## Optical homodyne tomography of information carrying laser beams

Optics Express, Vol. 3, Issue 4, pp. 154-161 (1998)

http://dx.doi.org/10.1364/OE.3.000154

Acrobat PDF (4852 KB)

### Abstract

Optical homodyne tomography (OHT) is a tool that allows the reconstruction of Wigner functions for each detection frequency of a propagating optical beam. It can measure probability distribution functions (PDF’s) of the field amplitude for any given quadrature of interest. We demonstrate OHT for a range of classical optical states with constant and time varying modulations and show the advantage of OHT over conventional homodyne detection. The OHT simultaneously determines the signal to noise ratio in both amplitude and phase quadratures. We show that highly non-Gaussian Wigner functions can be obtained from incoherent superpositions of optical states.

© Optical Society of America

## 1. Introduction

*W*(

*x*

_{1},

*x*

_{2}) [1

1. E. P. Wigner, Phys. Rev. **40**, 749 (1932). [CrossRef]

2. K. Vogel and H. Risken, Phys. Rev. A **40**, 2847 (1989). [CrossRef] [PubMed]

3. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. **70**, 1244 (1993). [CrossRef] [PubMed]

4. M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. **18**, 1259 (1993). [CrossRef] [PubMed]

5. G. Breitenbach, T. Müller, S. F. Pereira, J. -Ph. Poizat, S. Schiller, and J. Mlynek, J. Opt. Soc. Am. B **12**, 2304 (1995). [CrossRef]

6. G. Breitenbach, S. Schiller, and J. Mlynek, Nature **387**, 471 (1997). [CrossRef]

*. Each modulation frequency has its own Wigner function*

_{m}*W*

_{Ωm}(

*x*

_{1},

*x*

_{2}), and can vary dramatically from channel to channel [8

8. M. J. Collett and C. W. Gardiner, Phys. Rev. A **30**, 1386 (1984). [CrossRef]

*W*

_{Ω}, one for each frequency Ω.

3. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. **70**, 1244 (1993). [CrossRef] [PubMed]

*W*

_{Ωd}for squeezed and classical states were reported for specific detection frequencies Ω

*. In the case of squeezed light, Wigner functions with elliptical two-dimensional Gaussian quasi-probability distributions were demonstrated [5*

_{d}5. G. Breitenbach, T. Müller, S. F. Pereira, J. -Ph. Poizat, S. Schiller, and J. Mlynek, J. Opt. Soc. Am. B **12**, 2304 (1995). [CrossRef]

6. G. Breitenbach, S. Schiller, and J. Mlynek, Nature **387**, 471 (1997). [CrossRef]

9. H. P. Yuen and V. W. S. Chen, Opt. Lett. **18**, 177 (1983). [CrossRef]

10. B. Schumaker, Opt. Lett. **19**, 189 (1984). [CrossRef]

11. Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature **386**, 150 (1997). [CrossRef]

13. T. Felbinger, S. Schiller, and J. Mlynek, Phys. Rev. Lett. **80**, 492 (1998). [CrossRef]

## 2. Standard homodyne detection

*ν*is processed by an interferometric arrangement (Black components in Fig. 1). The majority of the optical power (

*P*) is split off by mirror M1 as the optical local oscillator beam and sent via mirror M3 to the combining beamsplitter M4. A small part of the optical power (

_{LO}*P*) reaches M4 via the mirror M2 as the test beam. The relative phase between the beams is controlled by moving M3 with a piezo position controller (PZT). The two optical outputs are converted into photocurrents

_{test}*i*

_{1}(

*t*) and

*i*

_{2}(

*t*) using matched, high efficiency detectors PD1 and PD2, respectively. The difference

*i*

_{-}(

*t*) =

*i*

_{1}(

*t*) -

*i*

_{2}(

*t*) between the currents is analysed. The fluctuations of the test beam can be described with the generalised quadratures

*x*̂(

*θ*) = (

*a*̂

*e*

^{-iθ}+

*a*̂

^{†}

*e*

^{iθ})/2. Here

*x*̂

_{1}=

*x*̂(0) is the amplitude quadrature,

*x*̂

_{2}=

*x*̂(

*π*/2) is the phase quadrature. Under the condition of

*P*≫

_{LO}*P*this device will provide a photocurrent

_{test}*i*

_{-}which contains information about the fluctuations of the test beam alone, and is immune to the fluctuations , or noise, of the local oscillator beam.

*V*

_{Ωd}(

*θ*) describes the properties of the test beam at one detection frequency Ω

_{d}and is normalised to the standard quantum limit so that for a coherent state,

*V*

_{Ω}(

*θ*) = 1. The difference photocurrent

*i*

_{-}(

*t*) is analysed using an RF spectrum analyser. The electric noise power

*P*

_{iΩd}is proportional to the optical power of the local oscillator

*P*and the variance

_{LO}*V*

_{Ωd}(

*θ*)

*V*

_{Ωd}< 1) or slightly noisy (

*V*

_{Ωd}> 1) the test beam would still have a noise spectrum close to the quantum noise limit of

*V*

_{Ωd}= 1. This device is not particularly sensitive to laser noise. However, any modulation or squeezing generated inside the interferometer is clearly detectable. For example, a phase modulation introduced by driving the electro-optical modulator (EOM) within the interferometer at frequency Ω

_{m}would increase

*V*

_{Ωm}(

*θ*=

*π*/2).

## 3. Modulation

*E*

_{out}=

*E*

_{0}cos[2

*πνt*+

*δ*sin(Ω

_{m}*t*)] and the phase modulation generates sideband pairs at

*ν*±

_{L}*n*Ω

_{m}. For

*δ*≪ 1, only the first order sidebands (

*n*= 1) are important. The linearized annihilation operator for the quantum mode after the modulator is given by

*β*=

*J*

_{1}(

*δ*), the first order Bessel function, and the operators contain the quantum fluctuations. The output current from the homodyne detector for a projection angle

*θ*is

*δX*̂(

*θ*,

*t*) describes the quantum fluctuation in this particular quadrature. In conventional experiments, the analysis of a particular Fourier component of the photocurrent is done by an RF spectrum analyser with the phase

*θ*of the local oscillator kept constant. The result is a spectrum as given in Eq. (1). As a consequence the phase variance

*V*

_{Ωm}(

*θ*=

*π*/2) increases proportional to the modulation depth while all other parts of the spectrum

*V*

_{Ω}(

*θ*=

*π*/2) with Ω ≠ Ω

_{m}remain unchanged.

## 4. Optical homodyne tomography

6. G. Breitenbach, S. Schiller, and J. Mlynek, Nature **387**, 471 (1997). [CrossRef]

*ψ*giving a mixed down difference current

*i*

_{Ωm}(

*θ*,

*ψ*;

*t*). Starting with Eq. (2) and using Fourier transforms we can derive the output current from the mixer as :

*δX*(

_{c}*θ*,Ω

_{m};

*t*) can be understood as the total quantum fluctuations centered around ±Ω

_{m}.

*δX*(

_{ci}*θ*,Ω

_{m};

*t*) and

*δX*(

_{cr}*θ*,

*Ω*;

_{m}*t*) are the imaginary and real parts of

*δX*(θ, Ω

_{c}_{m};

*t*) respectively. For

*ψ*= 0, we obtain

*θ*is repetitively scanned. By selecting data that correspond to the same value inside the vertical intervals

*δθ*, we obtain measurements of,

*w*

_{Ωd}(

*θ*,

*t*), for any given quadrature interval (

*θ*,

*θ*+

*δθ*). Next a histogram of this current is formed by binning the data in intervals

*δθ*for a coherent state. This results in the PDF

*w*

_{Ωd}(

*x*,

*θ*) of the quadrature amplitude for various projection angle

*θ*. Fig. 2(b) shows a series of such PDF’s for a full scan of

*θ*.

*V*(

*θ*) of the given quadrature. For any realistic, and thus linearisable state with photon number

*N*≫ 1, the shape of the PDF is a Gaussian. For a coherent state the width of the Gaussian is equal to the photon number, thus identical to a Poissonian distribution. Squeezed states have sub-Poissonian PDF’s at one particular angle

*θ*, the squeezing quadrature.

_{s}## 5. Wigner function reconstruction

2. K. Vogel and H. Risken, Phys. Rev. A **40**, 2847 (1989). [CrossRef] [PubMed]

*W*

_{Ωd}(

*x*

_{1},

*x*

_{2}) is shown in Fig. 2(c). For a coherent state the function is symmetric - with concentric contour lines. For a squeezed state the function is asymmetric, with elliptical contour lines. For all coherent or squeezed states, without modulation (

*β*= 0), the Wigner function is centered at the origin. This can be seen from equation Eq. (5) where only the second term contributes in this case. The orientation of the ellipse gives the squeezing quadrature. Since the Wigner function is normalized to the standard quantum limit, its position and size is independent of the optical power. Thus, a squeezed vacuum state has the same Wigner function as a bright squeezed state, provided that

*P*≫

_{LO}*P*. Note that the Wigner function should not be confused with the widely used picture of a “ball on a stick” which tries to describe several properties of an optical state simultaneously. The “stick” indicates the average, DC optical power while the “ball” represents high frequency, AC fluctuations. The Wigner function which can be measured at a particular frequency depends only on the noise and signals at that frequency. Hence, Wigner functions can only be displaced from the origin by introducing a modulation.

_{test}## 6. Experimental results

### 6.1 Varying the depth of phase modulation

*β*= 0 (no modulation, i.e. a coherent state) the Wigner function is circular and centred at the origin. As the frequency modulation depth is increased, the Wigner function is displaced along the “

*x*

_{2}” (phase variance) axis. For large modulation, it is apparent that the phase modulation process has introduced significant amplitude modulation, resulting in the Wigner function being displaced vertically from the

*x*

_{1}axis (amplitude variance). This is due to the imperfection of the phase modulator.

*ψ*. This ensures that the modulation component is detected with maximum efficiency and results in the optimum SNR being recorded on the Wigner function.

### 6.2 Switched phase modulation

### 6.3 Asynchronous detection (variable phase ψ).

_{1}as the demodulation signal, which is different from the modulation frequency Ω

_{m}used to drive the EOM. It is necessary to ensure that Ω

_{m}and Ω

_{1}differ in frequency by an amount small comparing with the detection bandwidth. Under these conditions the detected Wigner function then represents the weighted average of all possible demodulation phase values

*ψ*. The resulting Wigner function, for phase modulation, is shown in Fig. 5. The distribution is now centred on the origin and spread symmetrically along the phase quadrature axis. The peaks at the extreme of the modulation correspond to the turning points where the dwell time of the modulation, as a function of phase, is greatest.

## 7. Discussion and summary

*W*

_{Ωd}provides information on the quadrature and strength of the modulation and the width of

*W*

_{Ωd}describes the noise. The signal to noise ratio at any given quadrature can be read directly.

*W*

_{Ωd}(

*x*

_{1},

*x*

_{2}). In order to demonstrate the ability of measuring highly non-Gaussian Wigner functions, an asynchronous demodulation scheme is used.

## Acknowledgements

## References

1. | E. P. Wigner, Phys. Rev. |

2. | K. Vogel and H. Risken, Phys. Rev. A |

3. | D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. |

4. | M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. |

5. | G. Breitenbach, T. Müller, S. F. Pereira, J. -Ph. Poizat, S. Schiller, and J. Mlynek, J. Opt. Soc. Am. B |

6. | G. Breitenbach, S. Schiller, and J. Mlynek, Nature |

7. | T. Coudreau, A. Z. Khoury, and E. Giacobino, |

8. | M. J. Collett and C. W. Gardiner, Phys. Rev. A |

9. | H. P. Yuen and V. W. S. Chen, Opt. Lett. |

10. | B. Schumaker, Opt. Lett. |

11. | Ch. Kurtsiefer, T. Pfau, and J. Mlynek, Nature |

12. | H.-A. Bachor, M. Taubman, A. G. White, T. Ralph, and D. E. McClalland, |

13. | T. Felbinger, S. Schiller, and J. Mlynek, Phys. Rev. Lett. |

14. | U. Leonhardt, |

**OCIS Codes**

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

(270.5570) Quantum optics : Quantum detectors

**ToC Category:**

Focus Issue: Quantum noise reduction in optical systems

**History**

Original Manuscript: April 3, 1998

Published: August 17, 1998

**Citation**

Jinwei Wu, Ping Lam, Malcolm Gray, and Hans Bachor, "Optical homodyne tomography of information carrying laser beams," Opt. Express **3**, 154-161 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-4-154

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### References

- E. P. Wigner, Phys. Rev. 40, 749 (1932). [CrossRef]
- K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989). [CrossRef] [PubMed]
- D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993). [CrossRef] [PubMed]
- M. Beck, D. T. Smithey, J. Cooper, and M. G. Raymer, Opt. Lett. 18, 1259 (1993). [CrossRef] [PubMed]
- G. Breitenbach, T. Muller, S. F. Pereira, J. -Ph. Poizat, S. Schiller and J. Mlynek, J. Opt. Soc. Am. B 12, 2304 (1995). [CrossRef]
- G. Breitenbach, S. Schiller and J. Mlynek, Nature 387, 471 (1997). [CrossRef]
- T. Coudreau, A. Z. Khoury, E. Giacobino, Laser Spectroscopy XIII international conference, 305 (World Scientific, 1997).
- M. J. Collett, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984). [CrossRef]
- H. P.Yuen and V. W. S. Chen, Opt. Lett. 18, 177 (1983). [CrossRef]
- B. Schumaker, Opt. Lett. 19, 189 (1984). [CrossRef]
- Ch. Kurtsiefer, T. Pfau and J. Mlynek, Nature 386, 150 (1997). [CrossRef]
- H.-A. Bachor, M. Taubman, A. G. White, T. Ralph and D. E. McClalland, Proceedings of the Fourth International Conference on Squeezed States and Uncertainty Relations, 381 (NASA, Goddard Space Flight Center, Greenbelt, Maryland, 1996).

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