Abstract

A conical mirror offers a means of magnifying the image of an object on an opaque screen, so that it may be viewed from a position directly in front of the screen, and the projecting distance necessary to produce the magnification is much less than that required by present methods. An experiment shows that the conical mirror gives high resolution when used in a specific projecting system. It may provide a convenient means with which to obtain high magnification for many optical systems. The magnification is shown by theory and experiment to be equal to the secant of the conical mirror angle.

© 1965 Optical Society of America

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Figures (8)

Fig. 1
Fig. 1

Ray diagram showing a top view of a point object Q being projected to a point image P′ by means of a conical mirror. Point Q seen as point P of Fig. 2.

Fig. 2
Fig. 2

Ray diagram showing a side view of a point object Q being projected to a point image P′ by means of a conical mirror.

Fig. 3
Fig. 3

A perspective view showing light rays from point Q being reflected from the circular arc VMN of a conical mirror and, thus, brought to a focus at point P′ of the screen.

Fig. 4
Fig. 4

Ray diagram showing light rays from point Q being reflected from the circular arc VMN of a conical mirror and, thus, brought to a focus at point P′ of the screen.

Fig. 5
Fig. 5

Ray diagram showing a lens–cone system for projecting magnified images upon a screen. L—point source of illuminating lamp; C—condenser lens; O—object to be magnified; P—microscopic objective lens with attached low-power planocylinderical lens; A—line of the conical mirror axis; D—axial line of lens system; Q—primary focal point of astigmatic lens axial image; P′ —secondary focal point of astigmatic lens axial image; I—opaque light-diffusing screen; V—the conical mirror.

Fig. 6
Fig. 6

A photographic enlargement showing cell structure of a plant louse. This demonstrates a magnification of 87× and is the size of the image that is projected onto the conical mirror.

Fig. 7
Fig. 7

A photographic enlargement showing cell structure of a plant louse. This demonstrates a magnification of 304× and is the size of the image that is projected onto the screen from the conical mirror.

Fig. 8
Fig. 8

A photographic enlargement showing cell structure of a plant louse. This demonstrates a magnification of 304× and is produced by the objective lens only. The numerical aperture of the objective lens was 0.25.

Equations (20)

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O P O P = V P V P .
O P O P = sec ( conical mirror angle ) .
- r r = - M .
s = θ 1 θ 2 r 2 + ( d r d θ ) 2 d θ .
s = θ 1 θ 2 ( - M r ) 2 + [ d ( - M r ) d θ ] 2 d θ .
s = ( - M ) θ 1 θ 2 r 2 + ( d r d θ ) 2 d θ .
A E = ( A M ) 2 ( Q A ) / [ ( Q M ) 2 - ( Q A ) 2 ] .
( Q M ) 2 = ( Q V ) 2 / ( 1 - sin 2 ϕ ) .
A E = ( A M ) 2 ( Q A ) [ ( Q V ) 2 / ( 1 - sin 2 ϕ ) ] - ( Q A ) 2 .
V P Q V = A P Q A = V P - V F V F = M ,
V P Q V = V P - V F V F
1 Q V + 1 V P = 1 V F .
2 x - ( tan α / 2 ) y = 0.
sin β Q M = sin 2 ω Q A + A E ,
sin β 2 cos ω ( Q M ) = sin ω Q A + A E .
sin β A M = sin ω A E .
2 cos ω ( Q M ) A M = Q A + A E A E .
2 cos ω = [ ( Q M ) 2 + ( A M ) 2 - ( Q A ) 2 ] / ( Q M ) ( A M ) .
( Q M ) 2 + ( A M ) 2 - ( Q A ) 2 ( A M ) 2 = Q A + A E A E
( Q M ) 2 - ( Q A ) 2 ( A M ) 2 + 1 = Q A A E + 1.

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