Astigmatic mirror multipass absorption cells for long-path-length ...

Astigmatic mirror multipass absorption
cells for long-path-length spectroscopy
J. 6. McManus, P. L. Kebabian. and M. S. Zahniser
A multipass absorptton cell, based on an astigmatx variant of the off-axis resonator (Herrtott)
contiguratton. has been desrgned to obtain long path lengths in small volumes. Rotatron of the mtrror
axes 1s used to obtam an effective adjustabtlity in the two mirror radii. Thus allows one to compensate for
errors rn mirror radii that are encountered In manufacture, thereby generating the dewed reentrant
patterns wth less-precise mirrors. A combination of mirror rotation and separation changes can be used
to reach a variety of reentrant patterns and path lengths with a fixed set of astigmatic mu-rors. The
accessible patterns can be determined from trajectories, as a function of rotation and separation. through
a general map of reentrant solutions. Desirable patterns for long-path spectroscopy can be chosen on the
basis of path length. distance of the closest beam spot from the coupling hole, and tilt insensitivity. We
describe the mathematics and analysis methods for the asttgmatic cell with mtrror rotation and then
describe the design and test of prototype cells with this concept. Two cell designs are presented, a cell
tith 100-m path length in a volume of 3 L and a cell with 36-m path length in a volume of 0.3 L. Tests of
low-volume absorption cells that use mirror rotation. designed for fast-flow atmospheric sampling, show
the validity and the usefulness of the techniques that we have developed.
Key words: Multipass absorption cells, asttgmatic mirrors
1. Introduction
Multipass optical cells with the configuration presented
by Herriott and coworkers1.2 have found increasing
application in long-path-length spectroscopy.z-‘1
This cell configuration offers several
advantages: it is easy to construct and align, long
path lengths can be achieved in a low volume, and
interference fringes that arise in the cell can be
suppressed effectively.6.7 The possibility of folding a
long path length into a small volume is especially
useful in laser spectroscopy, in which one may need to
measure species at low concentrations and in which
one needs a fast ‘response or has a limited sample size.
The signal-to-noise ratio in an absorption experiment
generally increases with the path length, up to a limit
at which reflection losses*2 or interference fringes in
the cell become important. The volume of an absorption
cell for a given path length is a limiting factor in
the measurement response time. Low mode volume
and the resulting smaller mirrors also can permit a
The authors are with Aerodyne Research. Inc.. 45 Manning
Road. Billerica. MassachusettsO1821.
Received 21 July 1994; revised manuscript recerved 16 December
1994.
0003-6935/95/183336-13$06.00/O.
0 1995 Optical Society of America
lower overall system mass, which is important for
portable field systems and in airborne applications.
The basic Kerriott cell, also known as an off-axis
resonator, consists of two spherical mirrors separated
by nearly their radius of curvature.rd An optical
beam is injected through a hole in one mirror in an
off-axis direction, and it recirculates a number of
times before exiting through the coupling hole. The
path length through the cell is established by a
reentrant path, where the recirculating beam closes
on itself at the coupling hole: The beam pattern, and
likewise the path length, can be changed by one’s
adjusting the mirror separation. The beam spots
fall on elliptical patterns on the mirrors, and the
beams fill a volume within the cell with the shape of a
flattened hollow hyperboloid. The beam exits the
cell at an angle from the input direction, and the cell
can be treated simply as an optical element equivalent
to a reflection from a convex mirror in the position of
the coupling hole. Thus this type of cell is especially
easy to design into optical systems.
Further advantages can be derived from astigmatic
off-axis resonator cells, i.e. where each of the cell
mirrors have two different radii of curvature.2.‘3
The main advantage of this variant is that the
circulating optical beam more fully fills the whole
volume of the cell, with a pattern of spots on the cell
mirrors that more evenly fills their area. Thus, for a
3336 APPLIED OPTICS Vol 34. No. 16 / 20 June 1995

given number of passes through the cell, the beam
spots are more widely spaced than in other types of
cells (conventional Herriott cells or White ceils”),
permitting smaller mode volumes and smaller mirrors.
One difficulty that has discouraged the USC of
astigmatic cells has been the need for extreme preci-
SIOII In the manufacture of the mirrors. One may
need to define two different radii with an accuracy of
1 part in 10’ to ensure that the pattern will close
With a spherical mirror cell. changes tn the mirror
spacing can accommodate errors 111 the radll, and the
desired pattern always can be achieved With astigmatlc
mirrors, a reentrant condition must be establlshed
for two axes simultaneously, so adlustment of
the mirror spacing alone cannot compensate for both
axes. Circumventing this problem by one’s stressing
the mirror is possible, but this would be unwieldy
In many applications. We have Investigated a
method13 of adjusting for mirror radii errors by
rotating one mirror about its optical axis. This
allows us to compensate for manufacturing errors in
the radii, permitting the nlirrors to be made with
relaxed tolerances and reduced cost.
An additional advantage of the use of ihe mirrortwist
degree of freedom is that other reentrant patterns,
corresponding to different path lengths, become
accessible with a given set of mirrors. By
translating and rotating the mirrors, one can use a
variety of cell patterns and path lengths. This can
be especially useful during setup, when a shorter path
length may be desired. Small changes in mirror
separation and twist also may be used to establish
path lengths that are significantly greater than the
nominal design.
In this paper we describe the mathematics and
analysis methods for the astigmatic cell with mirror
rotation and then describe the design and test of
prototype cells with this concept. We review the
basic reentrant equations for astigmatic cells and
then discuss methods of selecting favorable reentrant
patterns. We show how mirror rotation can be used
to accommodate mirror-manufacturing errors and to
achieve different patterns. We then show how the
set of all possible patterns can be organized in a
coordinate space-and how one can move through that
space by changing mirror separation and rotation.
Two prototype low-volume absorption cells that use
mirror rotation have been designed for fast-flow
atmospheric sampling. The larger cell has a 100-m
path length in a volume of 3 L, and the smaller cell
has a 36-m path length in a volume of 0.3 L. Tests of
these cells show the validity and the usefulness of the
techniques we have developed.
2. Cell Analysis
A Basic Reentrant Equations
The circulation of the beams within the cell can be
described approximately with the equations presented
by Herriott and coworkers ’ L The location of
the centroid of the beam on each mirror is described
by sinusoidal equations, with an angular advance on
each pass, 0, that is a function of the mirror radii and
spacing. In this simplified analysis, the two mirrors
are assumed to be identical. \vlth no tilt or rotation
This analysis is an idealization of the beam propagation
that IS useful for an understanding of the basic
bchavlors of the cell. especially the rcenlrant beam
patterns A description with 1 x 4 matrIces, presentcd
below. IS used to account for the effects of
rotation of the mirror principal axes with respect to
each other, as well as mirror tilt and nontdentical
mirrors.
With astigrnatlc mirrors,? there are two lndepcndent
reentrant equations that describe the ctrculatlon
of beams wlthln the cavltv. _ A single integer 1 IS used
to count the beam reflections from both mirrors, as IC
the spot patterns were combined on a single plane
The s-v coordinates of the lth beam spot are gven b!
x, = X0 sinllex), y, = Y, sm(10,),
D
0, = cos-’ ( 1 - p 1 t e,=cos-ll--, (1,
I ( R, 1
where X0 and Y0 define the size of the overall spot
pattern, R, and R, are the mirror radii of curvature (if
we assume that the two mirrors are identical), and D
is the mirror separation. Thus the beam spots trace
out a path on the mirrors that is sinusoidal in x and y
but with different frequencies (i.e., a Lissajous pattern).
For cells close to confocal spacing (D = RI,
one has 0, = 7712 and 8, = ::;‘2. The reentrant
condition is that the beam exits after N passes, I.e..
xN=yh’= 0, which is satisfied for
NO, = M=T, N9,, = M+ (2)
which yields the following for reentrant patterns:
8, = nM,/N, 8) = srM,/N. (3)
The integers N, M,, and -M, define the pattern of
transits through the cell. A choice of (N, M,, M,j
and a base length for the cell will result in design
values for the mirror radii. In this basic description
of the reentrant conditions, with the entrance and
exit at the center of one mirror, N must be even (an
integral number of round trips). For these patterns
to be unique in terms of (N, M,, M,( there must be no
common factors (other than 21 within this set of
integers. If there are such factors, the pattern is
degenerate, and at this value of 8, and 8, a pattern
with a lower pass number (reduced by the common
factor) will be supported by the cell.
The size of a spot pattern on the mirrors is set by
the imtlal alming of the first beam Into the cell at
coordinates X0 and & That lnltlal spot will be near
the maxImum radius of the overall spot pattern. and
the rest of the beam-spot tocz:!nns will result from

the initial aiming; The pattern must be large enough
that the recirculating beams avoid exiting through
the coupling hole before intended. If the closest
spots to the coupling hole are relatively far away in a
given pattern, the whole pattern size may be reduced.
We find it convenient to define reduced angularadvance
variables +, and 4, where
4, = 0, - 5 ’ e$y = 0, - 5. (4)
For nearly confocal cells, 4, = 0 and bY = 0. We
have used the + variable in our earlier discussion of
cell algebra,6 and the results here are similar. In
terms of the mirror radii and spacing we have
+Iy = sin or, for D = R,
4 ry = g-1.
i SY 1
Thus the 4 variables are approximately the distance
in separation from confocal spacing for the separate
radii, normalized by those radii. The arguments of
the b’s are the negatives of the usual resonator
parameters,15 g,, = (1 - D/R,,). 4 ranges from
-u/Z (at D = 0 or R,, = =) to +u/2 at (D/R,, = 2).
For DIR,, > 2, hy is undefined, but this corresponds
to unstable resonator’conditions (g2 > l), so
this range is unimportant practically. The 4’s also
may be defined for reentrant patterns as
(5)
4ry = (6)
Because we usually deal with patterns in nearly
confocal cells, we have N = 2M, A new pair of
pattern variables, k,, KY, may be defined as
k, =N-2Mx, k,=N-2My (7)
The k’s are even integers, which are small compared
with MxJ, that are related to various pattern characteristics.
In this case the reentrant 4 variables
become :
4ry = -bG)k,y/N. (8)
There are numerous patterns of recirculating
beams, as defined by choices of (N, M,, My) and different
choices in this set can have beam-spot patterns
with very different appearances. All the patterns
are within a rectanguk.r boundary, but there will be
different distances for the closest approach of spots to
the coupling hole and diRerent sensitivities to misalignments
of the mirrors. Examples of a few of
these patterns (for the front mirror) are shown in
Figs. l-3. Figure 1 shows a symmetrical pattern
with even M, and MY and spots far from the coupling
o” 0
0 0
00
00
0 0
0 0 0
0 O 0 0
0 0
0 0 -
@
0 O 0
0 0
0
0
0 0 0
0 O 0 0
0 00
0 0”
0
00 0 0 00
0
00
OO 0
0 00 O0 0 0 O 0 OO
Fig. 1. Symmetrical pattern (N = 182. M, = 80, M,, = 76). with
even M,, and with spots far from the coupling hole. Only spots on
the input mirror are shown. For all figures, a circle indicates the
coupling hole.
hole. Figure 2 shows a pattern with spots close to
the hole. Figure 3 shows an asymmetrical pattern,
with odd My.
The coordinates of the beam spots fall on the
Lissajous curve given by Eq. (l), where the ratio of
frequencies is 0,/C+ = MJM,. The beam spots lie
widely separated on this curve, so the spot pattern
often looks unlike a familiar continuous Lissajous
figure. In some cases the beam spots appear to lie on
other Lissajous curves that are much coarser than
those defined by the ratio MJM,. One can understand
this effect by expressing the spot coordinates
with reduced angles. If one considers only the frontmirror
beam spots, their coordinates can be reduced
when multiples of IT are removed from the sine
moo0 0 0 0 0 0 0 0 ooow
Qx)-z.00 0 0 0 0 0 0 cl OOOOD
030000000@000000Qp
Qcx)oooo 0 0 0 0 0 0000~
moo0 0 0 0 0 0 0 0 000003
Fig. 2. Pattern (190, 80. 761, with even M,, but with spots close
to the hole Only spots on the input mirror are shown.
3338 APPLIED OPTICS / Vol. 34. No. 18 / 20 June 1995

go0 ‘oooo 0 08
O Oo 0 0 o" O
O 0 0 O
8" 0 0 "B
0
0
0
/a O
0
0 0
0 0
0 0
0 0
0
0
0
0 : o” Oo o
8 0 0 0
00 O
0
0
o 0”
0 0 00 0 00 0 o”
O
0
0
0
8
OO 8
Fig 3. AsymmetrIcal pattern 188. 86. 83:. with odd .4f, Only
spots on the input mirror are show-n
argument
in Eqs. (1) yielding
x, = (- 1)’ sin (j:h,/N), y, = ( - l)-‘ sin (j&,/N),
where j counts the returns to the front mirror, i.e.,
j = i/2. These coordinates lie on a Lissajous curve
defined by the frequency ratio k,/k,. Similar arguments
can be used to generate back-mirror reduced
coordinates. In some cases even coarser curves can
be generated by subtraction of multiples of 217 from
the arguments in Eqs. (9). The value of the necessary
multiple, which is different for x and y, can be
found from Diophantine equations.i6 Similarly, the
reduced frequency ratio fv/fx is found from integer
solutions to be k,f, - k,f, = 2N.
One usually cannot tell by simple observation of
the spot pattern what the pattern parameters are.
When one sets up a practical cell, changing the mirror
spacing causes a variety of nearly reentrant spot
patterns to appear without an obvious sequential
order. Identification generally requires that observed
patterns be matched with calculated ones and
that one possesses a detailed knowledge of the relation
of patterns to each other.
B. Properties of Spot Patterns and Advantageous Choices
Advantageous patterns may be chosen for absorption
experiments in which the path length is long (large
N), the beam pattern is compact, and the fringe levels
are small Generally we use cells that are close to
confocal, because this results in the minimum mode
sizes for the propagating beam, yielding a smaller cell.
Interference fringes produced in the cell can be made
to appear at higher frequency (for greater ease of
suppression) when the spots neighboring the coupling
hole are nearly midway through the beam’s
orbit.6 Patterns that are less sensitive to mirror tilt
may also be chosen, so that intermediate beam spots
(9)
will be less susceptible to being driven out the coupling
hole by misalignments.
Patterns with even M, and I?, correspond to whole
numbers of complete transverse oscillations of the
beams in the cell by the use of Eqs. (2). The output
beam for these patterns will exit from the diagonal
corner in the square pattern; i.e., the last spot in the
cell is at (x:y) = (- 1. - 1) for the first spot at (1, 1).
In this case the beam will act as if it is reflecting from
a mirror (equivalent to the back of the cell-input
mirror) in the position of the input plane of the cell.
This is a great advantage in the application of these
cells to laser systems. and we use these patterns
exclusively. These patterns also are more symmetrical
in the distribution of spots on the mirrors than
patterns with either :%I, or M, odd. The even M,,
patterns are insensItIve to trlt. as can be seen by the
total cell transfer matrix being unity [as explained
below). With M, and M, even. N/2 must be odd to
avoid common factors.
For application to low-volume cells, we select patterns
with the innermost spots on the input mirror
far from the center. Naturally, this permits the use
of smaller mirrors and smaller mode volumes.
There is a relatively large set of patterns with a
similar wide spacing from the center available, and
for -these patterns the spacing scales as
l/I jc No 18 ;\P=‘LIE3 OPTICS 3339

&. M&ix Analysis of Mirror Rotation and lilt -
A 4 x 4 matrix description of the cell allows us to
investigate the effects of rotation of the mirrors about
the optical axis, nonidentical mirrors, and mirror tilt.
Once we derive a matrix description of a single
traverse of the cell, the location and slope of a ray
after N traverses results from the cell matrix raised to
the Nth power, multiplying the input ray vector.
Because we allow several nonideal effects to occur, the
matrix descriptions tend to produce complex analytical
expressions. Therefore many of the results of
the matrix analysis are achieved numerically.
The geometry of the cell configuration is shown in
Fig. 4. Concave astigmatic mirrors are located at z =
0 and z = D. The principal radii of curvature, R, and
Rb, are nominally aligned with the x and the y
coordinate axes, respectively. A beam is injected
into this system at x = y = z = 0, and it propagates
between the two mirrors. We are concerned primarily
with the transverse (x-y) position of the beam on
each return to the mirror at z = 0. The basic
geometry may be modified by rotation of the axes of
the two mirrors with respect to each other by an angle
0!. This rotation provides the added degree of freedom
needed to correct for errors in the cell ‘radii,
when it is used together with an adjustment of cell
length.
A propagating ray is represented by a column
vector 2, with components [x, x’, y, y’], where x’ and
y’ are the slopes in thex andy directions, respectively.
The ray matrix for one cycle of the beam’s propagation
through the cell is constructed from free-space
propagation D(D) and mirror reflection matrices
WC, Rd:
t-2 = R(% Rb)D(D)R(&,, &)D(D). (10)
The coordinate of the beam spot after n cycles, Z,, is
given by CT&, where Z, is the initial vector with x =
Y = 0. For mirror spacing (D) and mirror principle
radii (R,, Rb) aligned with the coordinate axes, the
I \
Fig. 4. Ckom t e r-y of the cell with mirror axis rotation
component matrices are given by
D(D) =
[l
0
D
100
0 01
I
OOlD'
I
0 0 0 1
1 1 0 0 Ol
I
R(RcmRb) = o o 1 o . (11)
-2/R., 0 01 -2/f& 0 o 1 i
When the mirror principal axes are aligned with the
coordinate axes, the cell matrix is block diagonal, with
uncoupled 2 x 2 matrices for x and y motion. When
the mirror axes are rotated by an angle 0, to the
coordinate axes, the reflection matrix becomes
R’UL Rd = T(-WW,, R,)T(e,), (12)
where T(0,) is the rotation matrix,
i cos 0 8, cos 0 8, sin 0 Bc sin 0 0. 1
‘WV = -sin e o . cos e -’ . (13)
I I I 0
I
1 0 -sin Br 0 cos 8, J
The rotation matrices mix x and y components of the
beam coordinates. Nevertheless the behavior of the
cell is substantially similar to the unrotated case,
with essentially sinusoidal x and y spot motions.
The frequencies are altered and the overall pattern
becomes mildly distorted, but complete reentrant
patterns still can be achieved.
We have explored the behavior of the multipass cell
matrix by direct computation using a program written
in the MATLAB langt~age.~~ We directed our initial
efforts at finding the spacing and twist adjustments
necessary to make up for errors in the mirror radii.
For this type of calculation, we start with a set of
mirror radii and a spacing that corresponds to a given
reentrant, pattern. Then the mirror radii are assumed
to deviate from those values by a small
amount. We change the spacing by an amount S and
change the mirror twist 0, so that the beam again
exits the cell in the same pattern. For small deviation
in the radii, the solution (8,, 6) that causes the
beam to exit results in a slightly distorted version of
the nominal pattern. When 0, and D are found by
adjusting them to make the last spot in the pattern
return to the origin, the resulting cell matrix CNi2 is
the identity matrix (for even M, and M ), to within
numerical tolerances. As an example of this adjustment
procedure for the pattern (N = 182, M, = 80,
My = 76) (Fig. l), we assume mirror errors of 2 X
10A3, i.e., R, = RJ1.002 and Rb = RJ0.998. The
required adjustments are 8, = -17.02” and 6/D =
-3.93 x 10-4.
3340 APPLIED OPTICS / Vol. 34. NO. 18 / 20 June 1995

We have used the matrix description of the cell to
search systematically for advantageous patterns for
spectroscopic applications. The primary search has
been in terms of finding patterns with wide spacing
around the coupling hole. The search has been
restricted to even M,, My, and odd N/2. Figure 1
shows an example of one such pattern with good
spacing. Other properties can be included in the
search, as outlined above. The effect of our tilting
the mirrors with respect to the cell z axis also has
been investigated. Tilt can be accounted for by the
addition of a constant to the slopes, (x’ ory’), at each
reflection. As expected, in patterns with even M,
and MI the coordinates of the last spot are insensitive
to tilt.
In addition to compensating for manufacturing
errors, mirror rotation and separation changes can be
used to achieve a variety of reentrant patterns that
would not be possible with an unrotated set of
astigmatic mirrors. This gives a valuable degree of
flexibility to the astigmatic cell design. With mirror
rotation, we can select from a set of different path
lengths with a single set of mirrors, similar to the
selection of different reentrant patterns in the nonastigmatic
cell through separation changes alone.
.
D. Iterative Equations and Recirculation Frequencies in
the Astigmatic Cell
Although the above 4 x 4 matrix description gives us
the mirror spacing and rotation for pattern closure
through numerical simulation, it would be desirable
to have corresponding analytical expressions. The
4 x 4 matrix description may be reduced to a pair of
coupled difference equations18 that naturally yield
analytical expressions for circulation frequencies in
the cell. This approach, which we outline below, is
analogous to the 2 x 2 matrix description of nonastigmatic
(i.e., spherical) mirror off-axis resonators and
the reduction of such a matrix to a single difference
equation. I9 With the 4 x 4 matrix expressions for
one and two complete mund trips of the cell, the
coordinates of the beam spot on the nth return to the
front mirror (x,, y,,) are found in terms of the previous
coordinates:
where
X” = (A* - 2 - e*)x,-, - x,,w2 + Ey,Ml,
yn = (B* - 2 - e*)y,-, - yn..* - E xndl, (14)
A = 2 - [(a,D)cos* T + (d)sin* T],
B = 2 - [(a$)sin* T + (a&)cos* T]
e = (1/2)(a, - q)D sin(27),
E = -(l/4)1(%
- ab)D]*sin(4r),
and where a, = 2/R,, q = 2/R&, T = 6,/2; i.e., T is one
half the total rotation angle between the mirrors.
In the basis used here, the mirrors are rotated
symmetrically by +T and -1. The coupling term E.
usually will be small. If the coupling were zero in
Eqs. (14), the resultant motions would be*&usoidal
inx andy, with cosines of the angular advances in two .
passes given by (A’ - 2 - e’)/2 and (B’ - 2 - e*)/2.
The coupled difference equations can be expresssed
in the form of a 2 x 2 matrix with the A? operator,
Azx, = x,,, - %I,, + x, m2:
(151
with /, = (A’ - 2 - eZ) and /,. = (B’ - 2 - e’). This
pair of equations is analogous to the pair of differential
equations for coupled oscillators. The motions
of coupled oscillators are described by shifted normal
mode frequencies that are arrived by one’s diagonalizing
the matrix for the coupled equations. In a
similar fashion, the difference equation matrix can be
diagonalized to yield shifted normal mode frequencies
for circulation through the cell. The results are, for
the cosines of the angular advances in two passes,
cos or* = ( 1/2)(Az - 2 - e* - Q.
cos ey2 = (1/2)(B* - 2 - e* + 0, (16)
with 5 = (1/2)(B* - A*)(- 1 + [ 1 - 4Ez j(B* - A2)*]11 *.
When 5 -=x(B* - A*), 5 = -&*/(B2 - A’).
These angular advances describe the motion of
compound variables that are linear combinations of x
and y, although the amount of mixing is small.
However, for reentrant patterns both x and y will
simultaneously be zero when the beam exits, so the
mixed variables will be zero as well. Therefore the
angles given in Eqs. (16) can be used to find reentrant
solutions in the astigmatic cells with rotated mirror
axes.
E. Beam Escape and Maps of Pattern Location
When the mirror spacing is displaced from the exact
reentrant condition, the last beam spot is swept
across the coupling hole. The tolerance for spacing
changes, i.e., the range over which the beam still falls
within the coupling-hole diameter (dh), is a function
of the pass number and the various cell parameters.
To express the spacing tolerance, we define a transverse
spot velocity IJ, = dS/dD, where S is the lateral
position of the last beam spot on the input mirror.
The tolerance then is AD = dh/u,. Based on Eq. (l),
the (vector) spot velocity is
u, = 3?X,i cos(iO,)[D(2R, - D)]-’ *
+ yY,i cos(iCl,)[D(2R, - D)]-“*. (17)
For the last spot, i = N and cos(NB) = ~1.
Furthermore, if& = Y0 = ( 1/J2)Ro, and D = RzJ, the
scalar velocity is approximately
“* = (1/J2)NRo([D(2R, - D))-’ f [D(2R, - D)]-‘I’/*
= NR,/D. (18)
Thus, as the number of passes increases, the spot
20 June 1995 VOI 34, No If3 APPLIED OPTICS 334 1

velocity increases proportionately, and the mirrorspacing
tolerance becomes correspondingly tighter.
If the mirror spacing changes beyond the tolerance
range, the beam pattern will change, yielding a
greater or a smaller number of passes before the
beam exists. This behavior may be investigated
numerically if we assume a set of mirror radii, derive
cb, and 6, as a function of mirror spacing from
relations (5), and iterate the beam coordinates until
the spot falls within the coupling-hole radius, i.e., it
escapes. A plot of the number of passes as a function
of mirror spacing for a fixed set of mirror radii has an
irregular appearance. Indeed, in tests of actual cells.
changing the mirror separation permits the beam to
exit in many nearly reentrant patterns with a widely
varying number of passes, without apparent logical
progression.
A general map of the location of reentrant solutions
In the two-dimensional space of (4,. 6,) can provide a
more complete picture of astigmatic cell behavior.
At an arbitrary point (c$,, &, I. one may calculate how
many times the beam circulates before it escapes.
The recirculating beam escapes at cycle j, (N = 2,)
when the beam coordinate is within the coupling-hole
radius p (ignoring the finite beam size), I.e.,
sin*(2j,&,) + sin’(2j,,&,) < 13~.
Here the hole radius is normalized by the overall
pattern size, or X,, = Y. = 1. The result of iteration
of relation (19) is shown in Fig. 5, in which the escape
pass number is plotted for the region -0.2 < 6, <
-0.18, -0.27 < &I~ < -0.25. Forachoiceofmirror
radii and spacing within an appropriate range, one
can WxIate 4, and $ and find the corresponding
number of passes from Fig. 5. The “+-space” plot,
N(p, &:, 6,). is a general map of the location of
reentrant patterns that is independent of specific
choices of cell parameters. The plot is made up of
overlapping circular zones (or radius P;N), each
supporting a single spot pattern with a constant pass
number corresponding to nearly reentrant conditions.
For a zone with pass number N and zone center
coordinates (da,. $,.), the pattern parameters can be
recovered from Eqs. (8). When larger zones completely
or partially overlie smaller ones. the beam
esits in the lower pass-number pattern.
The b-space plot can be constructed by the use of
geometric methods to produce maps of nearly reen-
Leant solutions with higher accuracy and more quickly
than b!. numerical iteration of relation i 19). The
constructlon algorithm. described In Appendix A. also
elucidates the structure of the solution space. Such
3 constructed plot is shown in Fig. 6. The plot of
~L:IP< 6,. &I shown in Figs. 5 and 6 Includes beam
escapes In any direction from the cell. If we consider
only patterns In which the beam exits in the opposite
quadrant from the input [i.e., the ( - 1, - 1) quadrant],
then a more sparse &space plot results, as shown in
Fig. 7. These zones correspond to even M, and hi!,
and odd N/2, i.e., where the cell behaves like a
mirror, which is the subset that we use in practical
cell designs.
For a fixed set of mirror radii, changing the mirror
spacing generates a nearly linear trajectory through
this (6,. do,) space. Relations 5 may be considered a
-.ZSO
Phiy
-.270
-.200 ..;os -.ioo -.16S -.160
Phh
Fig. 5. Iterated solution for cell escape as a functron of 6.. br, with
normalized hole radius p = 0.1. The logarithm of the number of
passes is encoded as tone. The location of pattern ir\I = 182.
M, = 80. M, = 76) is indicated
Fig. 6 Plot of *ones of nearly reentrant patterns constructed
wl:h the algorithm described in Appendix A. Patterns with the
beam exlrlng m any dlrection up to a limiting number of 1000
round trips are shown The location of our pattern wth 182
passes IC indicated
3342 APPLIED OPTICS Vol 34 No 18 20 June 1095

Phiy
0.260
:;i; w !irV LOCP-
-l
1.6 1.65 2.3 2.66 a
0-J
- There are several approximations that -al’e built
into the use of these maps to find cell patterns. The
maps are idealized and general. without effects that
are specific to particular mirrors or implementations.
The maps do not include mirror tilt, which would
tend to smear odd-M zones. The maps do not include
the effects of pattern distortion that occur at
larger mirror-rotation angles, which would also lead
to distortion of the zones. The maps also do not
include the effect of re-aiming the input beam.
Despite these approximations, the maps are a useful
tool in cell setup.
3. Prototype Cell Designs and Test Results
4.270
Fig. 7. Plot of zones of nearly reentrant patterns constructed
with the algorithm described In Appendix A. Patterns with the
beam exiting in the f - 1. -l! quadrant only; up to a limiting
number of 1000 passes, are shown. Trajectories showing the
patterns accessible to the system as the mirror separation and
rotation angle are changed are drawn. The three lines are for
spacing changes (tics are placed at intervals of 0.2% of the base
length) with fixed mirror axis angles of O’, 22’. and 30’.
parametric equation for 6, and $J in terms of D.
The trajectory may be expressed directly as
4190
Phlx
$.,(b,) = sin-‘(7 sin 6, + (Y - l)], (20)
where y = R,/Rb. If b is small, then &CC&) =
[Ydh + (Y - 111.
The above analysis provides an expression [Eqs.
(16)] for the cosines of the angular-advance angles as
a function of mirror rotation and spacing. From
Eqs. (16), one can derive a trajectory through the
(&J,, 4) plane as the mirror rotation and spacing are
adjusted. Such a set of trajectories is shown in Fig. 6
for a set of mirrors that is designed to access the
pattern { 182;‘80, 76) at a rotation angle of 22”. As in
Eq. (20), these trajectories are nearly linear when the
rotation angle is fixed.
Together the maps and trajectories in 4 space
provide useful tools in setting up astigmatic cells.
One can locate desirable and accessible patterns and
move to them with the right combination of translation
and rotation of the mirrors. At the initial setup
of a cell, errors in the mirror radii can be seen as an
offset in the location of the patterns from their
expected values. Using the map, one can then determine
the actual mirror radii. If one wants to move
between patterns with different total lengths, the
map and trajectory plot shows what combination of
translation and rotation is needed.
A. Cell Designs. 100 and 36 m
We have produced astigmatic cells with the mirrortwist
concept for application to fast-response atmospheric
trace-gas sampling.‘0-22 The low mode volume
afforded by the astigmatic design is critical in
such applications. Two cell designs have been constructed:
the larger one has a path length of 100 m
in a base length of 0.55 m and a volume of 3.2 L, and
the smaller one has a path length of 36 m in a base
length of 0.2 m and a volume of 0.3 L. Both of the
cells are based on a nominal 182 passes in the pattern
(N, M,, My] = { 182, 80, 761. As described in Section
2, it is possible to align these cells to have pass
numbers between approximately 0.5 and 2 times the
nominal number of passes. Both of these cells have
been used in laboratory and field measurements.
The 100-m cell has been used in measurements of
HO2 line broadening*O and in combustion diagnostics.*’
The 36-m cell has been incorporated in a
tunable diode laser based system for fast (lo-20-Hz)
concentration measurements for eddy correlation
determinations of trace-gas fluxes to the atmosphere.**
We primarily describe the larger fast-flow cell,
because we have done the most extensive tests on it
and the smaller cell is very similar. The design of
the larger cell is shown in Fig. 8. The cell structure
is built on an aluminum base with a glass tube center
section. End assemblies hold the mirrors, provide
optical access, shape the-gas flow, and pressure-seal
the cell.’ The front mirror is bolted directly to the
front flange with no adjustments. The rear mirror is
mounted with a post-and-collar arrangement to permit
length and rotation adjustments. An alignment
jig is attached to the mirror support post during setup
and alignment. After setup, the mirror post is locked
to the collar, the alignment jig is removed, and the cell
cover plate is attached.
The mirrors are made of aluminum in the form of
octagons that are 7.6 cm across. The active cell
volume, defined by the cylindrical space between the
mirrors, is 8.6 cm in diameter and 55 cm long,
yielding a total volume of 3.2 L. The volume occupied
by the beams is defined to the first order by the
square spot pattern on the mirrors ( - 5 cm on a side).
yielding - 1.4 L. The mirror size in this design is
20 June 1995 ‘.‘Ol 34 r40 18 APPLIED OPTICS 3343

Exhaust ’ \ 0. Ring
Pap*
EX?UUSl
Plwlurn ‘*
Mlnw
SSll
1Ocm
t
FIN. 8. Cross-secttonal view of the 100.m-path.len@h. 3.2-L fast-flow cell. The cell IS shown lo scale. with an overall length of 66.7 cm
and a diameter of 15 cm.
based on considerations of mode size at the maximum
design wavelength of 10 pm. We have used the
criterion that the separation between the input beam
and the closest recirculated beam spot on the input
mirror should be 10 times the Gaussian beam radius
(intensity). The coupling-hole radius (0.44 cm) is
then one half the distance to the nearest beam spot.
This yields a conservative design that may be made
tighter in applications in which cell volume is at a
premium.-
The scaling of the cell volume with pass number,
base length, and wavelength can be estimated simply
if we consider the case of nearly confocal cells. A
recirculating beam matched to a confocal cell has all
the beam spots the same size, w,, at ,z times the
radius at the center of the cell, wg = (Ad:‘2~;)’ *, where
d is the mirror spacing. For a pattern width of W,
the maximum s acing between spots, S, is approximately
S = W/ P N. If we equate the pattern spacing
with some factor a times the spot radius w,, we arrive
at the design pattern width, W = a(kdN.‘T;)’ *, and
the volume V = dW* = a*hd*Ni-rr. Thus the volume
will scale as the square and the mirror width will scale
as the square root of the base length. The scaling
arguments will change somewhat for departures from
the confocal condition or if beams that are injected
are not matched to the cell. However, the above
argument shows the general trend of scaling. With
our small-cell base length (20 cm), which is 0.36 times
that of the larger cell, we designed the mirror width to
be somewhat smaller (a factor of 0.5 rather than 0.6)
than would result from the scaling argument. The
factor of 2.8 reduction in path length yielded a
factor-of-10 decrease in volume.
An inlet system was designed to supply a uniform,
low-pressure flow.of the sampled gas to the optical
measurement volume. Rapid response of the cell
depends on one’s maintaining as uniform a flow as
possible with a minimum of recirculation zones that,
by trapping gas and slowly releasing it to the main
flow, would degrade the response. We typically operate
the cell at a pressure of -30 Torr. This pressure,
at which the collisional and the Doppler widths
are approximately equal, is optimum for atmospheric
trace-gas analysis. We maintain the pressure in the
fast-flow cell pumping at a fixed volume flow (10 L. s),
with a sonic inlet nozzle at the end of a 1.73~cm-i.d.
sampling tube. The sonic inlet nozzle is a stainlesssteel
conical tip with an inlet orifice 0.16 cm in
diameter. The nozzle is tapered on the inlet side and
polished throughout the conical expansion to the
sampling tube to ensure smooth flow into the sampling
probe tube. The cell inlet is shaped to direct
the flow past the mirror and into the volume smoothly.
At 30 Torr, the flow is laminar, with a Reynold’s
number of - 400. The mean flow velocity in the
cylindrical optical volume is (10 L/s)/‘((n/4) (8.5
cm)*] = 176 cm/s. The mean residence time is the
length of the optical cell divided by the mean flow
velocity, (55 cm)/(176 cm/s) = 0.31 s. The gas exits
around the periphery of the input mirror and enters
an exhaust plenum, which helps to maintain uniform
circumferential flow around the downstream mirror.
B. Cell Tests
The cell-performance issues include ease of setup to
the desired pattern, optical loss, optical fringe levels,
and flow response. The flow response includes a
comparison between the expected and the actual
flowing-gas replacement times. The cell has been
tested with infrared light derived from a tunable
diode laser operating near 2200 cm-i (4.5 Frn) and
140Ocm-i (7.5 prn).*O The tunable diode laser spectrometer
allows for frequency-stabilized and rapidly
swept laser operation with rapid (up to 30 Hz) data
collection and display. The data acquisition is based
on rapid-sweep integration over the full infrared
transition line shape with synchronous measurement
of the transmitted infrared light intensity. Two
molecules were observed in the lests, nitrous oxide
(N20) and water vapor, as examples of nonsticky and
sticky molecules, respectively. The N20 line monitored
was at 2197.699 cm-’ and had a line strength of
5.4 x lo-i9 cm2 molecules-’ cm-i. The water transition
was at 1329.9 cm-i and had a line strength of
4 x lo-** cm2 molecules-’ cm-’ at 296 K.
3344 APPLIED OPTICS I( Vol. 34. No 18 20 June 1995

C. Optical Setup
A carefully coaligned visible trace laser beam makes
infrared beam alignment relatively simple, and setup
of the desired reentrant pattern can be accomplished
rapidly. We have typically aligned the cell to operate
on the 182-pass pattern but also have tested it at 90,
274. 366, and 454 passes by making small changes in
the mirror separation and twist. The maps of the
pass number as a function of spacing and twist (Fig.
7) make the setup operation much easier. Typically
one will not know the precise spacing of the mirrors,
and small manufacturing errors lead to further uncertainty
as to the initial C$ coordinates. Being able to
move among a set of reentrant solutions adds confidence
that the location in C$ space is known and that
the identificatidn of higher-order patterns is correct.
The final step in the setup is to confirm that the cell
spot pattern corresponds to the calculated patterns.
Generally we achieve a very close match between
observed and calculated patterns. An example is
shown in Fig. 9 for the 182-pass pattern with a 17”
mirror rotation.
The infrared beam in the tunable diode laser
spectrometer is at a high f-number (> 40), with a
focus near the center of the cell. This yields a beam
that is nearly matched to the cell, where.the -recirculating
beam maintains a nearly constant size at the
mirrors. This yields the minimum cell-volume con-
dition. If the focus is made closer to the coupling
hole, the beam spots on the outer edges of the mirrors
become larger. However, changing the input-focusing
condition within reasonable limits does not
strongly affect the transmission through the cell.
The cell transmission as a function of pass number
was measured to check reflection losses. The transmission
at 2200 cm-l for the 182-pass pattern is 20%,
yielding a mirror reflectivity of 99.2%. which is
consistent with specifications for the enhanced silver
coating. We have also tested the fringe levels for the
Cell near 2200 cm-‘. The high-frequency. (0.005-
cm-‘) fringe amplitude on the transmitted beam is
-5 x 10-S p.-p. This detected fringe amplitude is
probably lower than the actual amplitude because
there is a frequency jitter on the laser of - 1 x 10-J
cm-‘, which will wash out the highest-frequency
fringes ( > 10-m source length).
0. Flow-Response Tests
We performed flow-response tests by rapidly injecting
small amounts N20 or H20 into a stream of carrier
gas (dry nitrogen) Rowing through the cell and then
by measuring the spectral contrast of absorption lines
as a function of time. The gas was injected both as
an impulse and as a rectangular wave. The first
experiment measured the time response of the fast-
Row cell to an impulse of N20. supplied by a pulsed
valve (Veeco Model pv-10) with a response time of a
few milliseconds. The output of the valve mixed
rapidly with a carrier gas and entered the flow-cell
sampling orifice in less than 10 ms. The result of the
pulse-response experiment is displayed in Fig. 10.
The rise time of the absorption signal is 40 ms and is
limited by the response time of the data-acquisition
system. This is not surprising because the gas transit
time from the orifice to the optical volume is less
than 5 ms. Hence the total time delay between the
pulsed valve and the optical volume is no more than
15 ms, and the time blurring is presumably even less.
The absorption signal decays with a much longer time
constant (T - 300 ms), which can be fitted to a single
exponential decay over 2 orders of magnitude. The
time constant for this decay is equal to the expected
pump-out time for the 3-L cell with a lo-L/s pumping
speed. As Fig. 10 shows, at least 99% of the gas flow
can be accounted for by single exponential decay.
No long time tailing is apparent, which suggests that
the flow cell is well mixed on a 100-ms time scale.
A second N20 experiment with rectangular-wave
Fig. 9. Comparison of calculated and observed rear-mirror spot patterns for the patkrn (182, 80, 76). The caJcula& pattern IS on the
left and includes 17’ of rotation between the mirror axes. The spot diameter decreases exponentially 10 slmo!ate decreasing brightness,
.A photograph of the observed spot pattern. with a visible trace beam. is shown on the right
20 June 1995 : Voi 34, tic ‘: A-Z=_ ~2 ,Z?TICS 3345

i
“1,~: )
tim~” (second:;
20
Fig. 10. Response of the IOO-m-path-length. 3.2-L fast-flow cell to
an impulse of NsO. With a pumping speed of 5 L/s. the Rushing
time 010.33 s (a 3-s-r rate)
gas injection provides further evidence that the cell
flow is well mixed on short time scales. The result of
this experiment is displayed in Fig. 11. The optical
response signal is roughly a rectangular wave, but
both the rising and the falling edges are rounded.
As is shown in Fig. 11, the rising edge is I well
described by an exponential approach to the peak
signal, and the falling edge is well described by an
exponential decay. The time constants in each case
are approximately equal to the 300-ms pump-out
time for the cell. This is exactly the result expected
when the impulse response function of Fig. 9 is
convolved with a sharp rectangular wave. A careful
look at Fig. 11 reveals a small deviation between the
observed signal and the fit to it in the tail of the signal
decay. It is not clear whether this weak, long time
tail is a property of the cell or an irregularity of the
pulsed valve. Its appearance seemed to be intermittent
and is of little practical consequence.
Our final flow-response test with the larger cell was
with a sticky molecule, water. In principle, the
response of the cell to a sticky molecule such as water
vapor might be much more sluggish than for a
nonsticky molecule (N,O), because of surface adsorp-
Fig. 12. Response of the 100.m-path-length. 3.2-L fast-flow cell to
a rectangular wave of HsO. The recorded stream data are shown
by the dotted curve. and an exponential fit to the fall is shown as a
solid curve. With a pumping speed of 5 L/s, the flushing time is
0.33 s (a 3-s-r rate).
tion-desorption. In this experiment we tested the
step response by rapidly switching between ambient
(humid) air and dry nitrogen. The result, displayed
in Fig. 12, shows very little long time tailing from
surface effects. There may be a trace of a long time
component, but its amplitude is no more than 1% of
the total signal.
A similar set of tests was conducted for our smaller
cell design. This 36-m-path-length cell, with a volume
of 0.27 L, was tested at a pumping speed of 5
L/s. In Fig. 13 we show the result of injecting a
square wave of CH,. The response time of 0.04 s is
consistent with the cell volume and pumping speed.
These response-time tests have shown that the
response of the cell is strongly dominated by a single
exponential component whose characteristic time is
determined by the cell volume divided by the pumping
speed. This suggests that the cell is well mixed
on a fast scale and that further improvements in
response time may be achieved with smaller cells and
bigger pumps. This is an important consideration in
atmospheric-sampling appiications that require subsecond
response times, such as eddy correlation flux
measurements.22 Furthermore, the water responsetime
tests indicate that surface effects for sticky
molecules may be overcome by high-volume laminar
I
. :: : I---\
f L ‘\ \ i yorallme
CL
i?i 0.17 b&o me late =3.5 s"
3
c :
i
= l
L
g 1.
L
0.01
j
I
I
-3.4 s-’
\
,h
$%Yl
: ,L
0 S 10 1S
time (seconds)
Fig. 11. Response of the lOO-m-path-length, 3.2-L fast-flow cell to
a rectangular wave of NsO. The recorded data stream is shown by
the dotted curve. and exponential tits to the rise and fall are shown
as solid curves. With a pumping speed of 5 L/s, the Rushing time
is -0.29 s (a 3.4-3.5-s-t rate).
0 5 10 15 20 2s 30
TIME
(sea.ods)
Fig. 13. Response of the 36-m-path-length, 0.27-L fast-flow cell to
a square wave of CH, With a pumping speed of 5 L/S. the
flushing time is 0 04 s la 25-s-i rate)
3346 APPLIED OPTICS / Vol 34. No. 18 / 20 June 1995

flow. This is important in sampling a variety of
atmospheric species that either interact with surfaces
(e.g., NHS, HN03) or that are reactive (e.g., H02).*0
4. Conclusions
In conclusion, we have explored the application of
astigmatic mirrors to multipass absorption cells for
fast-flow spectroscopic applications. By rotating the
mirrors about their optic axis, we can make up for
finite manufacturing tolerances and can make a
variety of reentrant patterns accessible. Thus the
same set of mirrors can be used to provide a range of
path lengths by making small changes in the mirror
separation and rotation angle. We have developed
techniques for mapping the available patterns and
have shown how the patterns are reached with mirror
separation and rotation.
We have designed and tested absorption cells with
the mirror-rotation concept. The technique IS useful
and easy to apply. The cell designs have base
path lengths of 100 m (in 3.6 L) and 36 m (in 0.27 L),
w:th 182 passes in their nominal reentrant patterns.
Other patterns have been accessible with mirror
rotation and separation changes, which permits path
lengths between 0.5 and 3 times the nominal values.
The cells show flow-response times consistent with
their active-volumes and show low fringe levels.
Appendix A. Structure of “0 Space”
In this appendix we describe in more detail the
properties of the solution space for relation (19),
which gives us the number of passes before escape for
arbitrary 4 coordinates, N(p, +,, +,.). As noted above,
a plot of N(p, +,, $-“) shows overlapping circular zones
(of radius p/N), each supporting a single spot pattern
with a constant pass number corresponding to nearly
reentrant conditions. The radius of the zones can be
understood when the velocity of the last spot is
expressed in terms of the do coordinates. Considering
first +,, we obtain dX,/d$, = N cos N+, = N,
with similar results for 6,. The tolerance in 4 is
then A+ = p/N, where p-is the mirror-hole radius
normalized by the pattern size.
Plots of N(p, b,, by), as shown in Figs. 5 and 6, show
striking patterns of high complexity with structure
on a wide range of scales. N(p, +,, 9) is highly
symmetrical, with all the quadrants of ++,, *&
having the same structure, with the additional symmetry
of reflection about the 45” line within the
quadrants. If one starts at a point N(p, b,, +Y), with
a corresponding (N, K,, k,) spot pattern and moves to
a corresponding point N(p, &,‘, b,‘), the cell spot
patterns are related in specific ways. For coordinate
exchange, (b,‘, 4,‘) = (by, b,), the patterns of spots on
the mirrors are rotated by 90” but are otherwise
identical. For sign changes, (+,‘, $‘) = (-&,, -+,),
the new patterns are (N, (N - M,), (N - MY)), which
are very similar to the initial patterns. If one starts
from a given pattern, one will be able to use mirror
scparatlon and rotation changes to reach the vicinit)
of only one other equivalent pattern. the one with a
sign change in both 4 coordinates.
One way to understand the structure 0fN(p, 6,“, 6,)
is through a consideration of the exact reentrant
solutions to relation (19) which occur at
Nb, = I,n, Iv+, = I,:. (Al I
for some Integers I, and f, BCcauSC thC solutions
are slmultancous. we have
4,./d), = I, 1, (A2 1
Thus, for any point (&,, +,), the number of passes can
be found from Eq. (Al) and from the lowest integer
ratlo formed from the coordmates. provided that the
point IS not overlaid by another zone with lower N
One can construct a map of h’ip. d)., b,,) efficiently b>
organlzlng the reentrant solutions into lines through
the 16,. 6,) plane with slopes of Integer ratios. 11, =
il, I,)b, = ~f:,/k,&. Here we Identify the Integers
I, and I, with k, and k, in Eqs. (81 The set of line
slopes is formed by the set of k,.‘k,, where there are
no common factors (other than 2). For a given range
of id,, 4,) on a line, the values of 6, or by that yield an
even, integral N [from Eqs. (811 yields a set of
first-order solutions on the line. These solutions
produce zones of radius p/N at the (6,. by) coordinates.
All the spot patterns found on a gven k,/k, line fall
on the same reduced Lissajous curve. The number
of slope ratios needed is limited by the requirement
that the lowest-order solutions on the line, within a
given Q range, are less than some maximum N, which
is set by cell loss. Thus one need not consider the
infinite set of rational slopes in filling the (b,, 6,)
plane, but only up to a limiting size of numerator and
denominator.
One complication in this filling method for the
(+,, bYI) plane with nearly reentrant zones is that
there are higher-order patterns on each line of KY/k,.
The higher-order patterns are where k, and k, are
both multiplied by some integer and the resulting N’s
are nearly, but not exactly, multiplied by the same
integer. The higher-order patterns are interleaved
symmetrically between each pair of first-order patterns.
For an interval on-a line between first-order
patterns, N, and N,+i, the interleaved patterns have
N = f,(N) + 2f), w h ere Z[ is the interleave factor and -
f is a rational fraction in the range O-l formed as
f = j/If, with no common factors between integers j
and Ifi
The above procedure can be used to reproduce the
N(p, b,, 4,) zone map analytically, more rapidly and
with higher accuracy than the point-by-point stepping
through 4 space that was used to generate Fig. 5.
It is essentially a systematic way to fill the plane with
zones located at points where the coordinates are in
integer ratios to each other, up to some limiting size
of the integers used. The procedure is amenable to
an ordered construction from high to low denominators
so that zones of lower N generally overlie zones of
higher N. An additional sorting step may be needed
20 June 1995 : Vol 34. No 1.3 A?PLlED OPTICS 3347

to ensure that all the lower N zones overlie the higher
N zones. A synthetic N(p, &, $.) map is shown in
Fig. 6, which corresponds to the same region as Fig. 5.
The construction algorithm also shows the nature of
the solution space, which is controlled by the set of
irreducible rational fractions, the Farey sequence.23.2’
The Farey sequence describes both the set of lines of
solutions in + space and the interleaving of higherorder
solutions on the lines.
We are grateful to the National Science Foundation
for partial support of this work through grant ISI-
890768 1.
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