Modern
`Lens Design
`
`Warren J. Smith
`Chief Scientist
`Ks/ser Electro-Optics, Inc., Carlsbad, California
`and Consultant In Optics and Design
`
`Second Edition
`
`McGraw-Hill
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`
`96
`
`Chapter Five
`
`These aberrations are related by:
`
`TA=LA· tan U
`
`AA=TAll'
`
`OPD=fAA
`
`The optical path (OP = L,n · d) is related to the time of travel of light,
`which is equal to L,n · d!c. Ideally the OP from the object point to a
`reference sphere centered on the image point (and often located at the
`exit pupil-or at infinity) should be constant over the full aperture.
`The optical path difference, OPD = (OP ray- OPrer), where OPray is the
`path along a ray and OP ref is the path along the axis or along the prin·
`cipal ray. The pupil function is OPD (x, y); the wave (front) function
`is w(x, y) = OPD (x, y)/}., in waves; and the phase function 4>(x, y) is
`2nw (x, y) in radians.
`Aberrations may be intrinsic or induced. The intrinsic aberrations
`are those of a surface (or element) that are unaffected by the aberra·
`tions of the other surfaces. Induced aberrations are created by the
`aberrations (i.e., changes in the ray heights or angles) of the other ele·
`ments. Usually the lower-order aberrations of the other surfaces cause
`induced higher-order aberrations. For example, the third-order aber(cid:173)
`rations of preceding surfaces will induce fifth-order spherical in fol(cid:173)
`lowing surfaces. See Chap. 6, Sees. 6.3 and 6.4 for an example of how
`the third-order spherical and first-order chromatic aberration in the
`first element affect the zonal (fifth-order) spherical and spherochro(cid:173)
`matic of the lens.
`
`5.4 Scaling a Design, Its Aberrations, and its
`Modulation Transfer Function
`
`A lens prescription can be scaled to any desired focal length simply by
`multiplying all of its dimensions by the same constant. All of the linear
`aberration measures will then be scaled by the same factor. Note, how(cid:173)
`ever, that percent distortion, chromatic difference of magnification
`(CDM), the numerical aperture or {number, the ab~rrations expressed
`as angular aberrations, and any other angular characteristics remain
`completely unchanged by scaling.
`The exact diffraction modulation transfer function (MTF) cannot be
`scaled with the lens data. The diffraction MTF, since it includes dif(cid:173)
`fraction effects that depend on wavelength, will not scale because the
`wavelength is not ordinarily scaled with the lens. A geometric MTF can
`be scaled by dividing the spatial frequency ordinate of the MTF plot by
`the scaling factor. Of course, because it neglects diffraction, the geometric
`
`

`

`Lens Design Data
`
`97
`
`MTF is quite inaccurate unless the aberrations are large, with OPD to
`the order of one or two wavelengths (and the MTF is correspondingly
`poor).
`A diffraction MTF can be scaled very approximately as follows:
`Determine the OPD that corresponds to the calculated MTF value of the
`lens for several spatial frequencies. This can be done by comparing the
`MTF plot for the lens to Figs. 4.5 and 4.6, which relate the MTF to
`OPD. Then multiply the OPD by the scaling factor and, again using Figs.
`4.5 and 4.6, determine the MTF corresponding to these scaled OPD
`values. Obviously the accuracy of this procedure depends on how well
`the simple relationships of Figs. 4.5 and 4.6 represent the mix of aber(cid:173)
`rations in a real lens.
`In the event that a proposed change of aperture or field is expected to
`produce a change in the amount of the aberrations, one can attempt to
`scale the MTF as affected by aberration. This is done by determining the
`type of aberration that most severely limits the MTF, then scaling the
`OPD according to the way that this aberration scales with aperture or
`field, in a manner analogous to that described in Sec. 5.3. In general, OPD
`as a function of aperture varies as one higher exponent of the aperture
`than does the corresponding transverse aberration. For example, the
`OPD for third-order transverse spherical (which varies as Y) varies as
`the fourth power of the ray height. In a form analogous to Eqs. (5.3) and
`(5.4) which indicate a power series expansion of the transverse aberra(cid:173)
`tions as a function of aperture and field, Eq. (5.5) gives the relationship
`for OPD. As in Sec. 5.3, the terms of the equation refer to Fig. 5.4.
`
`OPD =A' 1s2 +A' 2sh cos 8
`+ B\s4 + B'2s3h cos 8+ B'3i h2 cos28+ B'4ih2 + B'6sh3 cos8
`+ C'1s6 + C'2s6h cos 8+ C'4s4h2 + C'6s4h2 cos28+ C'7s3h 3 cos38
`+ C'ss3h3 cos38+ C'l0s2h 4 + C'us2h4 cos28+ C'12sh5 cos 8
`(5.5)
`
`Note that although the constants h ere correspond to those in Eqs. (5.3)
`and (5.4), they are not numerically the same and are primed to so indi(cid:173)
`cate; however, (because rays are normal to wavefronts) the expressions
`are related by
`
`'=TA = l aOPD
`ay
`y N
`Y
`
`and
`
`x' = TA = __!_ aOPD
`X N ax
`
`(5.6)
`
`where lis the pupil-to-image distance and N is the image space index.
`
`

`

`98
`
`Chapter Five
`
`Note that the exponent of the semiaperture terms is larger by 1 in
`the wavefront expression than in the ray-intercept equations. Fourth(cid:173)
`order OPD terms correspond to third-order transverse aberration terms.
`
`5.5 Notes on the Interpretation of Ray
`Intercept Plots
`
`Intercept plots
`5.5.1
`When the image plane intersection heights of a fan of meridional rays
`are plotted against the slope of the rays as they emerge from the lens,
`the resultant curve is called a ray intercept curve, an H' - tan U curve,
`or sometimes (erroneously) a rim ray curve. The shape of the intercept
`curve not only indicates the amount of spreading or blurring of the image
`directly, but also can serve to indicate which aberrations are present.
`In Fig. 5.6 an oblique fan of rays from a distant object point is brought
`to a perfect focus at point P. If the reference plane passes through P, it
`is apparent that the H'- tan U curve will be a straight horizontal line;
`however, if the reference plane is behind P (as shown) then the ray
`intercept curve becomes a tilted straight line since the height H'
`decreases exactly as tan U decreases. Thus it is apparent that shifting
`the reference plane (or focusing the system) is equivalent to a rotation
`of the H'- tan U coordinates. A valuable feature of this type of aberra(cid:173)
`tion representation is that one can immediately assess the effects of refo(cid:173)
`cusing the optical system by a simple rotation of the abscissa of the
`
`REFERENCE PLANE
`
`----::::::::~~~!·
`I
`
`H'
`
`~ un~
`
`~---=~~~~TANU'
`RAY lfHERCEPT CURVE
`Figure 5.6 The ray intercept curve (H versus - tan U) of an
`image point that does not lie in the reference plane is a tilted
`straight line. The slope of the line (dH /d tan U) is mathe(cid:173)
`matically identical to the distance from the reference plane
`to the point P. Note that this distance is equal to X, the tan(cid:173)
`gential field curvature (if the reference plane is the paraxial
`focal plane).
`
`