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A lthough equations (5.18) and (5.19) include the effects of non-stationary shot noise, it is instructive to analyse the origins of this noise in a tim e dom ain approach.

Starting w ith the standard optical shot noise expression:

w here we have assum ed a single sided pow er spectrum . If Idc is a function of time then, using a double sided pow er spectrum , gives a noise spectral density of:

<in2> = 2 e Idc B (2.28)

As shot noise is spectrally flat ("white") we can assume a delta correlation as follows;

<in(t)in(t + x)> = e I(t) 5(t) (5.24)

w here 5(x) is the Dirac delta function.

Optical internal m odulation shifts the signal spectrum to higher frequencies at com ± cos. Baseband signals are recovered by feeding the pho to cu rren t (signal + noise) into a m ixer w here it is m u ltip lie d by a p erio d ic dem odulating function D(t). The autocorrelation of the shot noise yn(t) at the mixer o u tp u t is

<yn(t) yn(t + x)> = e D2(t) I(t) 8(x) (5.25)

The in stan tan eo u s pow er spectral density can be d eterm in ed from the Fourier transform of equation (5.25).

Spectral in fo rm atio n is extracted by tu n in g a receiver, w ith a finite integration time T, to a desired frequency f. To resolve spectral information, T m u st be m uch greater th an the dem odulation period. The shot noise pow er spectral density (again, assum ing a single-sided pow er spectrum ) is th e n

Pn(fzt) = 2 e D2(t) I(t) (5.26)

w here the bar denotes an average perform ed over the integration time T of the receiver. This spectrum is flat (independent of frequency f) b u t the average on the right hand side m ay vary slowly com pared to T.

Of particular interest is an interferom eter, w ith high fringe visibility, set to a dark fringe, and subject to deliberate internal phase m odulation. In the absence of signals, simple sinusoidal phase m odulation at a frequency fm Hz, c en tred ab o u t the q u a d ra tic tu rn in g p o in t (d ark frin g e) p ro d u c e s a p p ro x im ate ly sin u so id a l in te n sity m o d u la tio n , a t tw ice the p hase m odulation frequency. This can be seen from equation (5.03) by setting 0S = 0, 0o = 7T, R = 1 and V = 1 to give

using a low order expansion to simplify the Bessel functions Jo(0m) and J2(0m)/ valid for 9m « 1 gives

I(t) - p Einc2/2 { 1 - (l-em2/4) - 2(em2/8)cos[ 2comt + 2 Xm]} - P Einc2/2 (0m2/ 4) (1 - cos[ 20^1 + 2 Xm])

- io (1 - cos[ 4it fmt + 2 XJ ) (5.28)

where io is the average photocurrent incident on the detector. The same internal modulation at fm would also shift any baseband signal spectrum by fm, hence demodulation with any periodic waveform with a fundamental frequency fm will recover the baseband signal spectrum. The simplest demodulation waveform that performs this role is :

D(t) = V2 sin [ 2 it fmt] (5.29)

where V2 has been used to give the standard shot noise expression at the mixer output. The noise power spectrum is obtained by substituting equations (5.28) and (5.29) into equation (5.26):

Pn(f,t) = Pshot [1 + 0.5 COS(2 Xm) ] (530)

where Pshot = (2 e io/T) is the noise power at the standard shot noise limit for a constant photocurrent i0. The phase dependence is caused by intensity modulation resulting from the deliberate optical phase modulation. (In this treatment we have assumed that 0S = 0. However, typical signals have negligible effect on shot noise and so this analysis is valid for the general internal modulation case 0S <<0m.) Optimum baseband signal recovery requires = 0, which corresponds, unfortunately, to an increased noise power of 3Pshot/2. Setting the demodulation in quadrature with the modulation (xm = n/2) results in a reduced noise floor of Psql/2, and zero recovered baseband signal. Best signal-to-noise is obtained when = 0/ as seen in Fig. (5.6).

Figure 5.11 (a) Non-stationary noise in(t) = white noise x sin2[27ufmt]. (b) i(t) x D(t); (D(t) = sin[2nfmt].) (0 i(t) x D(t); (d(t) = cos[27ufmt].)

The reason for the phase dependence in equation (5.30) can be seen in a heuristic picture of the time domain process of demodulation (see Fig. (5.11)): A modulated noise function is multiplied by in-phase (Fig. (5.11B), Xm = 0) and quadrature (Fig. (5.11C), Xm = rc/2) waveforms. The in-phase waveform "picks out" non-stationary noise more efficiently than the quadrature waveform. Equations (5.26) to (5.29) imply that any modulated photocurrent exhibits phase-dependent shot noise when demodulated at half of the intensity modulation frequency.

Optical and electronic configuration for the observation of Non-stationary shot noise

The arrangement shown in Fig. (5.12) was implemented to demonstrate the phase dependent shot noise in equation (5.30). Light was strongly intensity modulated by locking a Michelson interferometer midway between dark and bright fringes, and driving a resonant phase modulator in one arm at 75MHz. This proved more effective, for the simple purpose of achieving deep optical intensity modulation, than operating around a dark fringe.

The interferometer was locked by minimising the second harmonic (150 MHz) in the intensity spectrum, via a piezo mounted mirror in one arm. The maximum detected optical power at 1064nm was around 5mW, with a minimum of 0.17mW, a fringe visibility of 0.935.

As described in section 2.5, an electronic mixer typically performs a multiplication by a waveform approximating a square wave at the local oscillator fundam ental frequency. In order to perform the simple sinusoidal multiplication operation described in equation (5.29) it was necessary to include a 50 MHz low pass filter in the photocurrent path prior to the mixer operation. The higher harmonics of the local oscillator (at 37.5MHz x 3, x 5, x 7, x 9 etc) then have effectively no photocurrent to operate on (the 50MHz low pass filter was a high order filter giving greater than 80dB suppression by 112.5MHz) and the fundamental alone, produces the baseband mixer output.

Another reason for the 50MHz low pass filter before the fundamental mixer is that due to strong intensity modulation at 75MHz, there is a large photocurrent component at 75MHz. In comparison to shot noise this 75MHz signal was so large that mixer saturation occurred if the signal wasn't filtered out to a large extent*.

As equations (5.27), (5.28) and (5.30) show, intensity modulation at 2 fm produces phase dependent shot noise when demodulated with an electronic local oscillator at frequency fm. In our experiment the intensity is modulated at 75MHz. It is therefore necessary to use a demodulation frequency of 37.5 MHz. Offsetting the demodulation frequency slightly from 37.5 MHz produces a slow monotonic variation in the phase in equation (5.30), effectively scanning %m(t) through all values between 0 and 2n

repeatedly. The RMS noise oscillates between the two extremes allowed by equation (5.30) at a rate equal to twice the demodulation frequency offset. The expected variation in shot noise is therefore 1.5/0.5 = 3 or 4.8 dB.

* Note that saturation occured for both the fundamental mixer and the third harmonic if adequate filtering wasn't performed. The third harmonic mixer used a 100MHz high pass filter to remove the 75MHz modulation.

Figure 5.12 Experimental arrangement for measuring phase dependence of non­ stationary shot noise.

N d :YAG LASER SIG GEN 37.5 MHz LP FILT 50MHz ISOLATOR MIXER MIRROR SPECTRUM ANALYSER PD 2 PD 1 VARIABLE ATTENUATOR HP FILT 100 MHz 75MHz SIG GEN T E Z O DELAY LINE 0.5 (IS [IRROR HP FILT 150MHz MIXER FILT 9 0MHz MIXER MIXER RF AMP

Experimental results for simple sine-sine modulation-demodulation functions

In Fig. (5.13), trace (a), the demodulated shot noise is plotted against time, for a demodulation frequency offset of a few Hz. The observed peak-peak variation is very close to the 4.8 dB predicted for completely sinusoidally modulated light, indicating that the effects of electronic noise e and V less than one are negligible in this measurement. The effect of sweeping the demodulation frequency offset through zero is shown in Fig. (5.13), trace (b). The receiver post-detection bandwidth has been deliberately reduced to average out oscillations above 100 Hz. When the demodulation frequency is sufficiently far from 37.5 MHz, the observed noise averages to the standard shot noise level, highlighting the 1.8 dB noise penalty in the modulation quadrature.

Figure 5.13: (a) Demodulated noise power at 10 MHz (using fundamental harmonic only). RBW = 100 kHz, VBW = 100 Hz. (b) As in (a), but sweeping demodulation frequency offset through 0 Hz.

Using more complicated demodulation waveforms

This excess noise can be partially cancelled by demodulating with higher (odd) harmonics. This does not affect the recovered baseband signal spectrum , as higher harmonics are orthogonal to the m odulation waveform, but the signal-to-noise ratio is improved by lowering the non­ stationary shot noise floor. The optimum demodulation waveform can be shown to be the inverse of the modulation waveform21' 61. A simple realisable waveform which reduces demodulated shot noise contains both fundamental and third harmonics:

D(t) = V2 [ sin(27i fm t + Xm (t)) + y sin[67t fm t+3%m (t) + <j)] ] (5.31)

where y and <J) are the relative amplitude and phase of the third harmonic with respect to the fundamental. Substituting equations (5.31) and (5.28) into equation (5.26) gives the demodulated shot noise power spectrum:

Pn(f,t) = Pshot [1 + 1/2 COS [2 Xm (t)] + T2 - ■ycos[2 Xm (t)+ <)>]] (5.32)

Again this spectrum is flat. The modulation and quadrature phases are defined by Xm = 0 and n/2 respectively. Setting y = 1/2 and (j) = 0 reduces the in-phase noise to its minimum possible value, 1.25 P sh o t/ down from 1.5 P s h o t using the first harmonic only, a noise suppression of 0.8 dB61.

Once again, a time domain description illustrates the process. Consider the modulated noise shown in Fig. (5.11), trace (a). Fig. (5.14) shows the resulting noise for various dem odulating waveforms consisting of combinations of the first and third harmonics at different phases Xm and <}). When <J) = 0, the phase dependence in the demodulated noise disappears. (Signal demodulation still requires Xm = 0.) In contrast, when <{) = n, the resulting dem odulating waveform is much more peaked, and more efficiently picks out the increased noise in the fundamental modulation quadrature, greatly exaggerating the phase dependence of the resulting noise.

The effect of rapidly scanning Xm, while slowly varying <j) is plotted from equation (5.32) in Fig. (5.15), trace (a). This spectrum reflects the time domain predictions of Fig. (5.14). In particular, the demodulated noise limits are +3.5 dB and -6 dB for <j) = n and +0.97 dB (constant) for <J) = 0.

Figure 5.14: D(t) and demodulated noise in(t) x D(t): (a) Xm = 0, <J> = 0 (b) Xm = rc/2, <j) = 0 (c) Xm = 0, <t> = n (d) Xm = n/2, $ = n.

Electronic configuration of 1st and 3rd harmonic demodulation experiment

Experimentally, a third harmonic is produced by doubling the fundamental and mixing it with the original (inside dashed box in Fig. (5.12)). The residual fundamental is removed by a high-pass filter. The third harmonic is mixed with the non-stationary shot noise in a separate mixer (to prevent mixer saturation). The two mixer outputs are added at a signal combiner, and the output fed to a receiver. This operation is equivalent to combining the demodulation waveforms prior to mixing with the noise and was chosen so that y and <|> could be set independently while maintaining the optimum local oscillator power required to operate the third harmonic mixer.

The demodulation phase Xm(t) is scanned by offsetting the demodulation frequency from 37.5 MHz. One mixer output is delayed by 0.48|is, enabling the waveform phase (j> in equation (5.32) to be varied slowly by sweeping the receiver frequency, f (A<f>/Af - 2n radians per 2.1 MHz). As the resulting noise is spectrally flat, scanning the receiver frequency simply changes the relative phase of the third harmonic at the final combiner. The third harmonic amplitude y was set by a variable attenuator after the mixer operation.

Experimental results of using a first and third harmonic demodulation function

The experimental noise spectrum obtained with the combined harmonics is shown in Fig. (5.15), trace (b) and measured spectra due to each harmonic alone are given in Fig. (5.15), trace (c) and (d). Good qualitative agreement is obtained with Fig. (5.15), trace (a), in particular, strong undulations of the envelope as (j) is varied, confirming that the two mixer outputs are correlated. Filter-induced spectral asymmetry is believed to be responsible for the incomplete nodes in the experimental spectrum in Fig. (5.15), trace (b). This asymmetry also altered the optimum choice of y. The use of optimised filters in future designs should allow the expected noise reduction of 0.8 dB to be attained using the combined first and third harmonic demodulation function.

Figure 5.15: (a) Theoretical demodulated noise, in dB relative to Pshot/ Y = 0.5, <p(t) scanned rapidly, 0 varied slowly, (b) Experimental noise using 1st & 3rd harmonics, y

= 0.36. (c) Experimental noise using 1st harmonic (37.5 MHz) only, (d) Experimental noise using 3rd harmonic (112.5 MHz) only.

Frequency in MHz

Optimum modulation-demodulation functions

As pointed out by Schnupp65, Niebauer et al61 and Meers et al21, the optimum modulation-demodulation waveform combination for internal modulation interferometry is one where

where M(t) is the m odulation function and D(t) is the dem odulation function. Any combination for which equation (5.33) is true will give optimum limiting sensitivity of

identical to that of the direct detection system.

For electro-optic modulators requiring significant modulation depth we are restricted to sinusoidal modulation functions M(t) = sin(27T fm t). Equation (5.33) then infers that the optimum demodulation function is an inverse sinusoid. Fig. (5.16) plots several choices of M(t) and D(t) associated with a sinusoidal modulation function. As can be seen, trace (a) goes to infinity on a periodic basis and is clearly impractical. However, the optimum choice of first and third harmonics (trace (b)) is seen to be a good approximation to an inverse sinusoid (significantly better than the standard sinusoid itself). Detailed calculations based on equation (5.26) show that the demodulation functions of trace (b) and (c) yield approximately the same result; ldB of excess shot noise in the signal quadrature. It is, in principle, therefore only necessary to use a broad band detector and a standard rf mixer to dem odulate the photocurrent rather than the first and third harmonic system used in this experiment. However, as mentioned previously, the dynamic range of standard mixers is insufficient to prevent the mixer saturating due to the 2fm component and still be sensitive enough to demodulate shot noise.

M(t) x D(t) = Constant (5.33)

Figure 5.16: Several possible modulation-demodulation waveform combinations, (a)

sinusoid - inverse sinusoid, (b) sinusoid - first and third harmonics, (c) sinusoid - square wave.

By including still higher harmonics (5th, 7th etc) it is, in principle, possible to reduce the excess shot noise still further. However, greater electronic complication is required and the possibility of added electronic noise cancelling any improvement in shot noise would suggest that the use of higher harmonics in the demodulation waveform is impractical.

It should be noted that as fringe visibility V is reduced from unity and electronic noise e becomes significant, the effects of non-stationary shot noise are greatly reduced. This is due to the fact that both poor V and large e increase the stationary noise in the final output. As stationary noise increases, the use of higher harmonics in the demodulation waveform quickly becomes counter productive as they shift both stationary and non- stationary noise down to baseband. The non-stationary noise components cancel a fraction of the correlated noise while the stationary components are uncorrelated and simply add noise power to the baseband output.

The use of higher harmonics in the demodulation waveform should therefore be restricted to high quality interferometers (V -» 1) with little electronic detector noise.