Index variations in cover sheets

Alan B. Marchant

When an optical system focuses light through a transparent cover sheet, refractive-index variations createspherical aberration which can be partially compensated by adjusting the cover-sheet thickness. In practice,a wide range of refractive index can be tolerated by a diffraction-limited objective lens given such thicknessoptimization. For example, a system with a numerical aperture of 0.6 and a nominal cover-sheet thickness of1.2 mm can tolerate a refractive-index range from 1.45 to 1.60.

1. Introduction

Cover sheets are used in a variety of applications toprotect surfaces while permitting their examination byhigh-resolution optical systems. Historically theirmost widespread application has been in microscopy.A new application of increasing importance is opticalrecording. The recording surface of an optical diskmust be protected from dust and atmospheric con-tamination. This protection may take the form of adiscrete protective element or a transparent sub-strate2 between the recording surface and the read/write head. The term cover sheet as used in thisdiscussion refers to all such protective elements.

To provide diffraction-limited performance througha cover sheet, an objective lens must be specially de-signed. Generally, the lens design will assume somenominal cover-sheet thickness and refractive index.Then the lens quality will be specified for operationthrough such a cover sheet. However, variations inmaterials, processes, or wavelength may lead to anactual cover-sheet refractive index and thicknesswhich differ from the nominal values. These differ-ences will create spherical aberration in the focusedbeam (ignoring possible cover-sheet tilt). To lowestorder, these aberrations can be corrected by refocusingthe objective lens. Given a known (but not nominal)refractive index, the aberration can be reduced furtherby reoptimizing the thickness of the cover sheet ac-cordingly. The optical effects of cover-sheet thicknessvariation have been described in a previous paper,3

The author is with Eastman Kodak Company, Mass MemoryDivision, Rochester, New York 14650.

Received 4 September 1985.0003-6935/86/040490-03$02.00/0.© 1986 Optical Society of America.

which also discussed the thickness tolerance con-straints for optical disks. The purpose of this paper isto describe how cover-sheet thickness should be ad-justed to correct optimally for spherical aberration dueto refractive-index variation and to calculate theamount of residual aberration.

II. Adjusted Cover-Sheet Thickness

When a cover sheet causes a small amount of spheri-cal aberration, its effects are best described in terms ofthe optical path difference (OPD), the maximum dif-ference in optical path between any two geometricallight rays passing through the lens aperture and inter-secting at the nominal focus. The optical path incre-ment for an arbitrary ray due to a cover sheet of thick-ness t and index n is given by

OPI = d cosO + t(n coso' - cosO/n), (1)

where 0 is the angle of incidence, 0' is the angle of theray within the cover sheet (sinG' = sinG/n), and d is thedistance of the observation plane below the currentparaxial focus.3 This expression is not approximatebut is exact to all orders in 0. As discussed above, theobjective lens should be designed and constructed withundercorrected spherical aberration which exactlycancels the optical path variation for nominal values oft and n in the nominal paraxial focus plane, d = 0.

Now suppose that the nominal cover sheet (n,t)through which the optical system is focused is replacedby one with index h and thickness . The net opticalpath (less a system constant) for an arbitrary ray willbe

OP(0) = . coso + t(h cosOr - coso/h) - t(n cosO' - cosO/n), (2)

where singr = sinG/h, similar to the definition of 0', anda is the distance of the observation plane below thenew paraxial focus. The OPD can be calculated nu-merically by finding the peak-to-peak variation of OPfor all possible light rays, 0 < 0 = arcsin(N.A.), where

490 APPLIED OPTICS / Vol. 25, No. 4 / 15 February 1986

DESIGN INDEX

-. .53

l . , . . I . . . . I . . . . I I - --

1.02

<-

II

Io

I

0.981.45 1.65

ACTUAL REFRACTIVE INDEX. a

Fig. 1. Recommended relationship between actual cover-sheetthickness and refractive index. See the text for an explanation of

the design index.

N.A. is the numerical aperture of the objective lens.Given h, the optimum cover-sheet thickness is thatvalue of Z which minimizes the OPD.

A conservative step to minimize the OPD is to setOP(0) = OP(0) and solve for i. A typical lens designstrategy might set OP(0) OP(0.7 0 0), but this canresult in fairly large aberrations for the marginal rays.Since the entire aperture of the objective lens is impor-tant in optical recording, the more conservative ap-proach is appropriate. Applying this prescription toEq. (2), we obtain

a + i(h - 1/h) - t(n - 1/n) = a cosO + i(h cosb'

- cosO/h) - t(n cosO'

- cosO/n), (3)

where 0' and e' are defined analogously to 0' and O'.Equation (3) is not sufficient to determine 1, because

a is not yet known. a represents the focus location ofthe- system in the presence of cover-sheet aberration.The actual focus location is determined by the opticalsystem. For example, the head in an optical recorderhas a focus sensor and servo which continually adjustthe vertical position of the objective lens. Ordinarilythe system will act so that light rays retroreflectedfrom the recording surface are collimated somewherein the objective lens aperture. That is, dOP/dG = 0 forsome value 0 = X, 0 < < <0. For example, a half-aperture focus sensor will set 0 - 0.50, and a large-spotparaxial sensor will maintain - 0. Assuming somefixed value of 0 and differentiating Eq. (2), we obtain asecond relationship involving and a:

a - i(1 - coso/cos&'/n + t(1 - cos4/coso')/n = 0, (4)

where k' and ' have the usual definitions.Equations (3) and (4) can easily be combined to

calculate the value of ilt (as a function of n, h, N.A.,and ), which approximately minimizes the OPD.Figure 1 shows the optimal thickness plotted as a func-tion of the actual refractive index for the case n = 1.53,N.A. = 0.6, and sin0 = 0.18. These particular parame-ter values were chosen for reasons discussed in Sec. III.Note that ilt is a highly nonlinear function of n: e.g.,

1.45 1.65

ACTUAL REFRACTIVE INDEX .

Fig. 2. Maximum aberration resulting when the recommendedthickness vs index relationship is applied to an optical system with anumerical aperture of < 0.6. The solid and dashed lines correspond

to different focusing techniques, as described in the text.

keeping the optical thickness constant, h t = n t, isnot a near-optimal strategy.

Ill. Applicability

Given a function (h) which optimizes the cover-sheet thickness, we still need to know how much resid-ual aberration is left. Suppose a relationship such asthe curve in Fig. 1 is followed in the cover-sheet fabri-cation, so that when a material of different refractiveindex is used, the sheet thickness is adjusted accord-ingly. If we assume a value of f (not necessarily thevalue used to construct the curve) we can compute adirectly from Eq. (4). Now we can evaluate Eq. (2) forall values of 0 and determine the OPD. Figure 2 plotsOPD/t as a function of f assuming the relationship ofFig. 1, a numerical aperture of 0.6, and servo values ofsino = 0 and 0.4.

Examination of Fig. 2 explains some of the parame-ter choices in the example calculation of tit. n = 1.53was chosen to balance the performance over the actual1.48-1.60 index range, a range which includes the actu-al refractive index of PMMA, polycarbonate, and mostimportant glasses. The value of IO was chosen so thatthe single /t curve would be nearly optimum for anyreasonable focus sensor and servo. We may concludethat if the prescription of Fig. 1 is followed and if thelens has a N.A. of < 0.6 and a cover-sheet design indexof 1.53, the OPD will be < 0.002% of the cover-sheetthickness for a very wide range of actual index includ-ing 1.48-1.60. For IR optical recording through a 1.2-mm substrate, the corresponding spherical aberrationwill be less than 1/30th wave.

Although the relationship of Fig. 1 was optimized forN.A. = 0.6, it can also be applied to any system with aslower lens resulting in even less aberration. Howeveraberration due to refractive-index variation increasesvery rapidly for higher N.A. Figure 3 shows ti/t opti-mized as in Fig. 1 but with N.A. = 0.7 The correspond-ing OPD calculation is also plotted in this figure. Theresidual aberration shows nearly a fourfold increasecompared to N.A. = 0.6.

15 February 1986 / Vol. 25, No. 4 / APPLIED OPTICS 491

NA : 0.6 NA = 0.6

OPD/ t I(X105)

The residual spherical aberration discussed above NA:0.7

results because the optical effects of thickness andrefractive-index variations are not identical. Howev-Der, small index variations can be very accurately com- n .53

pensated by corresponding thickness adjustments, as \,prescribed by t/t. In this way, small refractive-indexvariations can be thought of as equivalent thickness Xvariations, and refractive-index tolerances can betreated as part of an effective thickness tolerance,simplifying the optical specifications for systems 0which use cover sheets. ,

0.98 01.45 1.65

ACTUAL REFRACTIVE INDEX, f

Fig. 3. Optimized thickness vs index relationship for a system witha numerical aperture of 0.7. The dashed line describes the residual

I wish to acknowledge the ANSI subcommittee aberration, which is much greater than in the case of lower numericalX3B11 for motivating this research. aperture.

References

1. D. G. Howe, J. J. Wrobel, A. B. Marchant, and F. F. Geyer, "High-Density Optical Recording in Organic Materials," in SecondSPSE Symposium on Optical Data Display, Processing andStorage (1981), p. 97.

2. R. A. Bartolini, A. E. Bell, R. E. Flory, M. Lurie, and F. W. Spong,"Optical Disk Systems Emerge," IEEE Spectrum 15, 20 (Aug.1978).

3. A. B. Marchant, "Cover Sheet Aberations in Optical Recording,"Pro. Soc. Photo-Opt. Instrum. Eng. 421, 43 (1983).

Optical Propagation, Detection, and Communication

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492 APPLIED OPTICS / Vol. 25, No. 4 / 15 February 1986