Self-homodyne tomography of a laser diode

Transcription

1 J. Opt. B: Quantum Semiclass. Opt. 2 (2000) Printed in the UK PII: S (00) Self-homodyne tomography of a laser diode M Fiorentino, A Conti, A Zavatta, G Giacomelli and F Marin Università di Napoli, INFM unità di Napoli, Italy Università di Firenze, INO Firenze, Italy Università di Bologna, INO Firenze, Italy INO Firenze, Italy Università di Firenze, LENS Firenze, Italy Mfiore@na.infn.it Received 9 September 1999 Abstract. Considerable interest is growing around self-homodyne techniques. Such techniques allow one to avoid non-unitary detection efficiencies due to inefficient mode matching of the local oscillator. Galatola et al (1991 Opt. Commun ) proposed one possible scheme of self-homodyne detection of a field with a strong coherent component involving the use of a Fabry Perot cavity with a perfectly reflecting mirror. We extend the theory to arbitrary cavities, and we observe a remarkable agreement between the theory and the observed data. Using the same setup we performed a tomographic measure of the Wigner function of the field emitted by a laser diode. Keywords: Optical homodyne tomography, Fabry Perot cavities, inverse radon transform, laser diode 1. Introduction The interest in reconstructing the quantum state of light has grown together with the possibility of creating new nonclassical states of radiation. Among the possible methods of state reconstruction, optical homodyne tomography has proved to be useful in real experiments. For that reason the field of optical homodyne tomography, since the seminal idea of Vogel and Risken [10] and the pioneering experiment of Smithley et al [1], has been continually growing (see the extensive reviews [2] and [3]). One of the main requirements for efficient optical homodyne detection is to mode match the strong local oscillator with the signal to be analysed [4]. Sometimes, however, good mode matching is impossible or no strong local oscillator is available; in these cases it may be useful to use self-homodyne techniques to analyse the field and eventually carry out a tomographic reconstruction of the quantum state of the field. Self-homodyne techniques [5, 6] are based on the possibility of using a local oscillator which is generated in the same process as the signal field and is automatically mode matched. In 1991 Galatola et al [7] proposed a self-homodyne scheme which is well suited for the analysis of the fluctuating part of fields with a strong coherent component (such as coherent light or bright squeezed light) involving a singleended, lossless Fabry Perot cavity. The aim of this paper is to extend the theory of this self-homodyne setup in the presence of losses, and to explore the experimental possibility of using such a scheme for a tomographic analysis of the field emitted by a laser diode. The paper is divided into two sections: in the first one we extend the theory of [7] to a lossy, double-ended cavity; in the second section we report the experimental results confirming the validity of our theory and we describe a possible application of the self-homodyne scheme to the reconstruction of the Wigner function of the field emitted by a laser diode. 2. Self-homodyne analysis of the field The self-homodyne analysis of fields with a strong coherent component described in this paper is based on the possibility of varying the angle between the noise ellipse and the coherent component of the field. This allows one to transform the fluctuations in a quadrature of the field into amplitude fluctuations that permit a direct detection. The process of phase rotation is similar to FM spectroscopy where the phase noise is transformed in amplitude noise using a dispersive medium. In our case the dispersive medium is the analysing cavity (FP) as shown in the schematic drawing of figure 1. Each Fourier component of the fluctuations undergoes a different phase rotation depending on the ratio of its frequency to the cavity half-linewidth. To understand how the phase rotation mechanism works let us consider the complex amplitudes of the positive and negative frequency parts of the quasi-monochromatic input field of pulsation ω f : E (+) IN (t) = hω f 2ε 0 c [α IN + δa IN ]e iωf t E ( ) IN (t) = hω f 2ε 0 c [α IN + δa IN ]eiωf t /00/ $ IOP Publishing Ltd

2 Self-homodyne tomography of a laser diode Figure 1. Experimental setup. After being collimated by the anamorphic prisms (AP) and passing through a Faraday rotator (FR), the laser light is sent to the analysing cavity (FP), by means of the quarter-wave plate the reflected light is sent to the homodyne detection scheme (HD). Figure 2. Plot of the coupling coefficient ξ versus total cavity losses for various transmissivities of the front mirror. where we carried out the separation between the timeindependent part α and the slowly varying part δa. The Fourier-transform components of the input field E (+, ) IN (ω) and of the field reflected by a plane parallel to the Fabry Perot cavity E (+, ) OUT (ω) are linked by the relation E (+) OUT (ω) = F(ω)E(+) IN (ω) (1) E ( ) OUT (ω) = F (ω)e ( ) IN (ω) where F(ω)is the frequency-dependent reflection coefficient of the cavity; for a cavity of length L/2 and resonating pulsation ω 0, F(ω) is, in terms of the front (i = 1) and rear (i = 2) mirrors reflectivities R i, transmissivities T i and losses i, F(ω) = 1 T 1 1 T 1 1 T2 2 e iφ T T2 2 e iφ with φ = L(ω ω o )/c. It may be easily shown that the same formula is valid for a well-aligned, nonmode matched, confocal cavity of the same length. In the limit of high finesse (T 1,T 2, 1, 2 1),we have ξ iθ(ω) F(θ) 1+iθ(ω) with the coupling coefficient ξ given by (2) Figure 3. Plot of the normalized variance of amplitude fluctuations of the reflected field versus cavity detuning. The three curves refer to an analysis frequency = 6 and to different values of the coupling parameter. ξ = T 2 T T 2 + T where θ(ω) = 2φ(ω)/(T 1 + T ) is the detuning between the cavity and incoming field frequencies normalized to the cavity half-linewidth. In figure 2 we present a plot of the coupling parameter versus the total losses for different values of the input mirror transmissivity assuming T 2 = 0. As can be easily seen, for high-finesse cavities, losses as low as 0.2% may cause a significant deviation of the coupling parameter from unity. As already stated in [7] the coupling parameter is of paramount importance in the self-homodyne scheme. It is then worth studying how the deviations of such a parameter Figure 4. Plot of the quadratures rotation angle versus cavity detuning. The three curves refer to an analysis frequency = 6 and to different values of the coupling parameter. from the ideal, single-ended, lossless cavity value of 1 affect the self-homodyne scheme. Using (1) it is easily seen that the Fourier components of normalized frequency of the reflected fluctuations are δa OUT ( ) = F(θ f )δa IN ( ) 185

3 M Fiorentino et al Figure 5. Plots of the normalized variance of the reflected field for different values of the analysis frequency. The thick curves represent the experimental data, the thin curves the fit of equation (4). The acquisitions were made with the spectrum analyser in zero span, RBW = 2 MHz, VBW = 1 khz. The curves are normalized using an independent measure of the shot noise. δa OUT ( ) = F (θ f )δa IN ( ) with θ f = θ(ω f ). Introducing the quadratures of the fluctuations δp IN,OUT = 1 2 (δa IN,OUT( ) + δa IN,OUT ( )) δq IN,OUT = i 2 (δa IN,OUT ( ) δa IN,OUT( )) we can express the effect of the reflection by the cavity by means of a matrix R δpout δpin = R( ) (3) δq OUT δq IN where R( ) ROF ( ) R( ) = R OF ( ) R( ) and R( ) = 1 [ F(θf ) + F (θ f + ) 2 F(θ f ) F (θ f ) R OF ( ) = i [ F(θf ) F (θ f + ) 2 F(θ f ) F (θ f ) ] F(θ f ) ] F(θ f ) (see the appendix for the explicit expressions of R( ) and R OF ( )). Following the steps of [8] we introduce the rotation angle Re (R( )) ϕ( ) = arctan Re (R OF ( )) and the attenuation a( ) = R( ) 2 + R OF ( ) 2 which describe the effect of cavity on the quadratures of the fluctuations. Moreover, we introduce the variance matrices for the input and output fields [8] V ( ) IN,OUT δp( )IN,OUT = (δp( ) δq( ) IN,OUT δq( ) IN,OUT ) IN,OUT δp( )IN,OUT + (δp( ) IN,OUT δq( ) IN,OUT ). δq( ) IN,OUT The element [1, 1] of the variance matrix is the variance of the amplitude fluctuations (normalized to the shot noise) which is measured in a photodetection process; V OUT ( )[1, 1] may now be expressed in terms of the elements of the input variance matrix: for a diagonal input matrix (no phase-amplitude correlations) the relation reads V OUT ( )[1, 1] = V IN ( )[1, 1]R( )R( ) + V IN ( )[2, 2]R OF ( )R OF ( ). (4) This relation allows one to show, very clearly, the effect of losses on the quadrature rotation scheme. Let us consider, by way of example, an incoming field with normalized variances V IN ( )[1, 1] = 1 and V IN ( )[2, 2] = 100, and suppose that we scan the resonance frequency of the analysis cavity around ω f. Then we detect the amplitude fluctuations of the reflected field and analyse their components at the frequency (normalized to the half cavity linewidth) = 6. In figures 3 and 4 respectively, we report the plots of the output variance and of the quadrature rotation angle versus the normalized cavity detuning for different coupling parameters. While the latter seems essentially uninfluenced by the losses, the former is strongly affected by the deviations of the cavity coupling parameter from unity. We also analysed the behaviour of the reflected field in the presence of cavity misalignment. Our simulations show that it is sufficient to set the length of our cavity with a precision of 50 µm (which is easily achievable with commercial micrometers) to reduce the effect of misalignment on the reflected fluctuations to a negligible level. 3. Experiments The setup used in our experiments is shown in figure 1. The laser is a Mitsubishi ML5415N diode emitting at 830 nm, externally feed-backed with a holographic grating. At the heart of the experiment is a confocal 10 cm FP cavity with a high-reflectivity back-mirror (T < 0.1%) and a measured finesse of 150; the value of the coupling coefficient of the cavity is ξ = The homodyne detection system is based on two EG&G FND100 detectors mounted on home-made low-noise amplifiers followed by passive power combiners which allow one to sum/subtract currents from the 186

4 Self-homodyne tomography of a laser diode Figure 6. Experimental setup for tomographic analysis of the field fluctuations. The signal from the homodyne detector is mixed with the analysing frequency, amplified, filtered and finally recorded with a waveform recorder (CAMAC). The latter is connected via GPIB to a PC for data analysis. amplifiers. A photodiode detecting the small transmission leakage of the interferometer is used for alignment. Using this experimental setup we measured the amplitude fluctuations of the reflected field with a spectrum analyser. As a check of our extended theory we fitted the experimental data with (4), using as fitting parameters the amplitude and fluctuations of the incoming field V OUT ( )[1, 1] and V OUT ( )[2, 2], the coupling parameters ξ and the scaling parameters of the curve. The result is shown in figure 5. The results obtained with our setup may be easily compared with existing literature on laser diode linewidth measurements. In fact [8], the phase noise power V φ = V OUT ( )[2, 2] of a laser diode, normalized to the shot noise, is related to the analysis frequency ω (equal to the normalized frequency multiplied by the half-linewidth of the cavity) and to the laser linewidth ν, via the following formula: V φ = 1+ 16πI o ν, (5) ω 2 where I o is the power at the output of the laser expressed in photons s 1. The results presented in figure 5 clearly show the 1/ω 2 -dependence of the phase noise power. An accurate measure of the phase noise, and a correction for losses, gives V φ = at the analysis frequency of 30 MHz. With these values, and a laser power P = 15 mw, we obtain ν 200 KHz. This value is in good agreement with that reported in [9]. We also explored the possibility of using this selfhomodyne technique to make a tomographic reconstruction of the Wigner function. For such measurements we used the detection scheme shown in figure 6. The output signal of the passive sum/difference elements is mixed with a reference sine signal (30 MHz). In the sum configuration we measure the amplitude fluctuations of the incoming field while in the difference configuration we are able to measure the shot noise level used for calibration. The deconvolved signal is amplified and filtered (low-pass filter, cutoff frequency 70 khz) to remove RF components, then it is sent to a Lecroy 6810 CAMAC waveform recorder, which acquires up to bit points at 5 Msamples s 1. The CAMAC is connected via GPIB to a PC for data analysis. The data acquisition is carried out while the cavity frequency is scanned across the laser frequency; the analysis frequency is 30 MHz corresponding to = 6.8; the duration Figure 7. Surface plot of the reconstructed Wigner function of the laser s MHz. The amplitude is given in arbitrary units while the x and y axes units are normalized to the shot noise. The main feature of this Wigner function is the large disproportion of the two axes of the noise ellipse: while the amplitude is almost shot noise limited, phase fluctuations show a large excess noise. The artefacts of reconstruction are easily seen near the tails of the Wigner function. Figure 8. Enlargement of figure 7 showing the interesting features of the measured Wigner function. of the acquisition is 200 ms during which 10 6 points are acquired. From the data registered by the CAMAC we calculate the variance on 10 3 grouped points and fit the variance curve (4) to obtain an experimental evaluation of the cavity and laser parameters (coupling parameter, amplitude and phase noise). The fitted parameters are used to infer the rotation of the quadratures and to obtain the experimental marginal distributions P(x,θ). 187

5 M Fiorentino et al Figure 9. Contour graphic of figure 8. In the early work of Vogel and Risken [10], it is suggested that the homodyne sampled marginal distributions P(x,θ) are the Radon transform of the Wigner function W(x,y) of the radiation field: + P(x,θ) = W(xcos θ y sin θ,x sin θ + y cos θ)dy. (6) Following the reconstruction algorithm of backprojection, it is easy to obtain the inversion formula W(x,y) = π 0 P c (x cos θ,y sin θ)dθ (7) where P c denotes the marginal distributions convolved with a filter function with bandwidth. It is well known [11] that a backprojection algorithm able to obtain a 2π/ resolution reconstructed matrix requires a sampling grid with parameters strictly connected to the bandwidth of the function to be reconstructed. In our case, since the fluctuations of the two quadratures of the state to be reconstructed have a very different scale, the optimal sampling grid should be so fine that 10 6 collected points would not be sufficient to build up the histograms. In order to perform the tomographic reconstruction, we choose to undersample the signal and, by means of Monte Carlo simulations, we find a trade-off among the sampling grid parameters, which minimizes the reconstruction error. The overall error E is estimated according to [12] ( nm E = nm fnm r ) 1/2 2 nm f nm 2 (8) where fnm r are the reconstructed points and f nm are the expected values. From our optimal grid we expect an overall error E 0.5. This value is confirmed in figure 8, where one can see some artefacts of the backprojection algorithm, in particular the long tails in the phase quadrature. In figures 9 and 10 we plot the central part of the overall reconstruction, where the main features of the state are represented. Using the backprojection algorithm, it is possible to reconstruct the Wigner function only in a qualitative form. If one wants to reveal the quantum features of the source, high resolution is required, and the backprojection algorithm requires an amount of data that is experimentally difficult to achieve. For example, to detect a 1 db amplitude squeezed state with a phase noise of about 60 db, one requires at least an 18-bit electronic sampling, which means about 10 6 points are required for just one histogram. In addition, the cutoff, introduced by filtering the marginal distribution, sets the resolution of the algorithm in advance. This fact could hide the quantum features of the source. Such considerations regarding the reconstruction algorithm induced us to take into account different reconstruction schemes. In the field of quantum tomography, D Ariano et al [3] developed algorithms that are able to sample the density matrix elements directly from the homodyne data. These kind of algorithms should be used once the quantum effect of the laser source can be detected, to show, for example, the Wheeler Schleich oscillations [13] or, in general, to demonstrate the nonclassical nature of the light [14]. However, it is hard to reconstruct all the significant elements of the density matrix if the mean photon number of the field radiation is too large. In our case this difficulty prevents us from obtaining useful results with this technique. 4. Conclusions In conclusion, we have demonstrated that the self-homodyne technique proposed in [7] is well suited for a tomographic analysis of the field. In our system the overall efficiency of the apparatus, η = η D η C η O, inclusive of the detector efficiency (η D 0.7), coupling efficiency (η C 0.9) and optical losses, is poor (η 0.3) since optical losses are 188

6 rather high due to the required high level of isolation between the laser and FP cavity, and this prevents the observation of quantum effects in the source. Anyway, a detection efficiency 0.65 may be competitive with the efficiency of a standard homodyne detection if the signal is not easily mode matched with the local oscillator. Moreover, the technique proposed here may be of great interest in all the systems where no local oscillator is available. Acknowledgments This work was partially funded by INFM in the frame of the PRA CAT project. MF would like to acknowledge the Istituto Nazionale di Ottica and INFM for financial support during the experiment. AC and AZ thank the INFM PRA Cat project. Special thanks go to G Passalaqua for his strong encouragement during this work. Appendix The explicit expressions for R( ) and R OF ( ) may be easily calculated using (2) to give R( ) = θ f 4 ξ(i+ )(iξ + ) + θ f 2(1+ξ 2 i iξ 2 ) (θf 2 + ξ 2 )(θf 2 (i+ )2 ) θ f 2 + ξ 2 1+θf 2 Self-homodyne tomography of a laser diode R OF ( ) = iθ f (1 ξ)(1 ξ i ) θ f 2 + ξ 2 (θf 2 + ξ 2 )(θf 2 (i+ )2 ) 1+θf 2. References [1] Smithey D, Beck M, Raymer M G and Faridani A 1993 Phys. Rev. Lett [2] Leonhardt U 1997 Measuring the Quantum State of Light (Cambridge: Cambridge University Press) [3] D Ariano G M 1997 Quantum Optics and Spectroscopy of Solids ed Shumowsky and Hakigiouglu (Dordrecht: Kluwer) p 175 [4] Raymer M G, Cooper J, Carmichael H J, Beck M and Smithey D T 1995 J. Opt. Soc. Am. B [5] Kim C and Kumar P 1994 Phys. Rev. Lett [6] Vasyliev M, Choi S K, Kumar P and D Ariano G M 1998 Opt. Lett [7] Galatola P, Lugiato L A, Porreca M G, Tombesi P and Leuchs G 1991 Opt. Commun [8] Zhang T C, Poizat JP, Grelu P, Roch JF, Grangier P, Marin F, Bramati A, Jost V, Levenson M D and Giacobino E 1995 Quantum Semiclass. Opt [9] Wieman C E and Hollberg L 1991 Rev. Sci. Instrum [10] Vogel K and Risken H 1989 Phys. Rev. A [11] Natterer F 1986 The Mathematics of Computerized Tomography (New York: Wiley) [12] Faridani A 1996 The IMA Volumes in Mathematics and its Applications ed Chavent Papanicolaou Sacks and Symes (Berlin: Springer) [13] Schleich W and Wheeler J A 1987 Nature [14] D Ariano M G, Sacchi M F and Kumar P 1999 Phys. Rev. A