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The ultrahigh resolution and sensitivity by spectral measurement on the basis of the ring laser
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Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 302–313 www.elsevier.com/locate/jqsrt
The ultrahigh resolution and sensitivity by spectral measurement on the basis of the ring laser V.I. Denisova, V.V. Grishacheva,, V.N. Kuryatovb, E.F. Nasedkinb, V.G. Zhotikovc a Physics Department, Lomonosov Moscow State University, 1 Vorob‘evy gory, Moscow, 119899, Russian Federation Laser Gyro Division, POLYUS Research and Development Institute, 3 Vvedensky St., Moscow, 117342, Russian Federation c MIPT, 9, Institutskii per., Dolgoprudny, Moscow Region, 141700, Russian Federation
b
Received 28 November 2005; accepted 6 February 2006
Abstract The laser gyroscope technique on the basis of the gas ring laser at use in the spectral dispersive and absorptive analysis of substance allows attaining to the ultrahigh frequency resolution (about 1016 and less). Experimental check of a method is realized on the basis of air research on dispersive dependence. r 2006 Elsevier Ltd. All rights reserved. Keywords: Ultrahigh resolution spectroscopy; Laser intra-cavity spectroscopy; Counter-propagating modes; Ring laser; Laser gyroscope
1. Introduction Designing of new measuring technical equipment with ultrahigh parameters on sensitivity, accuracy is a basis for development of modern experiment. Not always such step can be made, only modernizing existing methods. It occurs more often on crossing ways of two or several measuring techniques. In the present work it is offered to unit advantages of laser intra-cavity spectroscopy and laser gyroscope and to create a unique analysis method of substance with sensitivity and the spectral resolution not having analogues. It is known, that laser intra-cavity spectroscopy [1–3] is one of the most sensitive methods of the spectral analysis of substances. Specifically, absorptive properties are determined on a passage spectrum through the resonator such as Fabri-Perot with the researched substance with its help, lines with absorption factor less than 1011 cm1 are registered in this case [3,4]. The laser intra-cavity spectroscopy technique is widely used at the chemical analysis of gases, chemical reaction kinetics, research of plasma [5,6]. In addition to high sensitivity, distinctive feature of a method is high enough frequency (wavelength) resolution, which reaches 1 MHz in the best optical devices. The achieved resolution can be increased on the 9 orders if for the spectral analysis is using counter-propagating waves of the ring laser which frequencies difference is fixed with accuracy 0.001 Hz as it is carried out in laser gyroscope [2]. Corresponding author.
E-mail address: [email protected] (V.V. Grishachev). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.02.057
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In laser gyroscope is used both passive and active resonators. Ring lasers with active resonators have a several advantages, which allow creating effective measuring systems. But application of active resonators for the spectral analysis of substances has a several difficulties which are connected to influence of the investigated material on the laser generation conditions and interaction of counter-propagating waves. However in some cases such complication of measurements technical equipment can prove to be correct by increase in resolution and accuracy. So carrying out of the spectral analysis on counter-propagating waves allows to achieve the ultrahigh frequency resolution which by relative value reaches 1012 for direct measurements, and at indirect measurements makes about 1016 and less. It is connected with high measurement accuracy of a close frequencies difference of counter-propagating waves in the ring laser and high sensitivity of generation conditions of the ring laser with respect to external influences. In recently published our work [7], it was analyzed join of laser intracavity spectroscopy and laser giroscopy opportunities theoretically as well as it was offered possible ways of realization of a method in practice. Consideration must be given to existence of several parameters by which it can be investigated substance properties in the ring laser and they are connected among themselves by complex correlations. It makes a choice of measured parameters and an establishment of their interrelation of principal problems, which can simplify a problem of a method realization. In the present work in contrast to earlier, we offer one more way of realization and we carry out the analysis of the fulfilled measurements according to new correlations between measured parameters. 2. Principles of measurements In this section, use capability of intra-cavity spectroscopy technical equipment by counter-propagating waves in ring lasers for carrying out differential dispersive and absorptive measurements of substances properties are discussing. And also, the parameters of the ring laser required for similar measurements are considering. 2.1. Differential spectroscopy on the basis of counter-propagating modes Ring lasers find wide application in measuring technical equipment [8]. It is based that in a stationary singlemode operation of the ring laser can be generated two contra-directional running waves with frequencies n+ (the right mode) and n (the left mode) [9]. Frequencies difference of counter-propagating modes F ¼ (n+n) can be precisely measured at their optical convergence that provides the ultrahigh spectral resolution. Beating frequency F of two optical modes is frequency deviation (shift). The placement of researched substance within the resonator allows realizing its spectral analysis. Research dispersive and absorptive material properties is reduced to measurement except for frequencies counter-propagating modes n+, n and their intensities I+, I on an input in researched substance, also intensities I þ DI þ ; I DI on an output from one (Fig. 1). Measurement of output parameters of the ring laser beam convergence allows determining five parameters,
Fig. 1. The ring laser optical arrangements with the investigated material (X).
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which are connected to the investigated material properties: Z ¼ ðI =I þ Þ;
kþ ¼ ðDI þ =I þ Þ;
k ¼ ðDI =I Þ;
F ¼ ðnþ n Þ;
n ¼ ðnþ þ n Þ=2.
On the basis of these parameters it is possible to carry out the spectral analysis of substance with the ultrahigh resolution. 2.1.1. The dispersive analysis Total frequency shift in the ring laser is defined by the non-reciprocal components of the laser. It can vary in limits from 102 to 105 Hz depending on active element and non-reciprocal components parameters. In the value of total frequency shift (F) is brought with the contribution not only the researched non-reciprocity material but also its dispersion that is connected to distinction of a refractive index for counter-propagating modes. As a rule, it is possible to neglect material non-reciprocity in most cases. Otherwise additional researches are carried out with the purpose of separation of these two contributions to a frequency shift as, for example, it is offered in work [7]. Let the researched environment with geometrical length S is placed in the ring resonator for which the refractive index for the right mode is equal n+ (on frequency n+), and for the left mode—n (on frequency n), then the size of the dispersion contribution of a refractive index in frequency deviation will be given by expression F X ¼ nðnþ n Þ
S , L
(1)
where L—average by counter-propagating modes of the cavity optical length. Allocation of the investigated material contribution in frequency shift allows determining a dispersion of refraction index by the formula qn nþ n 1 L F X ¼ ¼ . (2) qn nþ n n S F The method resolution is estimated in order value it is equal nð@n=@nÞ ¼ 105 2106 . It corresponds to the minimal measured change of beat frequency for contra-directional modes, which is equaled 102 Hz at the basic difference frequencies 105 Hz and at relative length of a researched part equal (L/S) ¼ 10. Resolution of the given method on some orders can be exceeding other methods. 2.1.2. The absorptive analysis Relative change of output laser intensity is connected with total absorption in the investigated material: kþ ¼ aþ S; k ¼ a S, here a+ and a—linear absorption factor of the investigated material on frequencies n+ and n, correspondingly. The dispersion of an absorption factor in substance can be calculated by the formula þ qa aþ a k k ¼ þ ¼ . (3) qn n n 2S F The estimation of a method resolution shows an advances possibility by absolute value less (@a/@n) ¼ 1011 s/cm without use of measurements special ways and data processing. 2.1.3. Modulation spectroscopy using Increase of method sensitivity can have been achieved by modulation of the investigated material parameters (for example, density change, external fields influence, etc.), that allows determining the material properties connected to these parameters. For example, in gases the linear absorption factor is determined by interaction cross-section between a photon and a molecule s+ and s, then aþ ¼ sþ C n S a ¼ s C n S , where Cn—molecules concentration. It is admitted to estimate a dispersion of interaction cross-section of a photon with a molecule on optical frequency n by the expression þ qs sþ s k k ¼ þ ¼ . (4) qn n n 2S2 C n F
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Measurements carrying out at various molecules concentration allows excluding necessity for accurate concentration measure, if to apply its relative change only. Modulation spectroscopy methods can increase essentially measure accuracy by means of the ring laser. 2.2. Measured parameters of the ring laser For realization of the differential spectral analysis of substance on the basis of counter-propagating close modes in the ring resonator is required to measure the output laser parameters, to determine its design features. 2.2.1. Laser intensity One from principal’s parameters influencing on the measurement accuracy is output laser intensity. On an output of the ring laser is registered the beam convergence intensity (IL) from time (t) which without taking into account an initial phase is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5) I L ¼ 2I þ 4I 2 ðDIÞ2 cosð2p F tÞ. Here we have introduced symbols for average intensity of counter-propagating modes on an output of the ring laser I ¼ ðI þ þ I Þ=2 and their difference DI ¼ ðI þ I Þ. It is necessary to note that counter-propagating modes intensities I+ and I by absolute value and relative value depend on an output mirror position in regard to active medium and investigated material. Further the same symbols will be used for definition of contradirectional wave’s intensities on all along cavity. In spite of the fact that these values are function from position of an exit point, but the knowledge of the ring laser characteristics can be determining contradirectional wave’s intensity on all points of the cavity uniquely. Therefore in each particular case they are defined by the self-procedure or their relative change obtain that also allows avoiding a pinpointing for light outlet. From measurements of laser intensity IL can be determined maximal Imax and minimal Imin of a signal. From these values it is possible to express I ¼ 0:25ðI max þ I min Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jDI j ¼ I max I min ;
(6)
on which the basic are defined absorption of the investigated material. 2.2.2. Active medium monitoring Work of the ring laser depends on the active medium parameters. Which can be characterized in gain factor (a threshold gain) G+ and G (G þ 0 and G 0 ) of the active medium for the right and left counter-propagating modes, respectively. In the ring gas laser with high-frequency discharge pumping, gain factors G+ and G depend on an excess voltage (UU0) over the high-frequency threshold voltage (U0), which is given by factor g¼
U U0 . U0
(7)
These measured parameters characterize a high-frequency pumping level of the active medium. The gain factors has non-linear dependence from g, but at small voltage variations dependence G(g) can be linearized. Achievement of the greatest stability of work occurs at ðdG =dgÞ ! 0, when small voltage changes poorly influence on the gain factor. 2.3. Dispersive curve of the ring laser The basic dependence on which investigated material are probed this is the dispersive curve—a dependence of counter-propagating modes frequency shift F from optical length L of the ring laser perimeter. Therefore we shall discuss what information about the investigated material can be included here.
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2.3.1. Frequency shift features Frequency deviation in the ring laser is defined by its design and component parts so, that it is given by semi-empirical expression [8] F ¼ FS þ FD þ FX F S ¼ F ac þ F bc þ F cc ¼ const F D ¼ bas I þ bad DI ¼ F D ðI; DIÞ.
ð8Þ
Here for a constant frequency shift are entered designation—FS that we shall denominate a full frequency bias. Next elements are part of full frequency bias: F ac —constant frequency shift of the counter-propagating modes connected to the active medium; F bc —constant by value deviation of the frequency caused by a special controlled non-reciprocal element which we shall name a frequency bias; F cc —constant frequency shift connected to a non-reciprocity under the frequency lock-in effect. And also the variable component of a frequency shift FD is entered which depends from bac and bad —material parameters of the active medium and before entered I ¼ ðI þ þ I Þ=2—average intensity of the counter-propagating modes at the point of entry the active medium and DI ¼ ðI þ I Þ—its difference. For the contribution of the investigating material to frequency deviation of counter-propagating modes is kept designation FX which characterizes not only the dispersive contribution, but also connected with non-reciprocity of the investigated material. 2.3.2. Conversion domain At the fixed optical length L of the ring laser perimeter, frequencies n of generated longitudinal modes should correspond to conditions (Fig. 2) c n ¼ nN ¼ N dn ¼ N ; nA pnpnB , (9) L where c—light speed in vacuum, N—the whole positive number equal to wavelength number being stacked on average optical length L of the cavity perimeter, nA 2nB —frequency band of an gain envelope on a lasing threshold level for the active medium. These conditions correspond to that frequencies should be in accord with resonant frequency (the first condition) and to fall within frequency band of a gain envelope, where gain exceeds losses (the second condition). Let us enter the relation Dn nB nA ¼ L, (10) dn c here Dn ¼ nA nB —a frequency bandwidth of a gain envelope on a lasing threshold level, dn ¼ ðc=LÞ— difference frequency between two adjacent longitudinal modes. Parameter of M shows in how many time frequency bandwidths more than drift between the adjacent longitudinal modes. If Mo1 then in the laser generation only one longitudinal mode is possible. At values 1pMo2 in the laser generation as one and two longitudinal modes is possible. In a case, when value MX2 then the number of generated modes can achieve three and more. For stable work of the ring laser are used the single-mode operation and small bandwidth where double-mode operation is possible. The laser operation with parameter Mo1 is not used as in this case there should be frequency bands in which a generation breakdown happen. Value of Dn is determined by an active ions type, mix proportion of breakdown plasma and a pumping level. The choice of a pumping level generation close to a threshold allows receiving single-mode operation. M¼
2.3.3. Typical dependence Tuning cavity perimeter one can to scan all frequencies band from nA to nB of a gain envelope, thus frequency deviation F of contra-directional modes forms a dispersive curve. The dispersive curve type depends on the laser design, the active medium properties, a pumping level and other parameters and ones the distinctive feature of the ring laser. In case of our ring laser the typical dispersive curve looks like as is shown on Fig. 3. On a dispersive curve can single out two characteristic domains (I and II). The first domain is in accord with single-mode operation, and the second-double-mode operation. Extent of the second domain l can vary a pumping level changing. When parameter M 1, then domain spread (I) tent to zero, also can be
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Fig. 2. Gain envelope of the ring laser for single-mode and double-mode. G(n)—gain envelope of the ring laser, nN , nNþ1 , nNþ2 — longitudinal mode of the ring laser, dn—frequency difference between the near longitudinal modes, Dn ¼ nA nB —gain envelope width on a threshold gain G0, F—frequency shift between the near counter-propagating modes (+,–).
converted to small domain extent with conversion from one operation to another by abruptly. On a dispersive curve can allocate characteristic points (A, B, C), in which there is a change of generation conditions by abruptly and a conversion from single-mode operation to double-mode (A, C) and back (B). Conversion domain near to a point A is characterized by achievement at perimeter tuning (L decreases) of generation frequencies for one longitudinal modes of an gain envelope edge that corresponds to a þ condition nN ¼ nN þ ðF =2Þ ¼ nA ; ðAÞ , nþ N —frequency of one contra-directional mode (+) of a longitudinal mode N (Fig. 4). It corresponds to the beginning of conversion in a double-mode operation. The laser completely will switch in double-mode operation by all counter-propagating modes when other contra directional wave (–) will achieve gain envelope edge nN ¼ nN ðF =2Þ ¼ nA ; ðAÞ . The further perimeter tuning will result in achievement by the first longitudinal mode of the second gain envelope edge nþ ðBÞ . The laser completely will switch in a single-mode operation, when Nþ1 ¼ nNþ1 þ ðF =2Þ ¼ nB ; n ¼ n ðF =2Þ ¼ n ; ðBÞ . The subsequent perimeter decrease will result in conversion repeat from a Nþ1 B Nþ1 single-mode in double-mode only for the following longitudinal mode— nN1 ¼ nA ; ðCÞ . Frequency shift is decreasing in the dispersive curve domain (II) since modes intensity will have fallen. It is observed because to occur a redistribution pumping between two pairs longitudinal modes (each mode includes pair
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Fig. 3. Typical plot of frequency shift F according to perimeter L (a dispersive curve) of KM-model ring laser. A, B, C—typical points of a dispersive curve, I—domain of single-mode operation, II—domain of double-mode operation, l—width domain of II, l—wavelength of a generated light.
Fig. 4. The ring laser spectrum on various domains of a dispersive curve. A–B—double-mode operation, B–C—single-mode operation.
counter-propagating waves). Falling take place steadily in a case enough the big domain extent l (II) and abruptly at small spread as instability small domain. The ratio between extents of these domains is determined by parameter M. If ME1.5, then both domains have equal extents and the dispersive curve to become close to a symmetric type. The form of a gain envelope defines the form of a dispersive curve on section A2B and B2C as mode intensity depends on position of a
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generated longitudinal mode concerning a gain envelope. It causes change of frequency shift F in turn. Also, the form of a dispersive curve depends on how there is a perimeter tuning. 2.3.4. Particular points On a dispersive curve can select three characteristic frequencies shift: the maximal frequency (FC, FA, y), the minimal frequency (FB, y) and mean frequency (/FS). First two are defined as extremely probable values of frequencies shift at smooth perimeter tuning and last frequency as mean value on all potential frequencies on one tuning period. Taking into account connection of frequency deviation with the cavity properties, the active medium and the investigating material it is possible to receive expressions for distinctive values F Y ¼ F S þ F X þ F DY ;
F DY ¼ bac I Y þ bad DI Y ;
hF i ¼ F S þ F X þ hF D i;
hF D i ¼ bac hIi þ bad hDIi,
Y ¼ A; B; C . . . (11)
here I A ; I B ; I C . . . and DI A ; DI B ; DI C ; . . .—values of a half-sum intensities of counter-propagating modes in characteristic points Y ¼ A,B,Cy and differences corresponding to them, hIi; hDIi—mean intensities on the period of a dispersive curve and their difference. Characteristic points are determined on breaks on a dispersive curve. Value of frequency shift component FS (the full frequency bias) whit perimeter tuning does not change. The contribution FX of the investigated material can be tested by influence on one. Generally, mean values are determined by a dispersive curve symmetry, which is defined by gain envelope symmetry of the active medium in turn, so hIi ¼ 0:5ðI A þ I B Þ ¼ const, hDIi ¼ 0:5ðDI A þ DI B Þ ¼ const.
ð12Þ
Mean values remain constants at change of a full frequency bias F S , as they are assigned by pumping level. From characteristic values of a dispersive curve can compose an expression F A þ F B ¼ 2hF i, if to take into account dependence symmetry. The obtained dependences /FS from F S allow calculating the contribution /FD S in frequency shift F at given pumping. That allows excluding influence of modes intensity. Full tuning loop from a point A–C corresponds to perimeter change on one wavelength. The relation of domain extents with a double-mode spectrum (l) by the repetition period (l) of a dispersive curve is given by formula hF i nA l 1oMp2 , (13) l ¼ nB M 1 þ dn ; if small values of the order ðF =2nA;B Þ 1 nA =nB pðF =dnÞ 1 may be neglected. As evident from its expression relative extent of a double-mode operation is function of frequency shift. At values ME1 small changes of the relation (l/l) are proportional to change of mean frequency shift /FS. At perimeter tuning, value M changes, that enables to calculate from dispersive curve of value Dn ¼ nA nB depending on pumping level g. 3. Approbation of the ring laser as measuring device Some laws revealed above in laser emission behavior with the investigating material have been used for carrying out of experimental researches with the purpose of approbation of new measuring system with the ultrahigh resolution and accuracy. 3.1. Laboratory research complex on the basis of the ring laser Performance check of a method was executed on the basis of fine differential dispersive and absorptive the analysis of air properties on the modernized ring laser of KM-model (POLYUS RDI, Russia). Four total reflection prisms for one pair contra-directional longitudinal cavity resonance have formed the laser ring cavity. The cavity had a mono-block glass-ceramics design in the square form with the length side 0.43 m and the total length perimeter 1.72 m within which channels have been drilled. The active medium was capturing one channel with helium–neon mix of composition He3/Ne20/Ne22 ¼ 14/1.1/0.9 and the one was pumped up
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by the cross high-frequency discharge. Laser generation stability was provided with pumping stability which was controlled by means of the voltage of the high-frequency discharge generator with accuracy of 0.1%. The channel opposite to the channel with the active medium was pumped out under rough vacuum. Other channels have been sealed off and filled with air. In one of these channels the optical cavity perimeter was reconstructed by pressure change (heating or change of volume). The ring cavity has Q-factor not less than 108. The frequency lock-in bandwidth signified smaller 200 Hz. Introduction in the optical cavity arrangement of additional optical elements will be inevitable to cause reduction of Q-factor of laser and increase of frequency lock-in bandwidth therefore for measuring system approbation we were used the optical cavity arrangement as is. The important system performance of experimental setup is the lasing counter-propagating modes difference, which named a full frequency bias (FS). As it was marked, it depends on active medium (Fac), a frequency lock-in effect (Fcc) and a special non-reciprocal element (Fbc). As a special non-reciprocal element frequency bias it was used total reflection prism on which the external magnetic field was imposed. In the issue of Faraday effect in prism have to be arising non-reciprocity for counter-propagating modes. Thus, the prism in an external magnetic field operates as Faraday cell. The external magnetic field was created by constant magnets with stray fields in the total reflection prisms within the limits of 0.01–0.02 T. Prisms have been made of quartz at which Verde constant has value of the order 5 1012 rad=T m. Light falls on a prism face under Brewster angle and in prisms linearly polarized light is propagated. The external magnetic field projection distinct from zero onto a light wave line in a prism results in polarization twisting and ellipticity formation of plane-polarized wave. Elliptic polarization twisting in a prism for counter-propagating modes has an opposite rotational direction that causes non-reciprocity in speeds of counter waves and as consequence frequencies shift. Orientating various ways permanent magnets concerning a beam it is possible to receive an initial total frequency bias from 200 up to 1400 Hz at the given magnets. In the experimental setup counter-propagating modes are extracted from cavity through one of total reflection prisms and by optical elements is re-united on a photo-detector. The electric signal from a photodetector is processed, result of that is frequency shift F. Accuracy of measurement of frequency shift achieved in a maximum 0.01 Hz. During researches it was not carried out measurements of laser output power.
3.2. Measurements Experimental check of measuring equipment performance capabilities was carried out on the basis of spectral researches of air under standard conditions, which placed in one from ring cavity channels. Researches have consisted in obtaining of dispersive dependence for various values of an initial total frequency bias produced by a magnetic field. Estimation of a total frequency bias value (FS) it was made by averaging of frequency shift received during measurements at a constant external magnetic field, that is in amount hF i which is determined by the formula (11). In the beginning, rough tuning perimeter was made by change of air volume in the measuring channel for the purpose of a selection of a working point. Further, smooth tuning optical perimeter of the cavity was carried out by heating from the sinusoidal voltage generator with frequency 0.003 Hz. Heating power changed with the double frequency in the sine curve therefore into dependences can allocate linear change portions of temperature. Here an optical perimeter length was linearly proportional to time heating. Typical experimental dispersive dependences are submitted on Figs. 5–8. The pumping power was chose in such a way that the domain with a double-mode operation practically was absent—it is case when ME1. It especially well is visible on dispersive dependence with a domain only conversion (Fig. 9). In the conversion domain the increase in instability of frequency shift is observed, but its type repeats from one conversion to another. In a conversion point an absolute value of frequency shift is precisely determined such that from the left of conversion point verge towards nA , and from the right—nB . And the nearer to a conversion point, the more exact a frequency shift is coincides with nA or nB . Thus, the peak points on dispersive curve (Figs. 5–8) are fixed by wave frequency (wavelength) about accuracy not smaller, than accuracy of measurement of frequency shift which compiling 0.01 Hz. It allows asserting about realization of the ultrahigh spectral resolution about 1016.
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Figs. 5–8. Dispersive curves of the ring laser for various total frequency biases that is specified by average frequency /F S.
Thus, on the one hand, achievement of peak points on dispersive curve (points A and C on Fig. 3) occurs at laser generation on wave frequencies nA 0:5 F A , nC 0:5 F C and nA ¼ nC , determined with accuracy 0.01 Hz. On the other hand, frequencies shift F A and F C for the nearest-neighbor peak points differ, and their difference F A F C is connected to change of air properties in the cavity channel. In our case one is connected with change of air density. For the total frequency bias characterized by average frequency shift /FS ¼ 1121 Hz, difference equals to FA–FCE6 Hz. 4. Discussion The submitted measurements have a complicated interpretation that is bound up with influence on a dispersive curve of many parameters. Therefore at the results analysis it is necessary to allocate the physical phenomena rendering the basic influence on results of measurements. At perimeter tuning by heating of researched gas (air), two nearest-neighbor peak (points A and C on Fig. 3) of a dispersive curve is differing increase in the optical cavity length on l that corresponds to a time interval on an experimental curve equal (tCtA). It allows finding experimental value of optical length variation 0 with temporal changes dt: dL ¼ ldt=ðtC tA Þ. Let Y and Y is two characteristic same points on dispersive dependence, which located on the near by a curve periods (for example, points A and C). On experimental dispersive dependence it is possible to measure a frequency shift changing at transition from
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Fig. 9. Dispersive curve in the transition region from single-mode to double-mode and back for total frequency biases that is specified by average frequency /FS ¼ 1121 Hz.
point Y to point Y 0 F Y 0 F Y F Y 0 F Y tC tA ¼ . LY 0 LY l tY 0 tY
(14)
On the other hand, a frequency shift change is determined by expression: F Y 0 F Y ¼ dF S þ dF x þ dF DY . The neglect change of a frequency bias and an air non-reciprocity at variation of its concentration (dF S ¼ dF x ¼ 0) allows accepting that frequency shift changing is interlinked only with change of air absorption (dF DY a0). The increase in optical perimeter on LY 0 LY ¼ dn S corresponds to increase in absorption on 2da S for both counter-propagating modes. If to accept, that frequency shift changing is bound up only with change of losses then we shall obtain FY0 FY da ¼ 2F DY . dn LY 0 LY
(15)
For gases is realized a ratio between a factor absorption and a refraction index which allows to find its value for air da n 1 F Y 0 F Y tC tA a ¼ ð n 1Þ ¼ . (16) dn 2l F DY tY 0 tY All values in this expression are calculated experimentally on one of dispersive dependence behind exception FDY. For its definition was carried out measurements of a dispersive curve at various total frequency bias /FS. Value for absorption factor a ¼ 3.12 m1 is resulted from experimental data that on four order more than the real value. Thus, it is possible to assert with high probability that derivable frequency shift changing at air molecules concentration variation is not connected to absorption change and, hence, dF X a0. The nature of frequency shift changing at molecules concentration variation can be connected to airflow in the cavity channel with researched air, which arises at heating. The volume with heated up air is connected to the cavity channel a small aperture in its middle, therefore airflow are directed from in the channel centre by its ends. The difference in streams speeds can cause additional frequency splitting of counter-propagating modes due to Fennel entrainment. Let’s estimate a possible difference of streams speeds. In experiment air heating
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continued during T/2E83 s (half-period of electrical signal). For this time occurs frequency shift changing to several conversions in a double-mode operation, and also growth of a streams speeds difference. Process should proceed with difference acceleration equal FY0 FY l . (17) Da 0 tY tY 4ðn 1Þ The estimation for a total frequency bias that is specified by average frequency /FS ¼ 1121 Hz gives value Da ¼ 0.57 mm s2. That should create a speeds difference of opposite directed symmetric streams at the period end exceeding 48 mm s2. Such speeds difference should arise in a tube in diameter some millimeters and long 0.43 m that it is difficult to assume. Also it is found out that with increase in a total frequency bias grows Da though heating remains constant. The most probable explanation of effect of frequency shift changing at perimeter tuning of the cavity ring for one laser wavelength is an optical length change of both beams in a total reflection prism on which the external magnetic field is affected. Air heating in an isolated cavity channel causes to decrease in a refraction index of air on border with a prism it results in change of a refraction angle. It causes the general displacement of beams in the cavity ring, thus will change the optical way and the optical non-reciprocity of the counterpropagating modes, which caused by Faraday effect in a rotary prism. An estimation of this contribution in existing experiment geometry cannot be executed. 5. The conclusion In realization of a suggested method it was used only the measurements executed on the basis of a dispersive curve that is the most informative part of measurements. Measurement of output intensity, pumping value of the ring laser allows adding these measurements in the important parameters thus that having increased capability of a method under the resolution, sensitivity and accuracy. Except for it full measurement of all parameters of the ring laser allows to carry out not only the absorptive and the dispersive analysis of investigated material but also to execute researches on phase non-reciprocity, non-reciprocal losses in the cavity and also gains non-reciprocity of a laser active medium for counter-propagating modes. Work is executed at partial support of the Russian Federal Property Fund, the Grant no. 04-02-16604. References [1] Luk‘yanenko SF, Makogon MM, Sinitsa LN. Laser intracavity spectroscopy. Bases of a method and application. Novosibirsk: Nauka; 1985. [2] Stedman GE. Ring-laser tests of fundamental physics and geophysics. Rep Prog Phys 1997;60:615–88. [3] Baev VM, Latz T, Toschek PE. Laser intracavity absorption spectroscopy. Appl Phys B 1999;69:171–202. [4] Vinogradov SE, Katchanov AA, Kovalenko SA, Sviridenkov AA. Nonlinear dynamics the multimode laser on a crystal with a changeable dispersion of the cavity and sensitivity of intracavity laser spectroscopy. Pis’ma JETF 1992;55:560–3. [5] Sinitsa LN. Laser intracavity spectroscopy of the excited molecules. Opt Atmos Ocean 1997;10:420. [6] Burakov VS, Raikov SN. Laser intracavity spectroscopy: diagnostics of plasma and the spectral analysis (review). J Appl Spectrosc 2002;69:425–47. [7] Grishachev VV, Denisov VI, Zhotikov VG, Kuryatov VN, Nasedkin EF. New potentialities of intracavity spectroscopy of matter using counterpropagating waves in a ring laser. Opt Spectrosc 2005;98:47–52. [8] Privalov VE. Discharge lasers in measuring complexes. Leningrad: Sudostroenie; 1989. [9] Klimantovich Ju L, editor. Wave and fluctuating processes in lasers. Moscow: Nauka; 1974.
Journal of Quantitative Spectroscopy & Radiative Transfer 103 (2007) 302–313 www.elsevier.com/locate/jqsrt
The ultrahigh resolution and sensitivity by spectral measurement on the basis of the ring laser V.I. Denisova, V.V. Grishacheva,, V.N. Kuryatovb, E.F. Nasedkinb, V.G. Zhotikovc a Physics Department, Lomonosov Moscow State University, 1 Vorob‘evy gory, Moscow, 119899, Russian Federation Laser Gyro Division, POLYUS Research and Development Institute, 3 Vvedensky St., Moscow, 117342, Russian Federation c MIPT, 9, Institutskii per., Dolgoprudny, Moscow Region, 141700, Russian Federation
b
Received 28 November 2005; accepted 6 February 2006
Abstract The laser gyroscope technique on the basis of the gas ring laser at use in the spectral dispersive and absorptive analysis of substance allows attaining to the ultrahigh frequency resolution (about 1016 and less). Experimental check of a method is realized on the basis of air research on dispersive dependence. r 2006 Elsevier Ltd. All rights reserved. Keywords: Ultrahigh resolution spectroscopy; Laser intra-cavity spectroscopy; Counter-propagating modes; Ring laser; Laser gyroscope
1. Introduction Designing of new measuring technical equipment with ultrahigh parameters on sensitivity, accuracy is a basis for development of modern experiment. Not always such step can be made, only modernizing existing methods. It occurs more often on crossing ways of two or several measuring techniques. In the present work it is offered to unit advantages of laser intra-cavity spectroscopy and laser gyroscope and to create a unique analysis method of substance with sensitivity and the spectral resolution not having analogues. It is known, that laser intra-cavity spectroscopy [1–3] is one of the most sensitive methods of the spectral analysis of substances. Specifically, absorptive properties are determined on a passage spectrum through the resonator such as Fabri-Perot with the researched substance with its help, lines with absorption factor less than 1011 cm1 are registered in this case [3,4]. The laser intra-cavity spectroscopy technique is widely used at the chemical analysis of gases, chemical reaction kinetics, research of plasma [5,6]. In addition to high sensitivity, distinctive feature of a method is high enough frequency (wavelength) resolution, which reaches 1 MHz in the best optical devices. The achieved resolution can be increased on the 9 orders if for the spectral analysis is using counter-propagating waves of the ring laser which frequencies difference is fixed with accuracy 0.001 Hz as it is carried out in laser gyroscope [2]. Corresponding author.
E-mail address: [email protected] (V.V. Grishachev). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.02.057
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In laser gyroscope is used both passive and active resonators. Ring lasers with active resonators have a several advantages, which allow creating effective measuring systems. But application of active resonators for the spectral analysis of substances has a several difficulties which are connected to influence of the investigated material on the laser generation conditions and interaction of counter-propagating waves. However in some cases such complication of measurements technical equipment can prove to be correct by increase in resolution and accuracy. So carrying out of the spectral analysis on counter-propagating waves allows to achieve the ultrahigh frequency resolution which by relative value reaches 1012 for direct measurements, and at indirect measurements makes about 1016 and less. It is connected with high measurement accuracy of a close frequencies difference of counter-propagating waves in the ring laser and high sensitivity of generation conditions of the ring laser with respect to external influences. In recently published our work [7], it was analyzed join of laser intracavity spectroscopy and laser giroscopy opportunities theoretically as well as it was offered possible ways of realization of a method in practice. Consideration must be given to existence of several parameters by which it can be investigated substance properties in the ring laser and they are connected among themselves by complex correlations. It makes a choice of measured parameters and an establishment of their interrelation of principal problems, which can simplify a problem of a method realization. In the present work in contrast to earlier, we offer one more way of realization and we carry out the analysis of the fulfilled measurements according to new correlations between measured parameters. 2. Principles of measurements In this section, use capability of intra-cavity spectroscopy technical equipment by counter-propagating waves in ring lasers for carrying out differential dispersive and absorptive measurements of substances properties are discussing. And also, the parameters of the ring laser required for similar measurements are considering. 2.1. Differential spectroscopy on the basis of counter-propagating modes Ring lasers find wide application in measuring technical equipment [8]. It is based that in a stationary singlemode operation of the ring laser can be generated two contra-directional running waves with frequencies n+ (the right mode) and n (the left mode) [9]. Frequencies difference of counter-propagating modes F ¼ (n+n) can be precisely measured at their optical convergence that provides the ultrahigh spectral resolution. Beating frequency F of two optical modes is frequency deviation (shift). The placement of researched substance within the resonator allows realizing its spectral analysis. Research dispersive and absorptive material properties is reduced to measurement except for frequencies counter-propagating modes n+, n and their intensities I+, I on an input in researched substance, also intensities I þ DI þ ; I DI on an output from one (Fig. 1). Measurement of output parameters of the ring laser beam convergence allows determining five parameters,
Fig. 1. The ring laser optical arrangements with the investigated material (X).
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which are connected to the investigated material properties: Z ¼ ðI =I þ Þ;
kþ ¼ ðDI þ =I þ Þ;
k ¼ ðDI =I Þ;
F ¼ ðnþ n Þ;
n ¼ ðnþ þ n Þ=2.
On the basis of these parameters it is possible to carry out the spectral analysis of substance with the ultrahigh resolution. 2.1.1. The dispersive analysis Total frequency shift in the ring laser is defined by the non-reciprocal components of the laser. It can vary in limits from 102 to 105 Hz depending on active element and non-reciprocal components parameters. In the value of total frequency shift (F) is brought with the contribution not only the researched non-reciprocity material but also its dispersion that is connected to distinction of a refractive index for counter-propagating modes. As a rule, it is possible to neglect material non-reciprocity in most cases. Otherwise additional researches are carried out with the purpose of separation of these two contributions to a frequency shift as, for example, it is offered in work [7]. Let the researched environment with geometrical length S is placed in the ring resonator for which the refractive index for the right mode is equal n+ (on frequency n+), and for the left mode—n (on frequency n), then the size of the dispersion contribution of a refractive index in frequency deviation will be given by expression F X ¼ nðnþ n Þ
S , L
(1)
where L—average by counter-propagating modes of the cavity optical length. Allocation of the investigated material contribution in frequency shift allows determining a dispersion of refraction index by the formula qn nþ n 1 L F X ¼ ¼ . (2) qn nþ n n S F The method resolution is estimated in order value it is equal nð@n=@nÞ ¼ 105 2106 . It corresponds to the minimal measured change of beat frequency for contra-directional modes, which is equaled 102 Hz at the basic difference frequencies 105 Hz and at relative length of a researched part equal (L/S) ¼ 10. Resolution of the given method on some orders can be exceeding other methods. 2.1.2. The absorptive analysis Relative change of output laser intensity is connected with total absorption in the investigated material: kþ ¼ aþ S; k ¼ a S, here a+ and a—linear absorption factor of the investigated material on frequencies n+ and n, correspondingly. The dispersion of an absorption factor in substance can be calculated by the formula þ qa aþ a k k ¼ þ ¼ . (3) qn n n 2S F The estimation of a method resolution shows an advances possibility by absolute value less (@a/@n) ¼ 1011 s/cm without use of measurements special ways and data processing. 2.1.3. Modulation spectroscopy using Increase of method sensitivity can have been achieved by modulation of the investigated material parameters (for example, density change, external fields influence, etc.), that allows determining the material properties connected to these parameters. For example, in gases the linear absorption factor is determined by interaction cross-section between a photon and a molecule s+ and s, then aþ ¼ sþ C n S a ¼ s C n S , where Cn—molecules concentration. It is admitted to estimate a dispersion of interaction cross-section of a photon with a molecule on optical frequency n by the expression þ qs sþ s k k ¼ þ ¼ . (4) qn n n 2S2 C n F
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Measurements carrying out at various molecules concentration allows excluding necessity for accurate concentration measure, if to apply its relative change only. Modulation spectroscopy methods can increase essentially measure accuracy by means of the ring laser. 2.2. Measured parameters of the ring laser For realization of the differential spectral analysis of substance on the basis of counter-propagating close modes in the ring resonator is required to measure the output laser parameters, to determine its design features. 2.2.1. Laser intensity One from principal’s parameters influencing on the measurement accuracy is output laser intensity. On an output of the ring laser is registered the beam convergence intensity (IL) from time (t) which without taking into account an initial phase is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5) I L ¼ 2I þ 4I 2 ðDIÞ2 cosð2p F tÞ. Here we have introduced symbols for average intensity of counter-propagating modes on an output of the ring laser I ¼ ðI þ þ I Þ=2 and their difference DI ¼ ðI þ I Þ. It is necessary to note that counter-propagating modes intensities I+ and I by absolute value and relative value depend on an output mirror position in regard to active medium and investigated material. Further the same symbols will be used for definition of contradirectional wave’s intensities on all along cavity. In spite of the fact that these values are function from position of an exit point, but the knowledge of the ring laser characteristics can be determining contradirectional wave’s intensity on all points of the cavity uniquely. Therefore in each particular case they are defined by the self-procedure or their relative change obtain that also allows avoiding a pinpointing for light outlet. From measurements of laser intensity IL can be determined maximal Imax and minimal Imin of a signal. From these values it is possible to express I ¼ 0:25ðI max þ I min Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jDI j ¼ I max I min ;
(6)
on which the basic are defined absorption of the investigated material. 2.2.2. Active medium monitoring Work of the ring laser depends on the active medium parameters. Which can be characterized in gain factor (a threshold gain) G+ and G (G þ 0 and G 0 ) of the active medium for the right and left counter-propagating modes, respectively. In the ring gas laser with high-frequency discharge pumping, gain factors G+ and G depend on an excess voltage (UU0) over the high-frequency threshold voltage (U0), which is given by factor g¼
U U0 . U0
(7)
These measured parameters characterize a high-frequency pumping level of the active medium. The gain factors has non-linear dependence from g, but at small voltage variations dependence G(g) can be linearized. Achievement of the greatest stability of work occurs at ðdG =dgÞ ! 0, when small voltage changes poorly influence on the gain factor. 2.3. Dispersive curve of the ring laser The basic dependence on which investigated material are probed this is the dispersive curve—a dependence of counter-propagating modes frequency shift F from optical length L of the ring laser perimeter. Therefore we shall discuss what information about the investigated material can be included here.
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2.3.1. Frequency shift features Frequency deviation in the ring laser is defined by its design and component parts so, that it is given by semi-empirical expression [8] F ¼ FS þ FD þ FX F S ¼ F ac þ F bc þ F cc ¼ const F D ¼ bas I þ bad DI ¼ F D ðI; DIÞ.
ð8Þ
Here for a constant frequency shift are entered designation—FS that we shall denominate a full frequency bias. Next elements are part of full frequency bias: F ac —constant frequency shift of the counter-propagating modes connected to the active medium; F bc —constant by value deviation of the frequency caused by a special controlled non-reciprocal element which we shall name a frequency bias; F cc —constant frequency shift connected to a non-reciprocity under the frequency lock-in effect. And also the variable component of a frequency shift FD is entered which depends from bac and bad —material parameters of the active medium and before entered I ¼ ðI þ þ I Þ=2—average intensity of the counter-propagating modes at the point of entry the active medium and DI ¼ ðI þ I Þ—its difference. For the contribution of the investigating material to frequency deviation of counter-propagating modes is kept designation FX which characterizes not only the dispersive contribution, but also connected with non-reciprocity of the investigated material. 2.3.2. Conversion domain At the fixed optical length L of the ring laser perimeter, frequencies n of generated longitudinal modes should correspond to conditions (Fig. 2) c n ¼ nN ¼ N dn ¼ N ; nA pnpnB , (9) L where c—light speed in vacuum, N—the whole positive number equal to wavelength number being stacked on average optical length L of the cavity perimeter, nA 2nB —frequency band of an gain envelope on a lasing threshold level for the active medium. These conditions correspond to that frequencies should be in accord with resonant frequency (the first condition) and to fall within frequency band of a gain envelope, where gain exceeds losses (the second condition). Let us enter the relation Dn nB nA ¼ L, (10) dn c here Dn ¼ nA nB —a frequency bandwidth of a gain envelope on a lasing threshold level, dn ¼ ðc=LÞ— difference frequency between two adjacent longitudinal modes. Parameter of M shows in how many time frequency bandwidths more than drift between the adjacent longitudinal modes. If Mo1 then in the laser generation only one longitudinal mode is possible. At values 1pMo2 in the laser generation as one and two longitudinal modes is possible. In a case, when value MX2 then the number of generated modes can achieve three and more. For stable work of the ring laser are used the single-mode operation and small bandwidth where double-mode operation is possible. The laser operation with parameter Mo1 is not used as in this case there should be frequency bands in which a generation breakdown happen. Value of Dn is determined by an active ions type, mix proportion of breakdown plasma and a pumping level. The choice of a pumping level generation close to a threshold allows receiving single-mode operation. M¼
2.3.3. Typical dependence Tuning cavity perimeter one can to scan all frequencies band from nA to nB of a gain envelope, thus frequency deviation F of contra-directional modes forms a dispersive curve. The dispersive curve type depends on the laser design, the active medium properties, a pumping level and other parameters and ones the distinctive feature of the ring laser. In case of our ring laser the typical dispersive curve looks like as is shown on Fig. 3. On a dispersive curve can single out two characteristic domains (I and II). The first domain is in accord with single-mode operation, and the second-double-mode operation. Extent of the second domain l can vary a pumping level changing. When parameter M 1, then domain spread (I) tent to zero, also can be
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Fig. 2. Gain envelope of the ring laser for single-mode and double-mode. G(n)—gain envelope of the ring laser, nN , nNþ1 , nNþ2 — longitudinal mode of the ring laser, dn—frequency difference between the near longitudinal modes, Dn ¼ nA nB —gain envelope width on a threshold gain G0, F—frequency shift between the near counter-propagating modes (+,–).
converted to small domain extent with conversion from one operation to another by abruptly. On a dispersive curve can allocate characteristic points (A, B, C), in which there is a change of generation conditions by abruptly and a conversion from single-mode operation to double-mode (A, C) and back (B). Conversion domain near to a point A is characterized by achievement at perimeter tuning (L decreases) of generation frequencies for one longitudinal modes of an gain envelope edge that corresponds to a þ condition nN ¼ nN þ ðF =2Þ ¼ nA ; ðAÞ , nþ N —frequency of one contra-directional mode (+) of a longitudinal mode N (Fig. 4). It corresponds to the beginning of conversion in a double-mode operation. The laser completely will switch in double-mode operation by all counter-propagating modes when other contra directional wave (–) will achieve gain envelope edge nN ¼ nN ðF =2Þ ¼ nA ; ðAÞ . The further perimeter tuning will result in achievement by the first longitudinal mode of the second gain envelope edge nþ ðBÞ . The laser completely will switch in a single-mode operation, when Nþ1 ¼ nNþ1 þ ðF =2Þ ¼ nB ; n ¼ n ðF =2Þ ¼ n ; ðBÞ . The subsequent perimeter decrease will result in conversion repeat from a Nþ1 B Nþ1 single-mode in double-mode only for the following longitudinal mode— nN1 ¼ nA ; ðCÞ . Frequency shift is decreasing in the dispersive curve domain (II) since modes intensity will have fallen. It is observed because to occur a redistribution pumping between two pairs longitudinal modes (each mode includes pair
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Fig. 3. Typical plot of frequency shift F according to perimeter L (a dispersive curve) of KM-model ring laser. A, B, C—typical points of a dispersive curve, I—domain of single-mode operation, II—domain of double-mode operation, l—width domain of II, l—wavelength of a generated light.
Fig. 4. The ring laser spectrum on various domains of a dispersive curve. A–B—double-mode operation, B–C—single-mode operation.
counter-propagating waves). Falling take place steadily in a case enough the big domain extent l (II) and abruptly at small spread as instability small domain. The ratio between extents of these domains is determined by parameter M. If ME1.5, then both domains have equal extents and the dispersive curve to become close to a symmetric type. The form of a gain envelope defines the form of a dispersive curve on section A2B and B2C as mode intensity depends on position of a
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generated longitudinal mode concerning a gain envelope. It causes change of frequency shift F in turn. Also, the form of a dispersive curve depends on how there is a perimeter tuning. 2.3.4. Particular points On a dispersive curve can select three characteristic frequencies shift: the maximal frequency (FC, FA, y), the minimal frequency (FB, y) and mean frequency (/FS). First two are defined as extremely probable values of frequencies shift at smooth perimeter tuning and last frequency as mean value on all potential frequencies on one tuning period. Taking into account connection of frequency deviation with the cavity properties, the active medium and the investigating material it is possible to receive expressions for distinctive values F Y ¼ F S þ F X þ F DY ;
F DY ¼ bac I Y þ bad DI Y ;
hF i ¼ F S þ F X þ hF D i;
hF D i ¼ bac hIi þ bad hDIi,
Y ¼ A; B; C . . . (11)
here I A ; I B ; I C . . . and DI A ; DI B ; DI C ; . . .—values of a half-sum intensities of counter-propagating modes in characteristic points Y ¼ A,B,Cy and differences corresponding to them, hIi; hDIi—mean intensities on the period of a dispersive curve and their difference. Characteristic points are determined on breaks on a dispersive curve. Value of frequency shift component FS (the full frequency bias) whit perimeter tuning does not change. The contribution FX of the investigated material can be tested by influence on one. Generally, mean values are determined by a dispersive curve symmetry, which is defined by gain envelope symmetry of the active medium in turn, so hIi ¼ 0:5ðI A þ I B Þ ¼ const, hDIi ¼ 0:5ðDI A þ DI B Þ ¼ const.
ð12Þ
Mean values remain constants at change of a full frequency bias F S , as they are assigned by pumping level. From characteristic values of a dispersive curve can compose an expression F A þ F B ¼ 2hF i, if to take into account dependence symmetry. The obtained dependences /FS from F S allow calculating the contribution /FD S in frequency shift F at given pumping. That allows excluding influence of modes intensity. Full tuning loop from a point A–C corresponds to perimeter change on one wavelength. The relation of domain extents with a double-mode spectrum (l) by the repetition period (l) of a dispersive curve is given by formula hF i nA l 1oMp2 , (13) l ¼ nB M 1 þ dn ; if small values of the order ðF =2nA;B Þ 1 nA =nB pðF =dnÞ 1 may be neglected. As evident from its expression relative extent of a double-mode operation is function of frequency shift. At values ME1 small changes of the relation (l/l) are proportional to change of mean frequency shift /FS. At perimeter tuning, value M changes, that enables to calculate from dispersive curve of value Dn ¼ nA nB depending on pumping level g. 3. Approbation of the ring laser as measuring device Some laws revealed above in laser emission behavior with the investigating material have been used for carrying out of experimental researches with the purpose of approbation of new measuring system with the ultrahigh resolution and accuracy. 3.1. Laboratory research complex on the basis of the ring laser Performance check of a method was executed on the basis of fine differential dispersive and absorptive the analysis of air properties on the modernized ring laser of KM-model (POLYUS RDI, Russia). Four total reflection prisms for one pair contra-directional longitudinal cavity resonance have formed the laser ring cavity. The cavity had a mono-block glass-ceramics design in the square form with the length side 0.43 m and the total length perimeter 1.72 m within which channels have been drilled. The active medium was capturing one channel with helium–neon mix of composition He3/Ne20/Ne22 ¼ 14/1.1/0.9 and the one was pumped up
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by the cross high-frequency discharge. Laser generation stability was provided with pumping stability which was controlled by means of the voltage of the high-frequency discharge generator with accuracy of 0.1%. The channel opposite to the channel with the active medium was pumped out under rough vacuum. Other channels have been sealed off and filled with air. In one of these channels the optical cavity perimeter was reconstructed by pressure change (heating or change of volume). The ring cavity has Q-factor not less than 108. The frequency lock-in bandwidth signified smaller 200 Hz. Introduction in the optical cavity arrangement of additional optical elements will be inevitable to cause reduction of Q-factor of laser and increase of frequency lock-in bandwidth therefore for measuring system approbation we were used the optical cavity arrangement as is. The important system performance of experimental setup is the lasing counter-propagating modes difference, which named a full frequency bias (FS). As it was marked, it depends on active medium (Fac), a frequency lock-in effect (Fcc) and a special non-reciprocal element (Fbc). As a special non-reciprocal element frequency bias it was used total reflection prism on which the external magnetic field was imposed. In the issue of Faraday effect in prism have to be arising non-reciprocity for counter-propagating modes. Thus, the prism in an external magnetic field operates as Faraday cell. The external magnetic field was created by constant magnets with stray fields in the total reflection prisms within the limits of 0.01–0.02 T. Prisms have been made of quartz at which Verde constant has value of the order 5 1012 rad=T m. Light falls on a prism face under Brewster angle and in prisms linearly polarized light is propagated. The external magnetic field projection distinct from zero onto a light wave line in a prism results in polarization twisting and ellipticity formation of plane-polarized wave. Elliptic polarization twisting in a prism for counter-propagating modes has an opposite rotational direction that causes non-reciprocity in speeds of counter waves and as consequence frequencies shift. Orientating various ways permanent magnets concerning a beam it is possible to receive an initial total frequency bias from 200 up to 1400 Hz at the given magnets. In the experimental setup counter-propagating modes are extracted from cavity through one of total reflection prisms and by optical elements is re-united on a photo-detector. The electric signal from a photodetector is processed, result of that is frequency shift F. Accuracy of measurement of frequency shift achieved in a maximum 0.01 Hz. During researches it was not carried out measurements of laser output power.
3.2. Measurements Experimental check of measuring equipment performance capabilities was carried out on the basis of spectral researches of air under standard conditions, which placed in one from ring cavity channels. Researches have consisted in obtaining of dispersive dependence for various values of an initial total frequency bias produced by a magnetic field. Estimation of a total frequency bias value (FS) it was made by averaging of frequency shift received during measurements at a constant external magnetic field, that is in amount hF i which is determined by the formula (11). In the beginning, rough tuning perimeter was made by change of air volume in the measuring channel for the purpose of a selection of a working point. Further, smooth tuning optical perimeter of the cavity was carried out by heating from the sinusoidal voltage generator with frequency 0.003 Hz. Heating power changed with the double frequency in the sine curve therefore into dependences can allocate linear change portions of temperature. Here an optical perimeter length was linearly proportional to time heating. Typical experimental dispersive dependences are submitted on Figs. 5–8. The pumping power was chose in such a way that the domain with a double-mode operation practically was absent—it is case when ME1. It especially well is visible on dispersive dependence with a domain only conversion (Fig. 9). In the conversion domain the increase in instability of frequency shift is observed, but its type repeats from one conversion to another. In a conversion point an absolute value of frequency shift is precisely determined such that from the left of conversion point verge towards nA , and from the right—nB . And the nearer to a conversion point, the more exact a frequency shift is coincides with nA or nB . Thus, the peak points on dispersive curve (Figs. 5–8) are fixed by wave frequency (wavelength) about accuracy not smaller, than accuracy of measurement of frequency shift which compiling 0.01 Hz. It allows asserting about realization of the ultrahigh spectral resolution about 1016.
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Figs. 5–8. Dispersive curves of the ring laser for various total frequency biases that is specified by average frequency /F S.
Thus, on the one hand, achievement of peak points on dispersive curve (points A and C on Fig. 3) occurs at laser generation on wave frequencies nA 0:5 F A , nC 0:5 F C and nA ¼ nC , determined with accuracy 0.01 Hz. On the other hand, frequencies shift F A and F C for the nearest-neighbor peak points differ, and their difference F A F C is connected to change of air properties in the cavity channel. In our case one is connected with change of air density. For the total frequency bias characterized by average frequency shift /FS ¼ 1121 Hz, difference equals to FA–FCE6 Hz. 4. Discussion The submitted measurements have a complicated interpretation that is bound up with influence on a dispersive curve of many parameters. Therefore at the results analysis it is necessary to allocate the physical phenomena rendering the basic influence on results of measurements. At perimeter tuning by heating of researched gas (air), two nearest-neighbor peak (points A and C on Fig. 3) of a dispersive curve is differing increase in the optical cavity length on l that corresponds to a time interval on an experimental curve equal (tCtA). It allows finding experimental value of optical length variation 0 with temporal changes dt: dL ¼ ldt=ðtC tA Þ. Let Y and Y is two characteristic same points on dispersive dependence, which located on the near by a curve periods (for example, points A and C). On experimental dispersive dependence it is possible to measure a frequency shift changing at transition from
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Fig. 9. Dispersive curve in the transition region from single-mode to double-mode and back for total frequency biases that is specified by average frequency /FS ¼ 1121 Hz.
point Y to point Y 0 F Y 0 F Y F Y 0 F Y tC tA ¼ . LY 0 LY l tY 0 tY
(14)
On the other hand, a frequency shift change is determined by expression: F Y 0 F Y ¼ dF S þ dF x þ dF DY . The neglect change of a frequency bias and an air non-reciprocity at variation of its concentration (dF S ¼ dF x ¼ 0) allows accepting that frequency shift changing is interlinked only with change of air absorption (dF DY a0). The increase in optical perimeter on LY 0 LY ¼ dn S corresponds to increase in absorption on 2da S for both counter-propagating modes. If to accept, that frequency shift changing is bound up only with change of losses then we shall obtain FY0 FY da ¼ 2F DY . dn LY 0 LY
(15)
For gases is realized a ratio between a factor absorption and a refraction index which allows to find its value for air da n 1 F Y 0 F Y tC tA a ¼ ð n 1Þ ¼ . (16) dn 2l F DY tY 0 tY All values in this expression are calculated experimentally on one of dispersive dependence behind exception FDY. For its definition was carried out measurements of a dispersive curve at various total frequency bias /FS. Value for absorption factor a ¼ 3.12 m1 is resulted from experimental data that on four order more than the real value. Thus, it is possible to assert with high probability that derivable frequency shift changing at air molecules concentration variation is not connected to absorption change and, hence, dF X a0. The nature of frequency shift changing at molecules concentration variation can be connected to airflow in the cavity channel with researched air, which arises at heating. The volume with heated up air is connected to the cavity channel a small aperture in its middle, therefore airflow are directed from in the channel centre by its ends. The difference in streams speeds can cause additional frequency splitting of counter-propagating modes due to Fennel entrainment. Let’s estimate a possible difference of streams speeds. In experiment air heating
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continued during T/2E83 s (half-period of electrical signal). For this time occurs frequency shift changing to several conversions in a double-mode operation, and also growth of a streams speeds difference. Process should proceed with difference acceleration equal FY0 FY l . (17) Da 0 tY tY 4ðn 1Þ The estimation for a total frequency bias that is specified by average frequency /FS ¼ 1121 Hz gives value Da ¼ 0.57 mm s2. That should create a speeds difference of opposite directed symmetric streams at the period end exceeding 48 mm s2. Such speeds difference should arise in a tube in diameter some millimeters and long 0.43 m that it is difficult to assume. Also it is found out that with increase in a total frequency bias grows Da though heating remains constant. The most probable explanation of effect of frequency shift changing at perimeter tuning of the cavity ring for one laser wavelength is an optical length change of both beams in a total reflection prism on which the external magnetic field is affected. Air heating in an isolated cavity channel causes to decrease in a refraction index of air on border with a prism it results in change of a refraction angle. It causes the general displacement of beams in the cavity ring, thus will change the optical way and the optical non-reciprocity of the counterpropagating modes, which caused by Faraday effect in a rotary prism. An estimation of this contribution in existing experiment geometry cannot be executed. 5. The conclusion In realization of a suggested method it was used only the measurements executed on the basis of a dispersive curve that is the most informative part of measurements. Measurement of output intensity, pumping value of the ring laser allows adding these measurements in the important parameters thus that having increased capability of a method under the resolution, sensitivity and accuracy. Except for it full measurement of all parameters of the ring laser allows to carry out not only the absorptive and the dispersive analysis of investigated material but also to execute researches on phase non-reciprocity, non-reciprocal losses in the cavity and also gains non-reciprocity of a laser active medium for counter-propagating modes. Work is executed at partial support of the Russian Federal Property Fund, the Grant no. 04-02-16604. References [1] Luk‘yanenko SF, Makogon MM, Sinitsa LN. Laser intracavity spectroscopy. Bases of a method and application. Novosibirsk: Nauka; 1985. [2] Stedman GE. Ring-laser tests of fundamental physics and geophysics. Rep Prog Phys 1997;60:615–88. [3] Baev VM, Latz T, Toschek PE. Laser intracavity absorption spectroscopy. Appl Phys B 1999;69:171–202. [4] Vinogradov SE, Katchanov AA, Kovalenko SA, Sviridenkov AA. Nonlinear dynamics the multimode laser on a crystal with a changeable dispersion of the cavity and sensitivity of intracavity laser spectroscopy. Pis’ma JETF 1992;55:560–3. [5] Sinitsa LN. Laser intracavity spectroscopy of the excited molecules. Opt Atmos Ocean 1997;10:420. [6] Burakov VS, Raikov SN. Laser intracavity spectroscopy: diagnostics of plasma and the spectral analysis (review). J Appl Spectrosc 2002;69:425–47. [7] Grishachev VV, Denisov VI, Zhotikov VG, Kuryatov VN, Nasedkin EF. New potentialities of intracavity spectroscopy of matter using counterpropagating waves in a ring laser. Opt Spectrosc 2005;98:47–52. [8] Privalov VE. Discharge lasers in measuring complexes. Leningrad: Sudostroenie; 1989. [9] Klimantovich Ju L, editor. Wave and fluctuating processes in lasers. Moscow: Nauka; 1974.