## Optical phase-space-time-frequency tomography |

Optics Express, Vol. 19, Issue 8, pp. 7480-7490 (2011)

http://dx.doi.org/10.1364/OE.19.007480

Acrobat PDF (1112 KB)

### Abstract

We present a new approach for constructing optical phase-space-time-frequency tomography (OPSTFT) of an optical wave field. This tomography can be measured by using a novel four-window optical imaging system based on two local oscillator fields balanced heterodyne detection. The OPSTFT is a Wigner distribution function of two independent Fourier Transform pairs, i.e., phase-space and time-frequency. From its theoretical and experimental aspects, it can provide information of position, momentum, time and frequency of a spatial light field with precision beyond the uncertainty principle. Besides the distributions of *x* – *p* and *t* – *ω*, the OPSTFT can provide four other distributions such as *x* – *t*, *p* – *t*, *x* – *ω* and *p* – *ω*. We simulate the OPSTFT for a light field obscured by a wire and a single-line absorption filter. We believe that the four-window system can provide spatial and temporal properties of a wave field for quantum image processing and biophotonics.

© 2011 OSA

## 1. Introduction

*x*and momentum

*p*, time

*t*and frequency

*ω*of a spatial light field cannot be measured simultaneously with high resolution. However, the distribution of

*x*and

*p*,

*t*and

*ω*of the spatial light field can be measured simultaneously with high resolution by using two local oscillator fields. The use of two local oscillator fields in a balanced heterodyne detection scheme is also called two-window technique [1

1. K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. **24**, 1370–1372 (1999). [CrossRef]

2. V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A **81**, 063826 (2010). [CrossRef]

1. K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. **24**, 1370–1372 (1999). [CrossRef]

3. F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. **95**, 143903 (2005). [CrossRef] [PubMed]

4. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. **40**, 749–759 (1932). [CrossRef]

5. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. **106**, 121–167 (1984). [CrossRef]

*𝒲*(

*x*,

*p*,

*ω*,

*t*), will offer the correlation information of

*x*,

*p*,

*t*, and

*ω*of a wave field through the distributions of

*x*–

*p*,

*ω*–

*t*,

*x*–

*t*,

*p*–

*t*,

*x*–

*ω*, and

*p*–

*ω*, where the two variables are plotted by fixing the other two variables.

6. Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature **386**, 150–153 (1997). [CrossRef]

*x̂, p̂*] =

*i*. Existing approaches of reconstructing Wigner function [9

9. K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A **40**, 2847–2849 (1989). [CrossRef] [PubMed]

10. J. Bertrand and P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. **17**, 397–405 (1987). [CrossRef]

11. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. **70**, 1244–1247 (1993). [CrossRef] [PubMed]

12. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. **81**, 299–332 (2009). [CrossRef]

13. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. **72**, 1137–1140 (1994). [CrossRef] [PubMed]

14. D. F. McAlister, M. Beck, L. Clarke, A. Meyer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. **20**, 1181–1183 (1995). [CrossRef] [PubMed]

15. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. **18**, 2041–2043 (1993). [CrossRef] [PubMed]

16. B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. **30**, 3365–3367 (2005). [CrossRef]

17. K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science **321**, 541–543 (2008). [CrossRef] [PubMed]

19. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science **321**, 544–547 (2008). [CrossRef] [PubMed]

20. M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. **71**, 1539–1589 (1999). [CrossRef]

21. W. H. Zurek, “Sub-Planck structure in phase space and its relevance for quantum decoherence,” Nature **412**, 712–717 (2001). [CrossRef] [PubMed]

22. F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A **73**, 023803 (2006). [CrossRef]

## 2. Theoretical approach: measurement of OPSTFT

2. V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A **81**, 063826 (2010). [CrossRef]

*x*=

_{a}*a*and broad optical spectrum with bandwidth of Δ

*ω*=

_{a}*α*. The collimated (large) LO Gaussian beam has spatial width of Δ

*x*=

_{A}*A*and narrow optical spectrum with bandwidth of Δ

*ω*=

_{A}*β*. The position resolution

*a*is provided by the focused LO Gaussian beam. The momentum resolution 1/

*A*is provided by the collimated LO Gaussian beam. The purpose of this arrangement is to obtain independent control of position and momentum (angle) resolution such that the product of (Δ

*x*· Δ

_{a}*p*) =

_{A}*a*· 1/

*A*≤ 1 clearly surpasses the uncertainty principle limit as shown in the shaped area in Fig. 1(a). Simultaneously, the method can provide independent control of frequency (spectra) and time (path-length) resolution such that (Δ

*ω*·Δ

_{A}*t*) =

_{a}*β*·1/

*α*≤ 1 to surpass the uncertainty principle limit as shown in the shaped area in Fig. 1(b). The spectra-resolved resolution

*β*is provided by the narrowband collimated LO Gaussian beam. The path-resolved resolution 1/

*α*is provided by the broadband focused LO Gaussian beam. In other words, we use position (time) window to obtain position (temporal) information of an optical field and simultaneously use momentum (frequency) window to reject noises due to other propagating angles (frequencies) from coupling into the detection system.

*d*) and transverse momentum (

_{x}*d*), and by tuning wavelength (

_{p}*d*) and path delay (

_{ω}*τ*). Our system employs balanced heterodyne detection of the probe signal field, which is overlapped with two strong local oscillator fields (LO). The beat amplitude

*V*is determined by the spatial and spectra overlapping of two local oscillator (LO) and signal (S) fields at the plane of the detector (

_{B}*Z*=

*Z*) as, where

_{D}*E*

_{LO}_{(}

_{S}_{)}(

*x′, ω, z*) is used to represent the two local oscillator fields (signal field), respectively. The

_{D}*x′*denotes the transverse position at the detector plane. As shown in Fig. 2, when the LO fields are translated off-axis for a distance of

*d*and the tunable filter is tuned

_{x}*d*away from optical center frequency, the LO field will have its spatial and spectra arguments shifted in Eq. (2) as given by, Note that the center frequency of the collimated LO beam is needed to be tuned because we assume the focussed LO beam has a broad optical spectrum. Using the Fresnel approximation and standard Fourier optics technique, we can relate the fields at the detector plane,

_{ω}*E*(

*x′, ω, z*), to the fields at the source planes,

_{D}*E*(

*x, ω, z*= 0), of lenses L1 and L2 as follow; (i) The LO and signal fields each will experience a spatially varying phase of

*d*and the path-length of the signal beam is delayed by

_{p}*τ*, the signal field will experience additional phase-shift of

*e*

^{−}

*, respectively. (iii) The LO and signal fields propagate to the detector plane through a distance of*

^{iωτ}*d*=

*f*, which is the focus-length of lenses L1 and L2, and will experience the phase-shift of

*z*= 0 as described in [2

2. V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A **81**, 063826 (2010). [CrossRef]

*x*

_{∘}→

*x*and

*ω*

_{∘}→

*ω*. The |

*V*|

_{B}^{2}is proportional to the phase-space-time-frequency convolution integral of the Wigner distributions for the two local oscillator and the signal fields in the planes of the input lenses L1 and L2, respectively.

*γ*and

*ϕ*are the relative amplitude and phase of the two LO fields, respectively. The spectrum of LO fields are assumed to be Gaussian function. This can be accomplished by using a single mode fiber with a tunable bandpass Gaussian filter from Newport. The phase-dependent part of Wigner function for the LO takes the form where we take the approximation of

*A*≫

*a*and

*α*≫

*β*. Then, the range of integration for the momentum, position, frequency and time coordinates in Eq. (11) is limited by the signal field. In this scheme (similar electronic components as in Refs. [1

1. K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. **24**, 1370–1372 (1999). [CrossRef]

**81**, 063826 (2010). [CrossRef]

*ϕ*kHz are phase-locked. The signal field is modulated such that the heterodyne beat signals with the focused LO field and the collimated LO field are about Ω

*MHz and Ω*

_{ω}*MHz +*

_{ω}*ϕ*kHz, respectively. The root mean square beat amplitude is measured with an analog spectrum analyzer with a bandwidth of 100 kHz (

*> ϕ*kHz) centered at Ω

*MHz. The output of the spectrum analyzer is squared in real time with a low noise amplifier and an analog multiplier, then the amplified signal is sent to the lock-in-amplifier. Substituting Eq. (13) into Eq. (11) and changing notations (*

_{ω}*d*→

_{x}*x*,

_{o}*d*→

_{ω}*ω*,

_{o}*τ*→

*t*), we find that the in- and out-of phase quadrature amplitudes in the lock-in-amplifier are directly corresponding to the real and imaginary parts of the quantity,

_{o}*𝒦*(x, p,

*ω*, t) is the Kirkwood-Rihaczek phase-space-time-frequency distribution. Eq. (14) is readily inverted to yield the Wigner phase-space and time-frequency function or the OPSTFT of the signal field by a linear transformation. We obtain,

*S*and

_{R}*S*are the real and imaginary parts of Eq. (14), i.e. the in- and out-of-phase quadrature amplitudes, which are simultaneously measured.

_{I}## 3. Simulation of OPSTFT

### 3.1. A thin wire

*a*, 1/

*A*, 1/

*α*and

*β*are chosen to be small to resolve scales of interest on the spatial and temporal properties of a wave field. Physical properties of an object can be extracted through measuring the OPSTFT of the scattered light field through the object. The OPSTFT can provide

*x*–

*p*and

*ω*–

*t*distributions, and other four distributions such as

*ω*–

*x*,

*t*–

*x*,

*ω*–

*p*and

*t*–

*p*distributions. These six distributions can provide new types of information for the light field under study. We numerically simulate the measurement of OPSTFT for the light field scattered through a thin wire with the diameter of 0.6 mm. We use a Gaussian beam with a wave field as given by, where

*σ*and

_{x}*σ*are the spatial and temporal bandwidths. R and

_{t}*ω*are the radius of curvature and center/carrier frequency of the light field. We are interested in looking at the light scattered right after the wire, where the scattered light field,

_{c}*ℰ*(

_{wire}*x,t*), can be written as the product of Eq. (16) and a wire function (slitfun[x]=If[–0.3mm ≤

*x*≤ 0.3mm, 0.0, 1.0] in Mathematica program). We have used this field for exploring phase-space interference analog to superposition of two spatially separated coherent states [2

**81**, 063826 (2010). [CrossRef]

*σ*= 0.85 mm,

_{x}*σ*=200 fs, and

_{t}*R*=−10000 mm and

*ω*= 0 for the simulation. We first obtain

_{c}*ℰ*(

_{wire}*p, ω*) by numerically Fourier transforming the

*ℰ*(

_{wire}*x,t*). This can be accomplished by numerically generating 20 points in position co-ordinate and 20 points in time coordinate from the product field of Eq. (16) and the wire function. Then, we generate the Kirkwood-Richaczek phase-space-time-frequency distribution,

*ℰ*(

_{wire}*x,t*). We first plot the real and imaginary parts of

*𝒦*(

*x, p,*0

*,*0) as shown in Fig. 3(a) and (b). We adopt the units for position (x) in mm, momentum (p) in rad/mm, frequency (

*ω*) in 10

^{13}Hz, and time in 10

^{−13}

*s*. The frequency (

*ω*) is plotted as frequency difference from the center frequency

*ω*. The time (t) is plotted as time difference from the zero delay (

_{c}*τ*= 0). The

*𝒦*(0, 0,

*ω, t*) is zero because there is zero beat signal (|

*V*|

_{B}^{2}) at the position x=0 and p =0. To explore the time-frequency distribution associated with nonzero beat signal (|

*V*|

_{B}^{2}) at the phase-space point(x=0.4, p=0), we plot the real and imaginary parts of

*𝒦*(0.4, 0,

*ω, t*) as shown in Figs. 3(c) and (d). From the

*𝒦*(

*x, p, ω, t*) distribution, we can obtain Wigner phase-space-time-frequency distribution by using linear transformation as in Eq. (15). We use Mathematica program for performing the numerical integration. Figure 4(a) is the plot of

*𝒲*(

*x, p*, 0, 0) where the tunable filter in the collimated LO beam is set at center frequency and the delay (

*τ*) in the signal beam is set to zero. The phase-space oscillation along the x=0 is due to the coherent phase-space interference of two spatially separated wave packets after the wire. This oscillation exhibits sub-Planck phase-space structure [21

21. W. H. Zurek, “Sub-Planck structure in phase space and its relevance for quantum decoherence,” Nature **412**, 712–717 (2001). [CrossRef] [PubMed]

22. F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A **73**, 023803 (2006). [CrossRef]

6. Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature **386**, 150–153 (1997). [CrossRef]

*𝒲*(0, 0,

*ω, t*) as shown in Fig. 4(b), where the phase-space point is at (0, 0). Note that the

*𝒦*(0, 0,

*ω, t*) is zero everywhere but not the Wigner function of

*𝒲*(0, 0,

*ω, t*). The reason is KR distribution is suitable for describing local properties of a wave function, while the Wigner function is suitable for describing wave properties of a particle wave function [2

**81**, 063826 (2010). [CrossRef]

*𝒲*(0, 2,

*ω, t*), which is negative, i.e, the inverse of Fig. 4(b). This is because the Wigner distribution of

*𝒲*(0, 2, 0, 0) has negative value. These properties are important to explore hyper-entanglement of intrinsic properties of single-photon spatial qubit states. Other four distributions such as

*𝒲*(0,

*p*, 0,

*t*),

*𝒲*(0,

*p*,

*ω,*0),

*𝒲*(

*x,*0, 0,

*t*), and

*𝒲*(

*x,*0,

*ω,*0) are plotted as shown in Figs. 4(d), (e), (f) and (g), respectively. The

*𝒲*(0,

*p*, 0,

*t*)) and

*𝒲*(0,

*p*,

*ω*, 0) exhibit oscillation/interference behavior along momentum coordinate. The phase-space interference due to the two spatially separated wave packets influenced the distribution of momentum (angle) in the time and spectra domains.

### 3.2. An absorption filter

*σ*is the spectra bandwidth. It is much easier to work on the spectra domain of the field so that the light field passing through the filter can be written as the product of Eq. (17) and an absorption filter function (slitfun[x]=If[–0.1 ≤

_{ω}*x*≤ 0.1, 0.0, 1.0] in Mathematica program). We use

*σ*= 0.85 mm,

_{x}*σ*=5.0 THz,

_{t}*R*=−10000 mm and

*ω*= 0 for the simulation. First, we generate the Kirkwood-Richaczek phase-space-time-frequency distribution,

_{c}*ℰ*(

_{filter}*p,t*) is obtained by numerically Fourier transformed the

*ℰ*(

_{filter}*x, ω*). The

*𝒦*(

*x*,

*p*, 0, 0) is zero because there is zero beat signal (|

*V*|

_{B}^{2}) at

*ω*= 0 and

*t*= 0. Figures 5(a) and (b) show the real and imaginary parts of

*𝒦*(0, 0,

*ω*,

*t*). To explore the position-momentum distribution associated with nonzero beat signal (|

*V*|

_{B}^{2}) at the time-frequency point(

*ω*=0.2, t=0), we plot the real and imaginary parts of

*𝒦*(0.2, 0,

*ω*,

*t*) as shown in Figs. 5(c) and (d). Since we have the

*𝒦*(

*x*,

*p*,

*ω*,

*t*) distribution, then we can obtain Wigner phase-space-time-frequency distribution by using linear transformation as in Eq. (15), by means of numerical integration.

*𝒲*(0, 0,

*ω*,

*t*) where the position

*d*of the mirror for the LO beam is set to zero and the momentum

_{x}*d*of the lens L1 in the signal beam is set to zero. The time-frequency oscillation along the

_{p}*ω*= 0 is due to the coherent time-frequency interference of two spectrally separated wave packets after the filter. This observation has been observed in Wigner time-frequency distribution [15

15. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. **18**, 2041–2043 (1993). [CrossRef] [PubMed]

*𝒲*(

*x*,

*p*, 0, 0) as shown in Fig. 6(b), where the time-frequency point is at (0, 0). Figure 6(c) shows an interesting result of

*𝒲*(

*x*,

*p*, 0, 3), which is negative, i.e, the inverse of Fig. 6(b). This is because the Wigner distribution of

*𝒲*(0, 0, 0, 3) has negative value. Other four distributions such as

*𝒲*(0,

*p*, 0,

*t*),

*𝒲*(0,

*p*,

*ω*, 0),

*𝒲*(

*x*, 0, 0,

*t*), and

*𝒲*(

*x*, 0,

*ω*, 0) are plotted as shown in Figs. 6(d), (e), (f) and (g), respectively. The

*𝒲*(0,

*p*, 0,

*t*)) and

*𝒲*(

*x*, 0, 0,

*t*) exhibit oscillation/interference behavior along p =0 and x=0, respectively.

*×*10

*×*10

*×*10 =10000 measurement points for constructing the real and imaginary parts of

*𝒦*(

*x, p, ω, t*) distribution as given in Eq. (14). Then, the Wigner distribution

*𝒲*(

*x*,

*p*,

*ω*,

*t*) is obtained through linear transformation of KR distribution as given in Eq. (15), where the numerical integration is used to transform 10000 measurement values of the real and imaginary parts of KR distribution.

## 4. Conclusion

## References and links

1. | K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. |

2. | V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A |

3. | F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. |

4. | E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. |

5. | M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. |

6. | Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature |

7. | D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: reconstructing quantum states of atom motion,” Phys. Today |

8. | K. Wodkiewicz and G. H. Herling, “Classical and non-classical interference,” Phys. Rev. A |

9. | K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A |

10. | J. Bertrand and P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. |

11. | D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. |

12. | A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. |

13. | M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. |

14. | D. F. McAlister, M. Beck, L. Clarke, A. Meyer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. |

15. | M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. |

16. | B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. |

17. | K. Wagner, J. Janousek, V. Delaubert, H. Zou, C. Harb, N. Treps, J. F. Morizur, P. K. Lam, and H. A. Bachor, “Entangling the Spatial Properties of Laser Beams,” Science |

18. | M. I. Kolobiv, |

19. | V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science |

20. | M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. |

21. | W. H. Zurek, “Sub-Planck structure in phase space and its relevance for quantum decoherence,” Nature |

22. | F. Toscano, D. A. R. Dalvit, L. Davidovich, and W. H. Zurek, “Sub-Planck phase-space structures and Heisenberg-limited measurements,” Phys. Rev. A |

23. | S. John, G. Pang, and Y. Yang, “Optical Coherence Propagation and Imaging in a Multiple Scattering Medium,” J. Biomed. Opt. |

24. | A. Ishimaru, |

**OCIS Codes**

(110.6960) Imaging systems : Tomography

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: January 27, 2011

Revised Manuscript: March 15, 2011

Manuscript Accepted: March 19, 2011

Published: April 4, 2011

**Virtual Issues**

Vol. 6, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Paul Rojas, Rachel Blaser, Yong Meng Sua, and Kim Fook Lee, "Optical phase-space-time-frequency tomography," Opt. Express **19**, 7480-7490 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7480

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### References

- K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, “Heterodyne measurement of Wigner distributions for classical optical fields,” Opt. Lett. 24, 1370–1372 (1999). [CrossRef]
- V. Bollen, Y. M. Sua, and K. F. Lee, “Direct measurement of the Kirkwood-Rihaczek distribution for the spatial properties of a coherent light beam,” Phys. Rev. A 81, 063826 (2010). [CrossRef]
- F. Reil and J. E. Thomas, “Observation of phase conjugation of light arising from enhanced backscattering in a random medium,” Phys. Rev. Lett. 95, 143903 (2005). [CrossRef] [PubMed]
- E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef]
- M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984). [CrossRef]
- Ch. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature 386, 150–153 (1997). [CrossRef]
- D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: reconstructing quantum states of atom motion,” Phys. Today 51(4), 22–28 (April1998). [CrossRef]
- K. Wodkiewicz and G. H. Herling, “Classical and non-classical interference,” Phys. Rev. A 57, 815–821 (1998).
- K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847–2849 (1989). [CrossRef] [PubMed]
- J. Bertrand and P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987). [CrossRef]
- D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993). [CrossRef] [PubMed]
- A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009). [CrossRef]
- M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994). [CrossRef] [PubMed]
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