Page 1

Finite conjugate spherical aberration compensation

in high numerical-aperture optical disc readout

Sjoerd Stallinga

Spherical aberration arising from deviations of the thickness of an optical disc substrate from a nominal

value can be compensated to a great extent by illuminating the scanning objective lens with a slightly

convergent or divergent beam. The optimum conjugate change and the amount and type of residual

aberration are calculated analytically for an objective lens that satisfies Abbe’s sine condition. The

aberration sensitivity is decreased by a factor of 25 for numerical aperture values of approximately 0.85,

and the residual aberrations consist mainly of the first higher-order Zernike spherical aberration term

A60. The Wasserman–Wolf–Vaskas method is used to design biaspheric objective lenses that satisfy a ray

condition that interpolates between the Abbe and the Herschel conditions. Requirements for coma by field

use allow for only small deviations from the Abbe condition, making the analytical theory a good

approximation for any objective lens used in practice.

OCIS codes:

080.1010, 210.4590.

© 2005 Optical Society of America

1.

In optical disc readout the beam is focused onto the

data layer of the disc through a substrate layer of

thickness d. The scanning objective lens is designed

in such a way that the spherical aberration result-

ing from focusing through this layer is compensated

for, so that the scanning spot at the data layer is

nominally free from aberrations. A mismatch of the

substrate thickness from the nominal value results

in spherical aberration. The sensitivity for thick-

ness mismatch-induced spherical aberration in-

creases strongly with the numerical aperture (NA)

of the objective lens. The wavelength is the natural

measure for aberrations, so that a decrease of wave-

length ? also results in greater aberration sensitivity.

The increase of NA and decrease of ? used to increase

the capacity of an optical disc (the size of the

focal spot scales as ??NA) thus deteriorates the tol-

erances of the system for substrate thickness mis-

match.

ForaCompactDisc(CD)(? ? 0.785 ?m, NAof0.50)

sensitivity is 0.4 m???m, for a Digital Versatile Disc

Introduction

(DVD) ?? ? 0.660 ?m, NA of 0.65) the sensitivity is

1.4 m???m, and for the Blu-ray Disc (BD) ?? ?

0.405 ?m, NA of 0.85? the sensitivity is 10.1 m??

?m.1(The aberration unit here is the milliwave;

1 m? ? ??1000.) With a spacing of the two layers of

a BD double-layer disc of 25 ?m the amount of spher-

ical aberration is ?126 m?, if we assume that the

objective lens is optimized for the focal position half-

way between the two layers. This is more than the

diffraction limit of ??8?3 ? 72 m? and is hence un-

acceptably large. Clearly, means to compensate for

spherical aberration must be introduced into the op-

tical system. The compensating system must be

switched into a first state when data are read from or

written to the upper layer of a double-layer BD, and

into a second state when data are read from or writ-

ten to the lower layer of a double-layer BD.

The most straightforward way of compensating for

spherical aberration is to use finite conjugate illumi-

nation of an objective lens. Spherical aberration de-

pends on the conjugate of the lens.2,3It follows that

this effect can be used to balance spherical aberration

arising from other causes. In the practice of optical

disc readout the lens is normally used at infinite

conjugate, i.e., the lens is illuminated with a colli-

mated beam. By changing to a slightly convergent or

divergent beam the spherical aberration that is due

to the thickness mismatch related to double-layer

discs can be compensated for. Although strictly

needed for the two discrete depths of a double-layer

The author (sjoerd.stallinga@philips.com) is with the Philips

ResearchLaboratories,ProfessorHolstlaan4,5656AAEindhoven,

The Netherlands.

Received 28 February 2005; revised manuscript received 4 May

2005; accepted 6 May 2005.

0003-6935/05/347307-06$15.00/0

© 2005 Optical Society of America

1 December 2005 ? Vol. 44, No. 34 ? APPLIED OPTICS7307

Page 2

disc, finite conjugate illumination can be used to com-

pensate for any thickness value in a continuous range

around the central value. In the field of microscopy

this method of spherical aberration compensation is

known as a change in the tube length (the distance

between object and image).4I report on how effective

the method is and which parameters determine the

conjugate change required to bridge a given amount

of thickness mismatch. The conclusions are reached

with exact analytical means and compared with nu-

merical ray-tracing methods. The question of the ef-

fectiveness of this method of spherical aberration

compensation has been analyzed before within the

context of confocal microscopy for the case of a small

refractive-index mismatch of the medium in which

the beam is focused with a nominal value.5,6This is in

contrast with the case considered here in which the

spherical aberration arises from a mismatch in thick-

ness of the layer through which the beam is focused.

The content of this paper is as follows. Section 2

deals with the analytical theory of finite conjugate

spherical aberration compensation for an aplanatic

objective lens. The effects of deviations of aplanaticity

are analyzed by numerical ray-tracing methods in

Section 3.

2.

Consideranobjectivelensthatfocusesabeamoflight

onto the data layer of an optical disc. There are two

ways to shift the focal point in the disc in the axial

direction, as illustrated in Fig. 1. The first way is to

change the conjugate of the optical system, i.e., by

axially shifting the source point, the image point will

be shifted in the axial direction over a distance ?zcon.

The second way is to bring the optical system as a

whole a distance ?zfwdcloser to the disc, thus reduc-

ing the free working distance (fwd) of the objective

lens. When the objective lens is used at infinite con-

jugate this is equivalent to bringing the objective lens

closer to the disc, while keeping the other optical

components at a fixed position. The diffraction focus,

Analytical Theory for an Aplanatic Objective Lens

i.e., the reference point for the aberration function

giving rise to a minimum rms value of the aberration

function, is in general a distance ?zreffurther in the

disc. The total focal shift ?d then follows as

?d??zcon??zfwd??zref.(1)

Clearly, given the total focal shift there are two de-

grees of freedom. These are fixed by the condition of

minimum rms aberration. The form of the aberration

function is determined by Abbe’s sine condition.2,3

According to this condition, if the optical system is

free from (spherical) aberration when the object and

image points are on the optical axis it is also free from

(comatic) aberration when the object and image

points are laterally displaced from the optical axis (to

first order in the field angle). Alignment tolerances of

an optical drive light path require that the design of

the objective lens cannot deviate too much from the

sine condition. Consequently, in many cases the ob-

jective lens may be approximated by an aplanat. The

accuracyofthisapproximationisdiscussedinSection

3. The sine condition imposes a relation between the

rays in object and image space, and this relation de-

termines the aberration function resulting from the

change in conjugate. In the folllowing an expression

for the aberration function is derived and values are

determined for ?zconand ?zfwdthat give rise to a min-

imum rms value of the aberration function.

Consider an axisymmetric optical system and a ray

passing through an object point and an image point,

both on the optical axis, such that the ray makes an

angle ?0with the optical axis in object space and an

angle ?1with the optical axis in image space. Abbe’s

sine condition relates the ray angles ?0and ?1in ob-

ject and image space by

n0sin ?0?Mn1sin ?1, (2)

where n0and n1are the refractive indices in object

and image space, and M is the (lateral) magnification

from object to image space. A scaled pupil coordinate

can be defined as

??n0sin ?0

NA0

?n1sin ?1

NA1

, (3)

where NA0and NA1are the NAs in object and image

space. It follows that ? has values between 0 and 1.

For a small axial displacement of object and image

points ?z0and ?z1, respectively, the aberration then

follows as

Wcon?n1?z1?cos ?1?1??n0?z0?cos ?0?1?. (4)

Note that a different sign convention is used in Refs.

2 and 3. The axial displacements are related by

?z1?M2n1

n0?z0,(5)

Fig. 1.

changing the conjugate of the objective lens (top) or by changing

the free working distance (bottom).

Changing the axial focal position can be achieved by

7308APPLIED OPTICS ? Vol. 44, No. 34 ? 1 December 2005

Page 3

so that

Wcon?n1?z1??1??2NA1

2

n1

2 ?

1?2

?1?

??z1??n1

n0

2

n1

2M2?1??1??2NA0

2

n0

2 ?

1?2??

2??2NA1

2?1?2?n1

?

?2NA1

2

n1??n1

2??2NA1

2M2n1

2?n0

2?1?2?. (6)

In the particular case in which we are interested we

have ?z1? ?zcon, and M ? 0 (infinite conjugate).

Using the abbreviations n1? n and NA1? NA it

then follows that

Wcon??zcon??n2??2NA2?1?2?n?p2NA2

2n?,(7)

which is in agreement with the expression derived by

Sheppard and Gu.7The decrease in free working dis-

tance introduces an aberration that is equivalent to

the insertion of a layer of refractive index n with a

thickness equal to the decrease in free working dis-

tance.1

Wfwd??zfwd ??n2??2NA2?1?2?n

???1??2NA2?1?2? 1??. (8)

The diffraction focus is found a distance ?zrefdeeper

into the disc, which gives an aberration1of

Wref??zref ??n2??2NA2?1?2?n?.(9)

Eliminating ?zrefin favor of ?d the total aberration

function follows as

W?Wcon?Wfwd?Wref

??d?(n2??2NA2)1?2?n???zcon

?2NA2

2n

??zfwd?(1??2NA2)1?2?1?. (10)

When the conjugate change ?zcon?0 the aberration

expression of Ref. 1 is recovered. For a given change

in focal position ?d the changes in conjugate and free

working distance are free parameters that can be

varied to determine the optimum focal spot. The op-

timum corresponds to a minimum rms wavefront ab-

erration. This minimization procedure can be done as

follows. First, write

W

?d?f? ?

i?1, 2?igi,(11)

with

f??n2??2NA2?1?2, (12)

g1???2NA2

2n

,(13)

g2?

1

n?1??2NA2?1?2,(14)

where the piston terms have been left out as they

cancel from the expression for the rms wavefront ab-

erration, and with

?1??con??zcon

?d,(15)

?2??fwd?n?zfwd

?d

. (16)

The rms wavefront aberration can now be written as

Wrms

(?d)2??W2???W?2

?frms

2

??d?2

2?2?

i?1, 2vi?i? ?

i, j?1,2Rij?i?j, (17)

where

frms

2??f2???f?2, (18)

vi??fgi???f??gi?,(19)

Rij??gigj???gi??gj?, (20)

the angle brackets indicate the pupil average defined

by

??

P

?A??

1

d2?A???,(21)

and the integration is over the unit circle. Explicit

analytical expressions for the relevant pupil averages

are given in Appendix A. Minimization with respect

to ?ileads to

?i? ?

j?1, 2Rij

?1vj,(22)

Wrms

??d?2?frms

2

2? ?

i, j?1, 2Rij

?1vivj,(23)

where R?1is the inverse of the 2 ? 2 matrix R. This

completes the analytical theory. The dependence of

the residual aberration sensitivity Wrms??d and the

parameters ?conand ?fwdon NA are discussed in the

following.

1 December 2005 ? Vol. 44, No. 34 ? APPLIED OPTICS 7309

Page 4

Figure 2 shows the residual aberration sensitivity

as a function of NA. For BD conditions ?NA of 0.85

NA, ? ? 405 nm, and n ? 1.6? the sensitivity is

0.41 m???m compared with the 10.07 m???m for the

case without conjugate change,1which amounts to a

reduction by a factor of 25. For a BD dual-layer disc

with a nominal spacer layer thickness of 25 ?m this

results in only a ?5.1 m? rms aberration, assuming

that the objective lens is optimized for the focal po-

sition halfway between the two layers. Clearly, con-

jugate change is an effective way to compensate for

spherical aberration.

The exact residual aberration sensitivity can be

expanded in powers of NA. This corresponds to an

expansion of the aberration function in terms of Sei-

del polynomials. The lowest-order terms are

Wrms

?d??n2?1?NA6

320?7n5

??n2?1??2n2?5?NA8

1280?7n7

.

(24)

The dominant term originates from the higher-order

Seidel spherical aberration term proportional to ?6.

Figure 2 shows the lowest-order term of the series as

a function of NA. Clearly, the ??6term seriously

underestimates the exact result. In fact, for a NA of

0.85 it is 61% too low. A better approximation is in

terms of the Zernike polynomials. It appears that the

dominant contribution is the lowest-order Zernike

spherical aberration term 20?6? 30?4? 12?2? 1

with coefficient A60. Figure 2 shows the A60coefficient

as a function of NA. Apparently, approximating the

exactaberrationfunctionbyuseofthissingleZernike

term is quite accurate, even for high-NA values. For

example, for a NA of 0.85 the error is only 4%.

Figure 3 shows the relative change in conjugate

and free working distance as a function of NA. It was

determined that ?con? ?fwdis close to unity through-

out the entire range of NA values. For example,

forBD conditionsit

? 0.80 and ?fwd? 0.22, giving ?con ? ?fwd? 1.02.

was foundthat

?con

The relative change in conjugate and free working

distance can also be expanded in powers of NA. The

lowest-order terms are

?con?n2?1

n2

?3?n2?1?NA2

4n4

?3?n4?51n2?50?NA4

280n6

, (25)

?fwd?

1

n2?3?n2?1?NA2

?3?3n4?22n2?25?NA4

140n6

4n4

, (26)

?con??fwd?1?3?n2?1?NA4

40n6

. (27)

It follows that the approximation ?con? ?fwd? 1 is

exact for small NA values.

3.

The change in conjugate is made in practice by an

axial translation of a lens in front of the objective

lens, for example, the collimator lens that converges

the beam emitted by the laser diode into a parallel

beam. The entrance NA0of this lens is much smaller

than the exit NA of the objective lens. Additional

sphericalaberrationcontributionsthatarisefromthe

change of conjugate for this lens can therefore be

neglected. It then follows that the required transla-

tion of this lens is given by

Effects of Deviating from the Sine Condition

?z0??zcon

nM2??con

?d

n

NA2

NA0

2. (28)

In practice it is advantageous to keep this axial

Fig. 2.

curve) and approximation by the single Zernike term A60(long-

dashed curve) and by the Seidel term proportional to NA6(short-

dashed curve).

Residual aberration sensitivity as a function of NA (solid

Fig. 3.

jugate (solid curve), the parameter ?fwdthat describes the relative

change in free working distance (long-dashed curve), and the sum

of the two parameters ?con? ?fwd(short-dashed curve) as a function

of NA.

Parameter ?conthat describes the relative change of con-

7310 APPLIED OPTICS ? Vol. 44, No. 34 ? 1 December 2005

Page 5

stroke ?z0as small as possible. This can be achieved

by deviating slightly from Abbe’s sine condition. This

would possibly allow a decrease in the parameter ?con.

This is investigated here by numerical means, in par-

ticular with automated lens design methods and ray

tracing.

An alternative to Abbe’s sine condition is the Her-

schel condition.2,3An imaging system that satisfies

the latter condition does not suffer from spherical

aberration when the conjugate of the imaging system

is changed (to first order in the shift of the object and

image points). In the present terms this means that

?con? 1 and ?fwd? 0, i.e., the conjugate change is

sufficient for a focal shift with minimum induced ab-

errations,theminimumbeingzerointhiscase.Alens

design condition that interpolates between Abbe and

Herschel is8,9

n0sin?

?0

q??Mn1sin?

?1

q?, (29)

where q is a real parameter. For q ? 1 the Abbe

condition is retrieved, whereas for q ? 2 the Herschel

condition is retrieved. Values of 1 ? q ? 2 therefore

interpolate between the two conditions. For values of

q ? 1 we are on the other side of Abbe, so to speak.

The trend in ?conas a function of q derived from

the two analytically available points ??con? 1 for q

? 2 and ?con ? 0.80 for q ? 1? suggests that the

wanted small values of?concan be found in the regime

q ? 1.

The q condition can be used to fully specify two

aspheric surfaces in an imaging system by use of an

algorithm that is based on the methods of Wasser-

man and Wolf,10Vaskas,11and Braat and Greve12for

the special case of the sine condition. This approach

to lens design has been used in Ref. 13 for the design

of lenses with a so-called flat intensity profile. The

objective lens was taken to be a biasphere operating

under BD conditions, i.e., the wavelength was taken

to be ? ? 0.405 ?m, the NA was 0.85, and the focus is

ata depth of87.5 ?m in

carbonate ?n ? 1.62231?. The glass refractive index

is nlens? 1.71055, lens thickness b ? 2.75 mm, the

free working distance is 0.75 mm, and the stop of

diameter 4.0 mm was taken to be at the vertex of the

first refractive surface of the lens. For a given q value

the surface sag of the two aspheres was determined

by use of an automated design tool that solves the

Wasserman–Wolf differential equations. The lens de-

sign thus obtained was futher analyzed with the Ze-

max ray-tracing software package14by considering a

cover layer that is 1 ?m thinner or thicker than the

nominal 87.5 ?m. This is sufficiently small to be in

agreement with the linearized treatment in axial dis-

placements of the analytical theory in Section 2. The

distance between the objective lens and the disc and

the location of the object point is then adjusted for a

minimum rms aberration. This allows for a numeri-

cal evaluation of parameters ?conand ?fwd. Figure 4

shows the resulting values as a function of the q

alayer ofpoly-

parameter. The agreement with the exact results for

q ? 1 and q ? 2 is quite good, with an estimated error

of less than 0.3%.

A deviation from Abbe’s sine condition results in a

nonzero sensitivity of coma for field use. Figure 5

shows the coma sensitivity as a function of the q

parameter. As expected, the sensitivity crosses zero

for q ? 1, i.e., when the lens satisfies the Abbe con-

dition. For a sufficiently small NA, this sensitivity is

proportional to 1 ? q?2. The numerical data fit quite

well with this function, even though the NA is as high

as 0.85. We found a coma sensitivity of 491 m??deg

? ?1 ? q?2?. The upper bound for this coma sensitiv-

ity is typically approximately 50 m??deg. It then fol-

lows that the lens design must satisfy 1.06 ? q

? 0.95, i.e., only small deviations from the Abbe con-

dition are allowed. As a consequence, the parameter

for relative conjugate change cannot deviate much

from the Abbe value ?con? 0.80. By use of the nu-

merically calculated values it was found that for the

range of q values 1.06 ? q ? 0.95 the conjugate

parameter is in the range 0.82 ? ?con? 0.78, i.e., the

Fig. 4.

?fwd(triangles) for a lens with 0.85 NA as a function of the q

parameter that interpolates between the Abbe and the Herschel

conditions.

Numerically determined parameters ?con(squares) and

Fig. 5.

the q parameter that interpolates between the Abbe and the Her-

schel conditions (squares) and a fit proportional to 1 ? q?2(solid

curve).

Sensitivity of coma for field use of the lens as a function of

1 December 2005 ? Vol. 44, No. 34 ? APPLIED OPTICS 7311

Page 6

decreaseinrequiredconjugatechangecomparedwith

the Abbe case is of the order of a few percent.

In conclusion, the analytical theory of finite conju-

gate spherical aberration compensation presented in

Section2isagoodaproximationforanyobjectivelens

used in practice. It allows for a simple, straightfor-

ward evaluation of the conjugate change required to

minimize the rms value of the aberration function

and, of this minimum, residual aberration sensitiv-

ity. A possible decrease of the required conjugate

change by non-Abbe objective lens designs is rela-

tively small. Finally, note that ways to break the sine

condition other than the q condition can be explored,

but it cannot be expected that such an approach will

result in an outcome substantially different from this

one.

Appendix A

All the pupil averages were evaluated by use of Math-

ematica15and yield

?f??

2

3NA2?n3??n2?NA2?3?2?,

(A1)

?f2??n2?1

2NA2,

(A2)

?g1???NA2

4n,

(A3)

?g2??

2

3nNA2?1??1?NA2?3?2?,

(A4)

?fg1???2n5??3NA4?n2NA2?2n4??n2?NA2?1?2

15nNA2

,

(A5)

?fg2??

1

4nNA2?n3?n??n2?1?2NA2?

??n2?NA2?1?2?1?NA2?1?2??n2?1?2

?log??n2?NA2?1?2??1?NA2?1?2

n?1

??, (A6)

?g1

2??

NA4

12n2, (A7)

?g2

2??2?NA2

2n2

, (A8)

?g1g2???2??3NA4?NA2?2??1?NA2?1?2

15n2NA2

.

(A9)

Some of these integrals can also be found in Ref. 1.

I am indebted to Teus Tukker for providing me

with his software tools for automated design of objec-

tive lenses that satisfies the q condition.

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7312 APPLIED OPTICS ? Vol. 44, No. 34 ? 1 December 2005