LENS (from Lat. lens, lentil, on account of the similarity of the form of a lens to that of a lentil seed), in optics, an instrument which refracts the luminous rays proceeding from an object in such a manner as to produce an image of the object. It may be regarded as having four principal functions: (1) to produce an image larger than the object, as in the magnifying glass, microscope, &c.; (2) to produce an image smaller than the object, as in the ordinary photographic camera; (3) to convert rays proceeding from a point or other luminous source into a definite pencil, as in lighthouse lenses, the engraver's globe, &c.; (4) to collect luminous and heating rays into a smaller area, as in the burning glass. A lens made up of two or more lenses cemented together or very close to each other is termed " composite or " compound "; several lenses arranged in succession at a distance from each other form a " system of lenses," and if the axes be collinear a " centred system." This article is concerned with the general theory of lenses; and more particularly with spherical lenses. For a special part of the theory of lenses see ABERRATION; the instruments in which the lenses occur are treated under their own headings.
The most important type of lens is the spherical lens, which is a piece of transparent material bounded by two spherical surfaces, the boundary at the edge being usually cylindrical or conical. The line joining the centres, Cl, C2 (fig. I), of the bounding surfaces is termed the axis; the points .S1, Si, at
which the axis intersects the surfaces; are termed the'" vertices " of the lens; and the distance between the vertices is termed the " thickness." If the edge be everywhere equidistant from the vertex, the lens is " centred."
Although light is really a wave motion in the aether, it is only necessary, in the investigation of the optical properties of systems of lenses, to trace the rectilinear path of the waves, i.e. the direction. of the normal to the wave front, and this can be done
by purely geometrical methods. It will be assumed that light, ~ so long as it traverses the same medium, always travels in a straight line; and in following out the geometrical theory it will always be assumed that the light travels from left to right; accordingly all distances measured in this direction are positive, while those measured in the opposite direction are
negative.
Theory of Optical Representation.—If a pencil of rays, i.e. the totality of the rays proceeding from a luminous point, falls on a lens or lens system, a section of the pencil, determined by the dimensions of the system, will be transmitted. The emergent rays will have directions differing from those of the incident rays, the alteration, however, being such that the transmitted rays are convergent in the " imagepoint," just as the incident rays diverge from the " objectpoint.' With each incident ray is associated an emergent ray; such pairs are termed " conjugate ray pairs." Similarly we define an objectpoint and its imagepoint as con
jugate points "; all objectpoints lie in the " objectspace," and all imagepoints lie in the " imagespace." The laws of optical representations were first deduced in their most general form by E. Abbe, who assumed (i) that an optical representation always exists, and (2) that to every point in. thethe two rays CD1, and EF1; the conjugate point P'i will be determined by the intersection of the conjugate rays C'D'1 and E'F'1, the points D'1, F'1, being readily found from the magnifications 1q31, $2. Since PP1 is parallel to CE and also to DF, then DF = D1F1. Since the plane 02 is similarly represented in 0'2, D'F' = D'IF'1; this is impossible unless P'P'i be parallel to C'E'. Therefore every perpendicular objectplane is represented by a perpendicular imageplane.
Let 0 be the intersection of the line PPI with the axis, and let 0' be its conjugate; then it may be shown that a fixed magnification S2 exists for the planes 0 and 0'. For PP1/FF1=001/0102, P'P'1/F'F'1=0'0'/010'2, and F'F'1=(32FFI. Eliminating FF1 and F'F'1 between these ratios, we have P'P'1/PP1$2=0'0'10102/001. 0'10'2, or 03 =$2.0'O'I.O102/001.0'10'2, i.e. $3=132Xa product of the axial distances.
The determination of the imagepoint of a given objectpoint is facilitated by means of the socalled " cardinal points " of the optical system. To determine the imagepoint 0'1 (fig. 3) corresponding to the objectpoint 01, we begin by choosing from the ray pencil proceeding from 01, the ray parallel with the axis, i.e. intersecting the axis at infinity.. Since the axis is its own conjugate, the parallel ray through 01 must intersect the axis after refraction (say at F'). Then F' is the imagepoint of ,an objectpoint situated at infinity on the axis, and is termed the "second principal focus " (German der bildseitige Brennpunkt, the imageside focus). Similarly if 0'4 be on the parallel through 01 but in the imagespace, then the conjugate ray must intersect the axis at a point (say F), which is conjugate with the point at infinity, on the axis in the imagespace. This point is termed the " first principal focus " (German der objektseitige Brennpunkt, the objectside. focus).
Let H1, H'1 be the intersections of the focal rays through F and F' with the line 010'4. These two points are in the position of object and image, since they are each determined by two pairs of conjugate rays (OiH1 being conjugate with H'1F', and O'4H'1 with H1F). It has already been shown that objectplanes perpendicular to the axis are represented by imageplanes also perpendicular to the axis. Two vertical planes through HI and H'1, are related as object and imageplanes; and if these planes intersect the axis in two points H and If', these points are named the " principal," or " Gauss points " of the system, H being the objectside " and H' the " imageside principal point." The vertical planes containing H and H' are the " principal planes." It is obvious that conjugate points in these planes are equidistant from the axis; in other words, the magnification $ of the pair of planes is unity. An additional characteristic of the principal planes is that the object and image are direct and not inverted. The distances between F and H, and between F' and H' are termed the focal lengths; the former may be called the " objectside focal length " and the latter the " imageside focal length." The two focal points and the two principal points constitute the socalled four cardinal points of the system, and with their aid the image of any object can be readily determined.
Equations relating to the Focal Points.—We know that the ray proceeding from the object point 01, parallel to the axis and intersecting the principal plane H in H1, passes through H'1 and F'.
Choose from the pencil a second ray which contains F and intersects the principal plane H in H2; then the conjugate ray must contain points corresponding to F and H2. The conjugate of F is the point at infinity on the axis, i.e. on the ray parallel to the axis. The image of H2 must be in the plane H' at the same distance from, and on the same side of, the axis, as in H'2. The straight line passing through H'2 parallel to the axis intersects the ray H'1F' in the point 01, which must be the image of 01. If 0 be the foct of the perpendicular from Oi to • the axis, then 001 is represented by the line 0'0'1 also perpendicular to the axis.
This construction is not applicable if the object or image be infinitely distant. For example, if the object 001 be at infinity (0 being assumed to be on the axis for the sake of simplicity), so that the object appears under a constant angle w, we know that the second principal focus is conjugate with the infinitely distant axispoint. If the object is at infinity in a plane perpendicular to the axis, the image must be in the perpendicular plane through the focal point F' (fig. 4).
The size y' of the image is readily deduced. Of the parallel rays from the object subtending the angle w, there is one which passes
objectspace there corresponds a point in the imagespace, these points being mutually convertible by straight rays; in other words, with each objectpoint is associated one, and only one, image .point, and if the objectpoint be placed at the imagepoint, the conjugate point is the original objectpoint. Such a transformation is termed a " collineation," since it transforms points into points and straight lines into straight lines. Prior to Abbe, however, James Clerk Maxwell published, in 1856, a geometrical theory of optical representation, but his methods were unknown to Abbe and to his pupils until O. Eppenstein drew attention to them. Although Maxwell's theory is not so general as Abbe's, it is used here since its methods permit a simple and convenient deduction of the laws.
Maxwell assumed that two objectplanes perpendicular to the axis are represented sharply and similarly in two imageplanes also perpendicular to the axis (by " sharply " is meant that the assumed ideal instrument unites all the rays proceeding from an objectpoint in one of the two planes in its imagepoint, the rays being generally transmitted by the system). The symmetry of the axis being premised, it is sufficient to deduce laws for a plane containing the axis.. In fig. 2 let 01, 02 be the two points in which the perpendicular objectplanes meet the axis; and since the axis corresponds to itself, the two conjugate points O'1, O'z, are at the intersections of the two imageplanes with the axis. We denote the four planes by the letters 0I, 02, and 0'1, 0'2. If two points A, C be taken in the plane Oi, their images are A', C' in the plane 0I, and since the p planes are represented similarly, we have 0'1A' : 01A=0'1C'1 :OIC =$i (say), in which th is easily seen to be the linear magnification of the planepair 01, 0'1. Similarly, if two points B, D be taken in the plane 0z and their images B', D' in the plane 0'2, we have O'2B':02B =O'3D':02D =02 (say), $2 being the linear magnification of the planepair 02, 0'2. The joins of A and B and of C and D intersect in a point P, and the joins of the conjugate points similarly determine the point P'.
If P' is the only possible imagepoint of the objectpoint P, then the conjugate of every ray passing through P must pass through P'. To prove this, take a third Iine through P intersecting the planes 01, 02 in the points E, F, and by means of the magnifications Pi, 132 determine the conjugate points E', F' in the planes O'1, O'2. Since the planes 01, 02 are parallel, then AC/AE=BD/BF; and since these planes are represented similarly in O'1, 0'2, then A'C'/A'E' =B'D'/B'F'. This proportion is only possible when the straight line E'F' contains the point P'. Since P was any point whatever, it follows that every point of the objectspace is represented in one and only one point in the imagespace. ,
Take a second objectpoint PI, vertically under P and defined by
corresponding image subtend the same angle at the principal points.
Equations Relating td the Principal Points.It is sometimes desirable to determine the distances of an object and its image, not from the focal points, but from the principal points. Let A (see fig. 3) be the principal point distance of the object and A' that of the image, we then have
A=HO=HF+F0=F0—FH=x—f,
A'=H'O'=H'F'+F'O'=F'O'—F'H'=x'—f,
whence x=A+f and x'=A'+f'.
Using xx'=ff, we have (A+f) (A'+f')=ff', which leads to AA'+ Af'+A'f=O, or
through the first principal focus F, and intersects the principal plane H in Hi. Its conjugate ray passes through H' parallel to, and at the same distance from the axis, and intersects the imageside focal plane in 0'1; this point is the image of 01, and y' is its magnitude. From the figure we have tan w = HH1/FH = y'/f, or f = y'/tan w; this equation was used by Gauss to define the focal length.
Referring to fig. 3, we have from the similarity of the triangles OOIF and HH2F, HH2/001=FH/F0, or O'O'1/001=FH/F0. Let y be the magnitude of the object OOi, y' that of the image 0'0'1, x the focal distance FO of the object, and f the objectside focal distance FH; then the above equation may be written y'/y=f/x. From the similar triangles H'1H'F' and O'10'F', we obtain 0'0'1/001 =F'O'/F'H'. Let x' be the focal distance of the image F'O', and f' the imageside focal length F'H'; then y'/y= x'/f. The ratio of the size of the image to the size of the object is termed the lateral magnification. Denoting this by /3, we have
9=Y /y=f/x=x'/f , (I)
and also
xx'=ff. (2)
By differentiating equation (2) we obtain
dx' = — (ff'/x2)dx or dx'/dx = (3)
The ratio of the displacement of the image dx' to the displacement of the object dx is the axial magnification, and is denoted by a. Equation (3) gives important information on the displacement of the image when the object is moved. Since f and f always have contrary signs (as is proved below), the product —ff.' is invariably positive, and since x2 is positive for all values of x, it follows that dx and dx' have the same sign, i.e. the object and image always move in the same direction, either both in the direction of the light, or both in the opposite direction. This is shown in fig. 3 by the object 0302 and the image 0'30'2.
If two conjugate rays be drawn from two conjugate points on the axis, making angles u and u' with the axis, as for example the rays OH1, O'H'1, in fig. 3, u is termed the " angular aperture for the object," and u' the " angular aperture for the image." The ratio of the tangents of these angles is termed the " convergence " and is denoted by y, thus y =tan u'/tan u. Now tan u = H'H'1/O'H' =H'H'1/(O'F'+F'H')=H'H'1/(F'H'—F'O'). Also tan u=HH1/OH =HH1/(OF+FH)=HH1/(FH= FO). Consequently y=(FH—FO) /(F'H'—F'O'), or, in our previous notation, y=(f—x)/(f'—x').
From equation (I) f/x=x'/f, we obtain by subtracting unity from both sides (f —x)/x = (x' —f) /f, and consequently
f —x  x  fy
f —x' f, x'
(4)
From equations (I), (3) and (4), it is seen that a simple relation exists between the lateral magnification, the axial magnification and the convergence, viz. ay =O.
In addition to the four cardinal points F, H, F', H', J. B. Listing,
" Beitrage ails physiologischen Optik," Gottinger Studien (1845)
introduced the socalled " nodal points " (Knotenpunkte) of the
system, which are
the two conjugate
points from which
the object and
image appear under
the same angle. In
fig. 5 let K be the
nodal point from
appears under the
same angle as the image y' from the other nodal point K'. Then 0O1/KO=O'O'1/K'O', or 001/(KF+FO) O'O'1/(K'F'+F'O'), or 001/(FO—FK)=O'O'1/(F'O'—F'K'). Calling the focal distances FK and F'K', X and X', we have y/(x—X)=y'/(x'—X'), and since y'/y=<3, it follows that 1/(x —X) = /3/(x' —X'). Replace x' and X' by the values given in equation (2), and we obtain
f
x I X=13/ (x  ff' ) or I =13X7.
Since $=fix =x'/f, we have f'= —X, f= —X'.
These equations show that to determine the nodal points, it is only necessary to measure the focal distance of the second principal focus from the first principal focus, and vice versa. In the special case when the initial and final medium is the same, as for example, a lens in air, we have f= f', and the nodal points coincide with the principal points of the system; we then speak of the " nodal point property of the principal points," meaning that the object and
I+A,+A=O;
this becomes in the special case when f = —f',
I I _I
A' A f
To express the linear magnification in terms of the principal point distances, we start with equation (4) (/—x)/(f'—x') = —x/f. From this we obtain A/A'= —x/f, or x= —f A/A'; and by using equation (I) we have /3= —fA'/f'A.
In the special case of f = —f', this becomes 13=A'/A=y'/y; from which it follows that the ratio of the dimensions of the object and image is equal to the ratio of the distances of the object and image from the principal points.
The convergence can be determined in terms of A and A' by substituting x= —f'A/A' in equation (4), when we obtain 7=A/A'.
Compound Systems.—In discussing the laws relating to compound systems, we assume that the cardinal points of the component systems are known, and also that the combinations are centred, i.e. that the axes of the component lenses coincide. If some object be represented by two systems arranged one behind the other, we can regard the systems as cooperating in the formation of the final image.
Let such a system be represented in fig. 6. The two single systems are denoted by the suffixes i and 2; for example, Fl is the first
H'2
principal focus of the first, and F'2 the second principal focus of the second system. A ray parallel to the axis at a distance y passes through the second principal focus F'1 of the first system, intersecting the axis at an angle w'1. The point F'1 will be represented in the second system by the point F', which is therefore conjugate to the point at infinity for the entire system, i.e. it is the second principal focus of the compound system. The representation of F'1 in F' by the second system leads to the relations F2F'1=x2, and F'2F'=x'2, whence x2x'2=f2f'2. Denoting the distance between the adjacent focal planes F'1, F2 by 0, we have d =F'1F2= —F2F'1, so that x'2= —f2f'2/0. A similar ray parallel to the axis at a distance y proceeding from the imageside will intersect the axis at the focal point F2; and by finding the image of this point in the first system, we determine the first principal focus of the compound system. Equation (2) gives xix'i and since x'1 = F'1F2 =Q, we have x1= f1 f1/A as the distance of the first principal focus F of the compound system from the first principal focus Fl of the first system.
To determine the focal lengths f and f' of the compound system and the principal points H and H', we employ the equations defining the focal lengths, viz. f=y'/tan w, and f'=y/tan w'. From the construction (fig. 6) tan w'i =y/f1. The variation of the angle w'1 by the second system is deduced from the equation to the convergence, viz. 7—tan w'2/tan w2=x2/f'2=A/f'2 and since w2=w'1, we have tan w'2= (A/f'2) tan w'1. Since w'=w'2 in our system of notation, we have
y _ yf'2
f tan w' Q tan w'1 0
By taking a ray proceeding from the imageside we obtain for the first principal focal distance of the combination
f = —f1f2/o•
In the particular case in which A=0, the two focal planes F'1, F2 coincide, and the focal lengths f, f' are infinite. Such a system is called a telescopic system, and this condition is realized in a telescope focused for a normal eye.
So far we have assumed that all the rays proceeding from an objectpoint are exactly united in an imagepoint after transmission through the ideal system. The question now arises so to how far this assumption is justified for spherical lenses. To investigate this it is simplest to trace the path of a ray through one spherical
H2
(5)
refracting surface. Let such a surface divide media of refractive indices n and n', the former being to the left. The point where the axis intersects the surface is the vertex. S (fig. 7). Denote the distance of the axial objectpoint 0 from S by s; the distance from
0 to the point of incidence P by p; the radius of the spherical surface by r; and the distance OC by c, C being the centre of the sphere. Let u be the angle made by the ray with the axis, and i the angle of incidence, i.e. the angle between the ray and the normal to the sphere at the point of incidence. The corresponding quantities in the imagespace are denoted by the same letters with a dash. From the triangle O'PC we have sin u=(r/c) sin i, and from the triangle O'PC we have sin u' _ (r/c') sin i'. By Snell's law we have n'/n =sin i/sin i', and also 4 = u'+2'. Consequently c' and the position of the image may be found.
To determine whether all the rays proceeding from 0 are refracted through 0', we investigate the triangle OPO'. We have pip' =sin u'/sin u. Substituting for sin u and sin u' the values found above, we obtain p'/p=c' sin sic sin i'=n'c'/nc. Also c=OC=CS+ SO=SC+SO=sr, and similarly c' =s's. Substituting these values we obtain
=n' (s'r) n(sr) n
='(s'r)
p n(sr) or p —• (6)
To obtain p and p' we use the triangles OPC and O'PC; we have p2 = (s .)2+r2+2r(sr) cos ,p, p'2 = (s'r)2+r2+2r(s'r) cos ¢. Hence if s, r, n and n' be constant, s' must vary as ¢ varies. The refracted rays therefore do not reunite in a point, and the deflection is termed the spherical aberration (see ABERRATION).
Developing cos 4, in powers of 4,, we obtain
P'=(sr)2+r2+2r(sr) I 14i68 ... ,
and therefore for such values of 4. for which the second and higher powers may be neglected, we have p2=(sr)2+r2+2r(sr), i.e. p=s, and similarly p'=s'. Equation (6) then becomes n(sr)/s= n'(s'r)/s' or
f having the opposite sign to ft. Denoting the distance F'1 F2 by 0, we have 0=F',F2 esF'1S1+S,Sz+S2F2=F'1S,+S,S2FZS2=f(,4d f2. Substituting for f, and f2 we obtain
0 =  nr1 +d + nr2
nI n
Writing R=0(nI), this relation becomes
R=n(r27'1)+d(nI).
We have already shown that f (the first principal focal length of a compound system) =flf2/ . Substituting for in f2, and 0 the values found above, we obtain
r1 r2n
f=(n—I)R(n—t)in(r2r1)+d(n—i)}' which is equivalent to
z f=(n_I)r1 r2)+( _r1rn
If the lens be infinitely thin, i.e. if d be zero, we have for the first principal focal length,
f= nt )}i ( 7'1 r i By the same method we obtain for the second principal focal length
, f'nr,ri,
f = o (n—i)K
The reciprocal of the focal length is termed the power of the lens and is denoted by sp. In formulae involving 4, it is customary to
rlr2n
s' = sn+n' r n (7) This relation shows that in a very small central aperture in which the equation p=s holds, all rays proceeding from an objectpoint
are exactly united in an imagepoint, and therefore the equations previously deduced are valid for this aperture. K. F. Gauss derived the equations for thin pencils in his Dioptrische Untersuchungen (284o) by very elegant methods. More recently the laws relating to systems with finite aperture have been. approximately realized, as for example, in wellcorrected photographic objectives.
Position of the Cardinal Points of a Lens.—Taking the case of a single spherical refracting surface, and limiting ourselves to the small central aperture, it is seen that the second principal focus F' is obtained when s is infinitely great. Consequently s'=f'; the difference of sign is obvious, since s' is measured from S, while f' is measured from F'. The focal lengths are directly deducible from equation (7)
:
f' /(n/n) (9) )
f=nr/((n'
By joining this simple refracting system with a similar one, so that the second spherical surface limits the medium of refractive index n', we derive the spherical lens. Generally the two spherical surfaces enclose a glass lens, and are bounded on the outside by air of refractive index x.
The deduction of the cardinal points of a spherical glass lens in air from the relations already proved is readily effected if we regard the lens as a combination of two systems each having one refracting surface, the light passing in the first system from air to glass, and in the second from glass to air. If we know the refractive index of the glass n, the radii r,, r2 of the spherical surfaces, and the distances of the two lensvertices (or the thickness of the lens d) we can determine all the properties of the lens. A biconvex lens is shown in fig. 8. Let F1 be the first principal focus of the first system of radius r1, and F1' the second principal focus; and let S, be its vertex. Denote the distance F, S, (the first principal focal length) by fi, and the corresponding distance F'1 S1 by f',. Let the corresponding quantities in the second system be denoted by the same letters with the suffix 2.
By equations (8) and (9) we have
ri : nr2 , r2
fi =n f ,1=—nnr, f2=—n—i' f 2 _FIG. 8.
denote the reciprocal of the radii by the symbol p; we thus have
4,=I/f, p=I/r. Equation (Io) thus becomes
= (n — I) (pi —pi) + (n  i )2dpipz n
The unit of power employed by spectaclemakers is termed the diopter or die pine (see SPECTACLES).
We proceed to determine the distances of the focal points from the vertices of the lens, i.e. the distances FS1 and F'S2. Since F is represented by the first system in F2, we have by equation (2)
f=flf',=— nr12
x1 x'1 0 (nI)R'
where x1=F,F, and x'1=F'1F2 =0• The distance of the first principal focus from the vertex S, i.e. S1F, which we denote by SP is given by SF=SIF=SIF,+F,F=F,S1+F1F. Now F1S1 is the distance from the vertex of the first principal focus of the first system, i.e. fi, and F,F=xi. Substituting these values, we obtain
nr12 (nr1 +R)
sF ni (n1)R (nI)R
The distance F'2F' or x'2 is similarly determined by considering F', to be represented by the second system in F'.
We have
x'2 .f212 _ _f2 ~z = nr22
x2 0 (nI)R'
sF _ x,2 — f,2 _ r(nnri ) R )
where sF' denotes the distance of the second principal focus from the vertex S2.
The two focal lengths and the distances of the foci from the vertices being known, the positions of the remaining cardinal points, i.e. the principal points H and H', are readily determined. Let s,., =S,H, i.e. the distance of the objectside principal point from the vertex of the first surface, and sH~ =SSH', i.e. the distance of the imageside principal point from the vertex of the second surface, then f = FH =FS1+S1H = S1F+S1H = sr +sH ; hence 5 H =ss+f
dri/R. Similarly SH/=sF,+f'= —dr2/R. It is readily seen that the distances SH and 5H, are in the ratio of the radii r1 and r2.
The distance between the two principal planes (the interstitium) is deduced very simply. We have S1Si=S1H+HH'+H'S2, or HH'=S1S2  S1H+S2H'. Substituting, we have
HH' =ds,,+sH, =d(nI) (r2 ri+d)/R.
The interstitium becomes zero, or the two principal planes coincide, if d=r1r2.
We have now derived all the properties of the lens in terms of its elements, viz. the refractive index, the radii of the surfaces, and the thickness.
Forms of Lenses.—By varying the signs and relative magnitude of the radii, lenses may be divided into two groups according to their action, and into four groups according to their form.
According to their action, lenses are either collecting, convergent
so that
and condensing, or divergent and dispersing; the term positive is sometimes applied to the former, and the term negative to the latter. Convergent lenses transform a parallel pencil into a converging one, and increase the convergence, and diminish the divergence of any pencil. Divergent lenses, on the other hand, transform a parallel pencil into a diverging one, and diminish the convergence, and increase the divergence of any pencil. In convergent lenses the first principal focal distance is positive and the second principal focal distance negative; in divergent lenses the converse holds.
The four forms of lenses are interpretable by means of equation (to).
(nI) In (r2ri) +d (nI) }
(I) If r1 be positive and r2 negative. This type is called biconvex (fig. 9, I). The first principal focus is in front of the lens, and the second principal focus behind the lens, and the two principal points
r1r2n
r 17
\al 1\ L__.
2 4 5 6
are inside the lens. The order of the cardinal points is therefore FS1HH'S2F'. The lens is convergent so long as the thickness is less than n(rir2)/(n–i). The special case when one of the radii is infinite, in other words, when one of the bounding surfaces is plane is shown in fig. 9, 2. Such a collective lens is termed pianoconvex. As d increases, F and H move to the right and F' and H' to the left. If d=n(r1–r2)/(n–i), the focal length is infinite, i.e. the lens is telescopic. If the thickness be greater than n(r1–r2)l(n–I), the lens is dispersive, and the order of the cardinal points is HFS1S2F'H'.
(2) If r1 is negative and r2 positive. This type is called biconcave (fig. 9, 4). Such lenses are dispersive for all thicknesses. If d increases, the radii remaining constant, the focal lengths diminish. It is seen from the equations giving the distances of the cardinal points from the vertices that the first principal focus F is always behind Si, and the second principal focus F' always in front of S2, and that the principal points are within the lens, H' always following H. If one of the radii becomes infinite, the lens is pianoconcave
(fig. 9, 5).
(3) If the radii are both positive. These lenses are called convexoconcave. Two cases occur according as r2> r1, or
End of Article: LENS (from Lat. lens, lentil, on account of the similarity of the form of a lens to that of a lentil seed) 

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