II Development of high – sensitivity dispersion laser
interferometer
Last years interest to sensitive seismographs was stimulated with possibility to
predict earthquake. A key to the understanding of the state of stress of the earth and
the association phenomena of tectonic activity and earthquakes is a knowledge of
the spatial distribution of the earth strain. The development of the laser as a source
of coherent optical radiation has permitted the application of interferometric
techniques to the problem of earth strain measurement. In other work authors
represent the laser interferometer technique that allows the displacements to be
measured with the instrumental resolution up to 1 pm.
Besides dispersion interferometers (DI) are currently used for determining the
nonlinearity of refraction of optical medium induced by the field of an intense light
wave.
Areas of application
The laser interferometers are found a wide using in measurements of space
movements of the objects in the mechanical engineering, geodesy and during
geophysical surveys. The high accuracy of positioning is required while processing
of sensitive devices for computer engineering and nanoelectronics.
Introduction
Traditional interferometric methods for measuring using different geometric
paths for interacting waves possess disadvantages typical of spatially separated
beams: beams propagate in different optical elements along paths with different
optical elements along their lengths. The advantage of dispersion interferometers
consists in the fact that interacting waves propagate along the same geometric path.
Thus, the requirements on both the optical properties of elements (quality,
transparency) and stability of individual elements of the optical layout of
interferometers drop out. The process of second-harmonic generation in nonlinear
crystal has permitted the application of the process of optical frequency conversion
to the problem of earth strain measurement and of determination of dispersion of
refractive index using dispersion interferometers. The operation of DI was
analyzed mainly in constant-field approximation.
In the given work it is offered to spend the analysis of process of frequency
conversion in DI using the constant-intensity approximation. The analysis of
harmonic generation process in the constant-intensity approximation allowed one
to suggest a new method of determination of displacement (for example, earth’s
strain) and refractive index dispersion. As a result of analysis in this
approximation, in contrast to the constant-field approximation, the minima of
intensity of the second-harmonic dependent on the phase mismatch. This fact
allows simple, reliable way with greater accuracy to define displacement and
refractive index dispersion. According to results of the theoretical analysis
reception highly sensitive DI with wide frequency and dynamic ranges is expected.
Description
The proposal offers two variants of the dispersion interferometer. The first variant
assumes the case when the between two crystals with quadratic susceptibility the
medium is placed. It is shown that second-harmonic intensity is a function of phase
difference of interacting waves. The change of length of the medium caused in
particularly by the earth strain induces an additional phase difference and this leads
to a corresponding shift of the interference function I 2out ( ) along the abscissa. By
the comparison of the interference patterns in the presence and the absence of the
earth's strain (or of investigated medium), this strain (or dispersion of refraction
index) can be determined from the displacement of these patterns. It is obtained
that in the constant-intensity approximation definition of zero of a curve of
synchronism more precisely, than in the constant-field approximation. Hence
expected accuracy in measurement should be above, than in the existing methods
of the measurement based on the constant-field approximation. The use of
dispersion interferometer with the crystal and medium inside the laser resonator
(second variant) allows to increase the interferometer sensitivity. At the exit from
the interferometer in the constant-intensity approximation for the first variant of
medium location we obtain with the corresponding boundary conditions the
following expression for the second-harmonic intensity
2
I
first
2, output
 I 2 ( )e
 2


i  i i 
e 
sin 2 ctg2   1 cctg1 
 ,

2

22 
2 
 2

(1)
2 2
2
, 2  222  , 12   1 2 I10 , 22   1 2 I1 (),  first    21 (d )  2 (d ) ,
4
4
are the nonlinear wave coupling coefficients, 1, 2 are the absorptions
where 12  212 
 1, 2
coefficients,   2k2  k1 is the phase mismatch. It can be seen from (1) that secondharmonic intensity is a function of phase difference  of interacting waves and
oscillates with varying of  . At the exit of interferometer the recorded secondharmonic intensity through the phase difference of the both waves depends on the
medium parameters. The change of length of the medium (for example, an air gap
or gas) caused in particularly by the earth strain induces an additional phase
difference and this leads to a corresponding shift of the interference function
I 2out ( ) along the abscissa. By the comparison of the interference patterns in the
presence and the absence of the earth's strain (or of medium studied) determined
from the displacement of these patterns.
One of the ways of increase in the conversion efficiency of frequency is the
location of a nonlinear crystal inside the optical resonator (see Fig.1). \\inIn this
case (the second variant of the DI), owing to the additional multifold passes of a
crystal (2), the effective length of a nonlinear interaction increases, and the energy
exchange between the waves is mainly determined by the phase relationships.
Fig.1
At the exit from the interferometer in the constant-intensity approximation for
the second variant the second-harmonic intensity expression has the form
2


i  i sec ond i 
ond
 2
e
(2)
I 2sec
sin 2  ctg2    1 ctg2  

 ,
,output  I 2 ()e
22 
22 
 2

 sec ond    21 2d   2 (2d )  21  2 ,
where 1, 2 (2d ) are the phase shifts of the fundamental frequency and harmonic
waves in the medium of length d . 1, 2 are the phase shifts experienced by the
waves on reflection from the mirror (M2) at frequencies 1 and 21 , respectively. It
can be seen from (2) that the second-harmonic intensity is a function of  sec ond and
oscillates with varying  sec ond . Analogously to first variant the change of the
medium length induces an additional phase difference and this leads to a
corresponding shift of the interference function I 2output( ) along the abscissa. In
intracavity frequency conversion, even small phase variations in interacting waves
lead to significant variations of the harmonic intensity. Therefore, displacements of
phase mismatch can be found by determining these variations. The comparison of
the both variants has been performed on the base of the analytical investigation in
the constant-intensity approximation. The results of the graphical analysis of the
DI function both for the case when the nonlinear crystal is out of the cavity and for
the intracavity case is shown (see Fig.2). The use of DI with the crystal and
medium inside the laser resonator allows to increase the interferometer sensitivity.
The comparison of the curves 1, 2 and 3 shows that by the choice of optimum
values of phase mismatching and basic radiation intensity, one can vary not only
the interference pattern period, but also the character of I 2output( ) dependence, i.e.
the steepness of the curve leading to increase of DI sensitivity.
Thus, if the amplification at the laser transition is enough for the intracavity case,
then it is reasonable to deal with this variant. Nevertheless, another (first) variant
takes place.
Fig.2. Dependencies of the second-harmonic intensity on the phase mismatch
at  2   21  0.1;   1(1,3,4,6), 0.6 (2,5) and 1  0.5(1,4),1(2,3,5,6) for
the intracavity case (1-3) and for the case when the medium is out of
the cavity (4-6).
Expected accuracy In this work the obtaining of measured values with greater
accuracy, rather than in existing ways of definition of displacements of
interference patterns is supposed. This statement is based on preliminary
researches:
-the existing methods of determination of displacements were mainly based
on traditional interferometric methods where for measuring using different
geometric paths for interacting waves possess disadvantages typical of spatially
separated beams: beams propagate in different optical elements along paths with
different optical elements along their lengths. The advantage of dispersion
interferometers consists in the fact that interacting waves propagate along the
same geometric path. Hence, the requirements on both the optical properties of
elements (quality, transparency) and stability of individual elements of the optical
layout of interferometers drop out. Thus, the new way of definition of
displacements, differing from earlier existing the technique is offered. The offered
method is more exact, simple and reliable.
- the displacements of interference patterns can be determined directly from
the experimental measurements of shifts of minima or maxima of secondharmonic intensity;
- the position of the zeros of a curve of the synchronism for first three zeros
of the dependences of intensity of a harmonic on length of the nonlinear medium,
calculated in the constant-intensity approximation are more exact and are
correct, than in widely used constant-field approximation. Hence expected
accuracy in measurement should be above, than in the existing methods of the
measurement based on the constant-field approximation.
One of the important stages of the proposal project is creation of pilot
equipment for displacements measurement of interference patterns. Within the
limits of this project it is necessary to solve a problem of experimental
measurement of interference patterns (foe example, an earth's strain).
In the scientific plan following steps in the future must be are executed for
decision of this problem:
1. Development of construction of experimental equipment for special
schemes of displacement measurement (for example, caused by the earth
strain)- first variant.
2. Development of construction of experimental equipment for special
schemes of displacement measurement (for example, caused by the earth
strain) - second variant.
3. Experimental definition of shifts of the interference function (secondharmonic intensity) when the location of the medium inside the laser
resonator.
4. Experimental definition of shifts of the interference function (secondharmonic intensity) when the location of the medium out of the cavity.
References:
1.Z.H.Tagiev, R.J.Kasumova, R.A.Salmanova, N.V.Kerimova, “Constant
Intensity Approximation in the Theory of Nonlinear Waves”, J. Opt. B:
Quantum Semiclas. Opt. 3, (2001), 84-87
2.Z.H.Tagiev, R.J.Kasumova, R.A.Salmanova, N.V.Kerimova, “The
Theory of Nonlinear Dispersion Interferometers in the Constant Intensity
Approximation" Opt.Spectrosk., 2001, v.91, 968-971 [Opt.Spectrosc.,
2001, v.91, 909-912].