Quantum noise is an important and frequently limiting criterion for the performance of optical systems. For example, the size of quantum fluctuations in the amplitude quadrature limits the ability to measure or communicate small amplitude signals with a beam of light1. The signal to noise ratio, SNR, defined as the ratio of the signal variance, VS
(Ω), and the noise variance, VN
(Ω), describes the quality of the information carried by the light at one particular detection frequency Ω. Our aim is to design and operate a system that can process optical information without a reduction of the SNR. In other words, a system that has a signal transfer coefficient from input to output Ts
= SNRout/SNRin close to unity.
In order to design and optimize the optical system, we require techniques that describe how the noise and the signals are generated and transferred. The main tool is the derivation of noise transfer functions which describe how the various noise sources, quantum and classical, contribute to the total output. Linearized transfer functions can be obtained provided the fluctuations due to the signals and noise are small compared to the coherent amplitude of the light. This tool was developed for passive systems, such as beamsplitters, interferometers, cavities and nonlinear media2. More recently it has been expanded to include active systems such as solid state lasers3,4 and electro-optic control systems5,6,7. Most systems behave quite differently near the quantum noise limit (QNL) than in the classical regime. As an illustration we show the example of electro-optic intensity feedback control which clearly demonstrate the special properties of the transfer functions at the QNL.
In principle, coherent light with fluctuations at the QNL, where VN
(Ω)=1, or even squeezed light with fluctuations below the QNL, would be ideal for the detection and transmission of small signals. However such signals are very fragile to losses. This is because loss inevitably introduces vacuum fluctuations. When both signal and noise are well above the QNL this effect is negligible and loss reduces signal and noise equally, preserving the SNR. However, close to the QNL the vacuum fluctuations prevent the noise floor from going below the QNL thus reducing the SNR. If the light is squeezed such that the noise floor is initially below the QNL the vacuum fluctuations introduced by loss increase the noise floor towards the QNL, reducing the SNR even more rapidly.
A solution to this problem is to amplify the signal, by a factor G = V
S,in, until it is much larger than the QNL and hence robust to losses8. However, this has problems as phase insensitive amplifiers (PIA’s), such as laser amplifiers, also inevitably introduce excess quantum noise. This excess noise reduces the SNR in the high gain limit by a factor VN
+ 1)9,10. For coherent light (VN
= 1) the SNR is halved. For squeezed light the penalty is greater. To avoid this noise penalty, amplification must be phase sensitive10. That is the amplifier must be able to amplify one quadrature (say the amplitude) whilst deamplifying the conjugate quadrature (the phase). The deam-plification must be at least as large as the amplification. Phase sensitive amplification can be realized by non-linear optical processes. For example, optical parametric amplification has been used to amplify intensity signals with almost no noise penalty11. Unfortunately such experiments are complex and have usually achieved only small gains. Another method of phase sensitive amplification is to simply detect the light electronically, amplify the resulting photo-current and then re-emit the light using an LED12,13 or a diode laser. This method is phase sensitive as only the intensity is measured and amplified. The drawback to this method is that all phase information is destroyed by the detection process. The amplified output has no temporal or spatial coherence with the input beam. Here we present an alternative method, based on electro-optical feedforward, that combines all the advantages; it allows large values of gain, G ≫ 1; is noiseless, (Ts
≈ 1) and maintains a coherent link between input and output. This method amplifies the signal carried on the field rather than the power of the input.
2. Electro-optic feedback near the QNL
It is an obvious idea that the intensity noise of a laser can be reduced using an electro-optic modulator and a feedback controller. Practical applications are the suppression of the large modulations associated with the relaxation oscillation and the suppression of technical noise due to mechanical imperfections of the laser. The schematic of a noise eater is shown in Fig. 1
. It uses a beam splitter (BS) to generate a beam that is detected by the in-loop photo detector D
and control electronics, with a combination of proportional, integral and differential (PID) gains, that drive an amplitude modulator (AM). The laser has a noise spectrum V
(Ω) which is noise suppressed by the feedback system to a much quieter spectrum V
Figure 1. Schematics of an electro-optic feedback noise eater.
Using standard control theory14
, the feedback system can be described by the open loop gain h
(Ω) = (1-ε
λ(Ω) where ε
is the beamsplitter transmission and η
is the in-loop detector efficiency. The open loop gain quantifies the action of the entire feedback system, including beamsplitter, detector, amplifier and modulator. The electronic gain λ(Ω) is a complex quantity representing changes in the magnitude and the phase of the modulation. The phase shifts are due to the delay in the loop, introduced by the physical layout of the system and the phase shifts inside the electronic components. For stable, noise suppression we need ℜ(h
(Ω)) < 0, that is negative feedback. Fig. 2
. shows typical Bode plots of the gain spectrum h
(Ω) plotted in separate diagrams for the magnitude |h
(Ω)| and the phase arg(h
(Ω)) as functions of frequency Ω.
Figure 2. Bode plot of a typical feedback open loop gain.
Classical control theory leads to the following simple transfer function relating the noise spectrum of the input light from the laser (V
las) to the noise spectrum of the output light (V
All spectra are normalized to the QNL. Without feedback (h
(Ω) = 0) the output is just the input scaled by the beamsplitter transmission. With feedback there is noise suppression of 1/|1 - h
. Noise can be made arbitrarily small with large gain. However for realistic situations (see Fig.2
) there will be frequencies where the noise is increased.
The behavior of the feedback system near the QNL is quite different. In this case there will be additional contributions from the vacuum field which enters at the empty port of the beamsplitter and vacuum noise due to the non unity efficiency of the in-loop detector. The complete quantum theory 6,7 produces the following sum of transfer functions relating the output spectrum to the various input noise sources
are the vacuum noise sources due to the beamsplitter and the detector inefficiency respectively. As they are vacuum noise V
= 1. This complete solution is valid at all levels of input noise. It is instructive to consider the extreme limits. First take the classical limit for a laser with large fluctuations, where V
(Ω) ≫ 1. In this limit, we obtain Eq. (1
). The other limit is the performance of the noise eater for a QNL laser. Here we set V
(Ω) = 1 and obtain
This is a rather interesting result in that it provides extra noise thus making V
out(Ω) > 1. As a consequence, the output noise variance is always larger than the QNL. A noise penalty appears for the noise eater which does not eat quantum noise instead it actually generates excess noise. The higher the gain and the smaller the fraction (1 - ε) of the light used for control, the more excess noise is generated. For example, if only 10% of the light is used for the control (1 - ε = 0.1) and the open loop gain is h = -5 (negative feedback) the output variance will be 1.7. It could be significantly larger if the feedback phase is not optimum. In order to avoid this noise penalty all the light would have to be detected (ε = 0) which is clearly impractical. One can minimize the excess noise by detecting most of the light, leaving only a small intensity at the output. Alternatively, the gain can be reduced.
The effect of a noise eater on input laser noise. Output noise spectra are shown in black. Various input noise levels (V
=300, 100, 30, 10, 3, 1) are shown. Regions where the black traces are below the reds are regions of noise suppression. [Media 1
To illustrate this quantum effect the performance of the noise eater with a gain as shown in Fig. 2
. has been evaluated for a wide range of input laser noise levels, from V
= 300 to V
= 1. All noise spectra in Fig. 3
. are normalized to the QNL. The first trace of V
=300, shows the classical noise eater response. The noise is suppressed from very low frequencies to 1MHz. The maximum gain results in the best noise suppression at 300kHz. This feedback loop functions well at medium frequencies where the gain is large. But at higher frequencies the phase has increased too much and excess noise is created. A clear hump is visible from 1MHz to 40MHz. This result is typical for a realistic feedback system. The subsequent traces show the transition from the classical to the quantum regimes. At noise levels of V
= 10 the response is almost flat since in this case noise suppression and noise penalty balance each other. The final trace, for a QNL laser V
= 1, clearly demonstrates the noise penalty incurred. The noise produces about 8 dB of excess noise.
The excess noise is required to ensure that no violation of the uncertainty principle is possible. Consider the situation in which the input laser light is very strongly phase squeezed. The feedback loop is (ideally) unaffected by and has no effect on the phase quadrature. Suppose the beamsplitter was 50/50. Then close to 50% (3dB) phase squeezing will still appear in the output. To maintain the uncertainty relations the amplitude quadrature of the output must not fall below twice the QNL. This is indeed the high gain limit (h
→ ∞) of Eq. 2
. with unit in-loop detection efficiency and ε
= .5. The origin of the excess noise is the vacuum entering the empty port of the beamsplitter.
A more physical explanation for the noise penalty is to observe that the beamsplitter can be considered a random selector of photons. That is photons are randomly directed to either the in-loop or out-of-loop detector. The noise measured by the in-loop detector thus contains a random component which is anti-correlated to the fluctuations transmitted to the output. This small fraction of the noise is detected, converted into a current, amplified, and added by the modulator to the remaining fluctuations of the light from the laser. Unlike the correlated classical noise, the photon partition noise which was detected cannot cancel the remaining noise since there is an anti-correlation between them. Instead, they add leading to the excess noise. That is, the feedback loop amplifies the quantum noise entering through the empty port of the in-loop beamsplitter15,16
. The more gain we apply the worse it gets. This picture suggests that a small amount of positive feedback could be used to reduce the noise introduced by the beamsplitter. Indeed from Eq. 2
. we see that if we choose h
= 1 - ε or equivalently λ = 1 then the beamsplitter noise will be exactly canceled. As the denominator is now less than one the effect will be to amplify the laser noise. This suggests that a low noise signal amplifier could be constructed from a positive feedback loop. For stability reasons such an amplifier is more conveniently demonstrated using a feedforward loop. In the following two sections we present this demonstration.
3. Noiseless amplification with electro-optic feedforward
Figure 4. Schematics of an electro-optic feedforward amplifier.
In this section, we investigate the electro-optic feedforward scheme shown in Fig. 4
. Similar to electro-optic feedback, a beam splitter is used to tap off part of the input light to an in-loop detector17
. However, the detected in-loop photo-current is instead used to feed information forward onto the transmitted beam after the beam splitter. The transfer function of the scheme, taking into account the quantum noise contributions in this case, is given by18
’s represent the transmittivities of various components and η
’s are the efficiencies of the photodetectors as shown in Fig. 4
. The vacuum noises V
are due to the beam splitter, inefficiencies of the in-loop detector and the output detector, respectively. As before V
= 1. Because the optical control in feedforward is not looped back onto itself as in the case of feedback, the open loop gain λ(Ω) only affect the output beam linearly and operation is always stable. Two plots of the signal transfer coefficient as a function of gain are shown in Fig. 5
(i) and (ii). Signal transfer coefficient for varying electronic gain, λ, at one detection frequency for different beam splitter reflectivity (10%, 30%, 50%, 70%, 80%, 90%, 95%). (i) Ideal system with no detector or modulator losses. The plots show that there is always a positive feedforward gain which yields Ts
= 1 corresponding to noiseless amplification. The negative feedforward values which give Ts
= 0 correspond to optimum noise-eating operation. (ii) Realistic system with ηi
= 0.90 and ηoεm
= 0.60. In this case the maximum Ts
values for each reflectivity describe a locus of points asymptoting to the in-loop detector efficiency. [Media 2
] [Media 3
Assuming the system is lossless, ηi
= 1, and taking λ as real, Eq. 4
then reduces to
Note that the electronic gain parameter λ(Ω) in Eq. 5
is positively multiplied to the input signal V
(Ω) whilst at the same time is negatively influencing the residual noise term (second term). This is the same anti-correlation property discussed for the feedback system. If we choose the open loop gain to be
corresponding to negative feedforward, the first term of Eq. (5
) is completely canceled leaving the output completely independent of the input signal.
This corresponds to the point of Ts
= 0 for the plots in Fig. 5(i)
. Again, we note that although all of the input signal has been ‘eaten’, we are still left with a noise floor that is much larger than the QNL, analogous to the characteristics of the feedback noise-eater.
Figure 6. Experimental results of the electro-optic feedforward scheme. The in-loop detector efficiency is ηi
= 0.92. (i) ε = 0.5. The maximum Ts
occurs at λopt in agreement with theory. (ii) ε = 0.1, for low transmittivity, the maximum Ts
occurs at high signal gain. Traces shown are limiting cases: (a) is the Ts
value corresponding to the in-loop signal. Points above this line are evidence of the cancellation of quantum noise; (b) is the T
s,max of the scheme limited mainly by the in-loop detector efficiency. (iii) Locus of T
s,max, obtained for various transmittivity ε. Also shown is the best possible performance of a phase insensitive amplifier.
However, if we now choose the open loop gain such that
corresponding to positive feedforward, then the beamsplitter noise term of Eq. (5
) is completely canceled instead. The output of the feedforward scheme becomes
This corresponds to the point of Ts
= 1 for the plots in Fig. 5(i)
. This is the ideal limit for this phase sensitive amplifier. Eq. (7
) is the transfer function of a noiseless amplifier with G
. In this scheme, it is the simultaneous ability to amplify the input noise (first term), while canceling the vacuum noise from the feedforward beam splitter (second term) which make the amplifier noiseless.
When losses are present, the maximum Ts
value is no longer 1 (see Fig. 5(ii)
). At high electronic gains Eq. 4
implies that Ts
will approach the in-loop detector efficiency. The gain which maximizes Ts
however, is now somewhat greater than the value of λ+
. Fig. 6
. shows experimental results of the feedforward scheme in excellent agreement with theory. These results show that a practical system can obtain Ts
much greater than that for an ideal, standard, phase insensitive laser amplifier. We measure these values of Ts
by amplitude modulating the input field with a fixed modulation depth and frequency. We measure the signal to noise ratio of the field before and after the feedforward loop.
4. Noiseless processing of squeezed light
In this section, we use our feedforward system to signal amplify amplitude squeezed light, which is notoriously sensitive to losses. As our squeezed light source, we use the second harmonic output from a singly resonant frequency doubler as described in19. A small input signal is obtained by amplitude modulating this light field at 10 MHz before it is incident on the in-loop beamsplitter. The signal was slightly above the QNL. Accounting for detection efficiency, SNRin = 1.10 ± 0.03. We used this as our test signal for the noiseless amplifier.
Experimental demonstration of lossy transmission and noiseless amplification of amplitude squeezed light [Media 4
The top half of Fig. 7
. shows the input measured noise spectra. This is obtained by using a balanced detector pair to perform self-homodyne measurements on all of the input light. The red trace shows the QNL, which is obtained by subtracting the photo-currents of the two detectors. The blue trace is the addition of the two photo-currents, which gives the noise spectrum of the input light. Regions where the blue trace is below the red are thus regions of amplitude squeezing. The maximum measured squeezing of 1.6 dB is observed in the region of 8-10MHz on a 26mW beam. The inferred value after taking into account the detection efficiency and electronic noise floor is 1.8dB. Other features of the spectra include the residual 17.5MHz locking signals of the frequency doubling system19
and the low frequency roll-off of the photo-detector, which was introduced to avoid saturation due to the large relaxation oscillation of the pump laser at ≈ 0.5MHz.
The bottom half of Fig. 7
shows the measured noise spectra obtained from the output detector. Frame 1. of the diagram shows the output spectrum when the input amplitude squeezed light is unprocessed. We simulate transmission losses by adding a total of 86% absorption (εoηm
= .14) before detecting the transmitted signal using the output detector. We observe a significant reduction of the SNR in the output beam. Only 0.4dB of signal is now above the surrounding noise floor. This corresponds to a Ts
=0.11 in agreement with that predicted for the amount of attenuation introduced. In order to perform signal amplification on the input, we reflect 90% (corresponding to ε
= 0.1) of the input light into the in-loop detector. The second frame of the figure shows that the output signal is now no longer visible since this is equivalent to having only 1.4% of light transmitted. Because of the large amount of attenuation present, trace (b) can be regarded as itself quantum noise limited to within 0.1dB over most of the spectrum. Finally, by choosing the optimum feedforward gain, the green trace in Frame 3. shows the amplified input signal with SNRout
= 0.82 ± 0.03 and G
= 9.3 ± 0.2dB. Note that the green trace has a different shape to the traces of previous frames. This is due to the transfer function of the in-loop electronics and the phase variation of the feedforward across the frequency spectrum. As can be seen, the bandwidth of the RF gain is from 7MHz to 21MHz. However, the optimum feedforward gain is only satisfied at limited regions of the spectrum, for example at 10MHz. Note that both Frame 2. and Frame 3. contain traces that are of the same intensity, hence the output signal is now significantly above the QNL. That is the reason why the amplified output is far more robust to losses then the input. In spite of the presence of large introduced losses the amplifier still performs better than an ideal phase insensitive amplifier. This result corresponds to a signal transfer coefficient of Ts
= 0.75 + 0.02, again in good agreement with the theoretically calculated result of Ts
= 0.77. In comparison, an ideal phase insensitive amplifier with similarly squeezed input can only achieve Ts
In conclusion, we have shown that an electro-optic feedforward scheme can be used as a noiseless signal amplifier. The scheme does not employ any non-linear optical process and preserves optical coherence. It is phase sensitive as it only amplifies the amplitude quadrature. The optimum performances are explained in terms of the cancellation of vacuum fluctuations that are introduced during the measurement process. We have demonstrated the effectiveness of our scheme by amplifying signals carried by squeezed light with minimal loss of signal to noise, even in the presence of large (86%) losses. The scheme does cause a reduction of the optical power of the signal beam, however, this is not in principle a disadvantage as injection locking can be used to restore or even increase the output intensity without affecting the fluctuations20. In fact, as the signal is well above the QNL after amplification, it is now robust and can be further amplified by standard phase insensitive amplifiers, such as a laser amplifier, with negligible degradation of the SNR.
We wish to acknowledge useful discussions with B. C. Buchler. We thank M. S. Taubman and A. G. White for the setup of the second harmonic generator. This research is supported by the Australian Research Council.
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