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SEMI ME1392-0305 © SEMI 2003, 2005 3 4.2.3 bidirectional ref lectance distributi on functi on, BRDF — the sample ra diance divi ded by the sample irradiance. 4.2.3.1 Discussion — The procedures given in this practice are…
SEMI ME1392-0305 © SEMI 2003, 2005 2
complicate its practical application at longer wavelengths. Diffraction effects can be properly dealt with in scatter
measurements,
3
but they are not discussed in this practice.
2.6 Any experimental parameter is a possible variable. Parameters that remain constant during a measurement
sequence are reported as header information for the tabular data set. Related Information 3 gives a suggested
reporting format that is adaptable to varying any of the sample or system parameters.
2.7 This guide applies to flat or curved samples of arbitrary shape. However, only a flat, circular sample is
addressed in the discussion and examples. It is the user’s responsibility to define an appropriate sample coordinate
system to specify the measurement location on the sample surface for samples that are not flat.
2.8 The apparatus and measurement procedure are generic, so that specific instruments are neither excluded nor
implied in the use of this guide.
NOTICE: This standard does not purport to address safety issues, if any, associated with its use. It is the
responsibility of the users of this standard to establish appropriate safety and health practices and determine the
applicability of regulatory or other limitations prior to use.
3 Referenced Standards
3.1 SEMI Standards
SEMI M20 — Practice for Establishing a Wafer Coordinate System
SEMI MF1048 — Test Method for Measuring the Effective Surface Roughness of Optical Components by Total
Integrated Scattering
3.2 ASTM Standards
E 167 — Practice for Goniophotometry of Objects and Materials
4
E 179 — Guide for Selection of Geometric Conditions for Measurement of Reflection and Transmission Properties
of Materials
4
E 284 — Terminology Relating to Appearance
4
3.3 ANSI Standard
ANSI/ASME B46.1— Surface Texture (Surface Roughness, Waviness, and Lay)
5
NOTICE: Unless otherwise indicated, all documents cited shall be the latest published versions.
4 Terminology
4.1 Definitions of terms not included here will be found in Terminology E 284 or ANSI Standard B 46.1.
Additional graphic information will be found in Figures A1-1 through A1-3 in Appendix 1.
4.2 Definitions
4.2.1 angle of incidence,
I
, — polar angle between the central ray of the incident flux and the ZB axis, normal to
the sample surface.
4.2.2 beam coordinate system, XB YB ZB — a Cartesian coordinate system with the origin on the central ray of the
incident flux at the sample surface, the XB axis in the plane of incidence (PLIN) and the ZB axis normal to the
surface.
4.2.2.1 Discussion — The angle of incidence, scatter angle, and incident and scatter azimuth angles are defined
with respect to the beam coordinate system. This coordinate system is illustrated in Figure A1-1.
3 Smith, Sheldon M., “Reflectance of AMES 24E, Infrablack and Martin Black,” SPIE 967, 251 (1988).
4 Annual Book of ASTM Standards, Vol 06.01, ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. Telephone: 610-
832-9500, Fax: 610-832-9555, Website:
www.astm.org
.
5 American National Standards Institute, New York Office: 25 West 43rd Street, New York, NY 10036, USA. Telephone: 212-642-4900; Fax:
212-398-0023, Website:
www.ansi.org
.
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).