Laser Diode Interferometry
Michael Hercher & Geert Wyntjes
OPTRA, Inc.
66 Cherry Hill Drive
Beverly, Massachusetts 0 191 5
(508) 921-2100
Paper presented at
SPIE's 33rd Annual International Technical Symposium on
Optical & Optoelectronic Applied Science and Engineering
6 to 11 August, 1989
San Diego, California, USA
Laser diode interferometry
Michael Hercher & Geert Wyntjes
OPTRA Inc.
66 Cherry Hill Drive, Beverly, MA 01915
(508) 921-2100
ABSTRACT
By using a multiphase detection technique, unambiguous shot-noise-limited
measurements of unwrapped (i.e. not Limited to modulo 2π) interferometric phase may
be made at data rates in excess of 1 MHz. This technique is ideally suited for use with
laser diodes. We describe the technique and associated algorithms which facilitate the
rapid processing of data, and we present experimental data obtained with different types
of laser diode interferometers--including an imaging interferometer in which phase is
digitally measured for each pixel at TV frame rates, and an interferometric linear encoder
head which achieves a 1-microinch resolution with a 1250 line/inch linear scale and a 1
MHz update rate.
1. INTRODUCTION
Laser diodes are ideal coherent light sources in all but two regards: they aren't very
visible, and their free-running wavelength stability and linewidth are not all that one
might wish. In all other regards --size, economy, lifetime, ruggedness, and cost--they
perform very well indeed. At OPTRA we had developed a variety of precise, highresolution interferometric measurement techniques--all of which were based on the use of
a stabilized 2-frequency HeNe laser. We have sought to replace these relatively
expensive lasers with laser diodes. This paper describes some of our progress to date.
2. THE GENERAL INTERFEROMETRY PROBLEM
In quantitative interferometric metrology, the key task to accurately and unambiguously
measure the interferometric phase. In virtually every case, the physical quantity of
interest (most usually displacement) is proportional to this phase. Measurements made
using real detectors, however, do not measure phase: they measure a signal intensity
level, I:
I = I1 + I2 cos Φ
(1)
where Φ is the unknown phase, I1 is a DC light level, and I2 is the amplitude of the
portion of the signal that varies with phase. Often the quantity to be measured is an
optical path difference, OPD, in which case the phase Φ can be written:
Φ = Φ0 + 2πσ * OPD,
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(2)
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where σ is the optical frequency measured in wavenumbers (i.e. the reciprocal of the
wavelength). In many applications of interferometry it is adequate to keep track of Φ
only to within π or so (i.e. to monitor the OPD with an accuracy on the order of 1/2wavelength). In those cases, the usual technique of counting the number of cycles of the
intensity modulation gives perfectly usable results (providing, of course, that one
develops an adequate technique for keeping track of the total number of fringes-including their direction or sign). If, however, one wants to measure interferometric
phase with high resolution (i.e. a small fraction of a wavelength), then the problem
becomes more difficult. In equation 2 above there is one measured quantity, I, and three
unknowns: I1, I2, and Φ. Thus a small change in the observed value of I might be due to
changes in either I1, I2, or Φ itself. Changes in I1 and/or I2 might be due to fluctuations in
the laser intensity (or background illumination), or to time- and/or position-varying
changes in the reflectivity of the target. Moreover, there are values of @ (e.g. any integral
multiple of π) for which small changes of phase in either direction produce the same
change in the observed intensity I.
Heterodyne interferometry (using a 2-frequency laser HeNe) and measurements based on
phase quadrature are established techniques for addressing this problem, but either do not
translate into the realm of diode lasers very gracefully (heterodyne detection), or offer
only limited precision (quadrature). Inexpensive diode lasers work very nicely as singlewavelength sources with moderate frequency stability (a frequency or wavelength
stability of 1 part in 106 can be achieved in a passive system). Given such a source, the
problem is to make accurate and unambiguous measurements of interferometric phase.
3. MULTIPHASE DETECTION
Based on extensive evaluation and experimental work we have chosen to solve this
problem using a technique, originally developed by Larry Mertz for application in stellar
interferometry1, which we call multiphase detection. (This approach is conceptually
similar to phase-stepping interferometry23, but differs in the simultaneity of the phasestepped measurements, the means for introducing the phase shifts and the technique for
processing the data.) As described in more detail below, by using multiphase detection
we generate three detector signals R, S, and T:
R = IR = I1 + I2 cos(ΦR),
S = IS = I1 + I2 cos(ΦS),
T = IT = I1 + I2 cos(ΦT),
ΦR = Φ
ΦS = Φ + 2π/3
ΦT = Φ + 4π/3
(3)
One could, without undue difficulty, solve the three equations above for sinΦ and cosΦ,
and then either use a look-up table or a Pythagorean Processor chip (such as that
manufactured by Plessey) in order to unambiguously determine the phase (modulo 2 π).
However, a more elegant algorithm has been developed. Consider the following ratios
1
Lawrence Mertz, “Complex Interferometry”, Appl. Opt. 22, 1530-1534, 1983.
L.M. Franz et al, “Optical phase measurement in real time”, Appl. Opt. 18, 3301-3306, 1979.
3
P. Hariharan et al, “Digital phase-shifting interferometry: a simple error-compensating phase calculation
algorithm”, Appl. Opt. 26, 2504-2505, 1987.
2
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involving the measured signals R, S, and T:
A = (R-S)/(T-S); B = (S-T)/(R-T); C = (T-R)/(S-R)
(4)
Concerning these ratios, two points are immediately obvious: (1) the ratios are
independent of the DC light level, I1 (since both numerators and denominators are
differences between pairs of signals), and (2) they are independent of the AC-amplitude,
I2. In fact, combining equations (7) and (8), we find that the ratios are simple functions of
the phase Φ alone:
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A = ½ (1 + √3/tan Φ), etc.
(5)
Figure 1 shows the ratios A, B, and C and the signals R, S, and T over a 2π range of
phase angles. The critical feature to observe is that over each π /3 (60º) phase interval,
one of the three ratios (A, B, or C) is a very nearly linear function of the phase angle Φ.
For example, over the first 60º interval the phase can be written as:
Φ .(π/3)*(1-C)
(6)
By combining 6 such simple functions, selected on the basis of the values of the Boolean
quantities (R>T), (S>R), and (T>S), an approximately linear representation of Φ can be
generated to cover the full 2π range. Figure 2 shows the segmented linear function
ΦAPPROX defined in this manner, together with the generating signals R, S, and T. This
linear function has a maximum error of less than 0.02 rad (just over 1º): in a conventional
interferometer a 1º phase error corresponds to a path difference of 0.003-wavelength, or
about 0.0025µ (for a wavelength of 0.8µ).
An critical aspect of this algorithm, first recognized by Larry Mertz4, is the realization
that the ratios (R-S)/(T-S) etc. can be generated by individual A/D-converters. An A/Dconverter works by digitizing the quantity (Vin-Vo)/(Vref-Vo), where Vin, is the input
voltage (which, in the normal functioning of an A-D converter, is converted to a digital
representation), Vref is a stable reference voltage to which the input voltage is to be
compared, and Vo is the ground level with respect to which Vin and Vref are defined. In
order to generate a digital representation of the ratio (R-S)/(T-S) it is necessary only to
make (for example) the following connections to the A/D:
R --> Vin ; S --> Vo and T --> Vref
Thus three signals R, S, and T can be efficiently processed to generate a highly accurate
digital signal which directly represents the measured interferometric phase. This digital
signal can be readily converted to an analog signal to facilitate recording, interfacing to a
spectrum analyzer, temporal averaging, etc. Moreover, the 1% departure from linearity
can easily be eliminated by using a small look-up table to generate corrected values of the
phase, Φ.
4. ELECTRONIC CONFIGURATION
Figure 3 shows the electronic configuration used for processing the signals IR, IS and IT in
order to obtain 8 ½ -bit phase resolution (1 part in 384), together with unwrapped phase
(i.e. accumulated phase vs. modulo 2π phase). Six separate A-D converters are used –
one to cover each 60º segment of a phase cycle, with the output selection being made
based on the values of the Boolean quantities (R>S) etc., which are actually generated as
overflow bits from three of the A-D's. The ALU (arithmetic logic unit) and MAC
(multiply and accumulate unit) are used to generate an accumulated phase (i.e.
4
Larry Mertz, private communication.
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unwrapped phase), using a version of the polar innovations recursive digital filter5. This
serves both to generate an accumulated phase, and to perform a digital filtering function
(analogous to the use of a time constant) which both reduces noise and, in the presence of
noise, improves precision (by reducing the measurement bandwidth).
5. OPTICAL IMPLEMENTATIONS
The usual technique for controlling the interferometric phase (so-called phase-stepping)
has been to use a piezoelectric transducer to change the optical path in one arm of the
interferometer. This approach was predicated on the serial switching of the phase and on
the use of an interferometer with physically separated beams. In the interests of speed and
versatility we wished to generate the three phase signals simultaneously and without the
restriction that the interfering beams be physically separated.
INTERFEROMETRIC SCALE READER: Different applications have led to different
optical configurations: Figure 4 shows an interferometer for measuring the lateral
displacement of a scale relative to the sensor head. The scale is illuminated at nearnormal incidence by a collimated beam from a laser diode. The ±1 diffracted orders
(reflected in this case, but the concept works equally well in transmission) are brought
together by a biprism (which can usually be replaced by a lens) in the vicinity of a calcite
wedge. A ½-wavepIate is inserted into one of the diffracted beams so as to rotate its
polarization vector by 90º - so that the two beams are orthogonally polarized. The
function of the birefringent calcite wedge is to differentially refract the two beams so that
they emerge in the same direction (the calcite prism works just like a Wollaston prism,
except that it does not work at zero order and hence requires a source with a narrow
spectreal bandwidth). Once the two beams are made parallel, they can be made to
interfere by the addition of a linear polarizer set at 45º to each of the polarization vectors.
The calcite prism is then rotated slightly (about an axis normal to the plane of the paper)
so as to introduce a slight angle between the two beams--causing an interference pattern
of parallel fringes to be formed. When the lateral position of the scale is changed by an
amount x, a differential phase shift Φ is produced between the two beams:
Φ(x) = 2π*(2x/d)
(7)
where d is the scale spacing. (When the scale moves it approaches the point of
observation for one of the beams, and recedes for the other: it is this differential change
in optical paths that accounts for the phase shift.) By positioning three detectors relative
to the fringe pattern as shown in Figure 4, three signals IR, IS and IT may be generated
with the requisite phase differences. These signals are the inputs to the multiphase
detection circuit (Figure 3), which has an 8-digital display (either inches or centimeters
with 6 decimal places). Data is updated internally at 4 MHzwith a resolution of 1/384cycle, and is output (in both digital and analog form) at 16 kHzwith a phase resolution of
cycle. (The maximum velocity for full on-the-fly resolution is 4-inches/sec; the phase can
be tracked at up to 60-feet/sec.) The digital display is updated at 1 kHz, and the display
resolution of 0.01-micron corresponds to a phase resolution of about 1/1000-fringe.
5
L. Mertz, “Optical homodyne phase interferometry”, Appl. Opt. 28, 1011-1014, 1989.
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Using a commercially available scale with 1250 lines/inch, scale position can be
monitored with a precision of better than 1 microinch and an accuracy of better than 10
microinches (the residual inaccuracy is systematic and can be removed with a short lookup table). The illuminated area on the scale is approximately 1 mm in diameter, so that
dust on the scale is not a factor. The accuracy of the measurement is tied to the scale and
not to the wavelength of the laser diode. If the laser diode drifts in wavelength, or if the
sensor head becomes misaligned with respect to the scale, the signal level decreases
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without having any other significant affect on the measurement. The position of the
sensor head relative to the scale has a tolerance of ±5 mm.
By using a crossed grating with about 1000 lines/inch in each direction, and by adding a
second detection channel to the sensor head, both X and Y coordinate measurements can
be made with 1 microinch precision. We should note that an advantage of using a
relatively coarse scale (rather than, say, a holographic diffraction grating) is that the
scales can be inexpensively replicated in long lengths using standard photolithographic
techniques.
2-DIMENSIONAL DIGITAL CONTOURER: In a quite different application, illustrated
in Figure 5, a contoured surface is obliquely illuminated with an interference fringe
pattern, using a laser diode source. When viewed from above (by means of diffuse
reflection of the light) the fringe pattern is distorted by the contour of the target surface: if
at the point of observation (x,y) the surface height is h, measured relative to a reference
flat surface, then the phase of the fringe pattern at that point, measured relative to that of
the reference surface, is:
Φ(x,y) = 2π*h(x,y) tanθ/d
(8)
where d is the spacing of the fringe pattern on a nominally flat surface and e is the angle
between the illuminating beam and the normal to the reference surface. Thus there is a
linear relationship between Φ (x,y) and h(x,y). We measure the phase Φ (x,y) using the
same multiphase technique described above. The surface, illuminated by the fringe
pattern, is imaged onto a standard CCD vidicon array with a magnification such that the
image of the nominal fringe spacing is equal to 3 pixel widths; this is illustrated in Figure
6.
We regard the pixels as being grouped into image-elements, each of which contains 3
adjacent pixels on a horizontal scan line (assuming the fringes run vertically). As is clear
from Figure 6, these three pixels generate signals IR, IS and IT given by equation (3)
where Φ is the mean phase of the fringe pattern within the image-element. This phase Φ
then represents the height of the surface (modulo d/tanθ) at the point on the object
corresponding to the image element. Using an image area of 512 x 256 pixels, the pixel
data rate (30 Hz frame rate) is 3.93 x 106 sec, which matches the 4 MHz data rate of our
multiphase electronics. We are thus able to obtain single frame digital images at 30 Hz
with (512/3)x(256) = 43520 image elements, each of which has an 8 ½ -bit number (in
decimal, 0 to 384)associated with it which is proportional to the height of the surface at
that point. For example, consider a 5 cm x 10 cm object surface: we want 170
interference fringes across the 10 cm dimension. Let us assume a 45º angle of incidence.
In this case the fringe spacing d will be 0.59 mm (100mm/170), and the height resolution
(for a 30 Hz measurement bandwidth) will be 0.59mm/384 = 0.0015 mm or 1.5-microns.
The spatial resolution is defined by the size of the image-elements; in this case each
image-element is 0.59 mm x 0.20 mm.
This system bears some superficial resemblance to existing contouring systems in which
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the image of a grating is projected onto the surface to be contoured, and the illuminated
surface is then imaged onto a second appropriately scaled grating to produce a moire
fringe pattern whose fringes are a contour map of the surface. Our approach differs
primarily in that it provides real-time digital data which quantitatively describes the
surface contours. A secondary advantage lies in the use of an interference fringe patternwhich has a far greater depth of field than a projected image of a grating. We can, of
course, use our digital frame data to generate a pseudo contour map of the surface simply
by assigning a grey level (proportional in level to the phase Φxy) to each image-element,
and then displaying the resulting image. In such an image, because the fringes have a
saw-tooth profile (due to the fact that the intensity is linearly proportional to phase, rather
than being proportional to cos2Φ), the slope of the surface can be directly inferred from
the contour map image. We have not yet dealt with the phase-unwrapping problem,
although this should be entirely straightforward (with the constraint that the surface slope
not exceed a maximum value of about ½ tanθ).
It should be noted that the height resolution and the spatial resolution are linearly related:
better height resolution can be obtained by reducing the lateral extent of the target surface
(and also by reducing the measurement bandwidth). The direct availability of digital
height data will allow many surface properties to be evaluated in real time--without
further image processing. For example, the volume beneath a contoured surface is simply
proportional to the sum of the heights of the individual image-elements. Starting with an
image of a contour map, this measurement would be far more complex.
ELECTRIC FIELD SENSOR: This sensor is illustrated in Figure 7, and is representative
of a broad class of common-path interferometric applications, most of which involve
birefringent transducers of one type or another. In this case the sensor element is an
electrooptic (EO) crystal. In the presence of an electric field, the birefringence of the
crystal is changed by an amount which is proportional to the electric field. One of the
specifications for this sensor is that it have an all-dielectric link to the readout station--to
minimize hazards associated with its use on high-voltage transmission lines. For this
reason, the light is brought in and returned by multimode optical fibers. Multimode fibers
are used in preference to single mode fibers in order to improve reliability and economy
by relaxing mechanical and optical tolerances.
As shown in Figure 7, the laser diode radiation from the input fiber is collimated and
linearly polarized, with the polarization vector at 45º to the axis of the EO crystal. The
EO crystal introduces a phase shift between the two orthogonal incident polarizations
which is proportional to the electric field across the crystal. A weak Wollaston prism (or
Nomarski plate, or simple calcite wedge) follows the EO crystal as shown. Its function is
to introduce an additional differential phase shift whose magnitude varies in the vertical
direction--ideally spanning just over one complete cycle of phase. The next element in
the optical train is a second linear polarizer which mixes the two orthogonal polarizations
so that they can interfere. This is followed by 3 adjacent lenses, arrayed vertically, which
form three point images on the ends of the three output multimode fibers. Because of the
combined effects of the EO crystal and the Wollaston prism, the signals detected at the
output ends of the output fibers can be represented by IR, IS and IT as defined by equation
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(3) above: the EO crystal provides the information-carrying phase Φ, and the Wollaston
prism provides the phase offsets of 0, 2π/3 and 4π/3. These signals are processed by the
multiphase electronics as described earlier to provide a precise measure of the electric
field across the sensor element. For a 1000 volt drop across a 2 mm thick, 24 mm long
transverse KD*P (deuterated di-hydrogen phosphate) crystal, and assuming a wavelength
of 0.8 microns, the voltage-induced phase difference is about 1.2 cycles. Thus, at the full
measurement bandwidth of 4 MHz we are in theory able to sense voltage changes as
small as a few volts (1000 volts/[1.2 x 3841); for a reduced measurement bandwidth of
4kHz, the sensitivity drops to a few millivolts. For reasonable laser diode power levels
(i.e. more than a few milliwatts) digitization noise (whose level is set by the number of
bits in the A-D) dominates and shot noise is not a factor.
6. ACKNOWLEDGEMENTS
We are pleased to acknowledge the support of the Department of Energy's SBIR
program, which funded portions of the research described here.
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