal length of the lens and 2a is the diameter of the circular aperture or the diameter of the lens, whichever is smaller.

Although the size of the spot is very small, it plays an important role in determining the limit of resolution of optical instruments like a telescope or a microscope. For the two stars to be just resolved

`fDeltathetaapprox(r_o)=(0.61lamdaf)/a`

`=>` `Deltathetaapprox(0.61lamda)/a......(1)`

Thus Δθ will be small if the diameter of the objective is large. This implies that the telescope will have better resolving power if a is large. It is for this reason that for better resolution, a telescope must have a large diameter objective.

Resolving Power of Microscope

In this case, the object is placed slightly beyond f, so that a real image is formed at a distance v [Fig.]. The magnification ratio of image size to object size is given by `mapproxv/f`. It can be seen from Fig. that

`D/fapprox2tanbeta.......(2)`

where 2β is the angle subtended by the diameter of the objective lens at the focus of the microscope.

When the separation between two points in a microscopic specimen is comparable to the wavelength λ of the light, the diffraction effects become important. The image of a point object will again be a diffraction pattern whose size in the image plane will be

`vtheta=v((1.22lamda)/D)......(3)`

Two objects whose images are closer than this distance will not be resolved, they will be seen as one. The corresponding minimum separation, `d_(mi n)`, in the object plane is given by

`d_(mi n)=[v((1.22lamda)/D)]//m`
`=(1.22lamda)/D*v/m`
`=((1.22lamdaf)/D).......(4)`

Now, combining Eqs. (2) and (4), we get

`d_(mi n)=(1.22lamda)/(2tanbeta)approx(1.22lamda)/(2sinbeta).....(5)`

If the medium between the object and the objective lens is not air but a medium of refractive index n, Eq. (5) gets modified to

`d_(mi n)=(1.22lamda)/(2nsinbeta).....(6)`

The product n sinβ is called the numerical aperture and is sometimes marked on the objective.
The resolving power of the microscope is given by the reciprocal of the minimum separation of two points seen as distinct.

`text(Resolving Power)` `=1/(d_(mi n)`

In this case, the object is placed slightly beyond f, so that a real image is formed at a distance v [Fig.]. The magnification ratio of image size to object size is given by `mapproxv/f`. It can be seen from Fig. that

`D/fapprox2tanbeta.......(2)`

where 2β is the angle subtended by the diameter of the objective lens at the focus of the microscope.

When the separation between two points in a microscopic specimen is comparable to the wavelength λ of the light, the diffraction effects become important. The image of a point object will again be a diffraction pattern whose size in the image plane will be

`vtheta=v((1.22lamda)/D)......(3)`

Two objects whose images are closer than this distance will not be resolved, they will be seen as one. The corresponding minimum separation, `d_(mi n)`, in the object plane is given by

`d_(mi n)=[v((1.22lamda)/D)]//m`
`=(1.22lamda)/D*v/m`
`=((1.22lamdaf)/D).......(4)`

Now, combining Eqs. (2) and (4), we get

`d_(mi n)=(1.22lamda)/(2tanbeta)approx(1.22lamda)/(2sinbeta).....(5)`

If the medium between the object and the objective lens is not air but a medium of refractive index n, Eq. (5) gets modified to

`d_(mi n)=(1.22lamda)/(2nsinbeta).....(6)`

The product n sinβ is called the numerical aperture and is sometimes marked on the objective.
The resolving power of the microscope is given by the reciprocal of the minimum separation of two points seen as distinct.

`text(Resolving Power)` `=1/(d_(mi n)`

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