Wills Plasma Physics Department ER.67 I A 337 ym HCN
Transcript Of Wills Plasma Physics Department ER.67 I A 337 ym HCN
ER.67
I
Wills Plasma Physics Department
1
A 337 ym HCN LASER INTERFEROMETER FOR PLASMA DIAGNOSTICS
N.R. Heckenberg, G.D. Tait and L.B. Whitbourn
June, 1972
Submitted for publication to Journal of Applied Physics.
ABSTRACT
A selfmodulating laser interferometer incorporating a 337 ym HCN laser is described and a simple theory of its operation is developed. This theory shorrs that the long lifetimes of levels involved in molecular laser transitions may cause a peaked frequency response in such an instrument. Experimental observations indicate a flat frequency response up to the maximum attained fringing rate of 10 kHz. The interferometer was used to make electron density measurements in the range 2 x 1012 cm~3 to 1014 cm— 3 for an effective path length of 150 cm in a decaying helium plasma. The interferograms obtained show a marked variation in fringe amplitude which is attributed to the refractive effects of electron density gradients in the plasma.
1. INTRODUCTION
In this paper we describe a far infrared selfmodulating laser interferometer which has been used to measure electron densities in a decaying plasma. The instrument described employs a 337 urn HCN laser permitting measurement of electron densities up to the 337 um cutoff density of 9.1 x 1015 cm— 3, while the lower limit depends on the plasma path length and the minimum detectable phase change (e.g. 3.1 x 1012 cm— 3 for ir phase change over a 1 m path length). Although other technique . can be used to measure electron densities in this range, interferometry using the 337 um HCN laser line offers a number of advantages For example, while conventional 2 mm microwave methods can be used to measure densities up to cutoff at 3 x 1014 cm3, diffraction effects limit operation to short path lengths. Decreasing the wavelength to 337 um can provide acceptable spatial resolution over longer pathlengths without seriously reducing sensitivity. Even better spatial resolution may be achieved by using multipass techniques in conjunction with visible and infrared lasers. However for comparable sensitivity, the effects of vibration are very much worse than at 337 um and interpretation of the results may be complicated by the contribution of neutral species to the refractive index of the plasma. The most common alternatives to interferometry for measuring electron densities in transient plasmas (not in LTE) are Sterk broadening of hydrogen (or hydrogen impurity) lines
and Thompson scattering. Even in relatively cold (less
than 1 eV) plasmas, Stark broadening is useful only above •v 3 x 1014 cm—3 . Thompson scattering, which is in any
2.
case rauch more complicated than laser interferometry, can only be extended below 1014 cm3 with extreme difficulty.
The operating principle of the selfmodulating laser interferometer (hereafter abbreviated to "laser interferometer") is simple. The output beam is passed through the plasma under study and reflected back on itself into the laser (Fig. 1 ) . The phase and amplitude of this return beam control the amplitude of the laser output, which is monitored by a detector. The plasma electron density can be deduced from the phase shift thus
1 ?3 measured. A number of workers '"' have operated such systems incorporating CW HeNe lasers operating in the. visible and infrared regions. The greatly increased sensitivity of plasma interferometry at 337 ym makes a laser interferometer based on a farinfrared HCN laser an attractive proposition, particularly in view of the simplicity and ease of operation of laser interferometers. As well as increasing sensitivity, the use of this long wavelength reduces problems of alignment and vibration, especially so in comparison with the techniques 2 '3'4used to extend the sensitivity of interferometry at shorter wavelengths.
It is possible to describe the laser interferometer on the basis of a simple theory which leads to some interesting conclusions. It is shown in Sec. 2 that under certain circumstances the laser interferometer can have a peaked frequency response. The factors affecting the shape and size of the laser interferometer fringes are also discussed. The fringe shape is of importance in the interpretation of fractional fringe measurements , while the possibility of a nonlinear dependence of fringe size on return beam amplitude
3.
limits the use of laser interferograms for determination of the plasma temperature from the magnitude of the attenuation. In practice, however, attenuation measurements may well be rendered worthless by the refractive effects ofplasma density gradients (Sec. 5) sothat this is not considered a serious disadvantage.
2. THEORY OF THE LASER INTERFEROMETER
To understand the behaviour of the laser interferometer, we consider first the factors which determine the operating point ofthe laser itself. The effect of returning part of the output beam into the laser cavity can then be treated as a modulation ofthe cavity loss.
The active medium of the laser is characterized by a gain per unit length, g, resulting from stimulated emission. That is
where I is the radiation intensity in the cavity. The gain falls off with increasing intensity because of reduction qf the population inversion (saturation). The steady state intensity is determined by the condition that the energy gained per pass is just equal to the energy lost. Consider ing one round trip of the cavity, this implies
Aexp(2gil)  1 = 0 ,
(24
where ./vis the (intensity) attenuation factor for a round trip and I is the length of the active medium.
Saturation Characteristic
The functional form of the saturation characteristic depends on a number of factors, but there is strong justification for writing it as '
g(I) =
«
.
(3)
(l + 3 D r
The unsaturated gain per unit length, a» and the saturation parameter, 3, are determined by the excitation and deexcitation rates of the energy levels involved in the laser transition. The value taken by the exponent, r, depends on the processes which broaden the corresponding spontaneous emission line. In the case of homogeneous broadening (e.g. when pressure broadening dominates), the radiation is emitted by an ensemble of a priori indistinguishable atoms (or molecules). Each emitter can contribute to the gain at any frequency, with a probability distribution which is simply the line shape function. Saturation then reduces the gain at all frequencies without affecting the spontaneous line shape. In so far as they overlap spatially in the active medium, different modes must compete directly with each other for gain, and normally only a single dominant mode oscillates to the exclusion of all others. Saturation of a homogeneously broadened transition is described by eq. (3) with r = 1.
On the other hand, when a line is inhomogeneously
broadened (e.g. when Doppler broadening dominates), only
that fraction of the population whose natural emission 7
frequency is shifted to within about the natural linewidth of the laser frequency can take part in stimulated emissiop.
Gain saturation involves the removal of these atoms from the
population distribution, "burning a hole" in the gain vs. frequency curve6.'8'9 At low intensities, the hole width
7 equals the natural linewidth and the hole widens as the intensity increases, due to induced transitions decreasing the lifetimes of the laser levels. Saturation of a single mode laser is now described by eq. (3) with r = \. However multimode operation is not only possible with inhomogeneously broadened transitions (as different modes can burn separate holes in the gain curve), but is quite normal for optical lasers unless special precautions are taken. Although only the onset of saturation in multimode lasers is well understood theoretically , there is evidence for the general applicability of the saturation formula (3), given appropriate choice of r. For example in the case of the multimode HeNe laser, it has been argued that the total intensity of
all modes tends to saturate the gain as if the laser transition was homogeneously broadened 11 '12 . Since pressure broadening dominates in the laser used in the experiments described in this paper (Sec. 3)^ we employ the form of eq. (3) appropriate to homogeneous broadening (r = 1) in subsequent discussion. The results obtained are readily extended to the general case of arbitrary r.
Loss Modulation by Return Beam
In what follows, it is convenient to write the attenuation factor in the form
A s (i <£) (i 
(4)
where o is the fraction of stored energy lost per round trip
I
i r
hi
of the cavity as output (i.e.^7 is the transmissivity of the

laser output mirror) and dL represents all other losses.
I
6.
From eqs. (2)  (4), the quiescent cavity intensity, IQ is given by
(1 <£) (11) exp L I"* 1  1 = 0 .
(5)
*•
o*
To simplify analysis and avoid some complications discussed
below, we now assume that the losses are small (say
oC +0" < 20%). This assumption is valid for the 633 nm and
1.15 um HeNe laser transitions which have been used
extensively for laser interferometry but not for the very high gain 3.39 um line13 . It is also valid for the 337 Urn
HCN laser used '.a our experiments. When the. losses are
small, the exponential term in eq. (5) must be close to
unity and so we can write
(6)
31o
When part of the output beam is returned into the laser, the losses represented by df are modified according to the phase and amplitude of the return beam. For the interferometer shown in Fig. 1, the intensity returned into the cavity may be written as e J I. The factor t7 arises
2 from the two passes made through the output mirror and e represents the additional attenuation of the return beam due to diffraction loss, reflection from optical components in the external path, etc. In the absence of the return beam, the amplitude of the wave reflected by the output mirror (amplitude reflectance r) is r/T. If the return beam undergoes a total phase change, the effective reflected amplitude becomes (coherent sum)
A = r/l + e7/T ej* ,
(7)
7.
and the corresponding intensity is
A2 = r2l (1 + 2e/jT cos ()>) ,
(8)
where the term in 5 has been neglected r.d r has been taken as unity in the term containing^. We now discuss the implications of eqs. (7) and (8).
The phase variation of the reflected amplitude, A,
is usually ignored in discussions on laser interferometry.
It has; the effect of varying the apparent length of the
cavity by a fraction of a wavelength, thereby pulling the
laser operating frequency. This has two consequences.
Firstly, the phase shift in the external path is in turn
changed, requiring that the equations above be reconstructed,

allowing for pulling, to produce a closed system for

solution. As this procedure is algebraically involved it
will not be attempted here. It suffices to establish
conditions under which pulling can be ignored. From eq.
(7), the maximum phase change in the effective reflected
amplitude is approximately e£T radians in which case the
frequency is pulled by an amount
6o) = ^ etT ,
(9)
where 6u is the shift in the laser wave angular frequency, (0,
and 2£/c is the transit time for radiation around the cavity
(for convenience, the cavity length is taken to be the same
as the length, A, of the active medium). The phase of the
return beam will then be changed by
:
6() = 5w — , c
(10)
8.
where L is length of the external return path. Clearly from eqs. (9) and (10), <5 will be insignificant if eO" L/& is small, as is the case for the HCN laser interferometer described in Sec. 5. Note that although such pulling can alter the fringe shape, the positions of maxima and minima remain unchanged since there :is no pulling for = 0 or IT.
The other effect of pulling is to shift the laser operating frequency on the gain profile. In the case of optical lasers where there are normally several longitudinal cavity modes within the gain width this is of little significance. In contrast, the gain width of the HCN laser is less than the spacing between low loss modes, as demonstrated by the fact that the laser can be extinguished by manipulation of the cavity length15 . However, from eq. (9), if e!J was as large as 0.1, the pulling would only be ± 1.3 MHz for our laser (£ = 1.8 m ) , compared with a gain width of about 13 MHz (see Sec. 3). We conclude that the effects of phase variation of the reflected amplitude are small in the case of our system and can be neglected
Nevertheless, a small change in the magnitude of the effective reflected amplitude results in a relatively large change in the total loss of the laser cavity and will cause significant variation of the laser output level. It is clear from eq^ (8) that to account for this effect the attenuation factor, J\, (given by eqn. (4)) must be replaced by
cos) . (11)
Using the same arguments as led to eq. ( 6 ) , we obtain
1 + 31
(12)
I
Wills Plasma Physics Department
1
A 337 ym HCN LASER INTERFEROMETER FOR PLASMA DIAGNOSTICS
N.R. Heckenberg, G.D. Tait and L.B. Whitbourn
June, 1972
Submitted for publication to Journal of Applied Physics.
ABSTRACT
A selfmodulating laser interferometer incorporating a 337 ym HCN laser is described and a simple theory of its operation is developed. This theory shorrs that the long lifetimes of levels involved in molecular laser transitions may cause a peaked frequency response in such an instrument. Experimental observations indicate a flat frequency response up to the maximum attained fringing rate of 10 kHz. The interferometer was used to make electron density measurements in the range 2 x 1012 cm~3 to 1014 cm— 3 for an effective path length of 150 cm in a decaying helium plasma. The interferograms obtained show a marked variation in fringe amplitude which is attributed to the refractive effects of electron density gradients in the plasma.
1. INTRODUCTION
In this paper we describe a far infrared selfmodulating laser interferometer which has been used to measure electron densities in a decaying plasma. The instrument described employs a 337 urn HCN laser permitting measurement of electron densities up to the 337 um cutoff density of 9.1 x 1015 cm— 3, while the lower limit depends on the plasma path length and the minimum detectable phase change (e.g. 3.1 x 1012 cm— 3 for ir phase change over a 1 m path length). Although other technique . can be used to measure electron densities in this range, interferometry using the 337 um HCN laser line offers a number of advantages For example, while conventional 2 mm microwave methods can be used to measure densities up to cutoff at 3 x 1014 cm3, diffraction effects limit operation to short path lengths. Decreasing the wavelength to 337 um can provide acceptable spatial resolution over longer pathlengths without seriously reducing sensitivity. Even better spatial resolution may be achieved by using multipass techniques in conjunction with visible and infrared lasers. However for comparable sensitivity, the effects of vibration are very much worse than at 337 um and interpretation of the results may be complicated by the contribution of neutral species to the refractive index of the plasma. The most common alternatives to interferometry for measuring electron densities in transient plasmas (not in LTE) are Sterk broadening of hydrogen (or hydrogen impurity) lines
and Thompson scattering. Even in relatively cold (less
than 1 eV) plasmas, Stark broadening is useful only above •v 3 x 1014 cm—3 . Thompson scattering, which is in any
2.
case rauch more complicated than laser interferometry, can only be extended below 1014 cm3 with extreme difficulty.
The operating principle of the selfmodulating laser interferometer (hereafter abbreviated to "laser interferometer") is simple. The output beam is passed through the plasma under study and reflected back on itself into the laser (Fig. 1 ) . The phase and amplitude of this return beam control the amplitude of the laser output, which is monitored by a detector. The plasma electron density can be deduced from the phase shift thus
1 ?3 measured. A number of workers '"' have operated such systems incorporating CW HeNe lasers operating in the. visible and infrared regions. The greatly increased sensitivity of plasma interferometry at 337 ym makes a laser interferometer based on a farinfrared HCN laser an attractive proposition, particularly in view of the simplicity and ease of operation of laser interferometers. As well as increasing sensitivity, the use of this long wavelength reduces problems of alignment and vibration, especially so in comparison with the techniques 2 '3'4used to extend the sensitivity of interferometry at shorter wavelengths.
It is possible to describe the laser interferometer on the basis of a simple theory which leads to some interesting conclusions. It is shown in Sec. 2 that under certain circumstances the laser interferometer can have a peaked frequency response. The factors affecting the shape and size of the laser interferometer fringes are also discussed. The fringe shape is of importance in the interpretation of fractional fringe measurements , while the possibility of a nonlinear dependence of fringe size on return beam amplitude
3.
limits the use of laser interferograms for determination of the plasma temperature from the magnitude of the attenuation. In practice, however, attenuation measurements may well be rendered worthless by the refractive effects ofplasma density gradients (Sec. 5) sothat this is not considered a serious disadvantage.
2. THEORY OF THE LASER INTERFEROMETER
To understand the behaviour of the laser interferometer, we consider first the factors which determine the operating point ofthe laser itself. The effect of returning part of the output beam into the laser cavity can then be treated as a modulation ofthe cavity loss.
The active medium of the laser is characterized by a gain per unit length, g, resulting from stimulated emission. That is
where I is the radiation intensity in the cavity. The gain falls off with increasing intensity because of reduction qf the population inversion (saturation). The steady state intensity is determined by the condition that the energy gained per pass is just equal to the energy lost. Consider ing one round trip of the cavity, this implies
Aexp(2gil)  1 = 0 ,
(24
where ./vis the (intensity) attenuation factor for a round trip and I is the length of the active medium.
Saturation Characteristic
The functional form of the saturation characteristic depends on a number of factors, but there is strong justification for writing it as '
g(I) =
«
.
(3)
(l + 3 D r
The unsaturated gain per unit length, a» and the saturation parameter, 3, are determined by the excitation and deexcitation rates of the energy levels involved in the laser transition. The value taken by the exponent, r, depends on the processes which broaden the corresponding spontaneous emission line. In the case of homogeneous broadening (e.g. when pressure broadening dominates), the radiation is emitted by an ensemble of a priori indistinguishable atoms (or molecules). Each emitter can contribute to the gain at any frequency, with a probability distribution which is simply the line shape function. Saturation then reduces the gain at all frequencies without affecting the spontaneous line shape. In so far as they overlap spatially in the active medium, different modes must compete directly with each other for gain, and normally only a single dominant mode oscillates to the exclusion of all others. Saturation of a homogeneously broadened transition is described by eq. (3) with r = 1.
On the other hand, when a line is inhomogeneously
broadened (e.g. when Doppler broadening dominates), only
that fraction of the population whose natural emission 7
frequency is shifted to within about the natural linewidth of the laser frequency can take part in stimulated emissiop.
Gain saturation involves the removal of these atoms from the
population distribution, "burning a hole" in the gain vs. frequency curve6.'8'9 At low intensities, the hole width
7 equals the natural linewidth and the hole widens as the intensity increases, due to induced transitions decreasing the lifetimes of the laser levels. Saturation of a single mode laser is now described by eq. (3) with r = \. However multimode operation is not only possible with inhomogeneously broadened transitions (as different modes can burn separate holes in the gain curve), but is quite normal for optical lasers unless special precautions are taken. Although only the onset of saturation in multimode lasers is well understood theoretically , there is evidence for the general applicability of the saturation formula (3), given appropriate choice of r. For example in the case of the multimode HeNe laser, it has been argued that the total intensity of
all modes tends to saturate the gain as if the laser transition was homogeneously broadened 11 '12 . Since pressure broadening dominates in the laser used in the experiments described in this paper (Sec. 3)^ we employ the form of eq. (3) appropriate to homogeneous broadening (r = 1) in subsequent discussion. The results obtained are readily extended to the general case of arbitrary r.
Loss Modulation by Return Beam
In what follows, it is convenient to write the attenuation factor in the form
A s (i <£) (i 
(4)
where o is the fraction of stored energy lost per round trip
I
i r
hi
of the cavity as output (i.e.^7 is the transmissivity of the

laser output mirror) and dL represents all other losses.
I
6.
From eqs. (2)  (4), the quiescent cavity intensity, IQ is given by
(1 <£) (11) exp L I"* 1  1 = 0 .
(5)
*•
o*
To simplify analysis and avoid some complications discussed
below, we now assume that the losses are small (say
oC +0" < 20%). This assumption is valid for the 633 nm and
1.15 um HeNe laser transitions which have been used
extensively for laser interferometry but not for the very high gain 3.39 um line13 . It is also valid for the 337 Urn
HCN laser used '.a our experiments. When the. losses are
small, the exponential term in eq. (5) must be close to
unity and so we can write
(6)
31o
When part of the output beam is returned into the laser, the losses represented by df are modified according to the phase and amplitude of the return beam. For the interferometer shown in Fig. 1, the intensity returned into the cavity may be written as e J I. The factor t7 arises
2 from the two passes made through the output mirror and e represents the additional attenuation of the return beam due to diffraction loss, reflection from optical components in the external path, etc. In the absence of the return beam, the amplitude of the wave reflected by the output mirror (amplitude reflectance r) is r/T. If the return beam undergoes a total phase change
A = r/l + e7/T ej* ,
(7)
7.
and the corresponding intensity is
A2 = r2l (1 + 2e/jT cos ()>) ,
(8)
where the term in 5 has been neglected r.d r has been taken as unity in the term containing^. We now discuss the implications of eqs. (7) and (8).
The phase variation of the reflected amplitude, A,
is usually ignored in discussions on laser interferometry.
It has; the effect of varying the apparent length of the
cavity by a fraction of a wavelength, thereby pulling the
laser operating frequency. This has two consequences.
Firstly, the phase shift in the external path is in turn
changed, requiring that the equations above be reconstructed,

allowing for pulling, to produce a closed system for

solution. As this procedure is algebraically involved it
will not be attempted here. It suffices to establish
conditions under which pulling can be ignored. From eq.
(7), the maximum phase change in the effective reflected
amplitude is approximately e£T radians in which case the
frequency is pulled by an amount
6o) = ^ etT ,
(9)
where 6u is the shift in the laser wave angular frequency, (0,
and 2£/c is the transit time for radiation around the cavity
(for convenience, the cavity length is taken to be the same
as the length, A, of the active medium). The phase of the
return beam will then be changed by
:
6() = 5w — , c
(10)
8.
where L is length of the external return path. Clearly from eqs. (9) and (10), <5
The other effect of pulling is to shift the laser operating frequency on the gain profile. In the case of optical lasers where there are normally several longitudinal cavity modes within the gain width this is of little significance. In contrast, the gain width of the HCN laser is less than the spacing between low loss modes, as demonstrated by the fact that the laser can be extinguished by manipulation of the cavity length15 . However, from eq. (9), if e!J was as large as 0.1, the pulling would only be ± 1.3 MHz for our laser (£ = 1.8 m ) , compared with a gain width of about 13 MHz (see Sec. 3). We conclude that the effects of phase variation of the reflected amplitude are small in the case of our system and can be neglected
Nevertheless, a small change in the magnitude of the effective reflected amplitude results in a relatively large change in the total loss of the laser cavity and will cause significant variation of the laser output level. It is clear from eq^ (8) that to account for this effect the attenuation factor, J\, (given by eqn. (4)) must be replaced by
cos
Using the same arguments as led to eq. ( 6 ) , we obtain
1 + 31
(12)