Several ways of measuring spot quality are compared. We examine in detail various figures of merit such as full width at half maximum (FWHM), full width at 1/(e exp 2) maximum, Strehl ratio, and encircled energy. Our application is optical data storage, but results can be applied to other areas like space communications and high energy lasers. We found that the optimum figure of merit in many cases is Strehl ratio

Milster, T. D.

Treptau, J. P.

English

NASA Technical Reports Server

T.D. Milster, Optical Sciences Center, University of Arizona, Tucson. Arizona 85721
J.P. Treptau, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721
Several wars of measuring spot quality are compared. We examine in detail various figures
of merit such as full-width-at-half maximum (FWHM), full-width.at-I/e: maximum, Strehl ratio, and
encircled energy. Our application is optical data storage, but results can be applied to other areas
like space communications and high-energy lasers. We found that the optimum figure of merit in
many cases is Strehl ratio.
f
Introduction
There are several methods useful for measuring laser spot quality, such as interferometry,
CCD cameras, and knife-edge scanners. Intefferometric methods commonly use a lens to recollimate
the light, as shown in Figure la. Measurements are highly precise with better than M100 wavefront
variation. However, there are certain difficulties involved in using an intefferometer. The numerical
aperture (NA) of the collecting optics must be greater than the NA of the beam. The collection
optics may introduce aberrations into the beam. Also,
it is tedious to align the interferometer. In some
situations, the physical size of the intefferometer is not
convenient. CCD cameras avoid some of the alignment
problems as,u_iated with intcrferometers. As shown in
Figure Ib, a laser spot can be focused directly onto the
CCD array. How_,'er, the sampling of the spot is
limitedby the pixelsize,which istypicallyabout 101_m
by 10 l_m. Spot diametersof leu thanseveralhundred
micronsare difficultto measure accuratelywitha CCD.
An auxiliary lens can be placed between the spot and
the CCD, as shown in Figure lc, but the NA of the
collection optics must be greater than the NA of the
beam. The alignment must also be done very carefully,
and the optics must be carefully considered. For
example, Bobroff et aL have comtructed, with some
difficulty, a very high quality system for measuring
spots from l/thographic lenses (1). A third option for
measuring spot quality is to use a knife-edge scanner.
The spot pmlile it estimated by scanning an opaque
edge through the spot, as shown in Figure Id. A
detector records transmitted power versus time,. The
derivative of the knife-edge wan with respect to time,
8i/8t, is the line scan of the spot, which is an estimate
of the spot profile. The line scan it equivalent to
scanning an infinitelythinslit,as shown in Figure 2.
The slitintestatesirtadiance in the y direction.
Therefore,the linescan itnot exactlyequivalentto the
truespotprofile.-However, because of itssmallsize
,[
To be publishedin Proc.SPIE 1414.\--
(b)
CCD
CCD
(c)
(d)
Figure 1. Measurement methods: (a)
interferometric; (b) CC'D direct; (c) CCD
with aux. lens; (d) knife-edge scanner.
.,f
https://ntrs.nasa.gov/search.jsp?R=19920005692 2020-03-17T14:13:32+00:00Z
182 Appendix H
and direct output, the knife-edge
technique h very convenient in the laboratory.
We can understand differences between the
knife-edge scan and the true spot profile by analyzing
the spot irradian¢_ distribution in Figure 3a. Various
amounts of aberration add to yield a wavefront standard
deviation , o, of a = 0.077, which is just at the
diffraction limit. The spot is slightly elongated in a
diagonal direction, which is most easily observed in the
5% irradiance contour level. The true spot profile and
the knife-edge scan (6i/bt) are shown in Figure 3b.
Since the knife--edge scan integrates the spot
distribution in one direction, it indicates a wider profile
than the true spot.
Each measurement method can produce various
figures of merit for the spot. The standard deviation of
the wavefront, o, can be found from interferometric
data. The full-width.at-half maximum (F'WHM) is
shown in Figure 4a. The full-width-at-l/e: maximum
(FWl/e:) is shown in Figure 4b. The Strehl ratio is
defined as the ratio of the maximum irradiance of the
aberrated spot to the maximum irradianceof an
unaberrated spot,as shown in Figure 4c. Encircled
energy isdefinedasthe ratioof the power deliveredin
L
x )
dx
di/dt
. t
Figure 2. The derivative of the knife-
edge scan is equivalent to scanning an
inYanitely thin slit across the spot. (a) slit
scan; (h) 8i/8t.
a circular area surrounding the maximum of the abet'rated spot m power in the same area around the
maximum of the unaberrated spot, as shown in Figure 4cl.
Our application is measuring sub-micron spot profiles for optical data storage devices. The
NA is typically between 0.45 and 0.60. Wavelengths are in the range of 780rim to 830rim. Spot sizes
are typically too small for direct CCD measurements, lnterferometric methods are commonly used
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Figure3. (a)An abet'ratedspot that shows an asymmetricirradianceprofileisanalyzed.The
value a - 0.077isnear the _n limit;(b)Differencesbetween the knife-edgescanthrough
the spot and the truespot profileare shown.
Append/x H 183
to test collimated light beam., in the optical path, but they
are not used for testing the focussed spot. lmtead, some
form of knife-edge scanner is used. Modified CCD devices
are of questionable utility due to the large NA. Size and
quality of the focused spot is very important because small
spots yield high data densities. Historically, FWHM has
been used as a metric for spot quality.
Other applications use similar figures of merit. In
high-energy laser work and space-laser communications, the
full width of the far-field divergence angle is typically used.
The motivation is to deliver as much energy as possible to
the receiver. However, it is often the case that a substantial
amount of energy falls outside of the central peak.
This paper addresses differences between various
figures of merit for measuring beams with small amounts of
aberration. We consider only spherical, coma, and
astigmatic aberrations. We restrict our attention to beams
that are near or under Marechars criterion for diffraction-
limited performance (2), that is a ,: 1/14 waves. For
larger amounts of aberration, ,rome figures of merit become
difficult to interpret. For example, it would be very difficult
to determine FWHM for Figure 5, which illustrates a spot
aberrated with 0.8 wave of coma, 0.8 wave of astigmatism,
and 1.6 waves of spherical. Piston, tilt, and defocus have
been added to minimize wavefront variance. This brings
the spot into best possible focus. Our results are based on
a computer simulation.
FWHM
FW I/e:
Figure 4. Various figures of
merit. (a) FWHM; (b) FW1/e2; (c)
Strehl ratio = IdI.; (d) encircled
energy = PjPo.
Si.tnulation
We analyze an optical data storage system that
focuses light from a laser diode to a disk medium. An
illustration of the optical path is shown in Figure 6. The
laser diode has different divergence angle, in the parallel
and perpendicular directions with respect to the junction.
Circularization optics are used after a collimator to make
the beam more uniform. A partially.polarizing beam
splitter is used to direct the reflected fight from the disk to
data and smt'_m detectors. The stop is located at the
objective hm_ which focuses light on the disk. Our system
parametem inttmle NA- 0.55, f. = 4.0 nun, and
_.- 780 rim. The amplitude dism'oution in the pupil is
Lorentzianin the directionperpendicularto the junction
and Gaussian inthe d/rectionparalleltothejunction.The
widthsof the amplitudedistn'butionwere adjustedforthe
best tradeoff between power throughput and spot size
(3). The unaberratedFWHM isapproximately0.9pro in
Figure5. 0.8 wave coma, 0.8
wave astigmatism, and 1.6 waves
sphericalatbestfocus.
the direction perpendicular to the junction and 0.8'71_m in the direction parallel to the junction.
Random amounts of aberration were added to the pupil for each trial, and the proper amount of
piston, tilt, and defocm were added in order to bring the spot into best focus. Standard deviation
184 Append/x H
of the wavefront was kept
between the limits: 0 < o <
0.10. Random parameters
included spherical, coma,
coma rotation angle,
astigmatism, and astigmatism
rotation angle. The
diffracted spot was
calculated with scalar
diffraction theory. Figures
of merit were calculated for
each trial, and the results
are plotted versus standard
deviation of the wavefront.
We calculate figures of
merit for both the true spot
profile and the knife-edge
scan.
,---f___,
Las4¢
D,o0e
,C.ollimilllr_ C#l_,/ll_zff'_
Let,s
SIoo D_sK
Plftllt_ Ob_cI_vQ
Po_l_zun(j Le_
I_bm S_n_
Figure 6. MO storage device (laser to disk).
R uln
Figure 7 displays results for FWHM.
For low values of o, the FWHM in x and y
directions (perpendicular and parallel to the
junction, respectively) are not equal. This is
due to Gaussian/Lorentzian filling of the
aperture. Also, the knife-edge scan is not
equivalent to the true spot profile. This is
became of the line-scan integration. As o is
increased, FWHM does not increase noticeably.
Near the diffraction limit, o - 0.07, FWHM
startsfluctuatingsignificantly,which indicatesa
poor correlationbetween _ and o. As o
isincreasedbeyond the dil_mctmn limit,the
IIItitfl'lll Ill_lblSl'll I_•lllllJl_lll Illltl)
Figure 7. FWHM from the computed spot
and knife-edgescans.
four different measures are very dependent on
the amount and rotation angle of the , .
aberration.
The fact that width measurements are *_
not very _ to small amounts of
aberrationmaybe imdemood by examining the | "
difference _ aberrated and unaberrated i ""spots. Figure 8 displ ys n aberrated spot with
0.955 waves of sphericaland the appropriate *."
amount of defocw, The differenceis alto
displayed, which shows that energy it taken
from the central lobe and placed in the outer ..... _ "7 h ,
rings.The differencefunctionpwaea through *" "*_' "' "_' " "_ ' "
zero at the first dark ring. Therefore, the width """ "" " " ""
of the central lobe it constrained. Any changes Figure & Aberration contribution to the spot
profile with 0.955 wave of spherical and the
appropriate amount of defocus.
C,31GINAL PAGE IS
OF POOR QUALITY
Appendix H 185
- { - , .
't¢_ __ 1=5_ ll'_l/edhl _i Y 8k,91r,_emI 3xl dlrllclluR OCEIi IFqJl/e.,]l
48v cllrectlon *cea rgl/•Zi
N'!
I 05 i
"t ,. ::
I 4S_" ¢ w
,I
I/
0 OOZ 004 0041 0041' O
govetro_¢ Stlm4e¢4 Detll¢_ol ¢gQveoi
Figure 9. FW1/e2 from the computed spot
and knife-edge scans.
OI
0e
p_
"k
,,t
• I
oS_Poilbl niuo rr_i e_4
2 •11 dlt_NpCCols _la Stre_l IllL/e
4_qql_o e8 1 Ill Y 41m_loa ._lll _.'_ll| hU4_
O01t 004 OOe 000 OI
gu,,tJhl'qHit |_la4iild INc_16t.iOell Iqhieosil
Figure 10. Comparison of Strehl ratio from
computed spot and knife-edge scans.
in width of the central lobe are secondary
effects. Similar results are obtained with other
aberrations.
Figure 9 displays results for FW1/¢:.
Differences between spot and knife-edge scan
are observed at low values of o, which are due
to integration of the line scan. Differences also
are observed between x and y profiles, which
are due to Gaussian/Lorentzian tilling of the
aperture. As o increases, FWlfe _ generally
increases, but it quickly becomes poorly
correlated with o. The FW1/e 2 is very sensitive
to the kinds of aberrations present and their
orientations.
Figure 10 displays results for Strehl
ratio. The Strehl ratio for the spot followed
Marechal's relationship (_, that is, SR - 1 -
(2_/,1.)zo z. Fluctuations in Strehl ratio for the
spot measurement ate very small 5trehl ratio
for the knife..edge scan in the x direction
fluctuates collsiderably. $trehi ratio for the
Oil0 " " v_ I
0 7'S _ ca I ./¢ i,ere*, _ \
e7 _ _4
O 0 " I 0104 0 ' I 0 _ _ 0 " I
Ill_t Sk4144q40, ev_ Ilhi'_eO!
Figure 11. Encircled energy: average of scans
and 0.25F_m diameter spot pinhole.
knife-edge _ in the y direction (not shown) also fluctuates considerably. A smoother curve is
generated by tanki_ the avenge of x and y scans. The average decreases with increasing o, which
indicates a reasonable figure of merit.
Figure 11 displays results for encircled energy. As with the Strehl ratio results, a
comparatively smooth curve is generated by taking an average of thex andy scant,. The averages for
21_m, 1Ftm and 0.51_m effective widths fzom the knife-edge scans are shown. The encircled energy
for a 0.25 micron diameter pinhole over the spot is also shown. The noi_-Iike variations in encircled
energy venus o are approx_nately equivalent for the data shown. The average of the 0.251_m
186 Appendix H
knife-edge scan is comparable to the average Strehl-ratio data of Figure 10. This suggests that an
adequate integration range around the central peak is about It.X3% x 0.25/0.9 = 28% of the
unaberrated spot FWHM.
Conclusions
We have illustrated several important points about measurement of laser spot quality. First,
results from knife-edge scans and actual spot profiles differ due to the integration of the
differentiated knife-edge signal. Secondly, width measurements are not adequate (by themselves) to
describe spot quality. This is because width of the central lobe is constrained by pupil filling for small
amounts of aberration, and, for large amounts of aberration, width measurements are difficult to
interpret. Thirdly, the most sensitive measure of spot quality for many applications is the Strehi ratio.
The Strehl ratio is difficult to measure directly, but it may be approximated by averaging the encircled
energy found from x and y scans of a knife-edge scanner. The integration range should be no more
than about 28% of the unaberrated spot FWHM.
Referer_ces
.
2.
3.
N. Bobroff, P. Fad/, A.E. Rosenbluth. and D.S. Goodman, "Bench evaluation of lithographic
lenses from measurements of the point spread function," Proc SPIE 922, pp. 376-386, 1988.
M. Born and F,. Wolf, Princioles of O0tjc_, 6th ed., Pergammon Press, New York, p 469
(1986).
H.M. HaskaL "Laser recording with truncated Gaussian beams," Appl. Opt., vol. 18 no 13, pp.
2143'2146 (1979).


Measurement of laser spot quality

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