Off-surface optic axis birefringent filters for smooth
tuning of broadband lasers
UMIT DEMIRBAS
1,2 *
1
Laser Technology Laboratory, Department of Electrical and Electronics Engineering, Antalya Bilim University, 07190 Antalya, Turkey
2
Center for Free-Electron Laser Science, The Hamburg Center of Ultrafast Imaging, Deutsches Elektronen Synchroton, Hamburg, Germany
*Corresponding author: umit79@alum.mit.edu
Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX
Transition metal-doped gain media such as Ce:LiCAF, Ti:Sapphire, Cr:ZnSe and Fe:ZnSe possesses wide gain
bandwidths that could provide ultra broadly tunable laser output with the usage of adequate intracavity tuning
elements. Birefringent filters (BRFs) are a low-cost and easy to use solution for tuning. However, for ultrabroad
gain media, regular on-surface optic axis BRFs could not provide smooth tuning of laser wavelength in the whole
emission range. Basically, regular BRFs could not accommodate a large enough free spectral range with acceptable
modulation depth variation while tuning due to their slow tuning rates. Motivated by this, in this study we have
numerically investigated the effect of optic axis orientation on filter parameters for magnesium fluoride
birefringent tuning plates. We have shown that, a magnesium fluoride BRF with an optic axis diving by 30 into the
plate could provide smooth tuning of ultra-broad laser gain media. A similar analysis has shown that for
broadband tuning applications the optimum optic axis diving angle lies around 25 for crystal quartz BRFs. The
proposed filters have the potential to be useful in tuning of broadband lasers in continuous-wave, long-pulsed and
femtosecond operation regimes. © 2017 Optical Society of America
OCIS codes: 230.7408: Wavelength filtering devices; 140.3600: Lasers, tunable; 140.3580: Lasers, solid-state.
http://dx.doi.org/10.1364/AO.99.099999
1. INTRODUCTION
Birefringent tuning plates (birefringent filters: BRFs) are routinely
employed in laser resonators for tuning of the output wavelength
and/or for narrowing down the laser emission spectrum [1-3]. Since
they can be inserted at Brewster's angle inside the laser cavity, they do
not require anti-reflective coatings, and their passive losses are very
low. This reduces their cost, enables broadband operation, and
increases their damage threshold.
Important parameters to consider while optimizing the usage of
birefringent tuning plates are free-spectral range (FSR), transmission
passband bandwidth (FWHM, full-width at half-maximum),
modulation-depth and tuning rate[4]. For optimization of filter
parameters for a specific application, designers first usually adjust the
thickness of the plate. Moreover, when one plate is not sufficient to
reach the desired performance pairs of birefringent plates with
different thicknesses could also be used.
Another less known knob that can be used to optimize the
properties of birefringent filters is the angle between the optic axis and
plates surface normal [1, 5-8]. In standard usage, the plates optic axis
lies on the surface of the plate for convenience in filter fabrication (can
be named as on-surface optic axis birefringent filters or as regular
birefringent filters). On the other hand, as several earlier studies with
quartz birefringent filters has already shown, this particular case is not
really optimum for many applications. Compared to regular
birefringent filters, BRFs with optic axis pointing out of its surface
provides a much broader set of filter parameters (named as diving
optic axis birefringent filters or off-surface optic axis birefringent filters
in the literature) [1, 3, 6, 9]. In particular, wisely designed off-surface
optic axis birefringent plates could: (i) generate larger FSR values, (ii)
provide a smoother variation of modulation depth as the wavelength is
tuned, (iii) enable faster tuning rates, and (iv) posses a larger range of
usable filter rotation angles. For example, in the case of wavelength
swept lasers , off-surface optic axis BRFs could provide faster tuning
rates. Moreover, for some other applications, such as tuning of
ultrabroad lasers (focus of this paper), it is not even possible to achieve
the desired tuning properties with regular BRFs, and off-surface BRFs
are the only solution (e.g., later see Fig. 11).
Unfortunately, the advantages of off-surface optic axis BRFs are not
well known by our community, limiting researchers ability to optimize
filters properties for their application. On top of this, as also pointed out
by Kobtsev et al. [1], some of the earlier papers on off-surface optic axis
birefringent filter design contain some errors, and optimal
performance of these elements has not been fully described in the
literature (in some cases not even correctly understood). Lastly, these
earlier work mostly focused on optimization of birefringent filter
properties for dye lasers which are relatively narrowband and to our
knowledge there is limited study on optimization of BRF properties for
today's broadly tunable lasers such as Ce:LiCAF [10, 11], Ti:Sapphire
[12], Cr:ZnSe [13, 14], and Fe:ZnSe [15].
Motivated by this, in this study we present detailed numerical
design considerations for off-surface optic axis magnesium fluoride
birefringent tuning plates that are suitable for ultra broad tuning. As
the birefringent material we have chosen to look at magnesium
fluoride since it posses a much broader transparency range than the
usually employed crystal quartz enabling its usage for the newly
emerged near to mid-infrared laser materials (transparency range for
quartz: 250 nm to 2500 nm, transparency range for magnesium
fluoride: 110 nm to 7500 nm) [16]. In our analysis, we have looked at
the effect of optic axis orientation and filter thickness on filter
parameters such as free-spectral range, modulation depth, tuning-rate,
and walk-off angle. We have shown that, for tuning of ultra-broad
lasers, magnesium fluoride birefringent filters with an optic axis
making 302 with the plates surface normal provides a good solution.
The optimum filter thickness value for this purpose has found to be
0.85 mm for Ce:LiCAF, 2.2 mm for Ti:Sapphire, 3.2 mm for
Cr:Forsterite, 6 mm for Cr:ZnSe, and 11.5 mm for Fe:ZnSe. We have
also looked at crystal quartz, and we have shown that quartz BRFs
with an optic axis making 252 with the plates surface normal could
also provide smooth and ultra-broad tuning performance.
The paper is organized as follows. Section 2 summarizes the theory
behind usage of intracavity birefringent filters as tuning elements in
standing wave optical cavities. Section 3 presents detailed simulation
results on the effect of optic axis orientation on filter performance.
Section 4 describes thickness optimization process for a few selected
laser gain media. Finally, Section 5 presents a summary of key findings
and gives a generalized description for filter parameter optimization
for wavelength tuning of other broadband laser gain media.
2. THEORY
Fig. 1 (a) shows a typical standing-wave laser cavity that includes a
magnesium fluoride birefringent filter (BRF) and a broadband laser
gain medium. Both the gain medium and the birefringent plate are
inserted at Brewster's angle; since, otherwise they will both require
broadband antireflective coatings, which increases complexity and
cost. Tuning of the laser wavelength is facilitated simply by rotation of
the birefringent plate about an axis normal to the surface (corresponds
to changing , in Fig. 1 (b)). The surfaces of the laser crystal and the
birefringent plate will create Fresnel reflection losses for the
transverse electric (TE/s) polarized part of the beam. Fig. 1 (b) shows a
detailed view of the BRF, where t is the thickness of the plate, i is the
incidence angle, is the internal Brewster’s angle (around 36.1 in
magnesium fluoride at 2.5 m),
s
is the direction of beam propagation
in the plate, is the rotation angle of the plate, is the angle between
the crystals axis and the beam propagation direction and is the angle
between the optic axis and the surface normal (when = 90, the optic
axis lies on the surface of the plate, which is the typical BRF).
To calculate the transmission characteristic of the laser system
[shown in Fig. 1 (a)] one needs to find the polarization eigenmodes
and eigenvalues of the overall Jones matrix of the cavity. This problem
has been studies in detail earlier and we will just provide a review of
the results here for the sake of completeness [2, 4-6, 17-19]. We know
form a physical point of view that, the polarization state should not
change after a round-trip over the laser cavity. This involves solution of
the following polarization eigenmode equation [5]:
EEM
cavity
, (1)
where
E
is the electric field vector (polarization eigenvector):
TETETMTM
eEeEE
ˆˆ
(2)
E
TE
and E
TM
are the transverse-electric (TE/s) and transverse-magnetic
(TM/p) polarized electric field vector components, and is the
polarization eigenvalue, which usually is a complex number. The
eigenvalues (
1
and
2
) of the 2x2 round-trip Jones matrix for the laser
cavity gives amplitude transmission coefficients for each polarization
modes of the cavity, and
2
21
),(
Max
gives the power transmission
for the relevant cavity polarization mode with the lowest loss [5].
(a)
Gain medium
Birefringent
filter (BRF)
(b)
c
t
Surface normal
s
Incident ray
E
TM
E
TE
i
E
”
TM
E
”
TE
Fig. 1. (a) Standing-wave laser resonator containing a gain medium
and a birefringent plate inserted at Brewster's angle. (b) Light beam
incident on the birefringent plate at Brewster’s angle [2]. : angle
between the optic axis and the surface normal,
c
: optic axis,
s
:
direction of beam propagation, i: incidence angle, : internal Brewster’s
angle, : rotation angle of the plate, t: thickness of the plate.
The overall Jones matrix for a round-trip in our standing wave cavity
can be written as:
BRFgainBRFgaincavity
MMMMM
, (3)
where M
gain
and M
BRF
are the Jones matrixes for the gain medium and
the birefringent plate, respectively. Note that the matrix above is for a
standing wave cavity, where we pass through each element twice. For
a ring cavity, the overall Jones matrix for the cavity will involve only
one pass through each element (M
ring cavity
= M
gain
.M
brf
). Also, when more
than one BRF is used, the overall Jones matrix should include the Jones
matrices for all the birefringent filters. The Jones matrix for the gain
medium (which acts as a partial polarizer) can be written as[5]:
2
0
01
gain
gain
q
M
, (4)
where
)1/(2
2
nnq
gain
is the transmission coefficient of the TE
polarized electric field at Brewster's angle for the gain medium, and n
is the refractive index (e.g., for Cr:ZnSe, refractive index around 2.5 m
is 2.44, Brewster's angle is 67.7, and q
znse
= 0.71). For the Brewster's
angle inserted BRF the Jones matrix can be written as [2, 5]:
)(1)(
1)()(
222
22
jExpSinCosqjExpSinCosq
jExpSinCosqjExpSinCos
M
brfbrf
brf
BRF
(5)
In (5), q
brf
is the transmission coefficient of the TE polarized electric
field at Brewster's angle (fore magnesium fluoride, refractive index is
around 1.37, Brewster's angle is 53.9, q
brf
=0.916), is the phase
retardation of the plate and can be calculated using:
22
22
Sin
Cos
nt
Sinnn
Cos
t
oe
. (6)
In Eq. (6), n
o
and n
e
are the ordinary and extraordinary indexes of
refraction for the BRF material, and for magnesium fluoride a Sell
Meier type dispersion equation given in Table I of [16] has been used
for calculation of the wavelength dependence of birefringence n=(n
e
-
n
o
). The angles
and
appearing in equations (5) and (6) can be
calculated in terms of the other more accessible angles using:
CosSinSinCosCosCos
, (7)
Sin
SinSin
Cos
. (8)
Figure 2 depicts the situation using an index ellipsoid for better
visibility. Note that is the angle between the optic axis of the
birefringent crystal and the beam propagation direction (angle
between
c
and
s
as shown in Fig. 1 (b) also), and
is the angle
between the TM polarized part of the electric field (E"
TM
) of the
incident light beam and the ordinary refractive axis of the birefringent
plate (
o
e
ˆ
). We note here that there is a mistake in the stated equation
for the calculation of in [2], and the correct from is given in Eq. (8).
E
”
TE
E
”
TM
x
y
z
c
o
e
ˆ
s
o
)(
e
n
e
e
ˆ
o
n
o
n
e
n
Fig. 2. Index ellipsoid of the magnesium fluoride crystal that posses a
positive uniaxial birefringence [2]. Projection of the incident electric
field onto the crystallographic axis of the filter has also shown. n
o
:
ordinary refractive index, n
e
: extraordinary refractive index, n
e
():
refractive index observed by the extraordinary wave in kDB
formalism,
c
: optic axis,
s
: direction of beam propagation, : angle
between
c
and
s
.
While using the birefringent filter inside the laser resonator, the
laser wavelengths that satisfy:
2m
(9)
relation will not be effected by the birefringent filter (will see it as a full-
wave plate). For our specific cavity described, then the solution of Eq.
(1) shows that, at these wavelengths the filter transmission is unity and
the polarization eigenmode has only the TM component. The
wavelengths at which Eq. (9) holds can be calculated using [5]:
mCos
nSint
m
2
. (10)
When the wavelength is
m
(m
th
resonance wavelength), the
polarization is TM at all the interfaces, and the beam will not observe
any loss. On the other hand, other wavelengths on both sides of
m
will
have elliptic polarization and will observe loss due to their TE
component of the electric field.
Tuning of the laser wavelength is facilitated by rotation of the plate
about an axis normal to the surface (corresponds to changing , which
will change and hence the peak transmission wavelength
m
). Hence
the tuning rate could be expressed as (d
m
/d):
mCos
nSint
RateTuning
2
_
, (11)
and using Eq. (10) can be re-written as:
2
2
_
2
Sin
Sin
RateTuning
. (12)
Achieving lasing only at a single wavelength at each instant during
tuning requires the filter to have only one transmission maxima in the
tuning range. In other words the filter free spectral range (FSR, spacing
in optic wavelength between two successive transmission maxima)
should be larger than gain bandwidth of the laser medium. One can
calculate the free spectral range of the filter using [5]:
2
2
nSint
Cos
FSR
. (13)
Note that, using Eq. (10), the equation for the FSR can also be rewritten
as [5]:
m
FSR
. (14)
Another important property of the birefringent filter is the
modulation depth. Basically, when the laser resonance is tuned to a
specific wavelength, for the neighboring wavelengths the loss should
be high enough to suppress lasing. The modulation depth (MD) of the
filter can be calculated using [2]:
22
4 SinCosMD
. (15)
The modulation depth could be maximized at values of 45 and
135 by exciting the ordinary and the extraordinary waves equally.
Also =0 and =90, excites only the ordinary or extraordinary waves,
and do not produce any modulation.
Lastly, as it is well-known, in birefringent crystals, incident ray is
divided into an ordinary and extraordinary ray, and the walk-off angle
(WOE, the angle between ordinary and the extraordinary rays) could
be estimated using [6]:
22
)()(2)(
)(
CosSinSinCosSinSinSin
Cos
Sin
n
n
WOE
o
, (16)
where
)(
)()(
Sin
CosSinCosSinCos
Cos
. (17)
Note than in the derivations for filter properties, weak birefringence is
assumed, n/ n
o
n
o
(n/ n
o
is less than 1% for magnesium fluoride),
but for the estimation of walk-off angle the first order effect is included
[6]. Lastly, the separation (D) between the ordinary and the
extraordinary rays at the exit face of the BRF could be estimated using:
WOETan
Cos
t
D
. (18)
3. DESIGN DISCUSSION
The aim of this design effort is to calculate the optimum cut angle ()
and optimum thickness (t) for a magnesium fluoride birefringent
tuning plate that could smoothly tune broad-bandwidth laser gain
media such as Ti:Sapphire, Cr:ZnSe, and Fe:ZnSe. In our discussion
below, we will try to speak in general terms. However, as a specific
example, we will work on Cr:ZnSe, which has a gain spectrum covering
the region roughly from
1
=1.8 m to
2
=3.4 m [13, 14]. A free-
spectral range of around 1.6 m is required for tuning Cr:ZnSe around
the central wavelength of 2.35 m (harmonic mean). Mathematica
has been used in all the calculations that will be presented below.
A. Effect of Optic Axis Direction on Free Spectral Range
Since we are interested with ultrabroad tuning of solid-state lasers and
hence large free spectral ranges (FSRs), we start our discussion by
looking at the effect of birefringent filter's crystal axis orientation ()
on FSR. Figure 3 shows the calculated variation of filter FSR with filter
rotation angle . The calculation has been performed at a central
wavelength of 2.35 m, for several different optic axis orientations in
the range from 0 and 90, for a magnesium fluoride birefringent
tuning plate with a thickness of 3 mm. Here, we take the thickness as 3
mm as a reasonable value which will enable us to start a discussion,
and optimization of the filter thickness will be discussed in Section 2.E.
0.1
1
10
0 45 90 135 180
Plate roration angle: (degrees)
Free Spectral Range (
m)
0 15 30
45 60 75
90 36.1
15
30
45
60
75
90
m=1
m=2
m=3
m=5
m=10
m=20
36.1
0
Fig. 3. Calculated variation of free spectral range for a standing-wave
Cr:ZnSe laser cavity around 2.35 m, as a function of birefringent plate
rotation angle (). The calculation has been performed for different
optic axis orientation values ranging between 0 and 90. The
magnesium fluoride birefringent plate was assumed to have a
thickness of 3 mm.
Note that in Fig. 3, the x-axis has been limited to plate rotation angle
values from =0 to =180, and the 180-360 range is a symmetric
copy. The solid lines have been calculated using Eq. (13), and the dots
on the solid lines are the resonance points. These are the points where
Eq. (10) is satisfied, and the plate produces a phase shift of m times 2.
Also, the dashed gray horizontal lines in the figure indicates the /m
values, at selected m values of 1, 2, 3, 5, 10 and 20 (Eq. (14)). Note that,
even though Eq. (13) predicts a continuous set of FSR values, in reality
at a central wavelength of , we can only achieve quantized FSR values
of /m. Hence, the largest FSR value that can be achieved from a BRF at
a central wavelength of is (for the case m=1), and calculated FSR
values that are larger than the central wavelength do not have physical
significance. Moreover, the thickness of the plate as well as its cut
angle () determines which orders (which m values) will be available
for a specific filter.
We see clearly from Fig. 3 that, birefringent filters diving angle ()
significantly effects the range of FSR values that can be obtained.
Basically FSR is proportional to Sin
-2
(), and is a function of , as well
as , so we are scanning a two dimensional surface (which enables
better optimization of filter properties). First of all, when =0, the
plate’s crystal axis is perpendicular to the plate surface, =90 and =
(the internal Brewster’s angle, 36.1), and is independent of (the
incident beam only excites the extraordinary polarization). Hence, the
BRF does not change the polarization state of the incident beam (which
is in an eigen polarization direction), and the calculated FSR values do
not have physical significance (transmission is independent of
wavelength (100%)). Here we are assuming any other possible
depolarization effect of the materials such as chirality is small.
When the plates optic axis lies on the surface of the plate (=90,
typical on-surface optic axis BRF), the obtainable FSR values are quite
small and varies in a very narrow range (130-200 nm). This is because
for this specific thickness (3 mm), at this specific central wavelength
(2.35 m), when =90 the BRF enables accessing plate orders (m)
between 12 and 17 only, and this limits the obtainable FSR values to
between /12196 nm and /17138 nm.
This is further illustrated in Fig. 4, where we plot variation of filter
resonance wavelength as a function of rotation angle (), for plate
optic axis orientation () values of 30 and 90. The horizontal gray
line in Fig. 4 indicates the central wavelength for Cr:ZnSe (2.35 m).
Note that, the rate of change of the resonance wavelength with rotation
angle is the tuning rate (Eq. 11) and for broadly tunable gain media a
large tuning range is generally desired. Therefore, the slope of the
curves in Fig. 4 gives the tuning rate of the filter for the specific order
(the steeper the better). The effect of plate optic axis orientation () on
tuning rate will be discussed in more detail in the next section.
1
2
3
4
0 45 90 135 180
Plate roration angle: (degrees)
Wavelength (
m)
=30
m=14
=90
m=12
=90
m=17
=30
m=1
Fig. 4. Calculated variation of filter resonance wavelength as a function
of birefringent plate rotation angle (). The calculation has been
performed for optic axis orientation values of =30 and =90 (the
later has been shown with dashed lines). The magnesium fluoride
birefringent plate was assumed to have a thickness of 3 mm.
For the specific case of Cr:ZnSe, we desire an FSR value of 1.6 m
around 2.35 m, which requires us to reach an m value of 1 (even m=2
which provides an FSR of 1.175 m will not be sufficient). As it is clear
from Fig. 4 also, a regular BRF with an on-surface optic axis (=90),
only has orders between m=12 and m=17 for 3 mm thickness, and is
not an option. By decreasing the filters thickness from 3 mm to 0.2 mm,
it is possible to achieve an order of m=1 for an on-surface optic axis
BRF also. However, such filters also posses very slow tuning rates,
limiting their usage in tuning (will be discussed later, e.g., Fig. 11).
We see from Fig. 3 that, very large values of FSR could be obtained
for small plate rotation angels (: 0-45), when the optic axis
orientations is in the 25-45 range.Especially, as gets closer to
(the internal Brewster’s angle, 36.1), things get quite interesting. At
this angle (=), for small values, the direction of beam propagation
s
gets closer to the direction of crystals c axis (
c
), and approaches
an angle of 0. Since the FSR scales inversely with Sin
2
(), as
approaches 0, the calculated FSR values increase sharply, and
diverges to infinity at =. However we should note here that, when
this happens (0), the phase retardation of the plate () also
approaches zero (Eq. (6)) indicating that, the obtained FSR values does
not have any physical significance (there is no transmission maxima
for 2.35 m). One can also see this in Fig. 2, when =0, n
e
()=n
o
, and
there is no phase difference between the ordinary and the
extraordinary waves.
To summarize, from Fig. 3 & 4, we see that optic axis orientations
() especially in the 20-45 range is interesting, since for a given
birefringent filter thickness, they enable resonance for a larger set of m
values. For example, when =30, the BRF plate provide resonances
for m values between 1 and 14, enabling FSR values between /12.35
m and /14168 nm. For our design aim, reaching m=1 is important
in attaining the largest possible FSR value. On a more general
perspective, a BRF that could provide a large variety of filter orders is
capable of producing a rich set of filter parameters. This is in general
very useful, since it enables optimization of performance for different
applications. As an example, a recent study demonstrated advantages
of off-surface optic axis BRFs in multicolor lasing [9, 20].
B. Effect of Optic Axis Direction on Tuning-Rate
In our discussion so far, we have focused our attention on optimization
of free-spectral range. As mentioned above, another important
parameter for a birefringent filter is its tuning rate. As we will see in
Section 3.D, as we rotate the filter (by varying ), the modulation depth
of the filter also changes. If the designed filter is not tuning fast enough,
then as one tunes the wavelength, this could result in significant
changes in modulation depth, and might result in undesired effects
such as wavelength jumps, etc... Hence, for our application here, in
tuning ultra-broad laser gain media, we require tuning rates roughly in
the order of FSR/10 per degree of plate rotation. Then, for the Cr:ZnSe
laser with a central wavelength of 2.35 m and a desired FSR of 1.6 m,
a tuning rate in the order of 100-200 nm/degree is desired.
1
10
100
1000
0 45 90 135 180
Plate roration angle: (degrees)
Tuning rate (nm/degrees)
15 30 45
60 75 90
36.1
15
30
45
60
75
90
36.1
Fig. 5. Calculated variation of tuning rate for a standing-wave Cr:ZnSe
laser cavity around 2.35 m, as a function of birefringent plate rotation
angle (). The calculation has been performed for different optic axis
orientation values ranging between 0 and 90. The results are
independent of filter thickness.
To investigate this issue, Fig. 5 shows the calculated variation of
filter tuning rate as a function of filter rotation angle (using Eq. 12, the
calculation is independent of filter thickness). Note that, similar to the
FSR, acquiring large tuning rates require usage of the filter at small
rotation angles (0-45 range). Moreover, the regular on-surface optic
axis birefringent filter (=90) provide very small tuning rates, and
again is not a good choice. On the other hand, off-surface optic axis
BRFs with a diving angle () between 25-45 enables large tuning
rates up to 150-200 nm/degree, indicating that the full tuning-range
(1.8-3.4 m) could be scanned by rotating the plate (changing ) 8-12
only. This finding will be important in our discussion of modulation
depth in the next section.
C. Effect of Optic Axis Direction on Modulation Depth
In the earlier sections, we have seen that filters with between 25-45
are good candidates for broadband tuning, when used in rotation
angles between 0-45 (m=1). However, for the BRF to work properly it
should also provide enough modulation depth to suppress neighboring
side wavelengths so that clean single wavelength operation is achieved
(might especially become an issue when the gain is high and output
coupling and other cavity losses are low). For that, first of all we desire
a modulation depth as large as possible. Moreover, as we tune the
wavelength, we want to keep the modulation depth as smooth as
possible to obtain the same filter rejection across the whole tuning
range (we need to rotate the filter around 10 to cover the full tuning
range, and the modulation depth should not change a lot as we tune).
0
0.25
0.5
0.75
1
0 45 90 135 180
Plate roration angle: (degrees)
Modulation depth (a.u.)
20 25 30
35 40 45
45
40
20
25
35
30
(a)
0
0.25
0.5
0.75
1
0 45 90 135 180
Plate roration angle: (degrees)
Modulation depth (a.u.)
0 15 30 45
60 75 90
15
30
45
60
75
90
0
(b)
Fig. 6. Calculated variation of modulation depth as a function of
magnesium fluoride birefringent plate rotation angle (). The
calculation has been performed for different optic axis orientation
values ranging between 0 and 90 (see differently colored graphs).
![Fig. 1. (a) Standing-wave laser resonator containing a gain medium and a birefringent plate inserted at Brewster's angle. (b) Light beam incident on the birefringent plate at Brewster’s angle [2]. : angle between the optic axis and the surface normal, c : optic axis, s : direction of beam propagation, i: incidence angle, : internal Brewster’s angle, : rotation angle of the plate, t: thickness of the plate.](/figures/fig-1-a-standing-wave-laser-resonator-containing-a-gain-2iz4swji.webp)
![Fig. 2. Index ellipsoid of the magnesium fluoride crystal that posses a positive uniaxial birefringence [2]. Projection of the incident electric field onto the crystallographic axis of the filter has also shown. no: ordinary refractive index, ne: extraordinary refractive index, ne(): refractive index observed by the extraordinary wave in kDB formalism, c : optic axis, s : direction of beam propagation, : angle](/figures/fig-2-index-ellipsoid-of-the-magnesium-fluoride-crystal-that-1oayz35l.webp)
