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*c
Strain tuning a Nonplanar Ring Oscillator by 3.5
GHz: Theory and Experiment
Thomas J. Kane, Member, IEEE, Kenji Numata, and Anthony Yu, Senior Member, IEEE
Abstract—We tuned the oscillating frequency of a nonplanar
ring oscillator by 3.5 GHz by applying strain to the monolithic
resonator using a piezoelectric element. The voltage applied to the
piezoelectric element was 192 volts, corresponding to a tuning
coefficient of 18.2 MHz/volt. Useful bandwidth was limited by a
resonance at 323 kHz and an anti-resonance at 420 kHz.
Performance improvement relative to previous work was achieved
primarily by using a resonator design with a small distance
between the piezoelectric element and the resonant laser beam, but
also by using a piezoelectric material with favorable properties
and reducing the thickness of the resonator. A th eoreti cal an aly sis
of strain tuning supported the design.
Index Terms—Laser, frequency-tuning, piezoelectric
I. INTRODUCTION
he nonplanar ring oscillator (NPRO), invented by Kane
and Byer [1] in 1984, sets the standard for narrow-
linewidth single-frequency lasers. Single-frequency
output at the several-hundred milliwatt to watt level is
commercially available at the two Nd:YAG wavelengths of
1064 nm and 1319 nm. At 1064 nm, amplification can take
power above 100 watts. Lasers based on the NPRO design are
critical elements in laser interferometers used for gravitational
wave detection [2].
The value of the NPRO is much enhanced by its convenient
tunability. Thermal tuning provides a multi-GHz tuning range
and a one-second response time. Strain tuning, accomplished
using a piezoelectric element bonded to one of the non-optical
surfaces of the NPRO, was reported by Kane and Cheng [3] in
1988. The reported tuning coefficient was 1 MHz/volt with
response time roughly one microsecond. This was adequate to
enable robust phase locking of NPROs [3] and hertz-level
locking to a reference interferometer [4].
NASA is designing an NPRO laser for use as the laser
oscillator for the Laser Interferometer Space Antenna (LISA)
project [5, 6]. This project will use three satellites in solar orbit
to detect gravitational waves at low frequencies inaccessible to
ground-based gravitational wave detectors [7]. Precise control
of laser frequency and phase is critical to the success of this
instrument. Only a tiny amount of light is detected due to the
2.5 gigameter separation of the three satellites which comprise
the LISA instrument. Interferometry will depend on the ability
to maintain a precise phase relationship between the detected
light and the output of the laser which produces the return beam.
*Manuscript received # #, 2024; revised # #, 2024; accepted # #, 2024. This work was supported by NASA under contract 80GSFC18C0120 and by the NASA
Physics of the Cosmos Study Office (Corresponding author: Thomas J. Kane)
Thomas J. Kane is unaffiliated (email tom.kane@ieee.org.)
Kenji Numata and Anthony Yu are with NASA Goddard Space Flight Center, Greenbelt MD USA
The NPRO design was selected not only for its low phase noise
but also for its precise and fast frequency control.
For use in space, it is desirable to keep voltages low. One of
the goals of the NASA effort is ±100 MHz of strain tuning
without the use of high-voltage amplifiers. Standard operational
amplifiers typically provide ±12 volts, so the goal was ±100
MHz tuning with ±12 volts, implying a strain tuning coefficient
of 8.33 MHz/volt or larger.
We re po rt str ai n t un in g w it h a t un in g c oe ff ic ie nt of 18. 2
MHz/volt and a 3 dB bandwidth of 214 kHz. Tuning of 3.5 GHz
was demonstrated at a 1 kHz drive frequency. The
improvements over previous designs are based on four changes:
1) We m ad e N PR Os w it h the i nt er na l b ea m ver y ne ar to t he
surface on which the piezoelectric element is bonded.
2) We u se d a piezoelectric material with a large strain tuning
figure-of-merit, which is primarily determined by the
product of the piezoelectric charge constant d31 and the
Yo un g ’s mo d ul u s Y.
3) We m ad e th e NP RO s ma ll. We w il l sh ow th at i f al l
proportions are maintained, then both the piezoelectric
strain tuning coefficient and the bandwidth increase as the
inverse of the dimensional scaling factor.
4) We m ad e th e PZ T el em en t as t hi n a s pr ac ti ca l.
This paper has two sections in addition to the introduction
and conclusion. Section II provides a theoretical analysis of
strain tuning which would apply to any monolithic single-
frequency laser. A simpl e equa tion giving the tunin g coefficie nt
in MHz/volt as a function of material properties and geometry
is compared with the result from finite element analysis. The
calculation of the first mechanical resonance of the monolithic
structure, which is what limits bandwidth, is dealt with in a
similar way. To our kn owl ed ge, t his is t he fi rs t publishe d
theoretical analysis of strain tuning for our geometry.
Section III presents our experimental results, which include
measurements of the tuning coefficient, the tuning range, and
the variation of the tuning coefficient with drive frequency.
II. THEORY
Figure 1 shows the geometry of the tunable laser which will
be analyzed in this section. A piece of piezoelectrical material,
which we will call PZT, is bonded to a piece of solid-state laser
material, which we will call YA G . (PZT stands for lead
T
This article has been accepted for publication in IEEE Journal of Quantum Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JQE.2024.3486164
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
zirconate titanite, the most widely used piezoelectric material,
and YAG stands for yttrium aluminum garnet, the most widely
used solid state laser material, most often doped with
neodymium. This analysis would apply to other materials as
well.) The monolithic resonator consists of a rectilinear YAG
block with length L, width w and thickness tYAG . The thickness
tYAG , which is the smallest of the three dimensions of the
monolithic block, is measured along the y coordinate axis. The
piece of PZT, with length L, width w and thickness tPZT, is
bonded to the resonator. The bonded interface is defined to be
the y=0 plane. The beam path in the monolithic resonator,
which may be either a linear path or a ring path, is defined to be
in the y=tLase plane. The YAG/PZT bond is considered to be
ideal. All other surfaces of the YAG and the PZT are considered
to be free. In reality, the surface at y=tYAG, the bottom surface,
will be bonded to a temperature-controlled pedestal, typically a
thermo-electric cooler. A soft adhesive for that bond will
approximate a free surface.
Fig. 1. The geometry of the strained monolithic laser used for
this analysis. A piezoelectric element (PZT) is bonded to the
monolithic laser resonator (YAG). The resonant laser beam is
in a plane at a distance tLase from the bonded interface. When
the PZT expands in the z dimension, it lengthens the resonant
beam path, and the resonant wavelength increases in the same
proportion.
For a nonplanar ring resonator, the distance tLase is considered
to be the average distance from the bonded interface to the
beam.
The PZT element is poled in the y dimension, perpendicular
to the bonding plane. The two large surfaces of the PZT with
dimensions L x w are metallized so that a voltage difference can
be applied. When voltage is applied, the PZT element attempts
to expand or contract in the x and z dimensions. Forces are
transmitted into the YAG which slightly change the round-trip
path of the resonant laser beam, thus changing the wavelength
and frequency.
There are other geometries whereby a PZT element may
apply strain to a monolithic resonator, for example, with a PZT
element or a screw acting as a piston to press the YAG against
its mount. Owyoung and Esherick [8] demonstrated 76.5 GHz
of strain tuning of a linear single-frequency Nd:YAG laser using
this approach. The advantage of the geometry of Fig. 1 is that
there is no mechanical connection created from the YA G to the
outside world. This eliminates a path for acoustic noise which
might modulate the resonator. Also, the compactness of this
structure allows high bandwidth because components are small
and stiff.
We will approximately calculate the tuning coefficient of the
laser resonator of Fig. 1 in units of megahertz of optical
frequency per volt applied to the PZT element. Equation (18)
will express this result.
When a laser is oscillating on a single mode and it is tuned
by lengthening the resonator by a small amount, the wavelength
λ and optical frequency f change by small amounts Δλ and Δf
according to the equation
∆"
"=∆#$%&'()*$(%+*),-.%+
#$%&'()*$(%+*),-.%+=/01
1. (1)
The fractional change in wavelength is the same as the
fractional change in the optical path length of the resonator.
This must be true since the same number of wavelengths fit into
the resonator before and after the change in resonator length.
For small changes, the fractional change in optical frequency is
simply the negative of the fractional change in wavelength.
The change in optical path length is due to two effects. First
is the simple dimensional change in the resonator, and second,
the change in the index of refraction due to the photoelastic
effect. Initially, we ignore the photoelastic effect. With this
simplification, the fractional change in the optical path length
is the same as the fractional change in dimension. The fractional
change in dimension is by definition the mechanical strain,
given by !. Positive values of strain correspond to lengthening.
The frequency change Δf is thus given by
"# $%&#! (2)
where ! is the strain in the YAG crystal in the direction parallel
to the beam path, averaged over the beam path.
A. Calculation using the Approximations of Euler Bernoulli
Beam Theory
We wi ll m ak e a simplifying assumption which enables a
calculation of the shape of the YAG under the force applied by
the PZT. Our calculation is based on the same assumption used
in Euler-Bernoulli structural beam analysis [9]. The assumption
is that strain is a linear function in y, the dimension in which the
beam is being loaded and thus bent. It is the same linear
function in both the PZT and in the YAG. We w il l further
assume that strain oriented in the two transverse directions x
and z are equal. These assumptions can be expressed as
!!! $%!"" $!'()$ !'*)%'+& #
$%&!"#), (3)
As with any linear equation in one variable, there are two
defining parameters. The two parameters we choose to use are
!'*)- the strain at the YAG/PZT interface, and .- which
specifies the location where strain goes to zero. These are
unknown values which we will solve for.
The strain in the PZT is the sum of two terms. The first is the
strain !'%created by an applied electric field due to the
piezoelectric effect, given by
y
z
PZT
YAG
Resonant laser beam
tYAG
tPZT
tLase
L
y: dimension of poling and electric field in PZT
z: dimension of relevant expansion / contraction of PZT
This article has been accepted for publication in IEEE Journal of Quantum Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JQE.2024.3486164
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
!'$%/()%0 $%/() %'
&$%& (4)
where d31 is a property of the PZT material and 0 is the electric
field inside the PZT, equal to the applied voltage V divided by
the thickness of the PZT tPZT. The sign of V is determined by the
orientation of the PZT. PZT elements have a defined positive
electrode, determined by the direction of poling. Vo l t a g e i s
defined to be positive when it is applied to the positive
electrode.
Since the PZT is bonded to the YA G, i t ca n n o t e x p a n d o r
contract by as large an amount as is expressed by (4). The YAG
will resist deformation by the PZT. There will be balanced
forces both in the YAG and in the PZT. The distribution of these
forces is expressed as a stress field, denoted by σ. For our case,
where !!! $!"", it will be true that 1!! $1"" $%1. When
!!! $!"" the relationship between stress and !*, the strain
associated with stress, is given by
!*$+),-.*
/$*
0 (5)
where Y is the Young’s modulus of the material and ν is
Poisson’s ratio for the material. Equation (5) defines a new
value 2 $ %34 '+& 5) which we will use from here on to keep
later equations from being too cluttered. There are, of course,
two relevant values of K, KYAG and KPZT.
The voltage-indued strain !'%has the same value everywhere
in the PZT and is zero in the YAG. The strain associated with
stress, !*, is a function of y such that together the two types of
strain add to the assumed linear function !'()%of (3). This is
expressed by
!'()$%!'6!*$%!'6*+#.
0 (6)
or, solving for stress, by
1'()$2%'!'()&!'), (7)
We n ow a pp ly t he t wo b as ic p ri nc ipl es o f st at ic beam theory.
First, there must not be any net force applied across any plane
cutting through the stressed structure. If there were, then the
two sides would either compress together or stretch apart in a
direction normal to the cutting plane until all forces through the
plane sum to zero. Second, there must not be any net rotational
or bending moment applied across any plane cutting through
the structure. If there were, then the two halves would bend until
the moments summed to zero. These two facts can be expressed
by the following two integrals. Equation (8) states that the net
force on the PZT, given by the first term, is equal and opposite
to the net force on the YAG, given by the second term.
71'()%/(671'()%/( $*,
&!"#
1
1
,&$%& (8)
Using (7), with !'$* in the YAG, this becomes
72234%'!'()&!')%/(672/56%!'()%/( $ *,
&!"#
1
1
,&$%& (9)
The second integral states that the rotational moments are
equal and opposite.
71'()%(%/(671'()%(%/( $*
&!"#
1
1
,&$%& , (10)
Again, as with (9), σ(y) can be eliminated using (7) so only
strain appears in the equation.
These integrals should also be carried out over the transverse
dimension, x, so that the integrated value is either a force or a
moment. We ha ve a ss um ed s tr ess a nd s tr ai n to b e co ns ta nt i n x,
so this change would only result in multiplication by a constant
which could then be divided out, resulting in the equations as
written.
These definite integrals can be carried out, yielding a pair of
equations which are linear in !'*) and a. These can be solved
for any values of tYAG and tPZT, and in Appendix A we provide
that solution. The result is more intelligible if we make the
further assumption that tPZT<< tYAG. With this assumption the
first term in (10) becomes negligible because y is small for all
values within the integral.
With the first term eliminated, and substituting according to
(3) and (5), and then dividing by KYAG , (10) becomes
*$71'()%( 7#
0!"# $7!'*)8+& #
$%&!"#9(
&!"#
1/( $
&!"#
1
!'*)8#'
8&#(
($%&!"#9:;/56
*$%!'*)8&!"#
'
8&&!"#
'
($ 9 (11)
Since !'*) and tYAG cannot be zero, it must be true that
a=2/3. (12)
Under our assumption of linear strain and a negligibly thick
PZT, the YAG will bend in such a way that the strain in the YAG
is zero 2/3 of its thickness away from the PZT.
With a known, (9) can be readily solved. The first term, the
integral of stress in the PZT, becomes, under the assumption
that tPZT is small,
72234%'!'()&!')%/( $%2234
1
,&)*+ '!'*)&!')%;234 . (13)
The second integral can be carried out according to
72/56%!'()%/( $
&!"#
172/56%!'*)8+& (#
8&!"#9
&!"#
1/( $%
2/56%!'*)8(& (#'
9&!"#9:;/56
*$0!"#%:+1.%&!"#
9 . (14)
Putting the terms of (9) together using the results of (13) and
(14), we see that
2234%'!'*)&!')%;234 6%0!"# %:+1.%&!"#
9$*, (15)
Solving for !'*) we get
!'*)$%!;890$%&&$%&
90$%&&$%&<0!"# &!"#9, (16)
This article has been accepted for publication in IEEE Journal of Quantum Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JQE.2024.3486164
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
Bringing everything together, using (3), (4), (12) and (16), we
get
!'()$%/() '
&$%& 890$%&&$%&
90$%&&$%&<0!"# &!"#98+& (#
8%&!"#9, (17)
The final equation of this subsection expresses the change in
optical frequency of a laser with a resonant beam in the plane
y=tlase when a voltage V is applied to the PZT:
<# $ &%/()%#%=%890$%&
90$%&&$%&<0!"# &!"#98+ & (&,-./
8%&!"#9, (18)
This equation, while approximate, gives clear guidance
toward achieving a large tuning coefficient. The guidance is:
1) The figure of merit for the PZT material is the product
/()2 $%/()%3 '+&5)
4.
2) The beam should be as close as possible to the PZT. The
strain at the PZT is four times the strain at the midplane.
3) The strain goes to zero at ;=$>? $>;/56 ?
4 and changes
sign for greater values of tLase so keep the beam out of that
region.
4) When @%2234;234 A2/56;/56 , as will typically be the
case, the tuning coefficient increases inversely with the
thickness of the YA G, so make the YAG as thin as
practical.
5) The tuning coefficient increases weakly as tPZT is
decreased once @%2234;234 A 2/56;/56.
We used both (18) and Appendix A as well as finite element
analysis (FEA) to calculate the tuning coefficient of a
monolithic resonator with dimensions and material properties
matching the NPRO being used by NASA. For the FEA the
modeled shape remained a rectilinear block; the angled facets
of the NPRO were not modeled. Figure 2 shows the tuning
coefficient in units of MHz/V as a function of tlase calculated
using these three approaches. Table I includes all the relevant
values along with the resulting calculated frequency tuning
coefficient. Note that all equations presented assume meters as
the unit of distance, while we use millimeters in our
descriptions.
TABLE I.
PA RA ME TE R S US E D A N D RE SULT S O BTA IN E D FO R
CALCULATION PRESENTED BY FIG. 2
!"#$%&"'()%*+$%,$-(
.&/$0-&*0-(
)%*+$%#1(
)23(
456 (
/&''&/$#$%-(
d317(+/89(
:;<=(
>(
tYAG
?@<A(
Y7(6)"
(
BC
(
;?=
(
tPZT
=@DA
(
ν(
=@;D(
=@;=(
tLASE
=@E;(
K7(6)"(
?=?(
EE;(
L x w
A@<(F(E@E(
G"H$'$0I#J(λ(
?=BE(0/(
K%$LM$0N1(f(
DOD(3PQ(
RLM",*0(S?OT(%$-M'#(
;=@A(!PQ89(
5++$0U&F(5(%$-M'#(
;D@;(!PQ89(
V$-M'#(W%*/(X0&#$($'$/$0#("0"'1-&-(
DD@D(!PQ89(
As will be shown in the experimental section, the measured
PZT response coefficient of our laser was 18.2 MHz/volt. The
result of (18) would be in much better agreement with FEA if
the long dimension of the resonator, L, were greater by an order
of magnitude or more than the bending dimension, tYAG . For our
design the ratio L / tYAG is 3.3. Equation (18) gives useful design
guidance and accuracy well within a factor of two, but for
accurate modeling, FEA is required.
Fig. 2. Calculated values of strain tuning coefficient of the PZT-
tuned monolithic oscillator specified by Table I.
Fig. 3 shows the distorted shape of the YAG as calculated
using FEA. The structure bends, which is why there is a region
in the YAG where strain has a sign opposite to that of the
voltage-induced strain !'. This bending is exactly what would
be experienced by a bimetallic strip made of metals with
differing thermal expansion when heated to a temperature away
from the temperature at which they were bonded. Our analysis
above closely follows the classical analysis of a bimetallic strip.
Fig. 3. Distorted shape of the YAG due to an expanding PZT
bonded to the top surface. The PZT is not shown. What is shown
is the YAG at the zy midplane, as calculated by FEA. The
vertical lines show the original positions of the end faces.
B. Correction due to photoelastic effect
The index of refraction changes with strain and that also leads
to tuning. This effect may increase or decrease the tuning
coefficient relative to the results above, depending on both the
polarization of the light and the orientation of the crystal. A
-40
-20
0
20
40
60
80
00.25 0.5 0.75 11.25 1.5 1.75
Tuning Coefficient (MHz / V)
Distance from PZT / YAG interface to beam , tLase (mm)
Equation (1 8)
Appendix A
Finite Element Analysis
YAG
PZT Attachment
Surface
Optical surface
at mid-plane of part,
strained, FEA
calculation
Optical Surface
plane, unstrained
y
z
This article has been accepted for publication in IEEE Journal of Quantum Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JQE.2024.3486164
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
complete calculation would be quite complex, because in YAG
the strength of the photoelastic effect varies a great deal as
crystal orientation changes relative to both the orientation of the
strain and the polarization of the light.
The photoelastic effect is a linear effect whereby strain
changes index of refraction. For a given orientation, the change
in index due to the photoelastic effect is exactly proportional to
the strain. The tuning coefficient calculated in the previous
section can be simply multiplied by a factor.
Appendix B calculates the effect for two relatively simple
cases that give an indication of both the magnitude and the
variability of the effect. The result is summarized in Table II.
YA G i s a c ub i c c r y s t al , a nd t he r e f o re h as t h re e eq u i v a le n t and
mutually perpendicular crystalline axes each of which could be
designated as <100>. We pre se nt two ca lc ul ati on s. For both,
we assume that one of the <100> axes is parallel to z, the long
axis of the monolithic block as defined by Fig. 1. We assume
beam propagation along z, which would imply a linear
resonator. The first orientation for which we did a calculation is
where the three <100> axes correspond to the three block axes,
xyz. We c al cu la te d th e ch an ge i n in de x fo r l ig ht p ol ar iz ed
“vertically,” meaning along y, and “horizontally,” meaning
along x. The result is shown in the first row of Table II as a
percentage change from the tuning coefficient calculated
considering only dimensional change. For one polarization the
tuning is reduced, for the other, increased, in each case by a
small amount.
For the second case we assumed that the YAG crystal was
rotated around the z, or beam propagation, axis. A 90° rota tion
would make no difference because the three equivalent axes are
separated by 90°. A rota tion o f 45° makes a big differ ence, as is
reported in the second row of Table II. The effect is larger,
approximately 20%. We numerically calculated several other
cases and found none that exceeded the magnitude of the 45°
case of Table II.
TABL E II.
PHOTOELASTIC CONTRIBUTION TO PZT RESPONSE
!"# $%&'()*+$
,&-./)*01/$
21+*&-3*01/$
4.&)5$ 6'7$
81&-35$697$
:;<<=$*9-($/1&>*+$
)1$2?@A!"#$-/).&B*C.$
DE5;FG$
HI5JKG$
:;<<=$*9-($IJL$)1$
2?@A!"#$-/).&B*C.$
DM<5;KG$
H;E5NIG$
It is clear that the YAG photoelastic effect is highly
anisotropic. It is also clear that the tuning due to the photoelastic
effect, while significant, is much smaller than the effect of
dimensional change. Unless there is a need for extreme unit-to-
unit consistency, it is not worth the trouble to orient YAG for
NPRO manufacturing. Variation from unit to unit in PZT
response will be dominated by the challenge of controlling the
distance tLase between the resonant laser beam and the
PZT/YAG interface.
C. Calculation of resonance limiting flat spectral response
Along with a high tuning coefficient, a large bandwidth is
useful. The bandwidth over which the PZT response is “flat” is
limited by the mechanical resonances of the PZT/YAG
structure.
We a gai n go to the theory of mechanical beams to obtain an
approximate but useful estimate of the lowest resonant
frequency of the monolithic resonator. As before, this provides
intuitive guidance which can be confirmed via finite element
analysis.
The monolithic resonator is modeled as a rectilinear block.
The PZT is ignored; only the YAG is considered. When the
longest dimension, L, is significantly greater than the shortest
dimension, tYAG , then the theory of mechanical beams provides
an equation useful for estimating the lowest order flexural
resonance. This equation, which yields a result in units of hertz,
is [10]
BCDE;%BFGHID.F%JGEKL.LMG%BDGN,$%)@18%&!"#
='O/!"#
A!"# (19)
where ρYAG, the density of the YAG, is 4560 kg/m3 and the other
parameters are provided in Table I. Equation 19 gives a value
of 453 kHz. It is seen that if the dimensional values L and tYA G
are scaled down together, then the first resonance will rise
inversely to the scaling factor.
Finite element analysis of the rectilinear block shows that the
resonance with the lowest frequency is a torsional resonance,
not flexural, and it gives a resonance frequency of 278 kHz. The
experimentally observed resonance was at 323 kHz.
D. Scaling argument
Equation (18) shows that, within the assumptions of that
model, the strain tuning response in MHz/volt increases
inversely with scale if all three dimensional values tYAG , tPZT and
tlase are scaled together. Equation (19) shows that the first
resonance frequency scales similarly with dimensions L and
tYAG .
By argument, it is possible to show that these two scaling
relationships are completely general, and will continue to apply
in the most sophisticated and complete model of an isolated
NPRO/PZT structure.
Imagine an NPRO/PZT structure with a given strain
distribution. Imagine it scaled up or down, but keeping the
same strain distribution, and thus the same amount of frequency
tuning, per (2.) Since strain is unchanged, the electric field in
the PZT must be unchanged, per (4.) But with PZT being scaled
in thickness, voltage is changed by the scaling factor. Thus,
while tuning per unit field is unchanged, tuning per volt scales
inversely with dimension.
Mechanical resonances, like musical instruments, resonate at
frequencies determined by the wavelengths that can find a
resonant mode. If a structure is scaled with fixed material
properties, the resonant wavelengths will scale with the
structure, and the resonant frequencies will scale inversely.
E. Thermal tuning
It is useful to compare the strain tuning to the thermal tuning
of a monolithic resonator. The rate of frequency change with
temperature is
7B
74 $P 6)
C%7C
749 (20)
This article has been accepted for publication in IEEE Journal of Quantum Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JQE.2024.3486164
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
where f is the optical frequency (282 THz for a 1064 nm laser),
α is the thermal expansion of the material (7.5x10-6 / K for
YA G ), n is the index of refraction (1.82 for YAG), and dn/dT is
the change in index with temperature (7.3x10-6 / K for YAG).
The resultant value is -3.2 GHz/K. It is interesting to note that
the change in index is responsible for 35% of the thermal
tuning, more than we calculate to be the case for strain tuning.
III. EXPERIMENT
A. The µNPRO monolithic resonator design
The NPRO designed for NASA, and used in this work, is
designated as a µNPRO [5]. Fig. 4 shows a CAD model of the
µNPRO, first as a transparent object and then with the PZT and
the pedestal upon which it is mounted. The pedestal is
controlled in temperature by a thermo-electric cooler. The
dimension of the PZT fits closely to the surface of the µNPRO
to which is bonded. A wrap-around electrode allows electrical
connection to the bonded surface.
Fig. 4. A CAD model of the µNPRO, as the NPRO designed for
NASA is designated. On the left is the µNPRO shown as a
transparent object. On the right, the PZT is attached to its top
surface, and it sits on a temperature-controlled pedestal. The
wrap-around electrode of the PZT is on its upper left corner.
The output power required for the NASA application is 200
mW. The optical pump power at 806 nm required to reach this
output is typically near 350 mW.
Tunability is critical to the NASA application, and both
thermal and strain tuning are used. Figure 5 shows a typical plot
of wavelength and optical frequency vs. temperature. The range
of continuous tuning is 20 GHz. The continuous tuning is
greater than one free spectral range, which for the µNPRO is 12
GHz. This is due to the fact that the Nd:YAG gain maximum
tunes also, and with the same sign as the thermal tuning.
Fig 5. Tuning the µNPRO wavelength with temperature.
Wave le ng th in cr ea ses wi th te mp er at ure in th e co nt in uo us ,
single-mode segments since the Nd:YAG monolithic resonator
is expanding.
Figure 6 is a mechanical drawing of the µNPRO. An inset
photograph shows a view through the front facet of the µNPRO
with the pump beam made visible by the fluorescence of the
pumped Nd ions. The µNPRO is designed so that the two
longest segments of the beam path are 0.25 mm from the
PZT/YAG interface. Ave ra ged o ver the entire 13.5-millimeter
round-trip beam path the distance to the PZT/YAG interface is
0.43 mm. There is an R=150 mm convex curvature on the front
facet. The e-2 radius of the TEM00 beam is calculated to be 77
µm at the convex facet and slightly smaller elsewhere. This is
small enough that there is effectively no clipping loss due to the
beam being too close to the edge of the part if the part is
manufactured per design.
The resonant beam path of an NPRO is determined by
geometry and cannot be altered once the NPRO is fabricated.
As can be seen by looking at Fig 6, three of the four reflection
points in the µNPRO are quite close to the edge of the part.
While essential to increasing PZT response, this creates the risk
that the geometrically defined beam path intersects the plane of
a facet beyond the edge of the facet, in which case the part is
useless. This concern is what causes most NPRO designs to
place the beam at the center of the crystal, near tYAG/2. That
choice results in relatively small strain tuning with high unit-to-
unit variability.
We use fabrication tooling which places the beam reliably
where we want it. Our most recent batch of 12 µNPROs,
designed for an 0.25-mm distance between the beam and the
surface where the PZT is attached, had an average value and a
standard deviation for this distance of 0.20 ±0.04 mm. Only one
of the batch of 12 was rejected because the beam came so close
to the edge that power was reduced.
Fig 6. Mechanical drawing of the NASA µNPRO monolithic
resonator. The dashed lines show the beam path. The inset photo
is a negative image looking into the front facet oriented
identically to the drawing view below it. The beam is visible
because the pumped ions are fluorescing.
This article has been accepted for publication in IEEE Journal of Quantum Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JQE.2024.3486164
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
B. Choice of PZT material
We used the PZT material that had the highest figure of merit
d31 YPZT / (1-ν) that we could find. The PZT material chosen was
3265HD available from CTS Corporation [11]. The thickness
tPZT which we used was 0.25 mm. For an 0.25-mm part the
maximum recommended voltage is 125 volts. The shape of the
PZT closely matches the top surface of the µNPRO, with no
overhang, as can be seen in Fig. 4. A wrap-around electrode is
used. It is important that the PZT element have no overhang.
Overhang will create undesirable low-frequency resonances in
the response. Table III summarizes the properties of this PZT
material and the specific part we used, along with
corresponding values for the widely used material PZT 5H.
TABL E III
PROPERTIES OF THE PZT MATERIAL AND PART
)%*+$%#1(
;DBAP.(
)23(AP(
4*M 0 I Y- (! *U M 'M - (YPZT *%(YE11(
BC(6)"(
BD(6)"(
)*&--*0Y-(%",*7(νPZT (
=@;D(
=@;?(
ZJ"%I$(N*0-#"0#7(d31(
:;<=(+/89(
:;D=+/89(
K&IM%$(*W(!$%&# d31 YPZT / (1-ν)(
;<AEE(
DO<AE(
3J&N[0$--(*W()237(tPZT(
=@DA(//(
5%$"(*W()237(A(
D?(//2(
.&$'$N#%&N(N*0-#"0#7(KT3(
BA==(
;O==(
Z"+"N&#"0N$(*W()237(
C = KT3 ε0 A / tPZT(
E@O(0K(
D@O(0K(
The Young’s Modulus of a PZT depends on both its
orientation and on whether it is measured “open circuit” (charge
held constant) or “short circuit” (voltage held constant.) The
relevant value of YPZT for the approach to strain tuning
described in this paper is designated by PZT manufacturers as
YE11, the short-circuit elastic constant perpendicular to the
poling direction.
The PZT is bonded to the YAG by the epoxy 353ND. This
epoxy has low viscosity right before it cures, so that a very thin,
stiff bond can be created. The µNPRO was mounted to its
temperature-controlled support using a compliant adhesive, so
that the YAG is free to deform, limited only by its own stiffness.
Electrically, the PZT is a capacitor, at least at frequencies
below the mechanical resonances. The last three rows in Table
III provide the equation and the parameters needed to calculate
its capacitance. If smaller values of capacitance are needed
either the PZT material can be changed to one with a smaller
dielectric constant, or more simply, the thickness of the PZT can
be increased, which will cause capacitance to decrease as the
inverse of tPZT. As shown in (18), an increase in tPZT will only
slightly reduce the tuning coefficient. An increase in tPZT will
also enable a higher voltage.
C. Measurement of strain tuning coefficient
We me as ur ed the st ra in -tuned frequency change of our
µNPRO by generating a beat note with another NPRO and then
digitizing the resultant beat signal on a fast oscilloscope. Figure
7 shows the experimental setup. The frequency as a function of
time was extracted from the digital record with a frequency
estimate being made every 0.4 µsec. The only adjustment of the
data was changing the sign of the frequency when the beat note
passed through zero frequency and offsetting the frequency so
that the minimum frequency is zero.
Fig. 7. Experimental setup for measurement of strain tuning
response by analysis of the beat frequency of the output of a
strain-tuned µNPRO with the output of a commercial NPRO
laser.
Fig. 8. Beat frequency of a µNPRO as a function of time with a
1 kHz triangle wave of amplitude 192 volts peak-peak applied
to the PZT. Time resolution is 0.4 µsec, so 7000 points are
displayed. No smoothing was applied to the data.
Fig. 8 shows frequency as a function of time for a PZT-tuned
µNPRO driven by a 1-kHz triangle wave with peak-to-peak
drive of 192 volts, symmetric around 0 volts. The peak-to-peak
frequency excursion is 3.5 GHz. The tuning coefficient is 3.5
GHz ÷ 192 V = 18.2 MHz/V. This is a reasonable match to the
FEA-based estimate of 22.2 MHz/V from Table I . The 20%
error is due to some combination of the following factors:
1. The FEA-based model contains simplifications. The
model assumes a laser in a single plane. The modeled shape
is a right-angled block, not the actual multi-faceted NPRO
shape.
2. This critical distance tlase is not known with accuracy.
3. The photoelastic contribution is assumed to be zero.
4. All material properties have a degree of uncertainty.
D. Effect of modulation frequency
We m ea su re d th e r es po ns e of t he P ZT a s a fun ct io n of
modulation frequency by driving the PZT with white noise,
creating the frequency vs. time record, and then calculating its
spectral density. This resulted in Fig. 9, which shows the tuning
coefficient from dc to 1 MHz. The response deviates by 3 dB
from the dc value at 214 kHz. The first resonance is at 323 kHz.
This is a reasonable match to the FEA estimate of 278 kHz. Ver y
Reference NPRO
Laser
(Lumentum 126)
NPRO Laser
under test
(µNPRO)
PZT Input
Fiber 3dB
coupler
Photodiode
Digitizing
Oscilloscope
Post-
processing,
(Matlab)
X
Signal Generator
(triangle wave or
white noise)
0
0.5
1
1.5
2
2.5
3
3.5
-0.5 0.0 0.5 1.0 1.5 2.0 2. 5
Optical Frequency change from
minimum (GHz)
Time (msec)
This article has been accepted for publication in IEEE Journal of Quantum Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JQE.2024.3486164
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
high response of 250 MHz/volt exists at the 908 kHz resonance.
There may be applications where this strong, fixed-frequency
modulation capability proves to be useful. There is an anti-
resonance near 420 kHz where virtually no modulation is
possible.
Fig. 9. Strain tuning response as a function of frequency. The
first resonance at 323 kHz limits the “flat response” region. At
214 kHz the response is 3 dB above the low-frequency
response.
IV. CONCLUSION
Commercial NPRO lasers utilize strain tuning, implemented
by applying a voltage to a PZT element bonded to a non-optical
facet of the monolithic resonator. A typical specification of the
strain tuning coefficient is near 1 MHz/volt. We have
demonstrated a strain tuned NPRO with a coefficient of 18.2
MHz/volt. This enables 100 MHz of tuning at low voltage, or
multi-GHz tuning within the allowable voltage range of the
PZT. The advantage of having a large tuning coefficient benefits
spaceborne applications. Large frequency tuning is now
possible without the use of a high voltage amplifier, thus
simplifying the laser drive electronics.
Our theoretical analysis of strain tuning shows the importance
of placing the resonant beam path close to the PZT element, and
of keeping the beam out of the 1/2 of the crystal farthest from
the PZT. In this region the strain goes to zero and then reverses
sign. Historical designs, seeking to keep the beam centered,
result in lasers with with small and variable tuning coefficients.
Our analysis provides a figure of merit to be used for choosing
PZT material. Finally, we have shown that scaling the
monolithic resonator down increases the PZT response as the
inverse of the scaling factor. The first mechanical resonance of
the monolithic resonator puts an upper limit on the useful
modulation frequency, and it scales upward with decreasing
dimensions in the same way. The design we demonstrated had
an ac response within 3 dB of the dc response up to 214 kHz.
The increased dynamic range and bandwidth of strain tuning
that we have demonstrated adds to the utility of the NPRO laser
design.
APPENDIX A
In the body of the paper, we presented equations for the strain within the PZT/YAG structure using the Euler-Bernoulli assumption
that strain is a linear function in one dimension. We then further assumed that tPZT << tYAG and provided a useful solution. A solution
that makes no assumptions about the relative values of tPZT and tYAG is presented here. These two equations would replace (12) and
(16.)
!'*)$% :0%0$%&%&$%&%+0$%&1&$%&1
(<%(0!"#%&$%&%&!"#
'%<%90!"#1&!"#
(.
0$%&
'%&$%&
2%<%90$%&10!"#%&$%&
(%&!"#%<%D0$%&%0!"# %&$%&
'%&!"#
'%<%9$%&%0!"#%&$%&%&!"#
(%<%0!"#
'%&!"#
2 (A1)
. $ 0$%&%&$%&
(%<%(0!"#%&)*+1&!"#
'%<%90!"#%&!"#
(
D%0!"#%&!"#
'%+&$%&<&!"#. (A2)
APPENDIX B
The orientation for which the calculation of the photoelastic
effect is easiest is where the <100> crystalline axis and the
equivalent <010> and <001> axes are aligned with the
propagation axis and the two polarization axes of the light, and
where those three axes also correspond to the natural axes of
the monolithic block. For this case a relatively simple equation
gives the amount of frequency tuning due to index change
relative to the amount that is due to dimensional change alone,
that is to say, relative to the value calculated using (18). We w il l
call that ratio R. The equation for light polarized along the y axis
(perpendicular to the plane of the PZT/YAG interface, or
“vertical”) is
J $%,C'+,%-%E33<+),%-.%E3'.
),%- , (A3)
where n is the index of refraction of YAG, ν is the Poisson’s
ratio of YAG, and p11 and p12 are elements of the elasto-optic
tensor [12]. Table A-I has the relevant values. For the values of
Tab le A-I, (A3) gives a value of R = +7.13%. The tuning
coefficient is increased due to the photoelastic effect.
0
50
100
150
200
250
300
0200 400 600 800 1000
PZT Response coefficient (MHz / V)
Drive frequency (kHz)
This article has been accepted for publication in IEEE Journal of Quantum Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JQE.2024.3486164
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
TABL E A -I:
PROPERTIES OF YAG USED TO CALCULATE THE
PHOTOELASTIC CONTRIBUTION TO STRAIN TUNING
\0U$F(*W(%$W%"N,*0(*W(4567(n
(
?@OD
(
)*&--*0Y-(%",*(*W(4567(ν
YAG(
=@;=
(
456 ($ '" -# *:*+,N(N*$]N&$0#(p
11(
:=@=DC=
(
456 ($ '" -# *:*+,N(N*$]N&$0#(p
12(
=@==C?
(
456 ($ '" -# *:*+,N(N*$]N&$0#(p
44(
:=@=B?A
(
For the other polarization, in the plane of the PZT-YAG
interface or “horizontal,” the corresponding equation is
J $%,C'F+),%-.%E33<+),(%-.%E3'G
8%+),%-., (A4)
For the values of Table 1, (A4) gives value of -4.59%. The
response is decreased.
If the crystalline axes of the YAG are re-oriented such that the
axis of propagation remains along one of the <100>
equivalents, but the other two <100> equivalents are at 45°
relative to the plane of the PZT/YAG interface, then the
deviation from the “dimensional effect only” case is
significantly larger. The equations corresponding to (A3) and
(A4), for this “45° rotated” case, are given below.
The equation for light polarized along the y axis
(perpendicular to plane of PZT-YA G i n te r f a c e, vertical) is
J $%45'6734(89)331:17(4;1891)3'1417':'891)22<
21734189 (A5)
Equation (A5) gives a value of +20.19%. For the other
polarization, in the plane of the PZT-YA G i nt e r f a c e, th e
corresponding equation is
J $%45'6734(89)331:17(4;1891)3':17':'891)22<
21734189 (A6)
Equation (A6) gives a value of -17.64%. Tab le II in the body
of the paper summarizes these four cases. Other cases have been
calculated, and fall within the range defined by these two.
ACKNOWLEDGMENT
Thanks to Jim Morehead for help with the elasto-optic tensor
calculations and to Dave Demmer and Will Drobnick of AVO
Photonics for valuable discussions on practical issues.
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[11] 3265HD is a form of lead zirconate titanate ceramic that was available
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Thomas J. Kane (Member, IEEE) was
born in Pleasanton, California in 1955. He
graduated from the University of
California at Davis in 1978 with a degree
in physics. In 1986 he completed his Ph.D.
in electrical engineering at Stanford
University, working under the supervision
of Prof. Robert L. Byer.
He was a founder of the pioneering diode pumped laser
company Lightwave Electronics. He spent 20 years working as
a laser engineer for Lightwave Electronics, and after its
purchase, for its buyer JDSU, now Lumentum. Dr. Kane is
currently an unaffiliated laser scientist and optical engineer.
Kenji Numata received his Ph.D. degree
in physics in 2003 from the University of
Tok yo , J apan. H e has been w or kin g for
NASA Goddard Space Flight Center since
then. He is playing key engineering roles in
precision space laser systems, including the
LISA mission and gas sensing lidars. His
research interests include precision laser
technology, laser metrology, control systems, solid-state and
fiber lasers, remote sensing, nonlinear optics, and thermal
noise.
Anthony Yu (Senior Member, IEEE) is a
laser scientist and branch technologist in
the Lasers and Electro-Optics Branch at
NASA Goddard Space Flight Center. He
has over 30 years of experience in space-
based laser development for Earth and
planetary science applications. He earned a
B.S. in physics from the University of
Central Florida in 1982 and an M.S. and Ph.D. in physics from
the Georgia Institute of Technology in 1984 and 1988. He is a
senior member of Optica.
This article has been accepted for publication in IEEE Journal of Quantum Electronics. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/JQE.2024.3486164
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/