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Asymmetric response profile of a scanning Fabry-Pérot interferometer
Optics Communications North-Holland
100 ( 1993 ) 240-246
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Full length article
Asymmetric response profile of a scanning Fabry-P&-ot interferometer Ziyuan Li, G.E. Stedman Department of Physics, University of Canterbury, Christchurch I, New Zealand
and H.R. Bilger School ofElectrical and Computer Engineering, Received
10 November
1992; revised manuscript
Oklahoma State Univetxty, Stillwater, OK 74078-0321, received
26 January
USA
1993
The finite scanning speed of a scanning Fabry-Perot interferometer induces asymmetry in the response profile to a monochromatic input beam. The amplitude of this profile is mathematically that of a fixed interferometer subjected to an input beam with suitable frequency scanning, and is related to the complex error function. The extent of the asymmetry thus depends on a single parameter, q=/3t:, where p is the frequency scanning rate and tc is the power decay time of the interferometer. To obtain an asymmetry less than 5% in the profile obtained from scanning a monochromatic beam requires a scanning time as much as 100 times of the decay time of the interferometer. Experimental data are in good agreement with our calculations.
1. Introduction In high resolution spectroscopy with a Fabry-Perot interferometer, the extraction of the input spectrum from the output profile is of great importance. Most of the literature on this subject uses the symmetric Airy function in the deconvolution algorithm [ I-31. However, the Airy function for the intensity distribution in the fringe pattern is strictly valid only for a source of constant brightness and spectral distribution and for a non-scanning interferometer. For a scanning interferometer, the limitations on rapid scanning were discussed by Greig and Cooper [4]. They estimated a time to establish the interference pattern as approximately 1.4F7, where F is the finesse and 7 is the round trip time of the interferometer. We can rewrite this expression as 1.4F7= 2.8nt, since F=S/Af,,, and 7= l/S, where S is the free spectral range of the interferometer and Afi,* is its power fwhm in frequency; Afiilz = 1/2x& where t, is the decay time of the power in the cavity. However, their analysis was based on the Lummer240
0030-4018/93/S
Gehrcke plate [ 5 1, and is relevant only for the steadystate multiple-beam interferometer. Optical ringing [ 6 ] occurs when the input frequency sweeps through an interferometer in a time which is not large compared to t,. The ringing profile is caused by the beating between the input frequency and the cavity mode frequency when either is swept in time. We find that the asymmetry in fringe profiles can occur in a scanning interferometer even when the time interval which is required for scanning over the probe frequency is as large as IOOt,. This is particularly important for spectral measurements using a supercavity interferometer. In using the Newport SR130 supercavity (whose free spectral range is 6 GHz and finesse - 60000 [ 61) to measure the mode spectrum in our 1 m2 ring laser, we can see the asymmetry of the fringe profiles of the interferometer output over a large range of interferometer scanning rates, from 3-50 GHz/s. When we further increase the scanning rate of the interferometer, the ringing profile discussed earlier [ 61 becomes conspicuous. We will present an analysis of this asymmetric fringe 06.00 0 1993 Elsevier Science Publishers
B.V. All rights reserved.
Volume 100, number 1,2,3,4
I July 1993
OPTICS COMMUNICATIONS
profile of a scanning interferometer in sec. 2. Generally, the response of a scanning interferometer can be expressed as a series in the intracavity field amplitudes. The asymmetry of the fringe profiles might be expected to depend on the three parameters: the reflectivity R of the mirrors, the round trip optical path L of the reflected beams in the interferometer and the scanning rate of the interferometer. While the influences of the three parameters on the asymmetric line profile could be determined by numerical calculations, its physical significance would be obscured. We will prove that the response of a scanning interferometer to a signal of constant frequency is equivalent to the response of a fixed interferometer to a frequency which is scanned in a certain linear manner. The asymmetry of the profiles will then depend on the single parameter q where q= fit:, The experimental results and discussion will be given in sec. 3.
2. Response of scanning Fabry-PCrot Let us consider a monochromatic wave Ai, ( t, z) = A0 exp i (k,z- oat) to be injected into a scanning interferometer (fig. 1). The mirrors are separated by a length I; one of these mirrors is moving at a constant speed u so as to increase 1. The frequency of the transmitted wavelet A,( t, z) will be Doppler shifted on each reflection; after the nth reflection, the wave has a frequency w,= w,,( 1 - 2nv/c) and wavevector k,=w,/c to first order, where c is the speed of light. The output of the scanning interferometer is the sum of the waves A,( t, z). Although the cavity length is modified by Al=vr within one round trip of a light pulse, we take a snapshot picture of the infinite wave trains at the given time t when the moving mirror is at z=O, and demand only that the boundary con-
Fig. 1. The scanning Fabry-Perot interferometer. Details are discussed in appendix A.
be satisfied at each reflection in order to determine the phases of the components. We relegate the essential mathematical development to the appendices. Applying the boundary conditions for reflection and transmission in each mirror rest frame, and Lorentz transforming where necessary to the laboratory frame, we obtain the transmitted field given by eq. (A. 1) : ditions
where ~~=oo~n[B,+(B,t+B,)v/c]+n2Cv/c})
(2)
and Bo= -C=21/c=r, B,=2, B2= -2(z+/)/c. B,, represents the phase shift from repeated internal reflections with. a stationary mirror, and would in isolation lead to the Airy function. B, arises from a combination of effects associated with the Doppler shift, including both its direct effect on the phases of the emerging partial waves and its indirect effects on the phase shift accumulated in internal reflections. On its own B, simply modifies the effective cavity length by a velocity-dependent amount, without affecting the symmetry of the Airy profile. Part of the indirect effect of the Doppler shift is to induce a term in the exponent which is quadratic in n (the term C) since on the nth bounce the frequency is shifted by n steps; it is this term which will induce asymmetry in the interferometer profile. Now the response of a fixed interferometer to a signal whose frequency is scanned at a rate p has a modulus of the form of eqs. ( 1) and (2 ) (see eq. (B.2)) in which v/c is replaced by /3r/2wo and the same values are obtained for B,, C, B1 and (to within a change of time origin) BZ. This proves the formal equivalence of the two problems: we can express the output of a scanning interferometer as that for a interferometer with a frequency scanned input beam. Neither eq. ( 1) nor its counterpart (eq. (B.2) ) for the fixed interferometer is easily solved; a series of the form I,,u”b”’ is involved. Instead, we use the equivalence described above to derive an alternative solution. Details are given in appendix C. The ratio of the outgoing and incoming intensities Z,,l ( lAout12), Z. ( JAo12> is then given by Z,,,lZo = (n/Q)
I w(i) I2 >
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where w(i) is given in eq. (C 4) and is related to the error function. Figure 2 shows this calculi ted intensity response (solid line) of a scanning interferometer for three values of the normalized scalming rate: ~=0.5, 0.1, and 0.01. The dashed line anll the dotted line in fig. 2a represent the real and irlaginary parts respectively of the transmitted amplitude. The definitions of the characteristic parame .ers of an asymmetric transmission profile as showr. in fig. 2b; 6 is the deviation of the profile peak fro n the ideal case. d, and A, describe the half-width of I he intensity profile on the left and right side, respectively. For convenience of discussion, we define the deviation parameter 1&=26/Af,,~ and the asymnetry parameter in the frequency domain of A,= 2(A,-A,),lAf,,,. For the case II= 1% as shown in fig. 2c, the corresponding scanning time over the probe frequency is 100 times the decay time of the interferometer, and the asymmetry parameter A, is less than 5%. As the normal-
ized scanning rate increases, the profile of the transmitted intensity of the scanning interferometer varies in four ways: the asymmetry and the peak shift of the profile change, the relative fwhm increases, and the transmitted intensity decreases. Figure 3 shows numerical calculations of the variations in the intensity ratio I,,,,/I, at the peak and the peak deviation 6 with the normalized scanning rate 4. An analytic approximation to the integral in eq. (3) is derived in appendix D, with the result Z,,,/Z,=exp[
- (2~)~‘~t’]/[
1 +{~t’2+fit’~3’2]
. (4)
A comparison in the case q= 1% of the results of eqs. ( 3) and (4) shows that even for this relatively large parameter value eq. (4) is accurate to - 10%. The asymmetry parameter is given from eq. (4) by a weighted average of the solutions T+ of I,,,/Z,= 5; on using a linear expansion of the exponent we find (appendix D) d,=4[2q3/(72q3+~)]“‘.
(5)
This equation (to be plotted in fig. 7) shows that 9 values of lo/o, 5% and 10% translate into asymmetry parameters of approximately 6%, 26% and 43%, respectively.
3. Experiment and discussion The experiment setup is shown in fig. 4. A Newport NL- 1 HeNe frequency-stabilised laser is used as
-100 (c)
-50 Normalized
0 scalming
50
100
time
Fig. 2. Amplitude and intensity prc#files from numerical evaluation of eqs. (C.7) and (3) at vario 1s normalized scanning rates q: (a) q=O.5, (b) q=O.l and (c) q =O.Ol. Throughout, the solid line represents the intensity profile, and in (a) the broken and dotted lines represent the real and imaginary parts of the amplitude. Here and in following figures the normalized time t’ is related to real time t by t’ =r,t.
242
0.6
10~’ Normalized scanning rate Fig. 3. The variation of the output intensity ratio Iou,/Io (broken line) and the peak deviation 6 (solid line) with the normalized scanning rate 9 from numerical calculation of eq. (3).
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Fig. 4. Experimental setup for measurement of the asymmetry the response of a scanning Fabry-Perot interferometer.
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in
a light source. Its rms frequency fluctuations are 100 kHz in short term (5 ms) and 5 kHz in medium term ( 100 s), respectively. An acousto-optic modulator (Newport N24080) inserted between the HeNe Laser and the scanning interferometer (Newport SR- 130 supercavity spectrum analyzer) has two functions. First it is an optical isolator to reduce the influence of the feedback of reflection from the interferometer mirror [ 61; we observed that the intensity of the NL1 laser output could have low frequency (a few Hz) fluctuations in the absence of the acousto-optic modulator. Secondly it supplies a frequency scanning calibration since the relation between voltage and expansion of the piezoelectric material is in general not exactly linear [ 8 1. In principle, we could use an AM or FM calibration of the scanning rate. Since the nonlinearity of the frequency sweeps of the driver of the acousto-optic modulator for FM is 15%, we chose the AM modulation in our experiment. A modulation signal of 1.5 MHz is used as input to the modulator driver and a frequency counter with a fractional precision of 10e7 ( 1 Hz in 10 MHz) is used to monitor the modulation frequency. The laser beam passing through the optical modulator is injected into the scanning interferometer with TEM,, mode matching by a conventional lens. Fringes from the scanning interferometer are detected by a photomultiplier (Thorn EM1 9214) with a bandwidth of 100 MHz and its output is sent to a digital storage oscilloscope (Hitachi Denshi Models VC-6275) with a maximum sampling rate of 200 million samples per
second and a fractional precision of time scale reading of 10w4. Frequency jitter in the output of the NL-1 frequency-stabilized laser is evident when the resonance of the interferometer is scanned over this probe frequency. The effect of this frequency jitter on the calibration of the scanning rate increases as the scanning rate slows down. We averaged runs to reduce this error. A single run with a scanning rate of 3.25 GHz/s gives sidebands whose fwhm is approximately that of the carrier. Figures 5a and 5b shows profiles from the average of 64 runs with the scanning rates 9.29 GHz/s and 3.46 GHz/s. The variation of the fwhm of the sidebands reflects the frequency jitter of the NL- 1. Figure 6 gives an extreme example of the effect of jitter for a very low scanning rate; the separations of the side bands from the central peak differ by more than 20%. The experimental data on profile asymmetry is compared with numerical calculations, both on the full and approximate models of eqs. (3) and (5), in fig. 7. Scales have been chosen for best fit, and this with the value of the known absolute frequency scanning rate /I gives a fwhm Afi,Z= (1/2n) (/3/n)‘/’ for
0.8
I
I
1
I
Scanning rate
Av64
9.29GHzk n
0.6
200
2
(4
c
0.8
400
400 (b)
800
1000
scanning raw 3.46GHds
Av64 t
0
600
800
1200
1
1600
Scanning time (microsecond)
Fig. 5. Joint effect on the profile of probe frequency jitter and frequency scanning rate. The averages of 64 runs at scanning rates of 9.29 GHz/s (a) and 3.46 GHz/s (b), with the oscilloscope triggering on the central peak. The larger width of the side peaks reveals the variation in scanning rate and so averaging of noncoincident peaks.
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Normalized
Scanning
rate
1.0
100 100
IO’
(I
Scanning 0.2
0.4
0.6
rate (GHz/s)
0.8 Fig. 8. The increase
Scanning time (ms) Fig. 6. A profile obtained at very 101~scanning rate (0.93 GHz/ s). (a) Unequal spacing of sidebands caused by jitter during collection of the profile. (b) Ragged shape of the profile for one such peak, reflecting shorter-term frequency jitter of the NL-I laser.
of the full width at half maximum (fwhm) of the profile with the scanning rate. The broken line is the theoretical prediction and the solid line is a fit to the experimental data.
time domain. The experimental agreement with the numerical
data are in excellent calculation from eq.
(3). Normalized sl:anning rate 10-2
10 1
100
0.7
Acknowledgements
0.6
We acknowledge gratefully the help of C. Rowe for much invaluable technical assistance and Ziyuan Li would like to thank P. Wells for much help in computing.
0.5 0.4 0.3 0.2 0.1 0.0 1
10
Scanning re te (GHz/s) Fig. 7. Variation of the asymmetry d, of the profile with the scanning rate (the normalized rate q used in the calculation being plotted on the upper axis, and the alrsolute value p for the experiment on the lower). The solid (broken) line is the numerical prediction from the full (approxima .e ) theory (eqs. ( 3 ) and ( 5 ), respectively).
the SR-I 30 supercavity in erferometer. Figure 8 shows the variation of the oaserved fwhm with the scanning rate. The squares rz present the experimental data, the fwhm being (determined by Afi,z= 3At,,2/At (MHz), where At is the time interval between the two sidebands and At, ,z is the fwhm in the 244
Appendix A Response of a scanning Fabry-PCrot interferometer to a monochromatic field The wave A0 exp [ i (k,z- coot) ] is incident on the back, stationary mirror of a scanning Fabry-Perot interferometer (fig. 1). Within the interferometer the waves reflected from the front (and moving) mirror have the form Ab=Aorqexp[-i(k,z+w,t) A’, =&r3qexp[ The transmitted
] ,
-i(kZz+co2t-2k,1)]
, etc.
wave has the components
A$ =Aoq2 exp[i(k,z-mot)]
,
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A;=Aor2q2exp[i(k,z-w,t+2k,I)], A;=Aov4q2exp{i[kzz-w2t+2(k,+k2)]},etc., where q is the transmission coefficient of the wave amplitude at each reflection. These give rise to a total transmitted amplitude of A,,,(t,Z)=Ai,(t,Z)(l-R){l+Rexp{i[(kr-k9)Z
form satisfies the wave equation with k=w/c. The output of the interferometer is given by the convolution of the input field and the impulse response of the cavity: Aout(t, z) = (1 -R)
2 R”A,,(t-m) ”
=(l-R)A,exp
[i(k,z-wot+/3tz/c-tPt*
-Pz’/~c’)]
1 R”exp(i@,) n
,
(B.2)
where =Ai”(t,Z)(l-R)
f
R”exp(i@,)
(A.1)
PI=0
(using X.,“m=n(n-1)/2) (1) and (2).
in agreement
with eqs.
Appendix B Response of a steady-state Fabry-PCrot to a frequency modulation field The amplitude transfer function of a fixed interferometer for a beam Aoexp[i(koz-~ot) ] has the formH(w)=(l-R)/(l-Rexpid),whered=(wO~)T is the phase delay in a round trip of the injected wave in the interferometer relative to that (a multiple of 271:oo= ~xN/T, NA=21) for exact resonance, andasbeforer=21/cH(o)=(l-R)C,,R”expinwr, and its Fourier transform is given by h(t) = (I-R)X,R* G(t-n7) where 6 is the Dirac delta function. For small detuning: w=:oo [ 61, this may beapproximatedbyH(w)=[l-2i(w-o,)/I’,]-’, where the power rate of decay in the cavity is r,= 1 / t,=2nAfr12; note that r, is twice the amplitude rate of decay yc used in ref. [ 61. The quality factor Q= o&, the free spectral range (FSR) is c/21, the finesse is F= rtR/ ( 1 -R) = Q;1/21, and the power full width at half maximum Af;12=fo/Q=c( 1 -R)/2nlR. Let us consider an input field which is ramped in frequency:
+
n2pTC/2wo}
.
(B.3)
Apart from the complex factor, which does not affect the magnitude of the temporal profile, this has the form of eq. (2) for the scanning interferometer under constant excitation, where (/3r/200) now plays the role of v/c and as before the parameters B. = - C= 7, B, = 2. B2 = - 2z/c and is now different to that for the scanning interferometer case; however this difference amounts merely to a change in the origin of time. Hence the cases of a linearly scanned interferometer with fixed input frequency and a fixed interferometer with a linearly ramped input frequency give indistinguishable intensity profiles at the output.
Appendix C Solution for the temporal profile We use the solution from appendix B for the scanning interferometer to solve for the cases of both appendices A and B. The input field Ain( t, z) has the Fourier transform Ai”(w,
z)=&
exp[i(koz-pz*/c’)]
CE X
I
dtexp{i[(o-w,+pz/c)t-tPt*]}
(C.1)
--03
Ai,(t,Z)=AoeXp{-i[Wo+jp(t-z/C)](t-Z/c)}.
(B.1) p is the frequency scanning rate; the instantaneous frequency o= wo+ p( t- z/c) and is related to the time derivative of the phase [ 91, and because of the dependence on t and z through (t-z/c) the above
and with the change of variable (w-wo)/p-z/c], this gives Ai,(w, ~)=&(2/lj)“* +(~-~o)z/cl}~,
exp{i[koz+
t+u=
(p/2)‘/*[t-
(~-00)*/2p (C.2)
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where
I=
Appendix D Analytical formulae for limiting cases
s -cc
du exp( -iu2)=n’/*
cxp( -k/4)
Multiplying this by the response function of the fixed interferometer (see appendix B; r,= 1/t,) and Fourier transforming, we obtain co A,“‘(L, z)=
2’,
s --m
Xexp{i[ (o-w0)*/2p+
00
O” dvexp( -v*) _-m c-& s
a= 1-v/X2,
fou, /IO = IA,,,/A,
b= &It, I2
dvw(-u2)
i-v
--m
=exp( -c2)
erfc( -ii)
( m [>O)
To this end we now change j,ariable [ (w-o+)/
-k/4)
.
(C.4)
w+v where
(2p)‘/*
-t(P/2)“2(Z’/C--f’)] ,
(C.5)
and the sign factor G= -sl:n[ 1+ (a/2)“*(z’/ct’ ) 1, where z’ =rcz, t’ = r,t, II= p/r:; this is chosen so that the expression l/Sq)‘/*[
(C.6)
always has a positive imaginary part. We then have the closed form for the resultant amplitude: A,,,(t,z)=-Gexp(-k/4) Xexp{i[kc’z-w,t-4j3(z/L~-t)*])
In the time domain the asymmetry parameter A,=2(2T,-T+-Tp)/(T+-T_), where the maximum of I,,’ is at t’ = To= - 8 ( 2q)‘12, and the halfpowers points are the solutions T, of I,,,= fIO, giving S~T<+3(2q)3/2Tk-1=0. Hence T++T_= - 12(8r])“*, (2T,-T+-T_)=8(2r])‘/*, and T+-T_=4(72r1~+)1)‘/~/q. We obtain A,=4[2q3/ whose form is plotted in fig. 7. (72$+~)1’/*,
References
-exp(in/4)
+exp( -in/4)(s/2)‘/*(z/c-t)]
(n/8~)‘/*A, w(c).
(C.7)
Hence the corresponding intensity ratio Z,,,, ( 1A,,,) *), I,, ( lAoI 2, of the outgoing s nd incoming beams is given by eq. ( 3 ) .
246
(8;)‘”
(C.3)
This can be cast into closed form using the complex error function in the form [; ]
v=Gexp(
“‘(‘)=
(al-wo)(z/c-t)
-u()t+k,z-7r/4]}.
‘I
‘/I
where <= Jq i. The denominator factor may be rewritten as exp(,/&/t+ fqv2/c2) to 0(vv2), and on completing the square of the gaussian integral and expanding the result to 0( qv2) we obtain
-‘x
w(i)= :,
We expand eq. (3) for small normalized scanning speed q. In the limit that the scanning rate B (and therefore ;rl) tens to zero, and omitting an unobservable overall phase,
dw cxp( - iot) Ai,(W, Z) H(o) __
[=G(
1 July 1993
[ I ] G. Hernandez, Appl. Optics 5 ( 1966) 1745. [2] D.M. Zipoy, Appl. Optics 18 (1979) 1988. [ 31 J.M. Vaughan, The Fabry-Perot interferometer (Hilger, Bristol, 1989) pp. 471-473. [4] J.R. Greig and J. Cooper, Appl. Optics 7 (1968) 2166. [ 5 ] M. Born and E. Wolf, Principles ofoptics (Pergamon, Oxford. 1964) p. 346. [6] Z. Li, R.G.T. Bennett and G.E. Stedman, Optics Comm. 86 (1991) 51. [7] M.A. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Dover, New York, 1965). [8] R. Aschauer, A. Asenbaum and H. Gerl, Appl. Optics 29 (1990) 953. [ 91 A.E. Siegman, Lasers (Univ. Science, Mill Valley, CA, 1986 ) p. 332.
100 ( 1993 ) 240-246
OPTICS
COMMUNICATIONS
Full length article
Asymmetric response profile of a scanning Fabry-P&-ot interferometer Ziyuan Li, G.E. Stedman Department of Physics, University of Canterbury, Christchurch I, New Zealand
and H.R. Bilger School ofElectrical and Computer Engineering, Received
10 November
1992; revised manuscript
Oklahoma State Univetxty, Stillwater, OK 74078-0321, received
26 January
USA
1993
The finite scanning speed of a scanning Fabry-Perot interferometer induces asymmetry in the response profile to a monochromatic input beam. The amplitude of this profile is mathematically that of a fixed interferometer subjected to an input beam with suitable frequency scanning, and is related to the complex error function. The extent of the asymmetry thus depends on a single parameter, q=/3t:, where p is the frequency scanning rate and tc is the power decay time of the interferometer. To obtain an asymmetry less than 5% in the profile obtained from scanning a monochromatic beam requires a scanning time as much as 100 times of the decay time of the interferometer. Experimental data are in good agreement with our calculations.
1. Introduction In high resolution spectroscopy with a Fabry-Perot interferometer, the extraction of the input spectrum from the output profile is of great importance. Most of the literature on this subject uses the symmetric Airy function in the deconvolution algorithm [ I-31. However, the Airy function for the intensity distribution in the fringe pattern is strictly valid only for a source of constant brightness and spectral distribution and for a non-scanning interferometer. For a scanning interferometer, the limitations on rapid scanning were discussed by Greig and Cooper [4]. They estimated a time to establish the interference pattern as approximately 1.4F7, where F is the finesse and 7 is the round trip time of the interferometer. We can rewrite this expression as 1.4F7= 2.8nt, since F=S/Af,,, and 7= l/S, where S is the free spectral range of the interferometer and Afi,* is its power fwhm in frequency; Afiilz = 1/2x& where t, is the decay time of the power in the cavity. However, their analysis was based on the Lummer240
0030-4018/93/S
Gehrcke plate [ 5 1, and is relevant only for the steadystate multiple-beam interferometer. Optical ringing [ 6 ] occurs when the input frequency sweeps through an interferometer in a time which is not large compared to t,. The ringing profile is caused by the beating between the input frequency and the cavity mode frequency when either is swept in time. We find that the asymmetry in fringe profiles can occur in a scanning interferometer even when the time interval which is required for scanning over the probe frequency is as large as IOOt,. This is particularly important for spectral measurements using a supercavity interferometer. In using the Newport SR130 supercavity (whose free spectral range is 6 GHz and finesse - 60000 [ 61) to measure the mode spectrum in our 1 m2 ring laser, we can see the asymmetry of the fringe profiles of the interferometer output over a large range of interferometer scanning rates, from 3-50 GHz/s. When we further increase the scanning rate of the interferometer, the ringing profile discussed earlier [ 61 becomes conspicuous. We will present an analysis of this asymmetric fringe 06.00 0 1993 Elsevier Science Publishers
B.V. All rights reserved.
Volume 100, number 1,2,3,4
I July 1993
OPTICS COMMUNICATIONS
profile of a scanning interferometer in sec. 2. Generally, the response of a scanning interferometer can be expressed as a series in the intracavity field amplitudes. The asymmetry of the fringe profiles might be expected to depend on the three parameters: the reflectivity R of the mirrors, the round trip optical path L of the reflected beams in the interferometer and the scanning rate of the interferometer. While the influences of the three parameters on the asymmetric line profile could be determined by numerical calculations, its physical significance would be obscured. We will prove that the response of a scanning interferometer to a signal of constant frequency is equivalent to the response of a fixed interferometer to a frequency which is scanned in a certain linear manner. The asymmetry of the profiles will then depend on the single parameter q where q= fit:, The experimental results and discussion will be given in sec. 3.
2. Response of scanning Fabry-PCrot Let us consider a monochromatic wave Ai, ( t, z) = A0 exp i (k,z- oat) to be injected into a scanning interferometer (fig. 1). The mirrors are separated by a length I; one of these mirrors is moving at a constant speed u so as to increase 1. The frequency of the transmitted wavelet A,( t, z) will be Doppler shifted on each reflection; after the nth reflection, the wave has a frequency w,= w,,( 1 - 2nv/c) and wavevector k,=w,/c to first order, where c is the speed of light. The output of the scanning interferometer is the sum of the waves A,( t, z). Although the cavity length is modified by Al=vr within one round trip of a light pulse, we take a snapshot picture of the infinite wave trains at the given time t when the moving mirror is at z=O, and demand only that the boundary con-
Fig. 1. The scanning Fabry-Perot interferometer. Details are discussed in appendix A.
be satisfied at each reflection in order to determine the phases of the components. We relegate the essential mathematical development to the appendices. Applying the boundary conditions for reflection and transmission in each mirror rest frame, and Lorentz transforming where necessary to the laboratory frame, we obtain the transmitted field given by eq. (A. 1) : ditions
where ~~=oo~n[B,+(B,t+B,)v/c]+n2Cv/c})
(2)
and Bo= -C=21/c=r, B,=2, B2= -2(z+/)/c. B,, represents the phase shift from repeated internal reflections with. a stationary mirror, and would in isolation lead to the Airy function. B, arises from a combination of effects associated with the Doppler shift, including both its direct effect on the phases of the emerging partial waves and its indirect effects on the phase shift accumulated in internal reflections. On its own B, simply modifies the effective cavity length by a velocity-dependent amount, without affecting the symmetry of the Airy profile. Part of the indirect effect of the Doppler shift is to induce a term in the exponent which is quadratic in n (the term C) since on the nth bounce the frequency is shifted by n steps; it is this term which will induce asymmetry in the interferometer profile. Now the response of a fixed interferometer to a signal whose frequency is scanned at a rate p has a modulus of the form of eqs. ( 1) and (2 ) (see eq. (B.2)) in which v/c is replaced by /3r/2wo and the same values are obtained for B,, C, B1 and (to within a change of time origin) BZ. This proves the formal equivalence of the two problems: we can express the output of a scanning interferometer as that for a interferometer with a frequency scanned input beam. Neither eq. ( 1) nor its counterpart (eq. (B.2) ) for the fixed interferometer is easily solved; a series of the form I,,u”b”’ is involved. Instead, we use the equivalence described above to derive an alternative solution. Details are given in appendix C. The ratio of the outgoing and incoming intensities Z,,l ( lAout12), Z. ( JAo12> is then given by Z,,,lZo = (n/Q)
I w(i) I2 >
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where w(i) is given in eq. (C 4) and is related to the error function. Figure 2 shows this calculi ted intensity response (solid line) of a scanning interferometer for three values of the normalized scalming rate: ~=0.5, 0.1, and 0.01. The dashed line anll the dotted line in fig. 2a represent the real and irlaginary parts respectively of the transmitted amplitude. The definitions of the characteristic parame .ers of an asymmetric transmission profile as showr. in fig. 2b; 6 is the deviation of the profile peak fro n the ideal case. d, and A, describe the half-width of I he intensity profile on the left and right side, respectively. For convenience of discussion, we define the deviation parameter 1&=26/Af,,~ and the asymnetry parameter in the frequency domain of A,= 2(A,-A,),lAf,,,. For the case II= 1% as shown in fig. 2c, the corresponding scanning time over the probe frequency is 100 times the decay time of the interferometer, and the asymmetry parameter A, is less than 5%. As the normal-
ized scanning rate increases, the profile of the transmitted intensity of the scanning interferometer varies in four ways: the asymmetry and the peak shift of the profile change, the relative fwhm increases, and the transmitted intensity decreases. Figure 3 shows numerical calculations of the variations in the intensity ratio I,,,,/I, at the peak and the peak deviation 6 with the normalized scanning rate 4. An analytic approximation to the integral in eq. (3) is derived in appendix D, with the result Z,,,/Z,=exp[
- (2~)~‘~t’]/[
1 +{~t’2+fit’~3’2]
. (4)
A comparison in the case q= 1% of the results of eqs. ( 3) and (4) shows that even for this relatively large parameter value eq. (4) is accurate to - 10%. The asymmetry parameter is given from eq. (4) by a weighted average of the solutions T+ of I,,,/Z,= 5; on using a linear expansion of the exponent we find (appendix D) d,=4[2q3/(72q3+~)]“‘.
(5)
This equation (to be plotted in fig. 7) shows that 9 values of lo/o, 5% and 10% translate into asymmetry parameters of approximately 6%, 26% and 43%, respectively.
3. Experiment and discussion The experiment setup is shown in fig. 4. A Newport NL- 1 HeNe frequency-stabilised laser is used as
-100 (c)
-50 Normalized
0 scalming
50
100
time
Fig. 2. Amplitude and intensity prc#files from numerical evaluation of eqs. (C.7) and (3) at vario 1s normalized scanning rates q: (a) q=O.5, (b) q=O.l and (c) q =O.Ol. Throughout, the solid line represents the intensity profile, and in (a) the broken and dotted lines represent the real and imaginary parts of the amplitude. Here and in following figures the normalized time t’ is related to real time t by t’ =r,t.
242
0.6
10~’ Normalized scanning rate Fig. 3. The variation of the output intensity ratio Iou,/Io (broken line) and the peak deviation 6 (solid line) with the normalized scanning rate 9 from numerical calculation of eq. (3).
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Fig. 4. Experimental setup for measurement of the asymmetry the response of a scanning Fabry-Perot interferometer.
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in
a light source. Its rms frequency fluctuations are 100 kHz in short term (5 ms) and 5 kHz in medium term ( 100 s), respectively. An acousto-optic modulator (Newport N24080) inserted between the HeNe Laser and the scanning interferometer (Newport SR- 130 supercavity spectrum analyzer) has two functions. First it is an optical isolator to reduce the influence of the feedback of reflection from the interferometer mirror [ 61; we observed that the intensity of the NL1 laser output could have low frequency (a few Hz) fluctuations in the absence of the acousto-optic modulator. Secondly it supplies a frequency scanning calibration since the relation between voltage and expansion of the piezoelectric material is in general not exactly linear [ 8 1. In principle, we could use an AM or FM calibration of the scanning rate. Since the nonlinearity of the frequency sweeps of the driver of the acousto-optic modulator for FM is 15%, we chose the AM modulation in our experiment. A modulation signal of 1.5 MHz is used as input to the modulator driver and a frequency counter with a fractional precision of 10e7 ( 1 Hz in 10 MHz) is used to monitor the modulation frequency. The laser beam passing through the optical modulator is injected into the scanning interferometer with TEM,, mode matching by a conventional lens. Fringes from the scanning interferometer are detected by a photomultiplier (Thorn EM1 9214) with a bandwidth of 100 MHz and its output is sent to a digital storage oscilloscope (Hitachi Denshi Models VC-6275) with a maximum sampling rate of 200 million samples per
second and a fractional precision of time scale reading of 10w4. Frequency jitter in the output of the NL-1 frequency-stabilized laser is evident when the resonance of the interferometer is scanned over this probe frequency. The effect of this frequency jitter on the calibration of the scanning rate increases as the scanning rate slows down. We averaged runs to reduce this error. A single run with a scanning rate of 3.25 GHz/s gives sidebands whose fwhm is approximately that of the carrier. Figures 5a and 5b shows profiles from the average of 64 runs with the scanning rates 9.29 GHz/s and 3.46 GHz/s. The variation of the fwhm of the sidebands reflects the frequency jitter of the NL- 1. Figure 6 gives an extreme example of the effect of jitter for a very low scanning rate; the separations of the side bands from the central peak differ by more than 20%. The experimental data on profile asymmetry is compared with numerical calculations, both on the full and approximate models of eqs. (3) and (5), in fig. 7. Scales have been chosen for best fit, and this with the value of the known absolute frequency scanning rate /I gives a fwhm Afi,Z= (1/2n) (/3/n)‘/’ for
0.8
I
I
1
I
Scanning rate
Av64
9.29GHzk n
0.6
200
2
(4
c
0.8
400
400 (b)
800
1000
scanning raw 3.46GHds
Av64 t
0
600
800
1200
1
1600
Scanning time (microsecond)
Fig. 5. Joint effect on the profile of probe frequency jitter and frequency scanning rate. The averages of 64 runs at scanning rates of 9.29 GHz/s (a) and 3.46 GHz/s (b), with the oscilloscope triggering on the central peak. The larger width of the side peaks reveals the variation in scanning rate and so averaging of noncoincident peaks.
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Normalized
Scanning
rate
1.0
100 100
IO’
(I
Scanning 0.2
0.4
0.6
rate (GHz/s)
0.8 Fig. 8. The increase
Scanning time (ms) Fig. 6. A profile obtained at very 101~scanning rate (0.93 GHz/ s). (a) Unequal spacing of sidebands caused by jitter during collection of the profile. (b) Ragged shape of the profile for one such peak, reflecting shorter-term frequency jitter of the NL-I laser.
of the full width at half maximum (fwhm) of the profile with the scanning rate. The broken line is the theoretical prediction and the solid line is a fit to the experimental data.
time domain. The experimental agreement with the numerical
data are in excellent calculation from eq.
(3). Normalized sl:anning rate 10-2
10 1
100
0.7
Acknowledgements
0.6
We acknowledge gratefully the help of C. Rowe for much invaluable technical assistance and Ziyuan Li would like to thank P. Wells for much help in computing.
0.5 0.4 0.3 0.2 0.1 0.0 1
10
Scanning re te (GHz/s) Fig. 7. Variation of the asymmetry d, of the profile with the scanning rate (the normalized rate q used in the calculation being plotted on the upper axis, and the alrsolute value p for the experiment on the lower). The solid (broken) line is the numerical prediction from the full (approxima .e ) theory (eqs. ( 3 ) and ( 5 ), respectively).
the SR-I 30 supercavity in erferometer. Figure 8 shows the variation of the oaserved fwhm with the scanning rate. The squares rz present the experimental data, the fwhm being (determined by Afi,z= 3At,,2/At (MHz), where At is the time interval between the two sidebands and At, ,z is the fwhm in the 244
Appendix A Response of a scanning Fabry-PCrot interferometer to a monochromatic field The wave A0 exp [ i (k,z- coot) ] is incident on the back, stationary mirror of a scanning Fabry-Perot interferometer (fig. 1). Within the interferometer the waves reflected from the front (and moving) mirror have the form Ab=Aorqexp[-i(k,z+w,t) A’, =&r3qexp[ The transmitted
] ,
-i(kZz+co2t-2k,1)]
, etc.
wave has the components
A$ =Aoq2 exp[i(k,z-mot)]
,
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A;=Aor2q2exp[i(k,z-w,t+2k,I)], A;=Aov4q2exp{i[kzz-w2t+2(k,+k2)]},etc., where q is the transmission coefficient of the wave amplitude at each reflection. These give rise to a total transmitted amplitude of A,,,(t,Z)=Ai,(t,Z)(l-R){l+Rexp{i[(kr-k9)Z
form satisfies the wave equation with k=w/c. The output of the interferometer is given by the convolution of the input field and the impulse response of the cavity: Aout(t, z) = (1 -R)
2 R”A,,(t-m) ”
=(l-R)A,exp
[i(k,z-wot+/3tz/c-tPt*
-Pz’/~c’)]
1 R”exp(i@,) n
,
(B.2)
where =Ai”(t,Z)(l-R)
f
R”exp(i@,)
(A.1)
PI=0
(using X.,“m=n(n-1)/2) (1) and (2).
in agreement
with eqs.
Appendix B Response of a steady-state Fabry-PCrot to a frequency modulation field The amplitude transfer function of a fixed interferometer for a beam Aoexp[i(koz-~ot) ] has the formH(w)=(l-R)/(l-Rexpid),whered=(wO~)T is the phase delay in a round trip of the injected wave in the interferometer relative to that (a multiple of 271:oo= ~xN/T, NA=21) for exact resonance, andasbeforer=21/cH(o)=(l-R)C,,R”expinwr, and its Fourier transform is given by h(t) = (I-R)X,R* G(t-n7) where 6 is the Dirac delta function. For small detuning: w=:oo [ 61, this may beapproximatedbyH(w)=[l-2i(w-o,)/I’,]-’, where the power rate of decay in the cavity is r,= 1 / t,=2nAfr12; note that r, is twice the amplitude rate of decay yc used in ref. [ 61. The quality factor Q= o&, the free spectral range (FSR) is c/21, the finesse is F= rtR/ ( 1 -R) = Q;1/21, and the power full width at half maximum Af;12=fo/Q=c( 1 -R)/2nlR. Let us consider an input field which is ramped in frequency:
+
n2pTC/2wo}
.
(B.3)
Apart from the complex factor, which does not affect the magnitude of the temporal profile, this has the form of eq. (2) for the scanning interferometer under constant excitation, where (/3r/200) now plays the role of v/c and as before the parameters B. = - C= 7, B, = 2. B2 = - 2z/c and is now different to that for the scanning interferometer case; however this difference amounts merely to a change in the origin of time. Hence the cases of a linearly scanned interferometer with fixed input frequency and a fixed interferometer with a linearly ramped input frequency give indistinguishable intensity profiles at the output.
Appendix C Solution for the temporal profile We use the solution from appendix B for the scanning interferometer to solve for the cases of both appendices A and B. The input field Ain( t, z) has the Fourier transform Ai”(w,
z)=&
exp[i(koz-pz*/c’)]
CE X
I
dtexp{i[(o-w,+pz/c)t-tPt*]}
(C.1)
--03
Ai,(t,Z)=AoeXp{-i[Wo+jp(t-z/C)](t-Z/c)}.
(B.1) p is the frequency scanning rate; the instantaneous frequency o= wo+ p( t- z/c) and is related to the time derivative of the phase [ 91, and because of the dependence on t and z through (t-z/c) the above
and with the change of variable (w-wo)/p-z/c], this gives Ai,(w, ~)=&(2/lj)“* +(~-~o)z/cl}~,
exp{i[koz+
t+u=
(p/2)‘/*[t-
(~-00)*/2p (C.2)
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where
I=
Appendix D Analytical formulae for limiting cases
s -cc
du exp( -iu2)=n’/*
cxp( -k/4)
Multiplying this by the response function of the fixed interferometer (see appendix B; r,= 1/t,) and Fourier transforming, we obtain co A,“‘(L, z)=
2’,
s --m
Xexp{i[ (o-w0)*/2p+
00
O” dvexp( -v*) _-m c-& s
a= 1-v/X2,
fou, /IO = IA,,,/A,
b= &It, I2
dvw(-u2)
i-v
--m
=exp( -c2)
erfc( -ii)
( m [>O)
To this end we now change j,ariable [ (w-o+)/
-k/4)
.
(C.4)
w+v where
(2p)‘/*
-t(P/2)“2(Z’/C--f’)] ,
(C.5)
and the sign factor G= -sl:n[ 1+ (a/2)“*(z’/ct’ ) 1, where z’ =rcz, t’ = r,t, II= p/r:; this is chosen so that the expression l/Sq)‘/*[
(C.6)
always has a positive imaginary part. We then have the closed form for the resultant amplitude: A,,,(t,z)=-Gexp(-k/4) Xexp{i[kc’z-w,t-4j3(z/L~-t)*])
In the time domain the asymmetry parameter A,=2(2T,-T+-Tp)/(T+-T_), where the maximum of I,,’ is at t’ = To= - 8 ( 2q)‘12, and the halfpowers points are the solutions T, of I,,,= fIO, giving S~T<+3(2q)3/2Tk-1=0. Hence T++T_= - 12(8r])“*, (2T,-T+-T_)=8(2r])‘/*, and T+-T_=4(72r1~+)1)‘/~/q. We obtain A,=4[2q3/ whose form is plotted in fig. 7. (72$+~)1’/*,
References
-exp(in/4)
+exp( -in/4)(s/2)‘/*(z/c-t)]
(n/8~)‘/*A, w(c).
(C.7)
Hence the corresponding intensity ratio Z,,,, ( 1A,,,) *), I,, ( lAoI 2, of the outgoing s nd incoming beams is given by eq. ( 3 ) .
246
(8;)‘”
(C.3)
This can be cast into closed form using the complex error function in the form [; ]
v=Gexp(
“‘(‘)=
(al-wo)(z/c-t)
-u()t+k,z-7r/4]}.
‘I
‘/I
where <= Jq i. The denominator factor may be rewritten as exp(,/&/t+ fqv2/c2) to 0(vv2), and on completing the square of the gaussian integral and expanding the result to 0( qv2) we obtain
-‘x
w(i)= :,
We expand eq. (3) for small normalized scanning speed q. In the limit that the scanning rate B (and therefore ;rl) tens to zero, and omitting an unobservable overall phase,
dw cxp( - iot) Ai,(W, Z) H(o) __
[=G(
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[ I ] G. Hernandez, Appl. Optics 5 ( 1966) 1745. [2] D.M. Zipoy, Appl. Optics 18 (1979) 1988. [ 31 J.M. Vaughan, The Fabry-Perot interferometer (Hilger, Bristol, 1989) pp. 471-473. [4] J.R. Greig and J. Cooper, Appl. Optics 7 (1968) 2166. [ 5 ] M. Born and E. Wolf, Principles ofoptics (Pergamon, Oxford. 1964) p. 346. [6] Z. Li, R.G.T. Bennett and G.E. Stedman, Optics Comm. 86 (1991) 51. [7] M.A. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Dover, New York, 1965). [8] R. Aschauer, A. Asenbaum and H. Gerl, Appl. Optics 29 (1990) 953. [ 91 A.E. Siegman, Lasers (Univ. Science, Mill Valley, CA, 1986 ) p. 332.