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Measurement of narrow linewidth with a Fabry-Perot interferometer of limited resolution
Volume 20, number 2
OPTICS COMMUNICATIONS
February 1977
MEASUREMENT OF NARROW LINEWIDTH WITH A FABRY-PEROT INTERFEROMETER OF LIMITED RESOLUTION G.W. BRADBERRY Physics Department, Exeter University, Devon, UK
J.M. VAUGHAN Appl. Physics Dept., Royal Signals and Radar Est., Great Malvern, Worcestershire, UK
Received 3 December 1976
It is shown that the high extinction of a muRi-passFabry-Perot interferometer may be exploited to measure the width of very narrow Lorentzian profiles. The technique is illustrated with measurementson re-orientational light scattering spectra of a nematic liquid crystal. Linewidths nearly two orders of magnitude narrower than the conventionally defined resolving limit are determined with an accuracy of a few percent.
The resolving power of classical spectroscopic instruments (the Fabry-Perot etalon, diffraction grating, etc.) is given by the number of wavelengths contained within the path difference of the interfering beams. Even with multiple reflections, as realised for example in a Fabry-Perot etalon, this path difference can hardly be made greater than some tens of meters. For visible light (of wavelength ~0.5/am, frequency "~ 6 X 1014 Hz) the resolving power is thus limited to ~ 108 and accordingly the conventionally defined resolving limit i~ restricted to a few MHz. In a number of laboratories interferometers offering such a narrow instrumental line width have been developed for laser scattering investigations (see e.g. [ 1]). However, the study of narrow spectral feature is made difficult by convolution, with the instrumental profile and intrinsic line width~ less than a few MHz can only be determined with limited accuracy. A recent advance in Fabry-Perot interferometry has b~en the development of multi-pass instruments providing an extinction (ratio of instrumental peak height: to mid-order background) greater than 106 [2]. Such instruments permit the study of very weak features-in the presence of very strong ones (e.g., weak, shifte~l, Brillouin scattering dominated by strong, elastic scattering). This article draws attention to another advantageous property of the multipass instrument. By exploiting the high extinction, intrinsic line widths
of lorentzian form two orders of magnitude narrower than the instrumental width may be deduced. The technique is illustrated by preliminary measurement of reorientational Rayleigh scattering of the nematic liquid crystal 4'-n-pentyl-4-cyanobiphenyl (5CB) over a small range of temperature in the isotropic phase. The instrumental form of a single pass Fabry-Perot instrumentis described by the well known Airy function which gives a train of high contrast fringes due to successive orders of interference. In practice the detailed shape of a fringe may be modified by lack of plate parallelism, departures from flatness, and, in the customary photoelectric Us% by the size of the scanning aperture. The apparent frequency interval, between successive orders, called the free spectral range (fsr), is given by c/2d Hz,:where c is the velocity of light and d the etalon plate separation. The width of a fringe depends on the plate reflectivity and the other factors outlined; the full width at half height (fwhh) is given by s[F where F is referred to as the finesse of the instrument. In multipass use, the finesse is slightly increased but most importantly the Airy shape is modified such that the extinction is greatly increased from typically 103 (single pass) to well over 106 (triple pass). In many scattering systems the Rayleigh line is of lorentzian form given by: I0 r2 I(v) = (v -- v0)2 + I-`2 ' 307
Volume 20, number 2
OPTICS COMMUNICATIONS
where I 0 is the peak intensity at frequency v0 and P is the half width at half height (hwhh). Convolution with a multipass instrument where F '< s/F gives an observed profile which at the centre is dominated by the instrument but in the wing is determined by the lorentzian form; over a wide frequency range between orders the instrumental contribution will be negligible. Thus scattered light is redistributed near the centre of the instrumental profile, while the wings are unaffected. Analysis of the observed wing intensity gives F as follows: The total integrated intensity T is given by oo
T=f
I(v) dv = rrloP.
_oo
The integrated intensity A within the frequency interval Av from (v 0 + k - ~i Av) to (v 0 + k + 1 Av)where k ~, F and Av < k is given by
A = AvloF2/k2.
February 1977
the contribution to the background due to e.g. phototube dark current, broad-band Raman scattering, fluorescence and Brfllouin scattering etc. The first of these can be resolved by applying correction factors computed by adding an infinite train o f lorentzian functions spaced at the free spectral range interval s. Table 1 shows the correction factors required for different fractions of an order. The problem of a broad uniform background can be dealt with by detailed fitting employing table 1. However we have found it convenient in practice to determine the background from two points, the quarter and mid point of an order, and then to use this value as a check on the lorentzian fitting over a specified range, usually between one eighth and one quarter of an order. For a summed sequence of lorentzians spaced s apart, Isum(V0 -+ ¼s) = 2/sum(V0 -+ ~ s). Given a uniform broad background of intensity b the observed intensity, lob s, is given by Iobs(V) =/sum(V) + b and iobs(V0 _+i s ) - b = 2 [Iobs(V0 +- i s ) - b ] .
Thus
p = nAk2/TAv and comparison of the intensity within a selected interval in the wing and the integrated intensity over the whole profile readily gives a value for F. However in practice two complicating factors arise: firstly the effect of overlapping orders, and secondly
Thus b is simply determined from the measured spectra. We have applied this analysis to a series of depolarised ( V - H ) spectra of 5 CB over a limited temperature range in the isotropic phase close to the nematic/ isotropic transition. The instrument was a triple pass interferometer with spacing set at 41.0 mm giving a
Table 1 The ratio of the intensity attributable to the nearest lorentzian profile compared to the sum of an inf'mite train of lorentzian functions spaced at the fsr interval, for different fractions of the fsr (for F < s/F, see text). OROM~ FRACTION
PATIO
ORDER ~PA~ION
RATIO
o~D~
PATIO
FRACTION
ORDIR
PATIO
ORDER FRACTION
RATIO
0.556
FRACTION
0.01
1.000
0.11
0.961
0.21
0.863
O.31
0,721
O=41
0.02
0.999
0.12
0.954
0.22
0.851
0.32
0.705
0.42
0=539
0.03
0.997
0.13
0.946
0.23
0.838
0,33
0,689
0=43
0.522
0.04
0.995
0.14
0.937
0.24
0.824
0,34
O=673
0.44
0.505
0.05
0.992
0.15
0.928
0.25
0.811
0.35
0.657
0.45
0.488
0.06
0.988
0.16
0,919
0.26
0.797
0.36
0.640
0.46
0=471
0.07
0.984
O.17
0.909
0.27
0.782
0.37
0.623
0.47
0.455
0.08 0.09 0,I0
308
,
0.979
0.18
0.898
0.28
0.767
0.38
0.607
0.48
0.436
0.974
0.19
0,887
0.29
0.752
0.39
0.590
0.49
0.422
0.968
0.20
0.875
0.30
0.737
0.40
0.573
0.50
0.405
Volume 20, number 2
OPTICS COMMUNICATIONS
fsr of 3.66 GHz and an extinction "-107. The instrumental finesse of'--40 thus provided a resolving limit of-,-100 MHz. The light source was a stabilised singlemode argon ion laser operating at 488 nm and <~10 mW; a low dark current photon counting photomultiplier and triggered, multi-channel recording was used. The collection of weak broad-band fluorescence Was reduced with a very narrow band (0.2 nm) fdter. Fig: 1 shows the inter-order intensity of a series of record, ings. The b~lckground'was fitted, as described, at the half and quarter order points and the integrated intensity (7') found by adding the channel contents (less background) within one order. For each spectrum 12 values of F were obtained from the individual channel contents (,4) over the one eighth to one quarter order region (between 0.45 and 0.9 GHz). These values were the same within statistical error. Mean data points are shown plotted against temperature in fig. 2; the smallest line width measured in this particular set was F = 1.10 + 0.04 MHz. The results fit approximately a straight line as expected over the small temperature range and are comparable to measurements in MBBA in the isotropic regime close to the isotropic-nematic transition [3]. Very much larger linewidths have been measured in binary solutions of MBBA and carbon tetrachloride [4]. It is worth listing a number of experimental precautions in using this technique. The spectra must be free from spurious, elastically scattered flare from cell walls, dust particles et.¢, otherwise the measurement of
1-
~""
3r'c_...
",~,..
~'~.:~....~ ;
o-5
Fraction
x~
of O r d e r 1
Fig, 1. Spectra of 5 CB at different temperatures. The integrated intensity within a single order of each spectrum is approximately 3 × 106 counts. Each channel corresponds to 39.7 MHz. For clarity, successive spectra have been displaced vertically by 800 counts per channel.
P
February 1977
8
CM-Iz)
/
/
/// I /
0 /
I
i
I
*
I 37 Temperature ( ' C )
Fig. 2. Lorentzian line width F plotted against temperature for 5 CB. The clearing temperature of the sample is indicated by the arrow. The error bars are approximately ~ 3%.
the total integrated scattering T is in error. Similarly the photodetector must provide uniformity of response over the whole intensity range - saturation or even minor dead time effects at the strong peak of the line would dearly lead to considerable error (for a treatment of dead time effects see e.g. [5]). Finally the technique requires a uniform flat background; in certain scattering situations frequency shifted Brillioun scattering would provide a complicating feature. These precautions are, not unexpectedly, closely similar to those required in photon correlation spectroscopy of Rayleigh lines. It is worth commenting that the present measurement of lorentzian line width from the tail of the spectrum is akin to the measurement of the slope at zero time delay in time correlation spectroscopy. In many systems, viscous materials for example (see e.g. [6,7]), it is found that the correlation functions are not of single exponential form - that is they are multi-lorentzian. In the present work the use of different etalon spacers might allow examination of the detailed form of the Rayleigh spectra over a wide range of frequency (corresponding of course to a wide range of delay times in correlation). It seems likely that many interesting departures from the simple hydrodynamic lorentzian form may be studied in this way. The present results fo~m part of our continuing program on Rayleigh line widths in cyanobipheriyl materials. The technique itself, with suitable choice of etalon spacer and adequate signal statistics, can ob309
Volume 20, number 2
OPTICS COMMUNICATIONS
viously be applied to considerably narrower lines than those reported here. We conclude that lorentzian line widths less than 500 kHz may readily be studied, which have hitherto only been accessible to post-detection methods o f analysis.
References [1] J.M. Vaughan, in: Photon Correlation and Light Beating Spectroscopy, eds. H.Z. Cummins & E.R. Pike (Plenum Press, N.Y., 1974), p. 429.
310
February 1977
[2] J.R. Sandercock, in: Light Scattering in Solides, ed. Balkanski (Flammarion, Paris, 1971), p. 9. [3] J.D. Litster and T.W. Stinson, J. App. Phys. 41 (1970) 996; Phys. Rev. Lett. 25 (1970) 503. [4] T.D. Gierke and W.H. Flygare, J. Chem. Phys. 61 (1974) 2231. [5] E.R. Pike, W.R.M. Pomercy and J.M. Vaughan, J. Chem. Phys. 62 (1975) 3188. [6] N. Ostrowsky, in: Photon Correlation and Light Beating Spectroscopy ed. H.Z. Cummins and E.R. Pike (Plenum Press, N.Y., 1974), p. 539. [7] J.C. Brown, P.N. Pusey and R. Dietz, J. Chem. Phys. 62 (1975) 1136.
OPTICS COMMUNICATIONS
February 1977
MEASUREMENT OF NARROW LINEWIDTH WITH A FABRY-PEROT INTERFEROMETER OF LIMITED RESOLUTION G.W. BRADBERRY Physics Department, Exeter University, Devon, UK
J.M. VAUGHAN Appl. Physics Dept., Royal Signals and Radar Est., Great Malvern, Worcestershire, UK
Received 3 December 1976
It is shown that the high extinction of a muRi-passFabry-Perot interferometer may be exploited to measure the width of very narrow Lorentzian profiles. The technique is illustrated with measurementson re-orientational light scattering spectra of a nematic liquid crystal. Linewidths nearly two orders of magnitude narrower than the conventionally defined resolving limit are determined with an accuracy of a few percent.
The resolving power of classical spectroscopic instruments (the Fabry-Perot etalon, diffraction grating, etc.) is given by the number of wavelengths contained within the path difference of the interfering beams. Even with multiple reflections, as realised for example in a Fabry-Perot etalon, this path difference can hardly be made greater than some tens of meters. For visible light (of wavelength ~0.5/am, frequency "~ 6 X 1014 Hz) the resolving power is thus limited to ~ 108 and accordingly the conventionally defined resolving limit i~ restricted to a few MHz. In a number of laboratories interferometers offering such a narrow instrumental line width have been developed for laser scattering investigations (see e.g. [ 1]). However, the study of narrow spectral feature is made difficult by convolution, with the instrumental profile and intrinsic line width~ less than a few MHz can only be determined with limited accuracy. A recent advance in Fabry-Perot interferometry has b~en the development of multi-pass instruments providing an extinction (ratio of instrumental peak height: to mid-order background) greater than 106 [2]. Such instruments permit the study of very weak features-in the presence of very strong ones (e.g., weak, shifte~l, Brillouin scattering dominated by strong, elastic scattering). This article draws attention to another advantageous property of the multipass instrument. By exploiting the high extinction, intrinsic line widths
of lorentzian form two orders of magnitude narrower than the instrumental width may be deduced. The technique is illustrated by preliminary measurement of reorientational Rayleigh scattering of the nematic liquid crystal 4'-n-pentyl-4-cyanobiphenyl (5CB) over a small range of temperature in the isotropic phase. The instrumental form of a single pass Fabry-Perot instrumentis described by the well known Airy function which gives a train of high contrast fringes due to successive orders of interference. In practice the detailed shape of a fringe may be modified by lack of plate parallelism, departures from flatness, and, in the customary photoelectric Us% by the size of the scanning aperture. The apparent frequency interval, between successive orders, called the free spectral range (fsr), is given by c/2d Hz,:where c is the velocity of light and d the etalon plate separation. The width of a fringe depends on the plate reflectivity and the other factors outlined; the full width at half height (fwhh) is given by s[F where F is referred to as the finesse of the instrument. In multipass use, the finesse is slightly increased but most importantly the Airy shape is modified such that the extinction is greatly increased from typically 103 (single pass) to well over 106 (triple pass). In many scattering systems the Rayleigh line is of lorentzian form given by: I0 r2 I(v) = (v -- v0)2 + I-`2 ' 307
Volume 20, number 2
OPTICS COMMUNICATIONS
where I 0 is the peak intensity at frequency v0 and P is the half width at half height (hwhh). Convolution with a multipass instrument where F '< s/F gives an observed profile which at the centre is dominated by the instrument but in the wing is determined by the lorentzian form; over a wide frequency range between orders the instrumental contribution will be negligible. Thus scattered light is redistributed near the centre of the instrumental profile, while the wings are unaffected. Analysis of the observed wing intensity gives F as follows: The total integrated intensity T is given by oo
T=f
I(v) dv = rrloP.
_oo
The integrated intensity A within the frequency interval Av from (v 0 + k - ~i Av) to (v 0 + k + 1 Av)where k ~, F and Av < k is given by
A = AvloF2/k2.
February 1977
the contribution to the background due to e.g. phototube dark current, broad-band Raman scattering, fluorescence and Brfllouin scattering etc. The first of these can be resolved by applying correction factors computed by adding an infinite train o f lorentzian functions spaced at the free spectral range interval s. Table 1 shows the correction factors required for different fractions of an order. The problem of a broad uniform background can be dealt with by detailed fitting employing table 1. However we have found it convenient in practice to determine the background from two points, the quarter and mid point of an order, and then to use this value as a check on the lorentzian fitting over a specified range, usually between one eighth and one quarter of an order. For a summed sequence of lorentzians spaced s apart, Isum(V0 -+ ¼s) = 2/sum(V0 -+ ~ s). Given a uniform broad background of intensity b the observed intensity, lob s, is given by Iobs(V) =/sum(V) + b and iobs(V0 _+i s ) - b = 2 [Iobs(V0 +- i s ) - b ] .
Thus
p = nAk2/TAv and comparison of the intensity within a selected interval in the wing and the integrated intensity over the whole profile readily gives a value for F. However in practice two complicating factors arise: firstly the effect of overlapping orders, and secondly
Thus b is simply determined from the measured spectra. We have applied this analysis to a series of depolarised ( V - H ) spectra of 5 CB over a limited temperature range in the isotropic phase close to the nematic/ isotropic transition. The instrument was a triple pass interferometer with spacing set at 41.0 mm giving a
Table 1 The ratio of the intensity attributable to the nearest lorentzian profile compared to the sum of an inf'mite train of lorentzian functions spaced at the fsr interval, for different fractions of the fsr (for F < s/F, see text). OROM~ FRACTION
PATIO
ORDER ~PA~ION
RATIO
o~D~
PATIO
FRACTION
ORDIR
PATIO
ORDER FRACTION
RATIO
0.556
FRACTION
0.01
1.000
0.11
0.961
0.21
0.863
O.31
0,721
O=41
0.02
0.999
0.12
0.954
0.22
0.851
0.32
0.705
0.42
0=539
0.03
0.997
0.13
0.946
0.23
0.838
0,33
0,689
0=43
0.522
0.04
0.995
0.14
0.937
0.24
0.824
0,34
O=673
0.44
0.505
0.05
0.992
0.15
0.928
0.25
0.811
0.35
0.657
0.45
0.488
0.06
0.988
0.16
0,919
0.26
0.797
0.36
0.640
0.46
0=471
0.07
0.984
O.17
0.909
0.27
0.782
0.37
0.623
0.47
0.455
0.08 0.09 0,I0
308
,
0.979
0.18
0.898
0.28
0.767
0.38
0.607
0.48
0.436
0.974
0.19
0,887
0.29
0.752
0.39
0.590
0.49
0.422
0.968
0.20
0.875
0.30
0.737
0.40
0.573
0.50
0.405
Volume 20, number 2
OPTICS COMMUNICATIONS
fsr of 3.66 GHz and an extinction "-107. The instrumental finesse of'--40 thus provided a resolving limit of-,-100 MHz. The light source was a stabilised singlemode argon ion laser operating at 488 nm and <~10 mW; a low dark current photon counting photomultiplier and triggered, multi-channel recording was used. The collection of weak broad-band fluorescence Was reduced with a very narrow band (0.2 nm) fdter. Fig: 1 shows the inter-order intensity of a series of record, ings. The b~lckground'was fitted, as described, at the half and quarter order points and the integrated intensity (7') found by adding the channel contents (less background) within one order. For each spectrum 12 values of F were obtained from the individual channel contents (,4) over the one eighth to one quarter order region (between 0.45 and 0.9 GHz). These values were the same within statistical error. Mean data points are shown plotted against temperature in fig. 2; the smallest line width measured in this particular set was F = 1.10 + 0.04 MHz. The results fit approximately a straight line as expected over the small temperature range and are comparable to measurements in MBBA in the isotropic regime close to the isotropic-nematic transition [3]. Very much larger linewidths have been measured in binary solutions of MBBA and carbon tetrachloride [4]. It is worth listing a number of experimental precautions in using this technique. The spectra must be free from spurious, elastically scattered flare from cell walls, dust particles et.¢, otherwise the measurement of
1-
~""
3r'c_...
",~,..
~'~.:~....~ ;
o-5
Fraction
x~
of O r d e r 1
Fig, 1. Spectra of 5 CB at different temperatures. The integrated intensity within a single order of each spectrum is approximately 3 × 106 counts. Each channel corresponds to 39.7 MHz. For clarity, successive spectra have been displaced vertically by 800 counts per channel.
P
February 1977
8
CM-Iz)
/
/
/// I /
0 /
I
i
I
*
I 37 Temperature ( ' C )
Fig. 2. Lorentzian line width F plotted against temperature for 5 CB. The clearing temperature of the sample is indicated by the arrow. The error bars are approximately ~ 3%.
the total integrated scattering T is in error. Similarly the photodetector must provide uniformity of response over the whole intensity range - saturation or even minor dead time effects at the strong peak of the line would dearly lead to considerable error (for a treatment of dead time effects see e.g. [5]). Finally the technique requires a uniform flat background; in certain scattering situations frequency shifted Brillioun scattering would provide a complicating feature. These precautions are, not unexpectedly, closely similar to those required in photon correlation spectroscopy of Rayleigh lines. It is worth commenting that the present measurement of lorentzian line width from the tail of the spectrum is akin to the measurement of the slope at zero time delay in time correlation spectroscopy. In many systems, viscous materials for example (see e.g. [6,7]), it is found that the correlation functions are not of single exponential form - that is they are multi-lorentzian. In the present work the use of different etalon spacers might allow examination of the detailed form of the Rayleigh spectra over a wide range of frequency (corresponding of course to a wide range of delay times in correlation). It seems likely that many interesting departures from the simple hydrodynamic lorentzian form may be studied in this way. The present results fo~m part of our continuing program on Rayleigh line widths in cyanobipheriyl materials. The technique itself, with suitable choice of etalon spacer and adequate signal statistics, can ob309
Volume 20, number 2
OPTICS COMMUNICATIONS
viously be applied to considerably narrower lines than those reported here. We conclude that lorentzian line widths less than 500 kHz may readily be studied, which have hitherto only been accessible to post-detection methods o f analysis.
References [1] J.M. Vaughan, in: Photon Correlation and Light Beating Spectroscopy, eds. H.Z. Cummins & E.R. Pike (Plenum Press, N.Y., 1974), p. 429.
310
February 1977
[2] J.R. Sandercock, in: Light Scattering in Solides, ed. Balkanski (Flammarion, Paris, 1971), p. 9. [3] J.D. Litster and T.W. Stinson, J. App. Phys. 41 (1970) 996; Phys. Rev. Lett. 25 (1970) 503. [4] T.D. Gierke and W.H. Flygare, J. Chem. Phys. 61 (1974) 2231. [5] E.R. Pike, W.R.M. Pomercy and J.M. Vaughan, J. Chem. Phys. 62 (1975) 3188. [6] N. Ostrowsky, in: Photon Correlation and Light Beating Spectroscopy ed. H.Z. Cummins and E.R. Pike (Plenum Press, N.Y., 1974), p. 539. [7] J.C. Brown, P.N. Pusey and R. Dietz, J. Chem. Phys. 62 (1975) 1136.