High-efficiency dielectric reflection gratings: design,
fabrication, and analysis
Karl Hehl, Joerg Bischoff, Ullrich Mohaupt, Martin Palme, Bernd Schnabel, Lutz Wenke,
Ragnar Bödefeld, Wolfgang Theobald, Eberhard Welsch, Roland Sauerbrey, and
Hartmut Heyer
We report on reflection gratings produced entirely of dielectric materials. This gives the opportunity to
enhance the laser damage threshold over that occurring in conventional metal gratings used for chirpedpulse-amplification, high-power lasers. The design of the system combines a dielectric mirror and a
well-defined corrugated top layer to obtain optimum results. The rules that have to be considered for the
design optimization are described. We optimized the parameters of a dielectric grating with a binary
structure and theoretically obtained 100% reflectivity for the ⫺1 order in the Littrow mounting for a 45°
angle of incidence. Subsequently we fabricated gratings by structuring a low-refractive-index top layer
of a multilayer stack with electron-beam lithography. The multilayer system was fabricated by conventional sputtering techniques onto a flat fused-silica substrate. The parameters of the device were
measured and controlled by light scatterometer equipment. We measured 97% diffraction efficiency in
the ⫺1 order and damage thresholds of 4.4 and 0.18 J兾cm2 with 5-ns and 1-ps laser pulses, respectively,
at a wavelength of 532 nm in working conditions. © 1999 Optical Society of America
OCIS codes: 050.0050, 050.1950, 320.0320, 320.5520.
1. Introduction
Contemporary developments of ultrashort highpower lasers that are based on chirped pulse amplification1,2 rely extensively on the dispersive
properties of diffraction gratings. Currently gratings in compressors of chirped-pulse-amplification laser systems are etched into a photoresist by use of
holographic techniques and are then covered by a
K. Hehl is with the Ing.-Büro Optimod, Ricarda-Huch-Weg 12,
D-07743 Jena, Germany. J. Bischoff is with the Technical University of Ilmenau, P.O. Box 100565, D-98684 Ilmenau, Germany.
U. Mohaupt and M. Palme are with the Fraunhofer Institut für
Angewandte Optik und Feinmechanik, Schillerstrasse 1, D-07745
Jena, Germany. B. Schnabel, L. Wenke, R. Bödefeld, W.
Theobald 共theobald@qe.physik.uni-jena.de兲, E. Welsch, and R.
Sauerbrey are with the Friedrich-Schiller-Universität, D-07743
Jena, Germany. B. Schnabel is with the Institut für Angewandte
Physik; L. Wenke is with the Institut für Angewandte Optik; R.
Bödefeld, W. Theobald, E. Welsch, and R. Sauerbrey are with the
Institut für Optik und Quantenelektronik. H. Heyer is with Layertec GmbH, Blankenhainer Strasse 169, D-99441 Mellingen, Germany.
Received 19 March 1999; revised manuscript received 22 July
1999.
0003-6935兾99兾306257-15$15.00兾0
© 1999 Optical Society of America
metallic layer. The gratings are fragile and can easily be damaged, especially by a high-intensity laser
beam.3 To minimize damage and to ensure that all
the incident laser light is reflected into only one diffraction order, with an efficiency as close as possible
to 100%, it is desirable to implement reflection gratings that are made entirely of dielectric optical
materials.4 – 8 Previously high-efficiency dielectric
gratings were produced by structuring a highrefractive-index hafnia layer on top of a multilayer
stack with a measured efficiency as great as 95%.8
Two optical functions must be incorporated into a
dielectric device. First, high reflectivity can be
reached by a dielectric multilayer stack, and, second,
diffraction is achieved by a lateral periodic groove
structure in an additional top layer. Besides the
advantage of almost no absorption in the dielectric
material the multilayer arrangement offers several
free parameters to optimize the performance of the
grating.
Here we describe in detail the design optimization
and fabrication of a dielectric grating produced by
structuring a low-refractive-index, fused-silica layer
on top of a dielectric multilayer stack. We performed a careful analysis by measuring the scattered
light and the efficiency of the various orders. The
measurement shows that the theoretical predictions
20 October 1999 兾 Vol. 38, No. 30 兾 APPLIED OPTICS
6257
of nearly 100% diffraction efficiency into the ⫺1 order
can also be achieved in practice, and in addition a
high damage threshold for a nanosecond laser pulse
can be obtained. The design parameters were optimized for wavelength ␭ ⫽ 532 nm in a classical Littrow mounting and an angle of incidence close to 45°.
The choice of wavelength was given only by the fact
that an existing damage test facility and the scattering setup were both equipped with lasers at ␭ ⫽ 532
nm and therefore provided a complete analysis of the
grating.
The paper is organized as follows. We start with
a general discussion of the design problem. The opportunities and the restrictions of an all-dielectric
grating are discussed and compared with the wellknown metallic grating in Section 2, especially in
Littrow mounting conditions. Both waveguiding in
the mirror layer stack and transmission in the substrate must be taken into account to prevent loss or
instability of the solution near the optimal condition.
In Section 3 we describe fabrication of the device.
The optimized mirror stack and a low-index top layer,
which were subsequently structured, were produced
by commercial sputtering techniques to achieve sufficient precision of the design parameters. The grating was produced in a fused-silica top layer by means
of an electron-beam lithographic technique described
in the text. In Section 4 we summarize the measured parameters of the grating and discuss the measured angular scattering distribution compared with
the calculated distribution. It is shown that our device works in Littrow conditions with 97% diffraction
efficiency in the ⫺1 order. The loss of 3% is due to a
substructure on the grating resulting in parasitic
ghost lines and much weaker diffuse scattering. Absorption is ruled out as a loss mechanism. The laser
damage threshold of the grating was measured with
nanosecond and subpicosecond laser pulses and was
found to be near the value of the flat layer system.
The damage threshold of our dielectric grating is a
factor of ⬃5 higher in the nanosecond regime,
whereas it is comparable with metallic gold gratings
for ultrashort pulses. The practical use of the dielectric grating is discussed in Section 5 with some open
problems and a look at further investigation.
2. Grating and Mirror Design
In the following we study dielectric gratings consisting of a multilayer mirror and a single corrugated top
layer. The objective is an optimal interplay of refraction in the top layer favoring transmission and
high reflectivity of an underlying dielectric quarterwave stack. This leads to additional constraints for
the optimization compared with a pure metallic grating. The conditions for an all-dielectric grating that
allows only reflection into the 0 and ⫺1 orders are
summarized under the so-called classical mounting.
For classical mounting the angle of incidence and the
grating period are restricted to a certain regime,
which will be discussed in more detail by use of a
scaled representation of effective indices. The conditions for avoiding unwanted reflection and trans6258
APPLIED OPTICS 兾 Vol. 38, No. 30 兾 20 October 1999
mission modes are discussed in general and then
specified for the specific design of a Littrow mounting.
Eventually the theoretical and the numerical treatment of the optimization process for the grating parameters are described.
A.
Low-Order Condition
The final goal of the optimization process is that all
diffraction orders in transmission and reflection vanish, except the backdiffracted ⫺1 order. We consider first the low-order reflection condition; i.e., we
allow only the ⫺1 and the 0 orders under reflection.
The grating equation n0 sin ⌰m ⫽ n0 sin ⌰0 ⫹ m␭兾p
for the reflected orders can be written in terms of the
so-called effective indices:
␤m ⫽ n0 sin ⌰m,
␤0 ⫽ n0 sin ⌰0,
␤m ⫽ ␤0 ⫹ m⌬␤,
m ⫽ 0, ⫾1, ⫾2, . . . ,
(1)
where n0 is the refractive index of the superstrate, ⌰0
is the angle of incidence, ⌰m is the diffraction angle,
and ␤m and ␤0 are the effective indices of the diffracted and the incident waves, respectively. The
term ⌬␤ ⫽ ␭兾p expresses the difference between the
effective indices of neighboring refractive orders excited by the grating with period p and wavelength ␭.
In fact, the effective indices ␤m are the wave-number
vector components of the different modes parallel to a
plane stack in units of the total air value. The two
dimensionality of these vector components can be neglected if a classical mounting condition is assumed.
With a conical mounting ␤0 and ⌬␤ have different
directions; i.e., the plane of incidence and the detection planes of the diffracted orders are different from
each other. Because of the periodicity of the grating
the ␤m values are equal in all parts of the stack,
whereas angles ⌰m are only defined for flat layers
and depend on the refractive index; i.e., they change
from layer to layer in accordance with Snell’s law.
We calculated the transmitted diffracted orders by
substituting, in the effective index, n0, with ns, the
refractive index of the substrate with the grating
structure. If for higher modes the total value of ␤m
exceeds the corresponding refractive index 共␤m ⬎ n
for positive orders and ␤m ⬍ n for negative orders兲, no
angle ⌰m is defined in such a plane layer. In contrast with radiative modes such modes are called
evanescent in the layer, which means that for the
superstrate or the substrate no reflection or transmission occurs, respectively. However, this can be
the case when reflected orders are suppressed while
the equivalent transmitted orders appear, owing to
ns ⬎ n0, which is mostly fulfilled.
From this consideration a first condition is derived,
which is that only the ⫺1 and the 0 orders shall occur
under reflection. The reflected ⫺1 order should be
radiative, leading to the condition ␤⫺1 ⬎ ⫺n0, and
when Eq. 共1兲 is used, this results in
⌬␤ ⬍ n0 ⫹ ␤0.
(2)
Fig. 1. Grating-index change ⌬␤ normalized to the refractive index n0 of the superstrate as a function of the effective index ␤0兾n0
of the incident light. The ratio ␤0兾n0 ⫽ sin ⌰0 is simply the sinus
of the angle of incidence, and the index change is given by ⌬␤ ⫽
␭兾p, where ␭ is the wavelength and p is the grating period. The
values of the abscissa are in the 0 ⬍ ␤0兾n0 ⬍ 1 range, while for the
ordinate 0 ⬍ ⌬␤兾n0 ⬍ 2. For this example the values for the
refractive indices are ns ⫽ 1.46 for the substrate and n0 ⫽ 1.0 for
the superstrate. Below line A⫺1 the reflected ⫺1 order is radiating, while above lines M⫹1 and M⫺2 the transmitted ⫹1 and ⫺2
orders are evanescent. The optimum area for the working condition is marked by the gray area.
Figure 1 shows ⌬␤兾n0, which is the grating-index
change ⌬␤ normalized to the refractive index of the
superstrate n0 as a function of the effective index
␤0兾n0 of the incident light. Line A⫺1 in Fig. 1 represents ⌬␤ ⫽ n0 ⫹ ␤0, and hence the ⫺1 order radiates below A⫺1 while above it is evanescent. We
consider only the area below A⫺1 in the 共␤0, ⌬␤兲 plane.
For the Littrow mounting the ⫺1 order is backreflected into the incident direction, i.e., 共⌰⫺1 ⫽ ⫺⌰0兲,
which results in
⌬␤ ⫽ 2␤0.
(3)
This condition is represented by the dashed diagonal
line in Fig. 1.
To prevent transmission into the substrate for the
0 and ⫺1 orders, a dielectric mirror must be placed
between the corrugated top layer and the substrate.
In our case for wavelength ␭ ⫽ 532 nm we used a
typical quarter-wave stack 共1H 1L兲 10, where L is a
SiO2 layer of thickness ␭兾4nL, nL ⫽ 1.46 and H is a
high refractive Nb2O5 layer of thickness ␭兾4nH, nH ⫽
2.375 on a fused-silica substrate. A reflectivity of
100% of the flat stack is reached for both diffraction
angles, ⌰0 and ⌰⫺1. Nevertheless we have to take
into account that total reflection is reached only for a
certain range of angles of incidence, i.e., in a limited
range of ␤0. The dielectric mirror provides total reflection for TE polarization 共s polarization兲 in the
whole range, 0 ⬍ ␤0, 兩␤⫺1兩 ⬍ 1. But for TM polar-
Fig. 2. Upper right: refractive-index profile of the flat layer system as a function of depth in the layer stack beyond the corrugated
top layer. The profile results from an optimized 共1H 1L兲10 1L
system enlarging the angular region of total reflection for TM
polarization. The excitation conditions for the grating with a period, p ⫽ 380 nm and ␭兾p ⫽ 1.4 for ␭ ⫽ 532 nm, are characterized
by the effective index regions 0 ⬍ ␤0 ⬍ 1 with n0 ⫽ 1 for the
different Rayleigh diffraction modes, ␤m ⫽ ␤0 ⫹ m␭兾p. To control
the negative index modes in the same way, the negative profile
image is also included in the lower right.
ization 共 p polarization兲 the transition to transmission
starts near ⌰0 ⫽ 45°. Since we use our dielectric
grating in the Littrow condition, which is close to
⌰0 ⫽ 45°, TM-polarized light will be partly transmitted. To use both polarization directions for an experimental analysis of our grating, which was
designed to achieve maximum efficiency for TE polarization, we had to modify the mirror design in such
a way that in TM polarization for larger angles of
incidence total reflection is also provided. We obtained this by changing slightly the thickness of the
layers to achieve a higher effective index ␤maxTM for
TM polarization. The value ␤maxTM is the effective
index where the reflectivity of the mirror drops to
50%.
In the upper right-hand part of Fig. 2 the
refractive-index profile of the optimized multilayer
stack is shown versus the depth of the layer stack.
As mentioned above the thickness of the layers differs
slightly from a pure quarter-wave stack. The lower
right and the left part of Fig. 2 are discussed below.
The measured and the predicted values for the reflectivity of the mirror are in fairly good agreement, as
can be seen in Fig. 3, where the reflectivity for TM
light is shown as a function of angle of incidence.
Now the maximum value is increased to ␤maxTM ⬃
0.93. For the Littrow mounting the condition ␤0 ⫽
兩␤⫺1兩 ⬍ ␤maxTM is fulfilled, which excludes transmission of these orders for both polarization states. The
transition value ␤maxTM is a property of the mirror
and therefore nearly independent from the superstrate and an additional L layer on top of the mirror,
which will be structured for the grating.
For the following calculation we assume that the
superstrate has a higher refractive index than any of
the materials used in the stack, which can be practically realized by coupling light through a prism or a
half-sphere of an appropriate material on top of the
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6259
B.
Exclusion of Higher Orders
To achieve evanescence for the higher-order modes
共⫺2 and ⫹1 orders兲 in reflection and transmission,
the condition that 兩␤⫺2兩, ␤⫹1 is larger than the corresponding refractive index n of the superstrate or the
substrate, respectively, has to be obeyed. The condition ␤⫹1 ⬎ n, which suppresses the ⫹1 order, is
given by
⌬␤ ⬎ n ⫺ ␤0.
Fig. 3. Comparison of the experimental and the calculated reflection curves of the optimized multilayer mirror for TM polarization.
The corresponding index profile is shown in Fig. 2.
mirror without a grating. The advantage of this
technique is that the effective index ␤0 can be varied
between 0 and the superstrate index, which we assume to be n0 ⫽ 3 in the following, which is larger
than nH. Therefore we can study the response of the
layer stack up to the appearance of evanescence in all
layers. For n0 ⫽ 3 Fig. 4 shows the calculated reflection coefficient of the mirror versus the angle of
incidence for TE and TM polarization. The curve for
TM shows nearly 100% reflectivity in the range up to
15° whereas for larger values transmission into the
substrate occurs. The transition point at 15° corresponds to ␤maxTM ⬃ 0.93. The TE-polarization band
edge for transmission occurs only for ␤maxTE ⯝ 1.29,
which cannot be observed when air is used as the
superstrate because of ␤maxTE ⬎ n0. For angles of
incidence larger than 29° total reflection is reached
again because the modes are evanescent in the substrate, and therefore the reflectivity rises to nearly
100%. The fluctuations beyond that value are observed because a small artificial absorption in the
layer stack is assumed in the calculation showing the
effect of waveguiding. This is explained in detail in
Subsection 2.C.
(4)
Line M⫹1 in Fig. 1 represents ⌬␤ ⫽ n ⫺ ␤0, the
transition of an evanescent ⫹1-order mode to a radiating mode. Since suppression of the ⫹1 order is
desired, the area above M⫹1 is considered. The
same argument holds for the ⫺2 order, which is evanescent when 兩␤⫺2兩 ⬎ n, and with Eq. 共1兲 this results
in
⌬␤ ⬎
1
共n ⫹ ␤0兲
2
(5)
and gives a further restriction to the allowed 共␤0, ⌬␤兲
region with M⫺2 being on the borderline of the gray
area in Fig. 1. If we substitute n with n0, the modes
vanish in reflection, and for n ⫽ ns, the refractive
index of the substrate, the modes are suppressed in
transmission. Because the refractive index of the
substrate is in general larger than that of the superstrate, the modes are automatically evanescent in
reflection when they vanish in transmission. In Fig.
1 all the lines were drawn assuming a quartz substrate with a refractive index of n ⫽ 1.46 at 532 nm
and air as the superstrate with n0 ⫽ 1. For a lower
substrate refractive index, M⫹1 and M⫺2 are shifted
downward and the allowed area increases. Since
M⫹1 and M⫺2 intersect at the Littrow line, substrates
with ns ⬎ 3n0 are not useful for the grating production because the gray area vanishes in that case.
The minimum value for ⌬␤ in the gray area is determined by the intersection of the M⫺2 and the M⫹1
lines, which is
⌬␤min ⫽
2
n,
3
(6)
with the corresponding effective index ␤0 ⫽ ⌬␤min兾2.
The lowest possible value for ␤0 ⫽ 共n ⫺ n0兲兾2 is
calculated from the intersection of the A⫺1 and the
M⫹1 lines. The optimum value for ⌬␤ is
⌬␤opt ⫽
Fig. 4. Effective reflectivity of the mirror versus angle of incidence for the excitation with a superstrate n0 ⫽ 3 for both polarizations. The mirror design that corresponds to Fig. 2 includes an
artificial weak absorption in the sixth H layer to enable visualizing
waveguide modes in reflection.
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APPLIED OPTICS 兾 Vol. 38, No. 30 兾 20 October 1999
n ⫹ n0
,
2
(7)
where the width of the gray area is maximal and the
M⫺2 line reaches the right border of the figure. At
that ⌬␤ ⫽ ␭兾p value the range of possible incident
angles is largest. If, for example, a larger angle of
incidence than the Littrow angle is chosen 共␤0 ⬎ ⌬␤兾
2兲, the ⫺1-order beam will be diffracted with an angle
lower than the Littrow angle 共兩␤⫺1兩 ⬍ ⌬␤兾2兲. In the
diagram the incident beam is then located at the
right side of the Littrow line, while the ⫺1 order lies
left of it and the same distance from the Littrow line.
Lowering the refractive index of the substrate will
increase the available range. A reasonable design
strategy is to choose first the refractive indices of the
multilayer stack yielding the highest reflectivity and
than an optimum value for ⌬␤ before starting the
optimization process. In the special case of the mirror shown in Fig. 2 we used ⌬␤ ⫽ 1.4, i.e., p ⫽ 380 nm
for ␭ ⫽ 532 nm, which is a little higher than the
optimum value ⌬␤opt ⫽ 1.23 derived from condition
共7兲 for the substrate index ns ⫽ nL ⫽ 1.46. The
reason for this is discussed in Subsection 2.C.
C.
Waveguiding Effects in the Stack
The gray area in Fig. 1 refers to the case in which
higher-order modes in the substrate and in the L
layers of the mirror are evanescent. As we now
demonstrate, free-wave propagation may occur in the
high-index H layer for ␤⫹1, 兩␤⫺2兩 ⬍ nH ⫽ 2.375 共see
Fig. 2兲. Transition from a radiative to an evanescent
character in a given layer is illustrated in Fig. 2 for
our special mirror stack. The effective indices for
the different orders vary between the indicated values if ␤0 changes between 0 and 1 as indicated by the
arrows at the left-hand side. At lower right, the
index profile is mirrored on the abscissa for the
negative-order modes. In the case in which the absolute value of ␤m of a diffraction order exceeds the
refractive index in the layer, the order becomes evanescent.
In the following we study in which conditions the
radiative character of higher orders in the mirror
might influence the coupling of the different orders.
The wave propagation of the different orders in the
mirror is now discussed separately by using the following technique. As mentioned above we are
studying the reflection behavior of the mirror without
a grating on top for a superstrate with n0 ⫽ 3 instead
of the excitation of the different orders. The main
difference is that the ␤0 value can now be varied
between 0 and n0 ⬎ nH. Therefore propagation of
the whole range of diffraction modes m ⫽ ⫺2, ⫺1, 0,
⫹1 in the multilayer stack can be studied separately
without a grating on top. As shown above, the ␤max
values for the two polarization states can be derived.
Figure 4 shows strong and very narrow variations of
the reflection curve in the ns ⬍ ␤0 ⬍ nH region. This
is known as the attenuated total reflection effect in
the case of plasmon excitation at a metal–air interface and coupling light through a prism into the mirror. But in our case the minima are connected with
the occurrence of guided waves that are bound in the
layer system between the substrate and the superstrate.9,10 At characteristic ␤0 values a minima occur in the reflectivity that are properties of the mirror
stack and will be referred to as ␤M values below. To
observe these minima it is necessary to introduce into
the calculation artificially a weak absorption in the
mirror stack, which was implemented in the sixth H
layer while all the other layers were assumed to be
perfectly transparent. The ␤M values that describe
the positions of the minima are not affected if absorp-
Fig. 5. 共a兲 Comparison of the reflection curve from Fig. 4 for TE
polarization in resonance region 1 ⬍ ␤0 ⬍ 2 with the different
complex eigenmodes ␤M described in the text in more detail. The
upper modes 共squares兲 correspond to the superstrate with n0 ⫽ 3
and the lower modes 共triangles兲 to the air case n0 ⫽ 1. 共b兲 Same
calculation for TM polarization.
tion in any other layer is allowed. However, the
depth and the width of the minima are influenced but
without any change of ␤M value.
In addition, we demonstrated waveguiding by calculating the reflectivity of the mirror versus the effective index for the two different systems with air
and n0 ⫽ 3 as a superstrate, respectively 共Fig. 5兲.
This is compared with the effective indices ␤M of the
eigenmodes of the mirror system. The solid curve in
Fig. 5共a兲 shows the reflectivity of the mirror for TE
polarization versus ␤0, which was already shown in
Fig. 4 as a function of ⌰0. The squares mark the
positions of the ␤ values of the complex eigenvalues
␤M, which were calculated by means of the computer
program11 TRAMAX for a superstrate n0 ⫽ 3. The ordinate values for the squares were chosen arbitrarily
to avoid overlapping the symbols. The filled squares
represent the real part Re共␤M兲 of the eigenvalues,
while the two open squares mark the positions
Re共␤M兲 ⫾ Im共␤M兲, where Im共␤M兲 is the imaginary
part of the eigenvalue, which is a measure for the
width of the resonance. The same was calculated by
using air 共n0 ⫽ 1兲 as a superstrate resulting in the
filled triangles as the real part and the open symbols
as the imaginary part. We note, first, that the number of modes that we define here as the number of the
eigenfunction modes is equal for both superstrates.
Second, for high Re共␤M兲 values the strongest bound
modes have equal Re共␤M兲 values, showing that the
superstrate does not affect these modes. The modes
20 October 1999 兾 Vol. 38, No. 30 兾 APPLIED OPTICS
6261
Fig. 6. 共a兲 Angular dependence of the calculated phase of the reflected TE light for the excitation with a superstrate n0 ⫽ 3. In contrast
to Fig. 4 we assumed for this calculation there is no absorption. 共b兲 The difference phase value of successive points from 共a兲 is plotted
versus ⌰0. 共c兲 Phase of the reflected TM light. 共d兲 Phase difference for TM polarization.
for n0 ⫽ 1 共triangles兲 with ␤M ⬎ 1.46 are sharp, which
means that they have no imaginary part. These
modes are evanescent in the superstrate; i.e., they
are real waveguiding modes. In contrast, for n0 ⫽ 3
all modes have a finite width since they are leaking
into the superstrate or substrate. A comparison of
the results for the two eigenmode systems and the
reflection curve shows good agreement for the lower
modes. The higher resonancelike modes show a
stronger dependence on the excitation conditions.
Figure 5共b兲 shows the calculation for TM polarization
where the strongest bound modes are located at lower
␤M values.9,10
Nevertheless, in the case in which no absorption is
assumed, the reflectivity of the mirror does not show
any waveguide effects in the ns ⬍ ␤ ⬍ nH region, but
the phase is strongly varying in this region as can be
seen in Fig. 6. Figures 6共a兲 and 6共c兲 show the calculated phase of the specular reflected light as a function of the angle of incidence 共n0 ⫽ 3兲 for TE and TM
polarization, respectively. To make the phase
change clearly visible, the phase value of successive
points is now subtracted and is shown as a function of
⌰0 in Fig. 6共b兲 for TE polarization and in Fig. 6共d兲 for
TM polarization. Figure 6 shows that the strongest
phase changes are observed only for certain ⌰0 values
that are related to the ␤M eigenmode values. This
influences the coupling between the different Rayleigh orders as is discussed below. It is clear that
this effect has to be avoided for the dielectric grating,
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APPLIED OPTICS 兾 Vol. 38, No. 30 兾 20 October 1999
and therefore the final condition for the allowed gray
area in Fig. 1 is defined by inserting for n the highest
value ␤M into the M-line conditions in inequalities 共4兲
and 共5兲. This value lies inside the ␤maxM, nH ⬎ ns
region, and the allowed area in Fig. 1 will be further
reduced compared with use of the refractive index of
the substrate. In our case ␤maxM ⬇ 2.0 for TE and a
value of ⌬␤opt ⫽ 1.5 is obtained from Eq. 共7兲. We
used a value of ⌬␤ ⫽ ␭兾p ⫽ 1.4, which is slightly
lower than ⌬␤opt. For TM polarization a smaller
value may be taken. Note that for higher stack
numbers the effective indices of the waveguide modes
increase to the maximum value nH in the limit of a
very thick mirror.9
D.
Theoretical and Numerical Treatment
To achieve high efficiency it is expected that the
grooves must be quite deep compared with the grating period, which is of the order of the wavelength.8
It is therefore important to use a rigorous method
based on the full-vector Maxwell equations. The
theoretical and computational solution of the diffraction in an arbitrary corrugated multilayer system is
described in more detail in Ref. 12. Briefly stated,
we idealize the grating profile as a set of discrete
slices that are rectangular in cross section. In each
slice the material consists of two separate dielectrics;
i.e., we approximate the surface profile by a steplike
profile. The propagation of the fields inside such a
corrugated slice layer is calculated by an expansion
into a full set of Bragg waves. The field in the flat
layers and in the semi-infinite substrate and superstrate is expressed as a Rayleigh expansion. The
field transition conditions lead to a linear system of
equations that yield the complex reflection and transmission matrices that are solvable by numerical computation.12
E.
Grating Optimization
Our dielectric grating with a period of p ⫽ 380 nm
and the mirror design shown in Fig. 2 work inside the
allowed area defined with n ⫽ 2 for the M-line conditions, and therefore only reflection into the 0 and
⫺1 orders occurs without waveguide influences.
The condition for total backreflection into the ⫺1 order 关RTE共⫺1兲兴 is obtained by minimization of the
specular reflection 关RTE共0兲兴, achieved by varying the
three free grating parameters’ groove depth tg, groove
width pg, and thickness tr of a residual plane layer of
the corrugated material above the mirror 共see also
Fig. 13兲. Following the arguments in Ref. 8, we start
with the case of a single corrugated layer on top of a
properly designed dielectric mirror. The alternative
route of producing a dielectric grating with a corrugated substrate, which is subsequently covered by a
multilayer stack, is much more complicated because
the surface profile at each layer is flattened more and
more during the growth process. This effect has to
be analyzed before optimization as shown, for example, by the theoretical simulations in Ref. 13. In
contrast to the investigations in Ref. 8 where the
structure was etched into the high-refractive-index
material, we corrugated a fused-silica top layer consisting of the low-index-refraction material. Deeper
grooves were expected but the reproducibility and
stability of the etching process were more controllable. The optimization process resulted in a groove
width of pg ⫽ 190 nm corresponding to a duty cycle of
1:1, defined as pg兾共 p ⫺ pg兲, a groove depth of tg ⫽ 435
nm, and a small residual thickness of tr ⫽ 20 nm,
which is also included in the index profile in Fig. 2.
The aspect ratio defined as tg兾共 p ⫺ pg兲 equals 2.29
and exceeds unity as supposed above. There are
other solutions where the residual layer tr is thicker
than 20 nm by the addition of multiples of about a
half a wave thickness, but the parameter stability of
our optimization is good enough to use the thinnest
corrugated L-top layer. Note that the structured top
layer has to be integrated into the mirror production.
A careful analysis of the design of the whole flat stack
must be carried out before etching. If necessary, a
small correction of the optimization parameters has
to be taken into account.
3. Device Fabrication
The optimized grating structure was fabricated in
two steps. First, a fused-silica substrate was coated
by subsequent H and L layers, finishing with an L
layer that was corrugated in the next step. The design parameters obtained in Section 2 are implemented as accurately as possible in the production
process.
A.
Mirror Fabrication
For mirror fabrication a plane fused-silica substrate
with high optical quality and negligible absorption in
the region of interest and without bulk inhomogeneities was alternately coated with a Nb2O5 highrefractive-index layer of nH 共532 nm兲 ⫽ 2.375 and a
SiO2 low-index layer of nL 共532 nm兲 ⫽ 1.46 with the
thicknesses shown in Fig. 2. All layers were produced by sputtering. The accuracy and the reproducibility of the layer parameters are of the order of
1% and are sufficient for our purpose. The packing
density is homogeneous and comparable with the
solid bulk properties inside each layer, as indicated
by measurement of the refractive indices and the
scattered light. Hence the mechanical stability is
very high, and the mirror provides a high resistance
against chemical and environmental influences.
The roughness of the layer interfaces does not increase remarkably in the spatial-frequency region
that is of interest for optical losses caused by light
scattering. This was checked with atomic force microscopy and light-scattering measurements such as
angle resolved scattering and total integrated scattering. As shown in Fig. 3 the mirror properties
agree with the predicted values of the optimized design with high accuracy. This was checked additionally by controlling the TM-reflection curve at 633 nm.
B.
Grating Fabrication
The fabrication procedure of the grating etched into
the top layer of the multilayer stack consists of two
basic steps: the lithographic generation of a grating
in a resist layer and the transfer of this grating into
the dielectric top layer by ion beam etching. For the
profile generation, electron-beam 共e-beam兲 lithography was preferred over holographic techniques because the grating design and optimization process
resulted in several gratings with different parameter
sets that had to be fabricated. The freedom of design in e-beam lithography, especially the opportunity for fast and easy variance of the grating period
and the duty cycle, is advantageous compared with
holographic techniques. Moreover grating areas of
some square millimeters 共which were feasible with
our e-beam writer without excessive writing time兲
were sufficient for optical testing. When the optimum grating parameters are fixed, holographic
methods will be applied for the grating fabrication,
which are superior to e-beam lithography if larger
grating areas of some 10 cm2 or even more have to be
produced.
Electron-beam exposure was performed with the
LION LV1 e-beam writer 共Leica Microsystems Lithography GmbH兲. We used the so-called continuous path control exposure mode. In this mode,
stitching errors 共a well-known effect in e-beam lithography and a serious problem especially for highquality gratings兲 are avoided as a matter of principle.
The resulting grating period and the profile quality
were uniform over the whole grating area. For the
ion-beam-etching step, a series of etching tests was
20 October 1999 兾 Vol. 38, No. 30 兾 APPLIED OPTICS
6263
duty cycle, whereas other grating features are mainly
determined by only one of the steps 共for example, the
grating period is determined by e-beam exposure
only兲. Finally, the actual duty cycle is difficult to
check during the fabrication process because the required accuracy is far beyond the resolution limit of
conventional optical microscopy. As a result, special
care was taken to control exactly the duty cycle in
each fabrication step. This was done mainly by
scanning electron microscopy inspection of the grating profiles.
4. Device Analysis
Fig. 7. SEM image of the dielectric grating.
performed even before starting the detailed grating
design process to find the top layer material convenient for both the fabrication process and the optical
function of the gratings. Material selection must be
performed before design optimization, because the
optical properties of the top-layer material crucially
influence the grating design. As a result the component with the lower refractive index, SiO2, was
chosen as the top-layer material because the whole
fabrication process was more stable than for a highindex component. This result was obtained in spite
of the deeper grating grooves that had to be fabricated in the low-index material to achieve the same
optical effect as in the high-index material. Moreover the experimental profiles were closer to the desired binary profile when an intermediate etching
mask made of chromium was used instead of etching
directly from the resist layer into the dielectric substrate.
The two-step etching process was performed with
two different machines, an Oxford Microfab 300 system with an electron-cyclotron-resonance ion-beam
source using pure argon as the etching gas and a
self-made ion-beam-etching system with a Kaufman
ion-beam source using a mixture of CF4 and CHF3 as
the etching gas. Owing to the stability of the chromium etching mask, a nearly binary grating profile
was obtained even for deep grooves. Therefore the
required groove depth of 435 nm and even more could
be realized for a grating period of 390 nm in the SiO2
layer. Figure 7 shows a scanning electron microscopy image of such a grating. In the entire fabrication process the duty cycle of the gratings 共or the
related groove width兲 turned out to be the most sensitive parameter for several reasons. First, the required accuracy is tight; even a deviation of 20 nm in
the groove width substantially reduces the optical
performance of the gratings. Second, each of the
several fabrication steps 共exposure, resist development, chromium etching, SiO2 etching兲 influences the
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The dielectric grating was investigated by measurement of the angular distribution of the specular and
the diffracted light to extract information about the
design properties and to compare the measurement
with the optimization calculations. The reconstruction of the groove profile and the waveguide effects
are analyzed in detail with theoretical curves obtained from our computer program. Subsequently
measurements were performed to determine the
damage threshold, i.e., the critical laser energy per
unit area for the onset of optical breakdown. The
following subsections characterize the investigation,
the methods, and the results.
A.
Diffraction Efficiency Measurements
The gratings produced were investigated and characterized by using scattering equipment that allows
measurement of the diffracted intensity for variable
polarization states and angles of the incoming and
the detected fields in a coplanar arrangement. The
setup of the scattering equipment is described in detail in Ref. 14. The automated scatterometer is
based on the method of angle resolved light scattering
for measuring diffraction efficiency in the 0 and ⫺1
orders in reflection for TE and TM polarization.
Measurement of the background light distribution
was necessary to investigate the behavior of scattering and losses caused by the parasitic excitation in
the corrugated multilayer system. The scatterometer works at a wavelength of ␭ ⫽ 532 nm. Because
every single measurement demands a different level
of sensitivity and a different dynamic range, two
types of detector heads were used. The background
level was measured with a dynamic range of ⬎108
with a photomultiplier tube and calibrated gray filters. The diffraction efficiencies were measured
with a dynamic range of ⬎104. For higher sensitivity an Ulbricht sphere with an integrated photodiode
was used. All measurements were normalized by
the total incident power determined separately for
every state of polarization and type of detector head.
To determine the background level, the grating
was adjusted in the Littrow mounting. In this case
angle of incidence ⌰0 and diffraction angle ⌰⫺1 of the
⫺1 order are equal, as can be seen in Fig. 8. For this
fixed incident angle the detector head was moved on
a circle around the sample between ⌰d ⫽ ⫺75° and
⫹75° in steps of 0.5°. A small off-plane angle ␦ ⫽ 1°
is necessary to move the detector head over the entire
Fig. 8. Principal setup of the scattering measurement in the Littrow mounting.
range of angles, which includes the Littrow angle.
The incident beam had a small divergence of 5 mrad.
The intensity in the reflected 0 and ⫺1 orders was
measured by varying the angle of incidence ⌰0 in
steps of 0.5°. The angular range of the incidence
beam for the 0-order measurement was between 0°
and 75° and for the ⫺1-order measurement between
25° and 75°. The angle of incidence ⌰0 and the angle
⌰⫺1 of the reflected diffracted light are different, and
the detector must be moved according to the grating
equation. Therefore the background distribution
measurement has to be performed first, since it provides an accurate measurement of the grating period
p, which is needed for calculating ⌰⫺1 as a function of
the angle of incidence. For this purpose a special
algorithm was implemented in the scatterometer
software. For a fixed range of angles of incidence
and a given polarization state the measurement process was carried out by the scatterometer in an automated mode. The measured values were stored as
the ratio of the power in the diffracted order to the
incident power. Figure 9 shows the measurement of
the scattering yield of our sample as a function of the
detector angle ⌰d for the TE polarization under ␪0 ⫽
45° near the Littrow mounting condition. The diffraction in the reflected ⫺1 order corresponds to ⌰d ⫽
Fig. 9. Measured light scattering yield 共dots兲 versus detection
angle ⌰d relative to the surface normal. The angle of incidence
under the so-called classical mounting condition is ⌰0 ⫽ 45° and
therefore near Littrow angle ⌰Litt ⫽ 44.7°. The predicted values
␤0 and ␤⫺1 are indicated by squares. Weak ghost lines of a superperiod P ⫽ 6p are marked by triangles.
45°, whereas the specular reflection is seen under
⌰d ⫽ ⫺45°. The theoretical prediction of nearly
100% for the ⫺1 order is confirmed by a value 2
orders of magnitude smaller for specular reflection.
The difference between the ␤0 and ␤⫺1 values corresponding to the exact angle ⌰d is in excellent agreement with ␭兾p ⫽ 1.4, which is shown by the filled
squares marking the calculated angular positions.
Nevertheless some ghost lines appear. A pronounced superperiod of P ⫽ 6p is observed marked by
the open triangles in Fig. 9 and which is also observed
by visible inspection of the grating by means of a
microscope. The low scattering noise is a hint of the
quality of the grating. The yield integrated over all
angles except the two main diffraction orders results
in a loss of approximately 1–2%, which is within the
measurement error caused by the detector resolution.
The 0- and ⫺1-order reflection efficiencies and the
loss functions are shown for TE polarization in Figs.
10共a兲–10共c兲, while Figs. 10共d兲–10共f 兲 contain the
curves for TM polarization. The loss function is defined by 关1 ⫺ R共0兲 ⫺ R共⫺1兲兴, i.e., the residual part of
light that is not contained in the reflected 0 and the
⫺1 orders. As a remarkable and predicted result
specular reflection RTE共0兲 vanishes completely in the
neighborhood of the Littrow mounting condition 关Fig.
10共a兲兴. The measured RTE共⫺1兲 shown in Fig. 10共b兲
rises to 97%, which is in fairly good agreement with
the scattering loss found above. This is correlated
with the observed behavior of the loss functions for
TE and TM, which are shown in Figs. 10共c兲 and 10共f 兲
as a function of ⌰0. Besides the small maxima in the
total loss function, the background value is always
nearly 3% in the angular range between 30° and 60°
independent of the polarization state, which is in
agreement with the background measurement shown
in Fig. 9. Note that the increase in the TM loss for
angles larger than 60° is caused by transmission of
the 0 order into the substrate as predicted in Fig. 3.
We now discuss the principal character of the diffraction efficiencies of the different orders for the excitation with a superstrate, n0 ⫽ 3. For this purpose
the reflection curve from Fig. 4 and the grating equation ␤m ⫽ n0 sin ⌰0 ⫹ m⌬␤ were used to calculate the
reflectivity of all diffraction orders, m ⫽ ⫺2, ⫺1, 0,
⫹1. The final result is shown in Fig. 11共a兲 for TE
and in 11共b兲 for TM polarization. It is obvious that
the general behavior shown in the measured diffraction efficiencies in Fig. 10 may now be understood in
terms of the mirror excitation. In the region around
30° ⬍ ⌰0 ⬍ 60° all the diffraction orders will be completely reflected as discussed above. The device
works like a metal grating without any loss in this
region. The efficiency is therefore a pure function of
the groove profile parameters and is discussed in
Subsection 4.B. The typical dielectric mirror effects,
especially waveguide effects, are described in more
detail in Subsection 4.C.
B.
Groove Profile Analysis
We have carried out calculations of the diffraction
efficiency starting with the optimized design param20 October 1999 兾 Vol. 38, No. 30 兾 APPLIED OPTICS
6265
Fig. 10. 共a兲 Measured efficiency of the reflected 0 order as a function of ⌰0 for TE polarization 共dotted curve兲. In comparison, the solid
curve shows the predicted values from the optimization process. 共b兲 Measured efficiency and calculation of the reflected ⫺1 order for TE.
共c兲 Loss function defined as the residual part of light that is not contained in the reflected 0 and ⫺1 orders for TE. 共d兲 Efficiency of the
0 order for TM polarization. 共e兲 Efficiency of the ⫺1 order for TM polarization. 共f 兲 Loss function for TM.
eters. The results referred to as Theory 共1兲 are
shown by experimental curves in Fig. 10. The general behavior of the curves agrees with the predictions discussed by the mirror excitation parameters
and follows the experimental curves. There is good
agreement for TE polarization, especially in the near
Littrow mounting region, but for TM polarization the
maximum of the ⫺1 order is somewhat overestimated
关see Fig. 10共e兲兴. We presume that this discrepancy is
due to a small deviation in the groove profile function
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from an ideal binary structure since the mirror does
not affect the efficiency as mentioned above. Therefore we used this discrepancy to fit the groove profile
function. The goal of this optimization consists of a
change in the profile function in such a way that only
the diffraction efficiency for TM is influenced. The
final result is Theory 共2兲 in Fig. 12 and now shows
good agreement for both polarization states.
The corresponding optimized profile function of the
structured fused-silica layer is shown in Fig. 13 com-
Fig. 11. 共a兲 Reflection coefficients of the different orders, m ⫽ ⫺2,
⫺1, 0, ⫹1, for TE polarization obtained from Fig. 4 and from using
␤m ⫽ n0 sin ⌰0 ⫹ m␭兾p, assuming a superstrate n0 ⫽ 3. 共b兲
Reflection coefficients for TM polarization.
pared with the initial profile. The tendency to smear
out the ideal binary profile corresponds to a flattening
of the edges and corners, which is expected during the
etching process. The stability of the TE diffraction
efficiency against profile fluctuations may be understood when we look at the differences in the electricfield components for both polarizations. In TE there
are only tangential-field components along the groove
direction that are continuous for the transition from
one side of the structured interface to the other. In
TM 共 p polarization兲 we have both a steady tangential
field perpendicular to the groove direction and a normal electric field that will be changed on the interface.
The splitting into field components is local and therefore is influenced by the profile function. Consequently it is expected that the interchange between the
RTM共0兲 and the RTM共⫺1兲 contribution is much more
sensitive against profile changes than for TE polarization. Thus a polarization-dependent measurement is
able to give additional information. An ellipsometric
detection and excitation system in conical mounting
conditions may be a good additional tool to determine
experimentally the profile function. On the other
hand, the Littrow mounted grating used for TE polarization is a good device with high stability against
profile fluctuations caused by fabrication errors.
C.
Waveguide Effects
The behavior in the diffraction curves shown in Fig.
10 outside the Littrow angle is explained in terms of
waveguiding and transmission into the mirror stack.
Two different effects must be taken into account.
On the left and the right side of the broad maximum
one observes that the efficiency strongly depends on
small changes in the angle of incidence. This strong
modulation occurs both in the experimental curve
and in the theoretical calculation. However, the loss
function 关see Figs. 10共c兲 and 10共f 兲兴 does not show
these modulations, which means that if, for example,
the reflectivity in the ⫺1 order drops, the reflectivity
of the 0 order rises by the same amount and vice
versa. There is a strong interchange of energy between the 2 orders in a very narrow angular range,
which becomes apparent as side loops left and right of
the maximum at the Littrow angle. There is no net
loss of radiation in connection with the narrow loops.
The theoretical curves show in principle the same
behavior. Only the corresponding angle of incidence
is slightly shifted compared with the experimental
curves. The behavior is connected to the phase effects shown in Fig. 6共a兲, and therefore the side loops
are created by the excitation of waveguide modes in
the mirror stack. The evanescent modes change
strongly the phase in the case of a resonance, and the
interplay between the radiative modes is disturbed.
The side loops left of the Littrow angle correspond to
resonances of the ⫹1 diffraction order. The side
loops for angles greater than the Littrow angle are
connected with waveguide excitation of the ⫺2 order.
According to the grating equation, Eq. 共1兲, the corresponding ␤M values for the waveguide modes that are
excited by the ⫹1 and ⫺2 orders are given by
␤M ⫽ ␤⫹1 ⫽ ␤0⫹1 ⫹ ⌬␤,
(8)
␤M ⫽ ⫺␤⫺2 ⫽ ⫺␤0⫺2 ⫹ 2⌬␤.
(9)
Subtracting Eqs. 共8兲 and 共9兲 from each other results in
␤0⫹1 ⫹ ␤0⫺2 ⫽ ⌬␤.
(10)
From Eq. 共10兲 it is obvious that the average value of
␤0⫹1 and ␤0⫺2 must be equal to the effective index
value of the incident beam in Littrow condition 关␤0 ⫽
⌬␤兾2 ⫽ 共␤0⫹1 ⫹ ␤0⫺2兲兾2兴. Therefore, if one adds the
␤0 values of the corresponding left- and right-hand
loops of the experimental and the theoretical curves
in Figs. 10 and 12, one gets the ⌬␤ value in both the
theoretical and the experimental case if the same
waveguide resonance is excited for both diffraction
orders. This relation is fulfilled in our case. The
discrepancy of the exact angular position of the loops
in the theoretical and the experimental curves can be
explained by Eqs. 共8兲 and 共9兲. Since the waveguide
resonances ␤M are highly sensitive against the mirror
design parameters, the exact positions of the corresponding effective indices of the ⫹1 and ⫺2 order
may differ.
The second effect consists of the occurrence of a net
loss, which is observed in Figs. 10共c兲 and 10共f 兲, especially for angles of incidence smaller than 25° when
the reflected ⫺1 order is evanescent. This is the
region where 1.4 ⬎ 兩␤⫺1兩 ⬎ 1 and 1.4 ⬍ ␤⫹1 ⬍ 2.1 and
20 October 1999 兾 Vol. 38, No. 30 兾 APPLIED OPTICS
6267
Fig. 12. Comparison of the measured efficiencies and an improved theoretical model, Theory 共2兲, which corresponds to the optimized
groove profile shown in Fig. 13. Theory 共3兲 includes an additional weak absorption in the sixth H layer. 共a兲 Measured efficiency and
calculation of the reflected 0 order as a function of ⌰0 for TE polarization. 共b兲 Measured efficiency of the reflected ⫺1 order for TE
polarization. 共c兲 The loss function for TE polarization. 共d兲 Efficiency of the 0 order for TM polarization. 共e兲 Efficiency of the ⫺1 order
for TM polarization. 共f 兲 TM loss function.
transmission in the substrate may occur in the regions with ␤max ⬍ 兩␤⫾1兩 ⬍ ns ⫽ 1.46. The ⫺2 order
does not play a role because of its evanescence in all
types of layers. This effect, however, causes losses
that are clearly visible in the calculations. The resonancelike behavior is due to so-called substrate
modes, also shown in Fig. 5. These modes depend
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APPLIED OPTICS 兾 Vol. 38, No. 30 兾 20 October 1999
much more on the mirror design parameters than on
the waveguide modes. In addition, the superstrate
and therefore the grating influences the exact values
of the resonances, and a small absorption in the stack
changes the character of the resonances, as can be
seen in Fig. 12. In Fig. 12 two theoretical curves,
Theory 共2兲 and Theory 共3兲, are compared, where for
Fig. 13. Comparison of the predicted groove profile and an optimized profile obtained by fitting the experimental curves in Fig. 10
around the Littrow condition.
the latter one the same design but an additional weak
absorption in the sixth H layer is assumed. A small
change in the direction of the experimental values
especially for TM polarization is noticeable. It is not
the intention to optimize all these modes since the fit
is very sensitive in all the design parameters. However, the consideration shows that one has to avoid
this region. A detailed study of waveguiding was
performed to understand the restriction on the Littrow mounting condition given by Eq. 共3兲. For example, an enlargement of the number of layers will
increase not only the ␤max values but also the effective index ␤M⫽0 to as high as the theoretical limit nH.
This restricts the allowed area in Fig. 1 and shows
the constraints of a dielectric grating design compared with a metallic grating.
D.
Damage Thresholds
Damage threshold measurements of the dielectric
gratings were performed with nanosecond and subpicosecond laser pulses. First we used a frequencydoubled Nd:YAG laser beam 共␭ ⫽ 532 nm兲 with a
pulse duration of 5 ns, which was focused onto the
grating with f# ⫽ 10 optics and an angle of incidence
equal to the Littrow angle. The beam profile was
determined by using the 共derivative兲 knife-edge scan
technique15 yielding a Gaussian profile with an intensity FWHM of 350 ␮m. The beam energy was
varied between 2 and 22 mJ in consecutive laser
shots, while for each shot a fresh target surface was
used. Therefore the diameter of the resulting damage spots, which were carefully inspected under the
microscope, increased with increasing energy to a
value as high as 340 ␮m. The laser beam energy
that was incident on the damaged area was calculated by integrating the beam profile function up to
the damage spot diameter and dividing by the total
incident energy. This allowed us to determine the
fluence at the margin of the damage spot, yielding for
each shot a damage fluence. Broadening of the damage spot due to heat diffusion was negligible in the
experimental conditions.16 We measured singleshot threshold fluences of 共8.8 ⫾ 2.6兲 J兾cm2 for the
unstructured top layer and 共4.4 ⫾ 1.3兲 J兾cm2 for the
grating itself, while all the values given here were
corrected with respect to the angle of incidence.
Each value is an average of approximately 80
single-shot damage thresholds, and the measurement error represents the standard deviation. The
damage threshold of the dielectric grating is high and
only a factor of 2 lower than for the unstructured
dielectric multilayer coating. The ratio between
both damage thresholds is surprisingly small taking
into account the depth of the grooves of ⬃420 nm in
relation to their width of 190 nm. In Ref. 17 multishot damage thresholds of 0.8 and 1.5 J兾cm2 were
measured for gold gratings and gold mirrors, respectively, irradiated with 1-ns, 1053-nm laser pulses.
Extrapolating to 5 ns using the well-established ␶1兾2
law results in 1.8 and 3.4 J兾cm2 for the grating and
the mirror, respectively. However, for irradiation at
532 nm we expect the damage fluence to be lower
than at 1053 nm. Measurements17 show that the
threshold of fused silica at 1053 nm is a factor of ⬃2
higher than at 526 nm for a pulse duration between
0.1 and 10 ps. We therefore estimate a damage
threshold higher by a factor of 4 –5 compared with
metallic gratings for nanosecond laser irradiation,
which is in agreement with Ref. 17. Laser damage
in metallic materials is dominated by the thermal
conductivity in the long-pulse regime leading to a
melting of the substrate when a certain threshold is
surpassed. The higher damage threshold in a dielectric material is attributed to the fact that the
conduction band is initially empty, and free electrons
have to be generated through multiphoton ionization
followed by an avalanche breakdown and energy
transfer to the lattice.18 When a dielectric grating is
exposed to an intense laser beam, the electric-field
strength at the groove edges is enhanced by a factor
of ⬃2 共Ref. 8兲, and therefore this might explain the
difference in the damage threshold between the unstructured dielectric multilayer and the grating itself. Note that the threshold of our dielectric
grating was 1 order of magnitude lower than stated
above when the laser beam impinged on areas with
scratches that were produced during the grating
manufacturing process. None of these scratches
was observed on the unstructured part of the mirror.
Similar measurements were eventually performed
with a subpicosecond hybrid dye– excimer laser system19 tuned to a wavelength of 532 nm. The pulse
duration was measured with a multishot autocorrelator using second harmonic generation in a 700␮m-thick ␤-barium borate crystal and yielded an
intensity FWHM of 共0.98 ⫾ 0.04兲 ps, assuming a
Gaussian pulse profile. The laser beam with maximum pulse energy of ⬃200 ␮J had to be focused much
harder onto the sample to reach the damage threshold. The beam was focused with f # ⫽ 10 optics and
an angle of incidence equal to the Littrow angle onto
the sample. The focus size was measured with a
microscope objective with 10⫻ magnification and a
CCD camera collecting the scattered light from a diffuse reflecting target. We carefully adjusted with a
micrometer-controlled stage the diffuse reflecting
20 October 1999 兾 Vol. 38, No. 30 兾 APPLIED OPTICS
6269
surface at the same position of the dielectric grating.
The precision was ⬃50 ␮m and is sufficient with
respect to a convocal parameter of ⬃2 mm of the
focusing optics. A nearly Gaussian intensity profile
with a minimum FWHM of 26 ␮m was measured.
However, a slightly larger spot size of 55 ␮m was
used for the damage tests. When applying a fresh
target surface for each shot, we observed through a
microscope no damage to the dielectric grating for
pulse energies smaller than 8.3 ␮J within a series of
⬃100 laser shots. This corresponds to a fluence of
0.25 J兾cm2 on target, while for the unstructured mirror a slightly higher value of 0.33 J兾cm2 was obtained.
Since the onset of damage is a statistical process,
we measured in several places the threshold when
100 laser pulses with a repetition rate of 1 Hz irradiated the same target area and obtained slightly
lower damage fluences of 共0.18 ⫾ 0.08兲 J兾cm2 and
共0.25 ⫾ 0.10兲 J兾cm2 for the grating and the unstructured surface, respectively. The corresponding fluences in the beam cross section were 共0.25 ⫾ 0.11兲
J兾cm2 and 共0.35 ⫾ 0.14兲 J兾cm2, respectively. To
cross check the obtained values we also measured the
single-shot damage threshold of a 1-mm-thick fusedsilica sample in the same experimental conditions,
which yielded a value of 2.7 J兾cm2 on target and was
in good agreement with other measurements for subpicosecond pulses.20,21 The damage fluence of
共0.18 ⫾ 0.08兲 J兾cm2 for the dielectric grating in the
ultrashort pulse regime was not as high as expected,
but it was consistent with other measurements for
dielectric gratings that yielded 0.21 J兾cm2 for 300-fs,
1053-nm laser irradiation.8 For gold gratings a
nearly constant threshold of ⬃0.4 J兾cm2 was determined for pulse durations between 0.1 and 100 ps at
1053 nm.17 If one takes into account a factor of 2
with respect to the wavelength, the dielectric and the
metallic gratings have approximately the same
threshold at 1 ps. The general trend is that the
difference between metallic and dielectric gratings
becomes smaller for ultrashort pulses. However,
more investigations in the femtosecond regime are
necessary to obtain the threshold of the gratings as a
function of both the pulse duration and the laser
wavelength.
5. Conclusion
We have shown that the design of an all-dielectric
grating demands a detailed analysis of the interplay
between the mirror properties and the constraints of
a transmission grating. This is in contrast to metallic gratings where only the grating design has to be
taken into account. We have described in detail especially the regime of an optimized working condition
to avoid transmission of higher orders into the substrate and the excitation of waveguiding modes in the
mirror stack. Although transmission into the substrate gives an additional parasitic loss, the excitation of waveguide modes leads to possible instability
for small parameter fluctuations. We have described all restrictions and degrees of freedom for the
design optimization of a dielectric grating and dem6270
APPLIED OPTICS 兾 Vol. 38, No. 30 兾 20 October 1999
onstrated their use for the special case of a totally
backreflecting grating under Littrow mounting.
Dielectric gratings were fabricated by using the
optimized parameters. The flat layers for the dielectric mirror stack as well as the top layer to be corrugated were deposited with a sputtering technique to
produce a dense multilayer system. Electron-beam
lithography and ion beam etching were applied to
produce nearly perfect binary groove structures in a
low-refractive-index SiO2 top layer.
The grating produced was investigated by a light
scatterometer. Both the diffuse background scattering intensity and the efficiency of the radiative diffraction orders were measured. The grating period
and a negligible superperiod were also derived from
the light-scattering measurement. There is good
agreement between the predicted and the measured
design parameters. The stability of the working
conditions under near Littrow mounting were also
derived from the analysis. The analysis showed
that transmission losses in the substrate and a strong
influence of waveguiding in the mirror stack become
apparent in certain conditions and must be prevented
by carefully designing the dielectric grating. The
average background loss of 3% corresponded to the
maximum reflectivity of 97% for the ⫺1-order diffraction intensity in TE polarization. The specular reflection vanished in Littrow conditions, and therefore
considerable absorption was excluded. Measurement of the efficiencies for TM polarization is a good
tool for investigating and determining the groove profile.
The measured damage threshold of 4.4 J兾cm2 for
5-ns laser pulses shows the expected increase compared with metallic gratings, whereas for 1-ps laser
irradiation the value of 0.18 J兾cm2 for the dielectric
grating was comparable with conventional gold gratings. Further investigation must be concentrated
on fabricating gratings with larger areas, which are
of practical interest for femtosecond–laser systems as
stretchers and compressors with sufficient base
length. Therefore the lithographic process must be
replaced by holography.
This research was supported by the federal state
of Thuringia under contract B503-95038 共Dielektrische Gitter zur Anwendung in der Spektroskopie兲 and by the European Commission within the
project Gratings for Ultrabright Lasers 共contract
ERBFMGECT980096兲.
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