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SEMI ME1392-0305 © SEMI 2003, 2005 4 furnish the i nstrument signature when reporting B RDF data so that the us er can decide at what scatt er direction t h e sample BRDF is lost in the signature. Prefe r ably the signat…
SEMI ME1392-0305 © SEMI 2003, 2005 3
4.2.3 bidirectional reflectance distribution function, BRDF — the sample radiance divided by the sample
irradiance.
4.2.3.1 Discussion — The procedures given in this practice are correct only if the field of view (FOV) determined
by the receiver field stop is sufficiently large to include the entire illuminated area for all angles of incidence of
interest. BRDF is a differential function dependent on the wavelength, incident direction, scatter direction, and
polarization states of the incident and scattered fluxes. In practice, it is calculated from the average radiance divided
by the average irradiance as follows:
1
cos
sr
cos
BRDF
si
s
A
P
A
P
e
e
P
P
E
L
i
s
s
(1)
The BRDF of a lambertian surface is independent of scatter direction. If a surface scatters nonuniformly from one
position to another then a series of measurements over the sample surface must be averaged to obtain suitable
statistical uncertainty. Nonuniformity may be caused by irregularity of the surface microughness or film, optical
property nonhomogeneity, or subsurface defects.
4.2.4 cosine-corrected BRDF — the BRDF times the cosine of the scatter polar angle.
4.2.4.1 Discussion — The cos
s
in the BRDF definition is a result of the radiometric definition of BRDF. It is
sometimes useful to express the scattered field as normalized scatter intensity [(watts scattered/solid angle)/incident
power] as a function of scatter direction. This is accomplished by multiplying the BRDF by cos
s
.
4.2.5 delta beta,
— the projection of
onto the XB-YB plane, that is, the delta theta angle measured in
direction cosine space.
4.2.5.1 Discussion — For scatter in the PLIN,
= sin
s
sin
i
. For scatter out of the plane of incidence (PLIN),
the calculation of
becomes more complicated (see §R1-2).
4.2.6 delta theta,
— the angle between the specular direction and the scatter direction.
4.2.7 incident azimuth angle,
i
— the fixed 180° angle from the XB axis to the projection of the incident direction
onto the XB-YB plane.
4.2.7.1 Discussion — It is convenient to use a beam coordinate system (refer to Figure A1-2), in which
i
= 180°,
since this makes
s
the correct angle to use directly in the familiar form of the grating equation. Conversion to a
sample coordinate system is straight forward, provided the sample location and rotation are known.
4.2.8 incident direction — the central ray of the incident flux specified by
i
and
i
in the beam coordinate system.
4.2.9 incident power, P
i
— the radiant flux incident on the sample.
4.2.9.1 Discussion — For relative BRDF measurements, the incident power is not measured directly. For absolute
BRDF measurements it is important to verify the linearity, and if necessary correct for the nonlinearity, of the
detector system over the range from the incident power level down to the scatter level which may be as many as 13
to 15 orders of magnitude lower. If the same detector is used to measure the incident power and the scattered flux,
then it is not necessary to correct for the detector responsivity; otherwise, the signal from each detector must be
normalized by its responsivity.
4.2.10 instrument signature — the mean scatter level detected when there is no sample scatter present expressed as
BRDF.
4.2.10.1 Discussion — Since BRDF is defined only for a surface, the instrument signature provides an equivalent
BRDF for the no-sample situation. The limitation on instrument signature is normally stray scatter from instrument
components and out-of-plane aperture position errors for receiver positions near the specular direction. For high
grade electronic detection systems, at large scatter angles, the limitation on instrument signature is normally
Rayleigh scatter from molecules within the volume of the incident light beam that is sampled by the receiver field of
view. As
s
approaches 90°, the accuracy of
s
becomes important because of the l/cos
s
term in BRDF. The
signature can be measured by scanning a very low scatter reference sample in which case the signature is adjusted
by dividing by the reference sample reflectance. The signature is commonly measured by moving the receiver near
the optical axis of the source and making an angle scan with no sample in the sample holder. It is necessary to
SEMI ME1392-0305 © SEMI 2003, 2005 4
furnish the instrument signature when reporting BRDF data so that the user can decide at what scatter direction the
sample BRDF is lost in the signature. Preferably the signature is several decades below the sample data and can be
ignored.
4.2.11 noise equivalent BRDF, NEBRDF — the root mean square (rms) of the noise fluctuation expressed as
equivalent BRDF.
4.2.11.1 Discussion — Measurement precision is limited by the acceptable signal to noise ratio with respect to these
fluctuations. It should be noted that although the detector noise is independent of
s
, the NEBRDF increases at large
values of
s
because of the 1/cos
s
factor. Measurement precision can also be limited by other experimental
parameters as discussed in Section 10. The NEBRDF can be measured by blocking the source light.
4.2.12 plane of incidence, PLIN, — the plane containing the sample normal and central ray of the incident flux.
4.2.13 receiver — a system that generally contains apertures, filters and focusing optics that gathers the scatter
signal over a known solid angle and transmits it to the scatter detector element.
4.2.14 receiver solid angle,
— the solid angle subtended by the receiver aperture stop from the sample origin.
4.2.15 sample coordinate system — a coordinate system fixed to the sample and used to specify position on the
sample surface for the measurement.
4.2.15.1 Discussion — The sample coordinate system is application and sample specific. The Cartesian coordinate
system shown in Figure A1-1 is recommended for flat samples. The origin is at the geometric center of the sample
face with the Z axis normal to the sample. A fiducial mark must be shown at the periphery of the sample; it is most
conveniently placed along either the X or Y axes. For silicon wafers, the fiducial mark is commonly placed on the
periphery of the wafer at the Y-axis as defined in SEMI M20.
4.2.16 sample irradiance, E
e
— the radiant flux incident on the sample surface per unit area.
4.2.16.1 Discussion — In practice, E
e
is an average calculated from the incident power, P
i
, divided by the
illuminated area, A. The incident flux should arrive from a single direction; however, the acceptable degree of
collimation or amount of divergence is application specific and should be reported.
4.2.17 sample radiance, L
e
— a differential quantity that is the reflected radiant flux per unit projected receiver
solid angle per unit sample area.
4.2.17.1 Discussion — In practice, L
e
is an average calculated from the scattered power, P
s
, collected by the
projected receiver solid angle,
·cos
s
, from the illuminated area, A. The receiver aperture and distance from the
sample determines
and the angular resolution of the instrument.
4.2.18 scatter — the radiant flux that has been redirected over a range of angles by interaction with the sample.
4.2.19 scatter azimuth angle,
s
, — angle from the XB axis to the projection of the scatter direction onto the XB-YB
plane.
4.2.20 scatter direction — the central ray of the collection solid angle of the scattered flux specified by
s
and
s
in
the beam coordinate system.
4.2.21 scatter plane — the plane containing the central rays of the incident flux and the scatter direction.
4.2.22 scatter polar angle,
s
, — polar angle between the central ray of the scattered flux and the ZB axis.
4.2.23 specular direction — the central ray of the reflected flux that lies in the PLIN with
s
=
i
and
s
= 0.
5 Apparatus
5.1 General — Non-specular reflectometers or instruments
6
used to measure scattered light utilize some form of the
five components described in this section. These components are described in a general manner so as to not exclude
any particular type of scatter instrument. To achieve (
i
,
i
;
s
,
s
) positioning the instrument design must
incorporate four angular degrees of freedom between the source, sample holder, and receiver assemblies.
6 Hsia, J. J. and Richmond, J. C., “A High Resolution Laser Bidirectional Reflectometer with Results on Several Optical Coatings,” J. Res. NBS-
A. Physics and Chemistry 80A, 189-205 (1976).
SEMI ME1392-0305 © SEMI 2003, 2005 5
5.2 Source Assembly — containing the source and associated optics to produce irradiance, E
e
, on the sample over a
specified spot area, A. If a broad band source is used, the wavelength selection technique should be specified.
Depending on the bandwidth and selection techniques, the detector assembly may affect the wavelength sensitivity.
If a laser source is used, it is usually sufficient to specify the center wavelength; however, it is sometimes necessary
to be more specific, such as providing the particular line in a CO
2
laser.
5.2.1 A source monitor is used to correct for fluctuations in the source intensity. If it is located at the source output
it only measures variations in the source power and is not sensitive to variations due to angular drift or downstream
transmission. The source monitor should monitor incident power as close to the sample as possible while
minimizing additional system scatter. Attention should be paid to possible laser mode hopping and consequent
wander of the beam on spatial filter pinholes and to fluctuations in source polarization.
5.2.2 Collimated or slightly converging source light can be used to measure BRDF. Most instruments use a
converging beam focused at the receiver. If the convergence angle is small, the uncertainty introduced by a non-
unique angle of incidence is usually negligible. The same considerations apply if a curved sample is measured. It is
the user's responsibility to assure that any spread in
i
does not compromise the results. Normal practice limits
convergence to f/20 or greater with a focus at the receiver to increase the angular resolution of measurements near
the specular beam or diffraction peaks.
5.2.3 Typically the source assembly is fixed in position and variations in
i
are made with the sample holder. Good
reduction of the instrument signature requires baffling around the source assembly and use of a spatial filter to limit
off-axis light. The final mirror (or lens) which directs light to the sample should have low scatter, since it
contributes directly to small angle scatter in the instrument signature.
5.2.4 A means should be provided for controlling the polarization state of the incident flux as this can impact the
measured BRDF. Orthogonal source polarization components (parallel, or p, and perpendicular, or s) are defined
relative to the plane defined by the source direction and the sample surface normal.
5.2.5 Absorbing samples may be heated by the incident flux and may change their scatter characteristics,
mechanically distort or burn. Special care must be taken with IR laser sources on absorbing samples.
5.3 Sample Holder — The sample holder should provide a secure mount for the sample that does not introduce any
warp. The rotation axes of the stages that achieve the (
i
,
i
,
s
s
) positioning must be relative to the sample front
surface; this can be accomplished by orienting the sample holder or the source, or both, and receiver assemblies.
Some sample mounts incorporate positioning stages for a raster scan of the sample surface at fixed incident and
scatter angles. The sample mount must be kept unobtrusive so that it does not contribute stray flux to the signature
or block large
s
scatter.
5.4 Beam Dump — It is important to trap any specular reflection from the sample so that it cannot contribute to the
scatter signal through lab/instrument reflections. Examples of beam dumps are black paper, a razor blade stack,
absorbing glass plates, or a tapered blackened glass tube.
5.5 Receiver Assembly — If the system design includes angular degrees of freedom at the receiver for achieving the
scatter direction, then the receiver assembly should normally have provisions for rotating about an axis on the front
face of the sample in order to vary
s
. If out of the PLIN measurements are required, the receiver assembly may also
rotate out of the PLIN. This capability may also be provided by pitch, yaw, and roll of the sample, but it becomes
more difficult to dump the specularly reflected beam.
5.5.1 The acceptance aperture for the receiver must be well defined, since the solid angle,
, subtended by the
receiver aperture stop from the sample, is used in the BRDF calculation and defines the angular resolution. The
field of view of the detector must include the entire irradiated area, A. There can be an exception to these
requirements if a relative BRDF or relative total reflectance normalization is used. In that case it is the user's
responsibility to ensure that the system parameters remain constant between measurements.
5.5.2 If the acceptance aperture is too small and a coherent source is used to irradiate the sample, speckle may
cause strong, unpredictable variations in the scatter. This is a common problem when measuring diffuse (that is,
rough) samples. It is sometimes desirable to spin a diffuse sample about its normal to average the effects of speckle
while making a measurement. It is the user’s responsibility to ensure that BRDF features are not due to speckle.
The user may wish to employ a variable aperture stop to trade sensitivity for angular resolution when measuring