Huygen's Eye-piece

Eye – piece

It is the system of two co - axial lenses having definite focal length and seperated one – another by a certain distance whose function is to produce magnified image without aberration. The lens which his exposed towards the object is called field lens and lens which is used to observe the final image is called Eye-lens. Threfore, Eye-piece = field image + eye lens

There are two types of eye-piece; they are:

(i) Huygen’s eye-piece

(ii) Ramsden’s eye-piece

Huygen’s Eye-piece

It consist of two plano convex lens having focal length in the ratio 3 : 1 and seperated by a distance which is equal to the difference between focal lengths. The convex surface of each lens exposed towards the object.

(i) Condition of chromatic aberration:

The system is said to from the chromatic aberration,if the two lenses with focal length \(f_1\) and \(f_2\) are placed apart by a distance of d, such that; \(d = \frac{f_1 +f_2}{2}\)

For the case of Huygen’s eye-piece, \(f_1\) = 3f and \(f_2\) = f,

then \(d = \frac{f_1 + f_2}{2} = \frac{3f + f}{2} = \frac{4f}{2} = 2f = d\)

hence, we can say that it is free from chromatic aberration.

(ii) Combined focal length:

The combined focal length of the two plano-convex lens which are seperated by distnce d is $$\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2}-\frac{d}{f_1 f_2}$$

$$=\frac{1}{3f}+\frac{1}{f}-\frac{2f}{3f^2}$$

$$=\frac{1+3-2}{3f}$$

$$=\frac{2}{3f}$$

$$\therefore \;\;F=\frac{3}{2}f$$

Hence,the equivalent lens should be placed at a distance \(\frac{3}{2}f\) from the image formed by objective in the absence of eye-piece or at a distance = \(\frac{F.d}{f_1} = \frac{\frac{3}{2}f.2f}{f} = 3f\) from field lens.

Hence, the position of equivalent lens should be at a distance 3f – 2f = f from the eye lens. Also, we have the distance of image formed by objective in the absence of eye piece from eye lens to be

\(\frac{3}{2}f – f = \frac{f}{2}\)

(iii) Condition of Spherical aberration:

The system of lens is said to be free from spherical aberration if difference in focal length is equal to the seperation between the lens.( i.e. d = \(f_1 – f_2\) )$$i.e. f_1 – f_2 = 3f – f = 2f = d$$

So, spherical aberrations is minimized in this type of eye-piece. If both spherical and chromatic aberrations are to be minimized simultaneously, then the following conditions must be satisfied: \(d=\frac{f_1+f_2}{2}\) for chromatic aberration

\(d = f_1 - f_2\) for spherical aberrations

Combining the two conditons, we have $$\frac{f_1+f_2}{2}=f_2-f_1$$

$$or,\;\;\;\frac{f_1+f_2}{2}=f_2-f_1$$

$$or,\;\;\;f_1+f_2=2f_2-2f_1$$

$$3f_1=f_2$$

$$thus,\;\;d=f_2-f_1$$

$$=3f_1-f_1$$

$$\therefore\;\; d=2f_1$$

Therefore, to satisfy the condition for minimum chromatic and spherical aberration, the focal length of field lens should be three times the focal length of eye lens and the distance of seperation between them should be equal to twice the focal length of eye lens. Huygen’s eye-piece is constructed depending on this principle. Each lens produces an equal deviation for the incident ray and satisfies the condition of achromatism.

(iv) Position of final image:

in the absence of eye-lens, the image formed by the object through field lens is \(II_1\), which is virtual. The field lens produces a real image at \(II’_1\) after placing the eye-lens. So that, the final image formed at infinity.

The position \(II_1\)is the virtual position of the object for field lens. It produces a real image at \(I’I’_1\) which is used asan object for eye-lens. The eye-lens produces magnified image at infinity.

(v) Position of cross-wire: Cross wire syste mis used to measure the magnification of final image mathematically. In the case of Huygen’s eye-piece, the cross-wire can’t be located at the initial position of the object due to it’s virtual property. If it is placed at \(I’I’_1\), the ratio of magnification for the object and cross-wire becomes unequal. It is so because the object is magnified by both field lens and eye- lens whereas cross-wire is magnified by eye-lens only. Therefore, cross-wire can’t be used in Huygrn’s eye-piece for mathematical measurement of image.

References:

Adhikari, P.B, Daya Nidhi Chhatkuli and Iswar Prasad Koirala. A Textbook of Physics. Vol. II. Kathmandu: Sukunda Pustak Bhawan, 2012.

Jenkins, F.A and H.E White. Fundamental of optics. New York (USA): McGraw-Hill Book Co, 1976.

wood, R.W. Physical Optics. New York (USA): Dover Publication , 1934.