Download Application of nonperiodic phase structures in

Application of nonperiodic phase structures in
optical systems
Benno H. W. Hendriks, Jorrit E. de Vries, and H. Paul Urbach
Wide, nonperiodic stepped phase structures are studied to correct various parameter-dependent wavefront aberrations in optical systems. The wide nature of these phase structures makes them easy to
manufacture with sufficient compensation of the wave-front aberrations. Wave-front aberration correction for both continuous and discrete parameter variations are studied. An analytical method is
derived for the discrete parameter variations to find the optimal phase structure. Both theoretical and
experimental results show that these nonperiodic phase structures can be used to make 共1兲 lenses
athermal 共defocus and spherical aberration compensated兲, 共2兲 lenses achromatic, 共3兲 lenses with a large
field of view, 共4兲 lenses with a reduced field curvature, and 共5兲 digital versatile disk objective lenses for
optical recording that are compatible with compact disk readout. © 2001 Optical Society of America
OCIS codes: 220.1000, 220.4830, 050.1940, 210.4770.
1. Introduction
Unwanted wave-front aberrations often arise in optical systems when a certain parameter changes.
Examples of such parameter changes are a change in
temperature, a change in wavelength, or, for optical
recording, a different substrate thickness. To reduce the unwanted wave-front aberrations, a structure is needed that has no optical power and is
aberration free in the nominal configuration. When
a certain parameter changes, however, that structure
must introduce a wave-front aberration that at least
partly compensates the unwanted one. An example
of such a structure is a diffractive periodic phase
structure studied in Ref. 1, based on notched lenses.
Each step in these notched lenses introduces an additional optical path equal to an integer multiple of
the wavelength ␭. Furthermore, the difference in
optical paths between two subsequent steps is the
same throughout the notched lens. These stepped
diffractive structures, therefore, can be considered to
be a combination of a refractive substrate and a
B. H. W. Hendriks 共[email protected]兲 and H. P. Urbach are with Philips Research Laboratories, Prof. Holstlaan 4,
5656 AA Eindhoven, The Netherlands. J. E. de Vries is with
Philips Optical Storage, P. O. Box 80002, 5600 JB Eindhoven, The
Netherlands.
Received 10 April 2001; revised manuscript received 16 July
2001.
0003-6935兾01兾356548-13$15.00兾0
© 2001 Optical Society of America
6548
APPLIED OPTICS 兾 Vol. 40, No. 35 兾 10 December 2001
blazed kinoform with zero combined power. The diffractive structure of these lenses is periodic in the
sense that traditional analysis and design techniques
of diffractive lenses can be used.2 As a result, various properties of these lenses were derived and used
to design achromatic lenses.1,3 In Ref. 4 the abovementioned structure was investigated to make lenses
athermal, and in Ref. 5 field curvature reduction was
studied. A drawback of these periodic diffractive
structures is that they generally lead to structures
that have a rather large number of small zones, making them difficult to manufacture. Furthermore, although these periodic notched lenses can be designed
to yield 100% efficiency, actual notched lenses never
attain such a high efficiency because of small manufacturing errors.
One way to avoid these problems is to let the abovementioned stepped structure be nonperiodic and
have relatively wide zones. Hence the difference in
optical paths between two subsequent steps may be
any value and may vary in any way throughout the
structure. Consequently, this class of phase structures allows a great degree of freedom in design.
Another aspect related to this freedom is that the
annular areas forming this nonperiodic pattern can
be made relatively wide, which significantly improves
the manufacturability and reduces stray-light losses,
at the expense of less perfect but still sufficient compensation of the wave-front deviation, as we show in
this paper. As a result, the conventional diffractive
description can no longer be applied to these wide,
nonperiodic phase structures 共NPSs兲. In fact, use of
nonperiodic wide zones ensures that the function of
the structure is based on refraction rather than diffraction. A NPS can be considered to be a limiting
case of a periodic diffractive structure with only one
zone. The distribution within one zone for a periodic
diffractive structure, normally used to optimize the
diffraction efficiency in one particular order, is now
used to compensate for the wave-front aberration.
In this paper we study the special case in which the
NPS, in the nominal configuration, gives rise to a
nonperiodic stepped wave-front distribution in which
each phase step in this wave front is equal to a multiple of 2␲. Hence, taking modulo 2␲, the structure
gives rise to a flat-phase wave front and therefore
adds no power and aberrations to the optical system.
When one of the nominal parameters changes, such
as the wavelength, this structure no longer gives rise
to phase steps equal to a multiple of 2␲, and, after we
take modulo 2␲, a stepped phase distribution remains. The effect on the optical properties of the
beam of such a deformation can be calculated in a
way similar to that used in Ref. 6. In Refs. 7 and 8
a NPS was used to make a digital versatile disk
共DVD兲 objective lens, designed for reading optical recording disks with a cover layer of 0.6 mm at wavelength ␭ ⫽ 660 nm, compatible with readout of
compact disks 共CDs兲 with a cover layer of 1.2 mm at
wavelength ␭ ⫽ 785 nm.
In this paper we show that the concept of a NPS
can be used more generally to partly compensate for
aberrations of optical systems when we correctly
choose the initial step heights and step widths of the
NPS. Two different classes of aberration correction
with such a NPS are studied. The first class consists
of aberrations induced by a parameter change, which
may vary continuously in a certain range, such as
temperature variations, wavelength variation
around a nominal value, or a distribution of field
angles. In the second class, aberration correction at
discrete values of a certain parameter is studied. An
example of this class is when the NPS must correct
spherical aberration at two discrete values of the
wavelength only. An analytical method to optimize
the wave-front compensation by the NPS is given.
Both theoretical and experimental results are presented for each of the above classes of wave-front
aberration correction with the aid of a NPS.
2. Aberration Correction with a Nonperiodic
Phase-Structure for a Continuous Range of
Parameter Values
A.
Temperature Stabilization
In general, when the temperature changes, lenses not
only generate unwanted defocus but also spherical
aberration. Objectives made of plastic, in particular
关and to a much lesser extent lenses made by the
glass兾photo-polymer 共glass兾2P兲 process9兴, suffer from
the above-mentioned aberration. In an optical
pickup, for example, this amount of spherical aberration can become too large in the specified operating
temperature range of 0 °C–70 °C for objectives with a
Fig. 1. Lens system with a NPS in front.
numerical aperture 共NA兲 greater than or equal to 0.6.
To compensate for both effects, we can make use of a
NPS plate in front of the lens system, as indicated in
Fig. 1. Note that instead of an extra plate, the proposed structure can also be incorporated in the aspherical layer of the objective lens. The NPS
consists of a number of annular zones with step
heights hj . We let the step height hj of each zone be
such that hj ⫽ mj h, with mj as the integer and h
equal to
h⫽
␭
,
n⫺1
(1)
where ␭ is the wavelength and n is the refractive
index of the material of the rings of the NPS at wavelength ␭ and design temperature T0.
As a result, this phase structure has no effect at the
design temperature. When the temperature
changes, the shape of the stepped phase structure
will change; the height of the rings will therefore also
change, the amount of change depending on the
linear-expansion coefficient ␣ of the material. Because the steps were chosen to be wide, the change in
width of the annular areas has a negligible effect on
the performance of the structure. The refractive index of the material of the structure will also change
as a function of temperature, the amount of change
depending on ␤ ⫽ dn兾dT. The length of the optical
paths through the annular areas will therefore depend on the temperature of the phase structure.
Note that the temperature dependence of the performance of a NPS is clearly different from that of a
grating. The temperature dependence of a grating
arises from the expansion of the zone width, whereas
the refractive-index change has no effect on the rays.
For a grating, a change in refractive index affects
only the efficiency of a particular diffraction order.
The phase change ⌬⌽j of the ring j of the phase structure, where the ring has a height hj , is now determined for a temperature change ⌬T and relative to
the phase of the structure at the temperature T0. If
10 December 2001 兾 Vol. 40, No. 35 兾 APPLIED OPTICS
6549
Table 1. Zone Widths and Height Distribution of the NPS for the
Athermalization of the Philips Glass兾2P DVD Objective
Fig. 2. Experimental results of the change in the lowest-order
spherical wave-front aberration as a function of the temperature
for the Philips glass兾2P DVD objective lens 共NA of 0.65兲 with and
without NPS present.
an isotropic expansion of the stepped structure is
assumed, the phase change is given by
冉
冊
⌬h
⌬n
⌬⌽ j ⫽ 2␲m j
,
⫹
h
n⫺1
(2)
where Eq. 共1兲 was used and quadratic terms in a
difference were neglected. In Eq. 共2兲 we also used
⌬n ⫽ n共T兲 ⫺ n共T0兲. Because
⌬h ⫽ ␣h⌬T,
(3)
⌬n ⫽ ␤⌬T,
(4)
the phase change is10
冉
⌬⌽ j ⫽ 2␲ ␣ ⫹
冊
␤
m j ⌬T.
n⫺1
(5)
When the temperature changes, the lens introduces a
wave-front aberration. To compensate for this
temperature-induced aberration, the values of the integer mj for each of the rings in the phase structure
must be chosen such that the phase structure will
introduce a wave-front deviation that approximates
the wave-front aberration of the lens but with the
opposite sign.
We use the NPS to reduce the temperature-induced
spherical aberration of the standard Philips glass兾2P
objective lens used for optical recording to illustrate
the above-mentioned athermalization method. This
f ⫽ 2.75-mm lens is a standard Philips glass兾2P DVD
lens 共NA of 0.65兲 with an entrance pupil diameter of
3.58 mm. It consists of a plano–spherical glass body
with a thin aspherical plastic layer on top of it.9 The
temperature-induced spherical aberration is measured with Twyman–Green interferometry. A special holder made of brass is placed around the lens to
heat it up. The temperature is measured with a
thermocouple. The measured change in the root
mean square of the optical path difference 共⌬OPD兲 as
a function of the temperature is shown in Fig. 2,
revealing that ⌬OPD兾⌬T ⫽ 0.5 m␭兾°C. We placed a
NPS plate made of poly共methyl methacrylate兲
6550
APPLIED OPTICS 兾 Vol. 40, No. 35 兾 10 December 2001
j
rj⫺1 共mm兲
rj 共mm兲
mj h 共␮m兲
mj
1
2
3
4
5
0.000
0.558
0.882
1.557
1.700
0.558
0.882
1.557
1.700
1.790
0.000
4.049
8.097
4.049
0.000
0
3
6
3
0
共PMMA兲 in front of the objective lens to reduce this
temperature dependence. The thermal expansion
coefficient of PMMA is ␣ ⫽ 62 ⫻ 10⫺6兾K, whereas the
change in refractive index is given by ␤ ⫽ ⫺12.5 ⫻
10⫺5兾K. The refractive index at ␭ ⫽ 660 nm is n ⫽
1.4891, hence h ⫽ 1.3495 ␮m. From Eq. 共5兲 it then
follows that
⌬⌽ j ⫽ ⫺0.001216m j ⌬T.
(6)
To compensate for the spherical aberration, we consider a NPS plate with five annular zones 共see Fig. 1兲.
The zone height and width distribution of the NPS is
given in Table 1. To design the structure, we first
chose the number of zones and their widths. Then
the zone heights are optimized such that the OPDrms
is minimal at an arbitrary elevated temperature.
Finally, the heights are rounded off to the nearest
fundamental height 关see Eq. 共1兲兴, giving rise to a multiple of 2␲ phase at the design temperature. A more
advanced design method is discussed in Subsection
3.A. Manufacturing the step height with a precision
of 20 nm would result in a wave-front aberration
OPDrms of 6 m␭ in the nominal configuration. This
is well within reach of a high-precision lathe. The
calculated spherical wave-front aberration is plotted
in Fig. 3 at an operating temperature of 50 °C, i.e.,
⌬T ⫽ 30 K with NPS and without NPS. The effect
introduced by the NPS only is also shown. Figure 3
shows that, according to the design, the NPS introduces a stepped wave-front contribution. This
stepped wave front of the NPS approximates the
Fig. 3. Plot of the OPD as a function of the relative pupil coordinate ␳ for the lens system when the temperature has changed by
⌬T ⫽ 30 K 共a兲 with no NPS present and 共c兲 with NPS present. 共b兲
The contribution to the wave front of the NPS.
spherical wave-front aberration introduced by the
lens but with the opposite sign. The resulting wave
front, when the NPS is present, is clearly reduced.
Because we used a phase structure with five relatively wide zones, the compensation is limited. In
this case, according to the calculations, the total
wave-front aberration, hence including higher-order
spherical wave-front aberration, is reduced by a factor of 3 共the lowest-order spherical wave-front aberration is reduced by a factor of almost 7兲. If we had
used more annular rings in the NPS, the reduction
factor would have been larger, at the expense of a
more complicated structure. This is one of the advantages of the NPS, because the nonperiodic nature
gives us the freedom to make the right balance between complexity of the structure and the required
reduction factor of the wave-front aberration. Measurements with the NPS present are shown in Fig. 2.
Figure 2 shows that the thermal dependence of the
lowest-order spherical wave-front aberration has
been reduced by a factor of more than 3. Taking also
the higher-order spherical aberration terms up to the
18th order into account, we then find the reduction
factor to be 2.2. The experimental results show that
thermal compensation with a NPS is possible and
thus agrees with the theory. Because the zones of
the NPS are large 共⬎140 ␮m兲, this structure can be
incorporated directly into the aspherical surface,
making it a cost-effective way to make lenses athermal.
B.
Defocus
In optical recording, when the laser is changed from
read to write power, the wavelength of the laser shifts.
As a result, when the objective lens is not achromatic,
the spot on the information layer of the disk will be out
of focus; hence we need to refocus with the actuator.
Because of the limited bandwidth of the actuator, this
will result in a time delay before we can write data on
the disk. Therefore it is desirable to have objective
lens systems that are achromatic. To make the above
objective achromatic, we can add an additional NPS
plate in front of the lens, in a way similar to that done
for the temperature compensation. As discussed in
Subsection 2.A, the NPS consists of a number of annular zones. Let each zone j have a height mj h,
where mj is an integer and h is defined by Eq. 共1兲, and
let n be the refractive index of the NPS material at the
design wavelength. When the wavelength changes
by ⌬␭, and the corresponding change in refractive index is ⌬n, then the above step of height hj now introduces a relative phase ⌬⌽j , which in the lowest order
in ⌬n and ⌬␭ is given by
⌬⌽ j ⫽ ⫺2␲m j
冉
冊
⌬␭
⌬n
.
⫺
␭
n⫺1
(7)
The defocus arising from the lens system can be compensated by the NPS by proper design of the zone
widths and heights.
To illustrate this defocus compensation, we consider an objective lens with a NA of 0.85 operated at
Table 2. Step Height and Zone Width Distribution of the NPS
Compensating for Defocus
j
rj⫺1 共mm兲
rj 共mm兲
mj h 共␮m兲
mj
1
2
3
4
5
6
0.0
0.5
0.73
0.99
1.23
1.40
0.5
0.73
0.99
1.23
1.40
1.50
0
6.400
16.800
30.400
43.200
54.400
0
8
21
38
54
68
␭ ⫽ 405 nm.9,11 Without additional measures, a
120-m␭ OPDrms defocus arises for 2-nm laser wavelength detuning. To compensate for this effect, we
put a NPS plate in front of the objective lens. The
NPS is made of PMMA, hence n ⫽ 1.5060, h ⫽ 0.800
␮m, and dn兾d␭ ⫽ ⫺0.000114 nm. This results in
⌬⌽ j ⫽ ⫺0.01693m j ⌬␭,
(8)
where ⌬␭ is expressed in nanometers. To exploit the
advantage of the NPS, we make a design consisting
of only six zones to compensate the defocus 共see
Table 2兲. The structure is clearly nonperiodic, as
can be determined from the step height distribution.
The NPS generates 59-m␭ OPDrms defocus wavefront aberration per nanometer wavelength shift at
the expense of 7-m␭ OPDrms residual 共higher-order兲
wave-front aberrations. This residual wave-front
aberration results from the nonperfect compensation
of the wave-front aberration by the NPS. As a result, the wave-front aberration of the objective for
⌬␭ ⫽ 2 nm reduces from 120- to 22-m␭ OPDrms when
the NPS is placed in front of it, which is well below
the diffraction limit. Note that use of a periodic diffractive structure as discussed in Ref. 3, with a height
increase at each ring of 1 ⫻ h, would result in a
periodic structure with approximately 80 zones.
To test the defocus compensator experimentally we
placed the NPS in a Twyman–Green interferometer.
Three different commercial blue laser diodes from
Nichia Corporation with ␭ ⫽ 400, 404, and 409 nm
were used to change the wavelength. Figure 4
shows measured interferograms of the NA of 0.85
objective with and without NPS present for ␭ ⫽ 404
and 409 nm without refocusing. The defocus is
clearly compensated by the NPS at the expense of a
smaller amount of higher-order aberrations 关zigzag
instead of straight-line fringes in Fig. 4共d兲兴. Figure
5 shows the wave-front aberration OPDrms 共the minus sign means that the Zernike coefficient is negative兲 of the NPS as tabulated in Table 2 as a function
of the wavelength. The amount of defocus generated by the NPS according to the theory is also
shown. Apart from an offset, which is probably due
to a small difference in refractive index of the actual
material and the one used in the calculations, the
results agree well with the experiment. Furthermore, the measurements showed a residual wavefront aberration 共when we took into account Zernike
terms up to the 18th order兲 of 3-m␭ OPDrms per nanometer wavelength shift, which is somewhat smaller
10 December 2001 兾 Vol. 40, No. 35 兾 APPLIED OPTICS
6551
Fig. 4. Interferograms of the NA of 0.85 objective without NPS
present for 共a兲 ␭ ⫽ 404 nm and 共b兲 ␭ ⫽ 409 nm and with NPS
present for 共c兲 ␭ ⫽ 404 nm and 共d兲 ␭ ⫽ 409 nm.
than the 5-m␭ OPDrms per nanometer wavelength
shift according to the theory 共when we also took into
account Zernike terms only up to the 18th order兲.
C.
Improved Field of View
In general, a lens set that does not fully comply with
the Abbe sine condition will have a limited field tolerance because of the coma wave-front aberration
arising as a function of the field. To improve the
field tolerance of such a lens, we superimpose a NPS
on the lens 共see Fig. 6兲. For ease of computation, let
the rays enter the lens parallel to the optical axis. It
can be derived that the stepped profile gives rise to an
additional OPD equal to
冋冉
OPD共␪兲 ⫽ h j n 1 ⫺
sin ␪
n2
2
冊
1兾2
册
⫺ cos ␪ ,
(9)
where hj is the height of the structure in the radial
direction as measured from the original base of the
lens surface, n is the refractive index of the lens, and
␪ is the angle the surface normal makes with the z
axis. When we choose the height of the step such
Fig. 6. Schematic ray trace through a lens containing a NPS.
The height of the steps is exaggerated; it is generally small compared with the physical dimensions of the lens.
that it introduces a phase equal to a multiple of 2␲,
the structure has no effect on the beam at zero field.
Now consider the case in which the rays enter the
lens with field angle ␺ or, equivalently, in which the
lens is rotated over an angle ␺ around the y axis,
where the origin of the coordinate system coincides
with the center of curvature of the best-fit radius R of
the aspherical surface of the lens 共see Fig. 7兲. Let 共x,
y, z兲 be a point on the stepped structure. This point
can be expressed in spherical coordinates as
共 x, y, z兲 ⫽ 共R sin ␨ cos ␸, R sin ␨ sin ␸, R cos ␨兲,
(10)
where ␨ and ␪ are related according to
␪ ⫽ ␲ ⫺ ␨.
(11)
After rotation around the y axis, this point is located
at 共 x⬘, y⬘, z⬘兲 and is given by
x⬘ ⫽ R sin ␨ cos ␸ cos ␺ ⫺ R cos ␨ sin ␺,
y⬘ ⫽ R sin ␨ sin ␸,
z⬘ ⫽ R sin ␨ cos ␸ sin ␺ ⫹ R cos ␨ cos ␺.
(12)
When we write
共 x⬘, y⬘, z⬘兲 ⫽ 共R sin ␨⬘ cos ␸⬘,
R sin ␨⬘ sin ␸⬘,
R cos ␨⬘兲,
(13)
we find in the lowest order in ␺ that
z⬘ ⫽ R sin ␨ cos ␸ sin ␺ ⫹ R cos ␨ cos ␺,
⬇ R␺ sin ␨ cos ␸ ⫹ R cos ␨,
⬇ R cos共␨ ⫺ ␺ cos ␸兲.
(14)
Hence we find that
Fig. 5. Experimental and theoretical results of the change in
defocus wave-front aberration OPDrms as a function of the wavelength for the NPS tabulated in Table 2.
6552
APPLIED OPTICS 兾 Vol. 40, No. 35 兾 10 December 2001
␨⬘ ⫽ ␨ ⫺ ␺ cos ␸,
(15)
␪⬘ ⫽ ␪ ⫹ ␺ cos ␸.
(16)
thus
Table 3. Step Width and Height Distribution of the NPS to Improve the
Field of View of a Lens
␳j⫺1
␳j
␳៮ j
mj
⌽j 兾共2␲兲
sin ␪j
hj 共␮m兲
0.0
0.2
0.7
0.8
0.92
0.2
0.7
0.8
0.92
1.0
0.1
0.45
0.75
0.86
0.96
0
⫺17
⫺4
2
10
0
⫺0.1224
⫺0.048
0.028
0.154
0
0.3228
0.5380
0.6170
0.6887
0
⫺19.3525
⫺4.2488
2.0439
9.7821
central radius of each zone is r៮ j . In each zone, we
introduce a step with a height such that it introduces
a phase equal to mj 2␲ when ␺ ⫽ 0. These steps
therefore have no influence on the properties of the
lens. When the lens tilts by an angle ␺, each zone j
gives rise to a relative phase ⌬⌽j equal to
⌬⌽ j ⫽ 2␲m j
Fig. 7. Schematic drawing of the coordinate system in which 共x,
y, z兲 is a point on the surface of the lens. Furthermore, ␺ is the
rotation angle of the lens around the y axis.
Substituting this into Eq. 共9兲, we find that this
small rotation gives rise to a change in the OPD, in
the lowest order in ␺, equal to
冤 冉
⌬OPD共␺兲 ⫽ h j ␺ cos ␸ sin ␪ 1 ⫺
cos ␪
sin2 ␪
n 1⫺
n2
冊冥
1兾2
.
(17)
Let the height hj be given by
hj ⫽ mj
␭
sin ␪
n 1⫺
n2
冉
2
冊
,
1兾2
(18)
⫺ cos ␪
where mj is an integer. Note that according to Eq.
共9兲 this height results in an OPD equal to mj ␭. For
simplicity’s sake, we assume that hj can be taken
constant within one zone while still, for a good approximation, introducing a phase mj 2␲ when ␺ ⫽ 0.
When ␺ is nonzero, this structure gives rise to a relative phase ⌬⌽rel 共hence phase modulo 2␲兲 equal to
⌬⌽ rel ⫽
2␲
␺ sin ␪ cos ␸
⌬OPD共␺兲 ⫽ 2␲m j
.
␭
n
(19)
Furthermore, using sin ␪ ⫽ r兾R 共see Fig. 7兲, we find
⌬⌽ rel ⫽ 2␲m j
r␺ cos ␸
.
nR
(20)
Note that this phase is linear in cos ␸ and therefore
generates a comatic aberration. The height distribution of the annular zones determines which
Zernike coma polynomial or combinations of Zernike
coma polynomials12 is approximated by the NPS.
Divide the pupil in a number of radial zones. The
r៮ j ␺ cos ␸
.
nR
(21)
We can sufficiently compensate the coma arising
from the tilted lens by these phase structures by
proper choice of the zones and the integer values mj .
In the nontilted case, these structures have no effect
on the wave front.
To illustrate the method to improve the field of
view, we consider a typical DVD lens made with a
glass replication technique. At a 2° field, such a lens
gives rise to an approximately 70-m␭ OPDrms coma
wave-front error. For the sake of simplicity, we assume that this is only the lowest-order coma. This
assumption simplifies the calculation somewhat but
is not essential for the above principle. The phase
⌽lens of the wave-front aberration introduced by the
lens set at a 2° field is then given by
⌽ lens ⫽ 2␲共0.4␳ ⫺ 0.6␳ 3兲cos ␸,
(22)
where ␳ is the normalized pupil radius. Note that
the radial distribution is proportional to the lowestorder coma Zernike term. The proportionality constant is such that the above phase term results in a
70-m␭ OPDrms wave-front aberration. We divide
the entrance pupil into five zones. We also let the
pupil radius be rmax ⫽ 1.65 mm, the best-fit radius of
the surface be R ⫽ 2.3 mm, ␭ ⫽ 660 nm, and the
refractive index of the replica layer in which these
steps are present be equal to n ⫽ 1.56. Because ␺ ⫽
2° ⫽ 0.035 rad, we find that each zone introduces a
relative phase equal to
⌬⌽ j
⫽ 0.016m j ␳៮ j cos ␸.
2␲
(23)
Table 3 shows the zones together with mj and the
corresponding phase. Figure 8 gives the OPD of the
wave-front error with and without the NPS present
at a 2° field. The phase introduced by the NPS is
also shown in Fig. 8. The wave-front error that is
due to the NPS reduces from 70 to 19 m␭; hence the
field of view is increased by a factor of more than 3.
Because both the phase of the comatic wave-front
10 December 2001 兾 Vol. 40, No. 35 兾 APPLIED OPTICS
6553
Fig. 8. Calculated coma wave-front error at a 2° field 共a兲 with no
NPS present and 共c兲 with NPS present. 共b兲 The phase introduced
by the NPS.
Fig. 9. Experimental and theoretical results of the change in
defocus wave-front aberration OPDrms as a function of the field
angle for the NPS tabulated in Table 2.
aberration and that of the NPS are linearly proportional to ␺, this reduction factor of 3 holds for a whole
range of ␺ near ␺ ⫽ 0. Although we have demonstrated the principle for third-order coma, higherorder coma terms can be incorporated in the
compensation in the same way.
imental results together with the theoretical results
for the defocus wave-front aberration OPDrms 共the
minus sign means that the Zernike coefficient is negative兲 are shown in Fig. 9, again revealing a good
agreement between the theory and the experiment.
D.
Field Curvature
A final example of the continuous parameter change
is field curvature compensation. To compensate for
field curvature, the NPS must generate defocus as a
function of the field angle. Consider the configuration shown in Fig. 1. Let the step heights hj of the
annular zones be integer multiples of the height h
defined in Eq. 共1兲. When the incoming parallel beam
enters the NPS with field angle ␪, the height hj now
gives rise to an OPD given by Eq. 共9兲. For small field
angles, the relative phase ⌬⌽j ⫽ 2␲兾␭关OPD共␪兲 ⫺
OPD共0兲兴 can then, when we use Eq. 共18兲, to the
lowest order in ␪, be written as
⌬⌽ j ⫽
␲m j 2
␪,
n
(24)
where n is the refractive index of the NPS material
and where the field angle ␪ is expressed in radians.
Because of this ␪2 dependence of the phase introduced
by the NPS, the NPS can be made to compensate for
the field curvature of the lens that is also proportional
to ␪2, if the zone width and height are properly designed.
The NPS tabulated in Table 2 to generate defocus
as a function of the wavelength can also be used to
generate defocus as a function of the field angle ␪.
Using Twyman–Green interferometry in a similar
way as in Subsection 2.B, we measured the amount
of defocus generated by the NPS tabulated in Table
2 as a function of the field angle. According to the
theory, the NPS generates 0.0073-m␭ OPDrms at a
1-mrad field angle at the expense of 0.0009-m␭
OPDrms residual 共higher-order兲 wave-front aberrations and scales proportionally with ␪2. The exper6554
APPLIED OPTICS 兾 Vol. 40, No. 35 兾 10 December 2001
3. Aberration Correction with a Nonperiodic Phase
Structure for Discrete Parameter Values
In the case of continuous parameter changes, the
effect introduced by the NPS must be proportional to
the change of the parameter to be able to compensate
for the aberrations for a whole range. As a result,
the step height distribution as a function of the radial
coordinate is a direct reflection of the shape of the
wave-front aberration to be corrected. This proportionality is no longer needed for the discrete parameter change, for example, for two different
wavelengths only. The only requirement is that the
stepped phase distribution introduced at each discrete parameter setting, modulo 2␲, approximates
the wave-front aberration of the lens system but with
the opposite sign. In general, this leads to more
design freedom than for the continuous case.
Another difference with respect to the continuous
case is that the parameter change in the discrete case
is generally large compared with that in the continuous case. As a result, a step height h, which introduces a phase step of 2␲ in the first parameter
setting, results in a significant phase step 共after we
take modulo 2␲兲 that is different from zero in the
second parameter setting. A further aspect related
to the discrete parameter setting is that the phase
共modulo 2␲兲 introduced at the second parameter setting by a step height m h becomes substantially the
same as that introduced by 共m ⫹ p兲h 共with m and p as
integers兲. Hence, when we take various integer
multiples of the step height h, only p substantially
different phase steps can be introduced. To determine the optimal distribution of the step heights of
the NPS is then rather straightforward because of
the limited number possibilities. It is not yet clear
how we can find the optimal zone width distribution,
with constraint
L i 共␳ 1, . . . , ␳ N兲 ⱖ 0,
i ⫽ 1, . . . , N ⫹ 1.
(31)
Note that
⳵F
⫽ ⫺ 4共a i ⫺ a i⫹1兲 f 共␳ i 兲␳ i ⫹ 2共a i2 ⫺ a i⫹12兲␳ i
⳵␳ i
⫹ 8共a i ⫺ a i⫹1兲␳ i
兰
1
f 共␳兲␳d␳
0
Fig. 10. Function f 共␳兲 and the stepped phase distribution that is
due to the NPS.
which is important if we are to have optimal compensation with the NPS. We first derive an analytical
relationship with which the optimal zone width distribution can be determined in the discrete case for a
given step height distribution. An explicit example
is then discussed.
A.
Optimal Zone Distribution
To determine the optimal zone distribution, we proceed as follows. Let 2␲f 共␳兲 be the phase that must
be compensated by the stepped phase distribution of
the NPS, where ␳ is the normalized pupil coordinate.
For example, 2␲f 共␳兲 is the phase arising because of
the spherical wave-front aberration W共␳兲, hence
N⫹1
兺 共a
⫺4
再
⫺1
⳵L i
⫽ 1
⳵␳ j
0
when j ⫽ i ⫺ 1
when j ⫽ i
elsewhere.
(25)
The rms value of the OPD that is due to wave-front
aberrations is defined as
OPD共 f 兲 ⫽ 2
2
兰
1
冋兰
f 共␳兲 ␳d␳ ⫺ 2
2
0
1
册
2
f 共␳兲␳d␳ ,
0
冋
N⫹1
兺 a1
i
⫽0
where ␳0 ⫽ 0 and ␳N⫹1 ⫽ 1.
L i 共␳ 1, . . . , ␳ N兲 ⫽ ␳ i ⫺ ␳ i⫺1,
i
共␳i⫺1,␳i 兲
i⫽1
共␳兲
i
o
1
␮ i ⱖ 0,
i ⫽ 1, . . . , N ⫹ 1,
i ⫽ 1, . . . , N ⫹ 1.
o
冋
4共a i ⫺ a i⫹1兲␳ io ⫺f 共␳ io兲 ⫹
a i ⫹ a i⫹1
⫹2
2
N⫹1
(27)
(35)
(36)
Hence we find
⫺
兺 a 共␳
j
o2
j
兰
1
f 共␳兲␳d␳
册
0
⫺ ␳ j⫺1o2兲 ⫽ ␮ i⫹1 ⫺ ␮ i ,
j⫽1
册
, . . . , ␳ No兲 ⫽ 0,
(34)
i ⫽ 1, . . . , N,
(37)
␮ i ⱖ 0,
i ⫽ 1, . . . , N ⫹ 1,
(38)
␮ i 共␳ io ⫺ ␳ i⫺1o兲 ⫽ 0,
i ⫽ 1, . . . , N ⫹ 1.
(39)
Suppose that all ␳i are different. Then ␮i ⫽ 0 for all
i, and Eq. 共37兲 implies that ␳io satisfies10
o
f 共␳ io兲 ⫽
a i ⫹ a i⫹1
⫹2
2
is minimal, where we defined
1 共a,b兲共␳兲 ⫽ 1
兺 ␮ ⵱L 共␳
i⫽1
(26)
where the OPD is expressed in wavelengths. The
problem can now be rephrased as follows. The NPS
introduces an annular stepped phase distribution
with phase step ⫺2␲ai at zone ␳i⫺1 ⱕ ␳ ⱕ ␳i 共see Fig.
10兲. For given values ai with i ⫽ 1, . . . , N ⫹ 1, we
determine the value of the pupil coordinates 0 ⬍ ␳1 ⬍
␳2 ⬍ . . . ⬍ ␳N ⬍ 1, such that
F共␳ 1, . . . , ␳ N兲 ⫽ OPD2 f 共␳兲 ⫺
(33)
N⫹1
⵱F共␳ 1o, . . . , ␳ No兲 ⫹
␮ i L i共␳ 1 , . . . , ␳ N 兲 ⫽ 0,
2␲
W共␳兲.
␭
(32)
From the Lagrange multiplier rule for inequality constraints 共Kuhn–Tucker theorem13兲, it follows that,
when ␳1o, . . . ␳No is a solution of the problem, Lagrange multipliers ␮1, . . . ␮N⫹1 exist such that
o
2␲f 共␳兲 ⫽
⫺ a i⫹1兲a j ␳ i 共␳ j2 ⫺ ␳ j⫺12兲,
i
j⫽1
兰
1
f 共␳兲␳d␳
0
N⫹1
⫺
a⬍␳⬍b
elsewhere,
j
o2
j
⫺ ␳ j⫺1o2兲.
(40)
j⫽1
(28)
Furthermore, we let
i ⫽ 1, . . . , N ⫹ 1.
(29)
The optimization problem is then to minimize
F共␳ 1, . . . , ␳ N兲
兺 a 共␳
(30)
If some of the ␳io are identical, Eq. 共40兲 holds for the
subset of ␳io that are different. Equation 共40兲 states
that the zone boundary ␳i is the position in which the
function f is equal to the averaged value of ai and ai⫹1
plus a correction factor that is equal to the difference
between the average value of f 共␳兲 over the pupil and
the averaged value of the step distribution over the
pupil. We can then find the ␳io by solving the coupled nonlinear system of Eq. 共40兲.
10 December 2001 兾 Vol. 40, No. 35 兾 APPLIED OPTICS
6555
Table 4. Added Phase for the CD Configurationa
Phase CD 共module 2␲兲
m
共rad兲
共⫻1兾2␲兲
1
2
3
4
5
6
7
5.250
4.217
3.184
2.151
1.118
0.085
5.336
0.8356
0.6712
0.5067
0.3423
0.1779
0.0135
0.8492
a
For the case of an extra height m ⫻ h made of PMMA, with h
given by Eq. 共1兲.
Fig. 11. Plot of the OPD in the CD configuration in the case 共a兲
with no NPS present and 共c兲 with NPS present. 共b兲 The contribution to the wave front of the NPS.
When we add an additional defocus term
⌬z␳2NA2兾␭ to f 共␳兲, it is also possible to derive relationships determining the optimal zone distribution
together with the best focus position 共see Appendix A兲
in a similar way.
1
2
B. Infinite Conjugate Objective Lens for Digital Versatile
Disk and Compact Disk
1. Design
To illustrate the design of a NPS for the discrete
parameter setting case, we consider a DVD objective
that reads DVD optical disks with a cover-layer thickness of 0.6 mm at a NA of 0.6 at wavelength ␭ ⫽ 660
nm. With the aid of a NPS in front of this objective,
it will also be made suitable to read and write CDs
with a cover-layer thickness of 1.2 mm and a NA of
0.5 at wavelength ␭ ⫽ 785 nm.7,8,10 Consequently,
we must correct the spherical aberration introduced
because of the cover-layer thickness difference and
the spherochromatism of the objective by the NPS,
using the fact that the wavelength of the incident
beam has two discrete values: for the DVD, ␭ ⫽ 660
nm and for the CD, ␭ ⫽ 785 nm.
This problem can be solved as follows: We start
with an objective lens design optimized for the DVD
configuration. The system suffers from a significant
amount of spherical aberration in the CD configuration 关see Fig. 11共a兲兴. For a typical f ⫽ 2.75-mm Philips glass兾2P DVD objective with a NA of 0.6 at
wavelength ␭ ⫽ 660 nm, and with a disk made of
polycarbonate 共nDVD ⫽ 1.5796 and nCD ⫽ 1.5733兲, a
wave-front aberration W of
W共␳兲 ⫽ 3.132共␳ 2 ⫺ ␳ 4兲␭
(41)
arises in the CD configuration with a NA of 0.5 and
␭ ⫽ 785 nm, where ␳ is the relative pupil coordinate
in the CD configuration 共hence ␳ ⫽ 1 corresponds to a
NA of 0.5 for the CD and ␳ ⫽ 1.2 corresponds to a NA
of 0.6 for the DVD兲. The corresponding phase is
then ⌽共␳兲 ⫽ 2␲W共␳兲兾␭. Twenty-eight percent of the
aberration is due to spherochromatism. To compen6556
APPLIED OPTICS 兾 Vol. 40, No. 35 兾 10 December 2001
sate for this spherical aberration, we add a plate in
front of the objective, with a NPS on top of it 共see Fig.
1兲. We divide the area of the NPS into nine annular
zones for rays corresponding to a NA less than or
equal to 0.5. Let the height h be defined by Eq. 共1兲,
where ␭ is in this case the wavelength in the DVD
configuration and n is the refractive index of the material of which the NPS is made at this wavelength.
In a similar way as above, we let the height of each
zone j be mj times the height h, where mj is an integer, which may differ from zone to zone. As a result
of this choice, each zone gives rise to an additional
phase to the wave front equal to 2␲mj for the DVD
configuration. Consequently, these zones have no
effect for the DVD configuration. For the CD configuration, however, these zones introduce a phase
difference that is not equal to a multiple of 2␲, and
they lead to a wave-front variation. For a NPS made
of PMMA with refractive index n ⫽ 1.4861 for ␭ ⫽
785 nm and n ⫽ 1.4891 for ␭ ⫽ 660 nm, we have h ⫽
1.349 ␮m. For the CD configuration, the phase
共modulo 2␲兲 introduced by a step m ⫻ h is tabulated
in Table 4 for various values of m. Note that there
are only six substantially different phase steps possible 共see also Appendix B兲.
When the step height distribution is chosen to be
0h, 5h, 4h, 3h, 2h, 3h, 4h, 5h, and 0h, the NPS will
give rise to a stepped wave-front distribution in the
CD configuration that approximates the one introduced by the lens– disk combination but with the opposite sign, as can be seen in Fig. 11. Note that, in
contrast to the continuous case, the highest phase
contribution in this discrete case originates from the
smallest step height. From Table 4 it follows that
the corresponding values for ai for the abovementioned step height distribution are a ⫽ 共0,
0.1779, 0.3423, 0.5067, 0.6712, 0.5067, 0.3423,
0.1779, 0兲. Substituting this into Eq. 共40兲 while using Eqs. 共41兲 and 共25兲 and then solving the resulting
coupled equations, we obtain the optimal zone distribution expressed in the relative pupil coordinates ␳ ⫽
共0.239, 0.352, 0.449, 0.556, 0.830, 0.893, 0.936, 0.971,
1兲. The OPDrms reduces from 234 m␭ 共without NPS兲
to 44 m␭ 共with NPS兲, which is well below the diffraction limit.
As a second example, let us consider the case in
Fig. 12. Plot of the OPD in the CD configuration with ⌬z ⫽
2.065-␮m defocus in the case 共a兲 with no NPS present and 共c兲 with
NPS present. 共b兲 The contribution to the wave front of the NPS.
which, in addition to the wave-front aberration W
given in Eq. 共41兲, we allow an additional amount of
defocus to be present, given by
W add ⫽ ⫺21 ⌬z␳ 2NA2,
(42)
where ⌬z is the focus shift. Choosing the zone
height distribution 0h, 5h, 4h, 3h, 4h, 5h, 0h, 1h, 2h
of the NPS made of PMMA, we find from Eqs. 共40兲
and 共A3兲 that the optimal zone width and defocus are
given by ␳ ⫽ 共0.261, 0.385, 0.505, 0.800, 0.864, 0.909,
0.945, 0.974, 1兲 and ⌬z ⫽ 2.065 ␮m, respectively.
The OPDrms reduces from 253 m␭ 共without NPS but
with ⌬z ⫽ 2.065-␮m defocus present兲 to 44 m␭ 共with
NPS兲 共see Fig. 12兲.
2. Experimental Results
To verify the compatibility solution experimentally,
we made a NPS plate out of PMMA using the diamond turning process. The shape of the NPS is
given by the first example in Subsection 3.B.1 共see
also Fig. 11兲. The plate was mounted together with
the DVD objective lens 共as discussed in Subsection
3.B兲 in a Philips two-dimensional actuator 共see Fig.
13兲. The aim of our experiments was to perform CD
reading and writing and DVD reading with a DVD
objective lens with NPS.
The aberrations of the NPS and objective lens combination were measured in a Twyman–Green interferometer at both wavelengths. The results indeed
confirmed that the presence of the NPS reduces the
spherical aberration to well below the diffraction
limit in the CD configuration 关total OPDrms spherical
wave-front aberration was 38 m␭ 共when we took into
account Zernike terms up to the 16th order兲兴. In the
DVD configuration, the spherical aberration was only
slightly affected by the presence of the NPS, showing
that the phase steps in the manufactured NPS were
not exactly equal to a multiple of 2␲. Furthermore,
the measurements also revealed that an amount of
coma and, to a much lesser extent, astigmatism were
present in the system 共not present in the case without
Fig. 13. Photograph of the actuator with DVD objective and NPS
used in the experiments.
NPS兲, indicating that the PMMA plate was somewhat deformed during the diamond turning process
and the mounting. In the experiments, the disk was
tilted to compensate for the coma introduced by the
lens and NPS system. The total OPDrms wave-front
aberration including the asymmetrical contributions
共except lowest-order coma兲 was 42 m␭ in the CD configuration and 46 m␭ in the DVD configuration.
We constructed an optical pickup unit using an f ⫽
11-mm collimator 共NA of 0.13兲. With a spherical
and cylindrical sensor lens, a spot of least confusion of
approximately 60 ␮m in diameter was generated on
the 共100-␮m兲2 detector. As a result, rays in the CD
configuration that are highly aberrated, hence corresponding to a NA greater than 0.5, miss the detector.
Because of this spatial filter, no additional aperturelimiting means are necessary in the CD configuration. The laser was a 80-mW Sharp infrared diode
laser 共FWHM 9° ⫻ 19°兲. The spot was elongated
perpendicularly to the tracks. The residual higherorder spherical aberrations in both NPS versions appeared to have only a limited effect on the CD readout
quality. Figure 14 shows examples of the readout
eye pattern, after equalization, obtained for both
DVD readout and CD reading and writing. A bottom jitter of 8.9% was found for DVD readout,
whereas for CD readout, 4.0% jitter was measured.
These values are well below the maximum allowed
values 共DVD jitter ⬍ 15% and CD jitter ⬍ 10%兲,
showing that CD and DVD compatibility can indeed
be achieved with a NPS. For CD readout, one important parameter is the so-called tilt window, i.e.,
the amount of disk tilt that can be allowed in the
complete system so that the decoder can still decode
the eight-to-fourteen modulation 共EFM兲 signal. We
used 10% data-to-clock jitter as the maximum jitter
value allowed by the CD decoder, when measured on
a Leader jitter meter at 1⫻. Disk tilt generates
coma, which is also proportional to the third power of
the NA. By definition, radial tilt involves a tilt
around the track, and tangential tilt involves a tilt of
the track.
10 December 2001 兾 Vol. 40, No. 35 兾 APPLIED OPTICS
6557
direction, a smaller tangential tilt window than radial can be allowed. The tilt windows found are sufficient for good CD readout.
Using the same optical pickup, we also performed
compact disk-recordable 共CD-R兲 writing 共1⫻兲 experiments. Figure 14 shows a typical eye pattern for a
CD-R 共after equalization兲, written with the pickup
with NPS present, with a jitter of 7.5%, which is
rather good for CD-R.
4. Summary and Conclusions
Fig. 14. Readout eye patterns, after equalization, obtained with
the NPS: 共a兲 DVD eye pattern with 8.9% jitter, 共b兲 CD eye pattern
with 4.0% jitter, 共c兲 CD-R 1⫻ eye pattern with 7.5% jitter.
Jitter values at different radial and tangential tilt
angles are shown in Fig. 15 for the case when NPS is
present. The radial tilt window is found to be 2.4°.
According to the CD standard, a disk can be curved
up to ⫾0.6° in the radial direction. Consequently,
for the system with NPS, there is enough margin left
for a tilt of the turntable or play of the shafts in the
bearings, for example. In a similar way as for the
radial tilt window, the tangential tilt window 共Fig.
15兲 is 1.4°. Because CDs reveal a much smaller curvature in the tangential direction than in the radial
In this paper we studied the application of wide
NPSs in optical systems. These NPSs give rise to
a nonperiodic stepped wave-front phase. In particular, we studied NPSs in which each phase step
in the nominal configuration is equal to a multiple
of 2␲ and therefore have no net effect on the optical
properties of the system studied. When a certain
parameter changes, the steps in the NPS no longer
result in a phase equal to a multiple of 2␲. The
resulting phase distribution can be used to compensate for various wave-front aberrations of the lens
system.
A NPS differs in two ways from a conventional
periodic structure: 共1兲 the structure is nonperiodic,
allowing more design freedom than conventional periodic diffractive structures, and 共2兲 the step widths
in the NPS are wide. As a result, certain wave-front
aberrations of the lens system are only partly compensated for, unlike periodic diffractive structures
whose compensation can be considered continuous
because of the relatively large number of zones. The
complexity of a NPS is thus reduced compared with
conventional periodic structures with sufficient compensation of wave-front errors.
The various examples in this paper showed that
sufficient compensation of the aberrations can be obtained despite the wide nature of the steps of the
NPS. In particular we showed that a DVD objective
lens can be made CD-R compatible in optical recording with the aid of a NPS. We presented a novel,
simple analytical way of optimizing the NPS for this
application. The theoretical results were confirmed
experimentally. Furthermore, we discussed making
lens systems athermal, achromatic, and with reduced
field curvature with a NPS. We verified this experimentally. Finally, we have shown that it is possible
to improve the field of view of a lens system with such
a structure.
In conclusion, NPSs form an interesting class of
phase structures that offer significant design freedom
and ease of manufacturing with sufficient compensation of wave-front aberrations.
Appendix A. Optimal Zone Distribution and Defocus
Adding a defocus of ⌬z to the system results in an
additional phase distribution on top of the aberration
W共␳兲兾␭ already present, which is given by
Fig. 15. Radial and tangential tilt windows for the DVD objective
with NPS in the CD configuration.
6558
APPLIED OPTICS 兾 Vol. 40, No. 35 兾 10 December 2001
1
f 共␳兲 ⫽ W共␳兲兾␭ ⫺ 2 ␳ 2⌬zNA2兾␭.
(A1)
Proceeding along the same lines as in Subsection 3.A,
we now obtain the additional constraint
⳵F
⫽ 0.
⳵⌬z
兰
1
兺 a 关共␳
j
4
j
Appendix B. Number of Substantial Different Phase
Steps for the Discrete Parameter Case
Consider two discrete parameter settings q1 and q2,
where q can be any parameter affecting the phase of
a step height h. Let h1 be the height of a phase
structure that introduces a phase step of 2␲ at parameter setting q1 and, similarly, h2 be the height
introducing a phase step of 2␲ at parameter setting
q2. Consider a step mh1, where m is an integer
number. At parameter setting q2 this step introduces a phase equal to 2␲h1兾h2. To find the number
p of substantially different phase steps 共modulo 2␲兲
for parameter setting q2, we write the ratio h1兾h2 as
a continued fraction 共CF兲.14 In general a CF is defined by
1
1
b3 ⫹
⬅ b0 ⫹
1
b4 ⫹ · · ·
1
1
1
· · ·.
b1 ⫹ b2 ⫹ b3 ⫹
1
1
1
1
···
b1 ⫹ b2 ⫹ b3 ⫹
bk
⬅ 共b 0, b 1, b 2,· · ·, b k兲 ⫽
Ak
,
Bk
(B9)
Proceeding in this way, we obtain
冉冊
(B10)
1
⫺ b k,
ak
(B11)
b k ⫽ Int
a k⫹1 ⫽
1
,
ak
and the CFk is uniquely defined. To find the number
p, we must determine the CFk corresponding to h1兾h2
such that, for that integer value of k, the CFk satisfies
the relation
冏
CFk ⫺
冏
h1
ⱕ 0.005
h2
h1
Ak
⬇ CFk ⫽
,
h2
Bk
(B1)
When the numbers bk are integer numbers, the CF
always converges. As a result, we can define the
truncation of this CF to the kth order to be CFk, which
can be written as
CFk ⫽ b 0 ⫹
1
⫺ b 1.
a1
1
,
a1
(B12)
for the first time. Note that demanding that the CFk
approximates the ratio h1兾h2 within 0.005 is taken as
an example only. Although this is in general a reasonable choice, it can be chosen differently depending
on the particular application. The rational approximation is then
1
b2 ⫹
(B8)
b 1 ⫽ Int
a2 ⫽
(B7)
冉冊
(A3)
Equation 共40兲 together with Eq. 共A3兲 determines the
optimal zone width ␳i and the corresponding optimal
defocus ⌬z.
b1 ⫹
(B6)
then
⫺ 共␳ j⫺14 ⫺ ␳ j⫺12兲兴 ⫽ 0.
CF ⫽ b 0 ⫹
b 0 ⫽ Int共a 0兲,
a 1 ⫽ a 0 ⫺ b 0,
⫺ ␳ j2兲
j⫽1
0
(B5)
where Int共 兲 means that we take the integer part of
a0. If we define
N⫹1
f 共␳兲共␳ ⫺ 2␳ 3兲d␳ ⫹
h1
,
h2
a0 ⫽
(A2)
As a result, apart from Eq. 共40兲 that remains unaltered, the additional relationship is found to be
2
with A⫺1 ⫽ 1, A0 ⫽ b0, B⫺1 ⫽ 0, and B0 ⫽ 1. The
coefficients bk can be determined as follows. Let
(B13)
and from this we find that the number p of substantially different phase steps for the parameter setting
is given by p ⫽ Bk.
In Table 5 an example is given corresponding to the
case discussed in Subsection 3.B.
Table 5. Example of the Determination of CFk for the Case Discussed
in Subsection 3.Ba
(B2)
where Ak and Bk are integers determined by
A k ⫽ b k A k⫺1 ⫹ A k⫺2,
(B3)
B k ⫽ b k B k⫺1 ⫹ B k⫺2,
(B4)
k
CFk
Ak兾Bk
兩CFk ⫺ 0.8356兩
Bk
1
2
共0,1兲
共0,1,5兲
1 兾1
5 兾6
0.164
0.002
1
6
a
The parameter q1 corresponds to the wavelength ␭1 ⫽ 660 nm
and q2 to ␭2 ⫽ 785 nm. The corresponding heights are h1 ⫽
1.3494 mm and h2 ⫽ 1.6149 mm, resulting in h1兾h2 ⫽ 0.8356.
The number of substantial different phase steps p is found to be
six.
10 December 2001 兾 Vol. 40, No. 35 兾 APPLIED OPTICS
6559
We thank L. van den Broek, F. van Gaal, M. de
Jongh, C. van der Vleuten, and J. Wijn from Philips
Enabling Technology Group for manufacturing and
assembling the various NPSs used in the experiments. We also thank M. van As from Philips Research Laboratories for conducting the defocus and
field curvature experiments, B. Jacobs from Philips
Research Laboratories for performing the DVD readout experiments, and A. Oudenhuysen from Philips
Optical Pickup Lenses for performing the thermal
experiments.
References
1. A. I. Tudorovskii, “An objective with a phase plate,” Opt. Spectrosc. 6, 126 –133 共1959兲.
2. H. P. Herzig, ed., Micro-optics 共Taylor & Francis, London,
1998兲.
3. K. Maruyama, M. Iwaki, S. Wakamiya, and R. Ogawa, “A
hybrid achromatic objective lens for optical data storage,” in
International Conference on Applications of Optical Holography, T. Honda, ed., Proc. SPIE 2577, 123–129 共1995兲.
4. Y. G. Soskind, “Novel technique for passive athermalization of
optical systems,” in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 43 of OSA Trends in Optics and Photonics 共Optical Society of America, Washington, D.C., 2000兲,
pp. 194 –204.
5. J. M. Sasian and R. A. Chipman, “Staircase lens: a binary
and diffractive field curvature corrector,” Appl. Opt. 32, 60 – 66
共1993兲.
6560
APPLIED OPTICS 兾 Vol. 40, No. 35 兾 10 December 2001
6. M. Born and E. Wolf, Principles of Optics, 6th ed. 共Pergamon,
Oxford, UK, 1980兲, Chap. 9.1.3, p. 463.
7. T. Shimano and A. Arimoto, “Objective lens and optical head
using the same,” European patent application EP 0865037A1
共16 September 1998兲.
8. R. Katayama, Y. Komatsu, and Y. Yamanaka, “Dualwavelength optical head with a wavelength-selective filter for
0.6- and 1.2-mm-thick-substrate optical disks,” Appl. Opt. 38,
3778 –3786 共1999兲.
9. B. H. W. Hendriks and P. G. J. M. Nuyens, “Design and manufacturing of far-field high-NA objective lenses for optical recording,” in 18th Congress of the International Commission for
Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther,
and T. Asakura, eds., Proc. SPIE 3749, 413– 414 共1999兲.
10. B. H. W. Hendriks, J. E. de Vries, and H. P. Urbach, “Application of non-periodic phase structures in optical systems,” in
Proceedings of the Second Conference on Optical Design and
Fabrication 2000 共Optical Society of Japan, Tokyo, 2000兲, pp.
325–328.
11. P. Smulders, J. P. Baartman, J. W. Aarts, and B. H. W. Hendriks, “Two-element objective lens and spherical aberration
correction for digital video recording 共DVR兲,” in Optical Data
Storage, D. G. Stinson and R. Katayama, eds., Proc. SPIE
4090, 302–308 共2000兲.
12. Ref. 6, p. 464.
13. D. G. Luenberger, Optimization by Vector Space Methods
共Wiley, New York, 1969兲.
14. M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions 共Dover, New York, 1970兲.