PART 2 :   BALANCED HOMODYNE        DETECTION             Michael G. RaymerOregon Center for Optics, University of Oregon ...
OUTLINEPART 11. Noise Properties of Photodetectors2. Quantization of Light3. Direct Photodetection and Photon CountingPART...
DC-BALANCED HOMODYNE DETECTION I          Goal -- measure quadrature amplitudes with high                  Q.E. and tempor...
DC-BALANCED HOMODYNE DETECTION II               integrator circuit                             n1                        d...
DC-BALANCED HOMODYNE DETECTION IIIΦS = signal amplitude; ΦL = laser reference amplitude                                   ...
DC-BALANCED HOMODYNE DETECTION IV           ∫                    ˆ (− ) (x,0,t − τ d ) ⋅ ∑ ak v k (x,0,t) + h.c.          ...
DC-BALANCED HOMODYNE DETECTION V                                                           wave-packetsignal : ΦS (r,t) ∝ ...
ULTRAFAST OPTICAL SAMPLING                   Conventional Approach:         Ultrafast Time Gating of Light Intensity by   ...
LINEAR OPTICAL SAMPLING IBHD for Ultrafast Time Gating of Quadrature Amplitudesdetected                   ˆ               ...
LINEAR OPTICAL SAMPLING II     Ultrafast Time Gating of Quadrature Amplitudes LO mode:          v L (x,0,t) ∝ α L v L (x) ...
LINEAR OPTICAL SAMPLING IIIM. E. Anderson, M. Munroe, U. Leonhardt, D. Boggavarapu, D. F. McAlister and M. G. Raymer, Proc...
LINEAR OPTICAL SAMPLING IV                                 LO                                                        scan ...
LINEAR OPTICAL SAMPLING V  Mean Quadrature Measurement - sub ps Time Resolution                                       Samp...
LINEAR OPTICAL SAMPLING VI      Phase Sweeping for Indirect Sampling of Mean     Photon Number and Photon Number Fluctuati...
LINEAR OPTICAL SAMPLING VII    Phase Sweeping --> Photon Number Fluctuations detected          ˆ                  N D (θ )...
LINEAR OPTICAL SAMPLING VIII   Phase Sweeping --> Photon Number FluctuationsVariance of Photon Number in Sampling TimeWind...
LINEAR OPTICAL SAMPLING IX                  Photon Number Fluctuationsif the signal is incoherent, no phase sweeping is re...
LINEAR OPTICAL SAMPLING X        Superluminescent Diode (SLD) Optical Amplifier                                           ...
LINEAR OPTICAL SAMPLING XI                                 (no cavity)                                    1.0             ...
LINEAR OPTICAL SAMPLING XII                         SLD in the single-pass configuration         3.0                   <n(...
LINEAR OPTICAL SAMPLING XIII                 SLD in the double-pass with grating configuration                            ...
Single-Shot Linear Optical Sampling I           -- Does not require phase sweeping.         Measure both quadratures simul...
Fiber Implementation of Single-shot Linear Optical           Sampling Of Photon NumberMFL: mode-locked Erbium-doped fiber ...
Measured quadratures(continuous and dashedline) on a 10-Gb/spulse train.Waveform obtained bypostdetection squaringand summ...
Two-Mode DC-HOMODYNE DETECTION I LO is in a Superposition of two wave-packet modes, 1 and 2    ˆ (+ ) (r,t) = i c | α L |e...
Two-Mode DC-HOMODYNE DETECTION II   ultrafast two-time number correlation measurements using dual-   LO BHD; super lumines...
Two-Mode DC-HOMODYNE DETECTION III                Alternative Method using a Single LO.             Signal is split and de...
Two-Mode DC-HOMODYNE DETECTION IV  Single-time, two-polarization correlation measurements on                              ...
Two-Mode DC-HOMODYNE DETECTION V      Single-time, two-   polarization correlation      measurements on   emission from a ...
Two-Mode DC-HOMODYNE DETECTION VI      Single-time, two-   polarization correlation      measurements on   emission from a...
SUMMARY: DC-Balanced Homodyne Detection1. BHD can take advantage of: high QE and ultrafast timegating.2. BHD can provide m...
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Balanced homodyne detection

  1. 1. PART 2 : BALANCED HOMODYNE DETECTION Michael G. RaymerOregon Center for Optics, University of Oregon raymer@uoregon.edu M.G.Raymer_TTRL2b_V2_2005 1 of 31
  2. 2. OUTLINEPART 11. Noise Properties of Photodetectors2. Quantization of Light3. Direct Photodetection and Photon CountingPART 24. Balanced Homodyne Detection5. Ultrafast Photon Number SamplingPART 36. Quantum State Tomography M.G.Raymer_TTRL2b_V2_2005 2 of 31
  3. 3. DC-BALANCED HOMODYNE DETECTION I Goal -- measure quadrature amplitudes with high Q.E. and temporal-mode selectivity ES = signal field (ωO), 1 - 1000 photons EL = laser reference field (local oscillator) (ωO), 106 photons n1 E1 = dtES (t) ES + E L PD ND BS n2 PD dtEL (t) θ E2 = ES - EL ND ∝ ∫ E1(− )(t − τ d ) E1(+) (t) dt τd delay − ∫ E 2(− )(t − τ d ) E 2(+) (t) dt M.G.Raymer_TTRL2b_V2_2005 3 of 31
  4. 4. DC-BALANCED HOMODYNE DETECTION II integrator circuit n1 dt PD ND n2 PD dt θ M.G.Raymer_TTRL2b_V2_2005 4 of 31
  5. 5. DC-BALANCED HOMODYNE DETECTION IIIΦS = signal amplitude; ΦL = laser reference amplitude n1 dt ΦSES (t) ND BS n2 dtΦEL (t) L θ τd delay overlap ∫ dt ∫ Det d x ΦL TˆND = ˆ (− ) (x,0,t − τ d ) ⋅ Φ(+) (x,0,t) + h.c. 2 ˆS 0 integral ˆ (+ ) (r,t) = i c ΦS ∑ ˆ ak v k (r,t) k v k (r,t) = ∑ Ck j u j (r) exp(−iω j t) j wave-packet c ∫ 0 dt ∫ Det d x v *k (x,0,t) ⋅ v m (x,0,t) = δ k m T 2 modes M.G.Raymer_TTRL2b_V2_2005 5 of 31
  6. 6. DC-BALANCED HOMODYNE DETECTION IV ∫ ˆ (− ) (x,0,t − τ d ) ⋅ ∑ ak v k (x,0,t) + h.c. dt ∫ Det d x ΦL T ˆ ND ∝ 2 ˆ 0 k wave-packet modesAssume that the LO pulse is a strong coherent state of a particularlocalized wave packet mode: LO phase ˆ (+ ) (r,t) ∝ | α | exp(i θ ) v L (r,t) + vacuum ΦL L N D (θ ) = | α L | ( a e−iθ + a† e iθ ) ˆ ˆ ˆ a = ∑ ak c ∫ 0 dt ∫ Det d 2 x v *L (x,0,t − τ d ) ⋅ v k (x,0,t) = ak= L T ˆ ˆ ˆ k The signal field is spatially and temporally gated by the LO field, which has a controlled shape. Where the LO is zero, that portion of the signal is rejected. Only a single temporal-spatial wave- packet mode of the signal is detected. M.G.Raymer_TTRL2b_V2_2005 6 of 31
  7. 7. DC-BALANCED HOMODYNE DETECTION V wave-packetsignal : ΦS (r,t) ∝ a v L (r,t) + ∑ ak v k (r,t) ˆ (+ ) ˆ ˆ k modesquadrature operators: q = ( a + a† ) / 21/2 ˆ ˆ ˆ p = (a − a† ) / i21/2 ˆ ˆ ˆ detected N D (θ ) a e−iθ + a† e iθ ˆ ˆ ˆ LO phase qθ ≡ ˆ = quantity: |αL | 2 2 ˆ N D (θ ) qθ ≡ ˆ = q cosθ + p sin θ ˆ ˆ |αL | 2 ⎛qθ ⎞ ⎛ cos θ sin θ ⎞⎛ q ⎞ ˆ ˆ ⎜ ⎟=⎜ ⎟⎜ ⎟ ⎝ pθ ⎠ ⎝ −sin θ cos θ⎠⎝ p⎠ ˆ ˆ M.G.Raymer_TTRL2b_V2_2005 7 of 31
  8. 8. ULTRAFAST OPTICAL SAMPLING Conventional Approach: Ultrafast Time Gating of Light Intensity by NON-LINEAR OPTICAL SAMPLING strong short pump (ωp )delay sum-frequency (ωp + ωs ) weak signal(ωs ) second-order NL crystal M.G.Raymer_TTRL2b_V2_2005 8 of 31
  9. 9. LINEAR OPTICAL SAMPLING IBHD for Ultrafast Time Gating of Quadrature Amplitudesdetected ˆ N D (θ ) qθ ≡ ˆ = q cosθ + p sin θ ˆ ˆquantity: |αL | 2 LO phase q = ( a + a† ) / 21/2 ˆ ˆ ˆ p = (a − a† ) / i21/2 ˆ ˆ ˆa = ∑ ak c ∫ 0 dt ∫ Det d 2 x v *L (x,0,t − τ d ) ⋅ v k (x,0,t) = ak= L Tˆ ˆ ˆ k LO signal t θ M.G.Raymer_TTRL2b_V2_2005 9 of 31
  10. 10. LINEAR OPTICAL SAMPLING II Ultrafast Time Gating of Quadrature Amplitudes LO mode: v L (x,0,t) ∝ α L v L (x) f L (t − τ d ) ∫ T ˆ N D (τ d ) = −i c α * dt f L* (t − τ d ) φS (t) + h.c. L 0 φS (t) = ∫ Det d x v L * (x) ⋅ ΦS 2 ˆ (+) (x,0,t)if signal is band-limited and signalLO covers the band, e.g. LO f L (t) ∝ (1 / t)sin(B t / 2) ν−Β/2 ν+Β/2 ω ˆ D (τ d ) ∝ α * f˜L* (ν ) ∫ ν +B /2 dω exp(−i ω τ d ) φ S (ω ) + h.c. N ˜ L ν −B /2 2π ∝ α L f˜L* (ν ) φ S (τ d ) + h.c. * exact sampling M.G.Raymer_TTRL2b_V2_2005 10 of 31
  11. 11. LINEAR OPTICAL SAMPLING IIIM. E. Anderson, M. Munroe, U. Leonhardt, D. Boggavarapu, D. F. McAlister and M. G. Raymer, Proceedings ofGeneration, Amplification, and Measurment of Ultrafast Laser Pulses III, pg 142-151 (OE/LASE, San Jose, Jan.1996) (SPIE, Vol. 2701, 1996). Ultrafast Signal Laser (optical or Source elect. synch.) Spectral Signal Filter Signal Reference (LO) Time Phase LO Balanced Delay Adjustment Homodyne Detector τd θ n1 n2 Computer mean quadrature amplitude in sampling ˆ qθ (t) ψ window at time t M.G.Raymer_TTRL2b_V2_2005 11 of 31
  12. 12. LINEAR OPTICAL SAMPLING IV LO scan LO840 nm, 170 fs θ delay τdSample: Microcavityexciton polariton coherent signal Balanced Homodyne detector ˆ qθ (t) ψ M.G.Raymer_TTRL2b_V2_2005 12 of 31
  13. 13. LINEAR OPTICAL SAMPLING V Mean Quadrature Measurement - sub ps Time Resolution Sample: Microcavity ˆ q (t) 10000θ ψ exciton polariton 5 1000 4mean 100 3quadrature g < n(t) > (2) 10 2 (t,t)amplitude<q> at 1 1time t 0.1 0 0.01 -1 0 2 4 6 8 10 12 Time (ps) LO delay τd (ps) ˆ coherent field --> qθ + π /2 (t) ψ = pθ (t) ψ ≅ 0 ˆ M.G.Raymer_TTRL2b_V2_2005 13 of 31
  14. 14. LINEAR OPTICAL SAMPLING VI Phase Sweeping for Indirect Sampling of Mean Photon Number and Photon Number Fluctuationsdetected ˆ N D (θ ) qθ ≡ ˆ = q cosθ + p sin θ (θ = LO phase) ˆ ˆquantity: |αL | 2 Relation with photon-number operator: 1 1 n = a a = ( q − i p )( q + i p ) = q + p + ˆ † ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 ˆ 2 2 2 Phase-averaged quadrature-squared: 1 π 2 1 π 1 2qθ θ = ∫ 0 qθ dθ = ∫0 ˆ(q cosθ + p sin θ ) dθ = (q + p 2 ) 2 2ˆ ˆ ˆ ˆ ˆ π π 2 1 ensemble 1 n = qθ ˆ ˆ 2 − n (t) ψ = qθ (t) ˆ ˆ 2 − θ 2 θ ψ 2 average works also for incoherent field (no fixed phase) M.G.Raymer_TTRL2b_V2_2005 14 of 31
  15. 15. LINEAR OPTICAL SAMPLING VII Phase Sweeping --> Photon Number Fluctuations detected ˆ N D (θ ) quantity: qθ ≡ | α | 2 = q cosθ + p sin θ ˆ ˆ ˆ L Richter’s formula for Factorial Moments: ∞ n (r ) ψ = ∑ [n(n −1)...(n − r + 1)] p(n) = ( a† ) r ( a) r ˆ ˆ ψ n= 0 (r!) 2 2 π dθ = r 2 (2r)! ∫ 0 2π H 2r (qθ ) ψ ˆHermite Polynomials: H 0 (x) = 1, H1 (x) = 2x, H 3 (x) = 4 x 2 − 2 1 2π dθ ∫ 1 n (1) = a ˆ a = ˆ † ˆθ 2 − 2 4q ˆ (t) ψ = qθ 2 (t) n ˆ − 4 0 2π ψ θ ψ 2 2π dθ 2 4 1 n (2) = a a ˆ ˆ †2 2 = ∫ 0 2π 3 qθ − 2 qθ + ˆ ˆ2 2 ψ M.G.Raymer_TTRL2b_V2_2005 15 of 31
  16. 16. LINEAR OPTICAL SAMPLING VIII Phase Sweeping --> Photon Number FluctuationsVariance of Photon Number in Sampling TimeWindow: var(n)=< n 2 > - < n >2 2π dθ ⎡ 2 4 1⎤ ∫ 2 var(n) = qθ − qθ − qθ ˆ ˆ2 ˆ2 + ⎥ 0 2π ⎢ 3 ⎣ 4⎦Second-Order Coherence of Photon Number inSampling Time Window: g(2)(t,t )=[< n 2 > - < n >]/< n >2g(2) (t,t) = 2 corresponds to thermal light, i.e. light producedprimarily by spontaneous emission.g(2) (t,t) = 1 corresponds to light with Poisson statistics, i.e., lightproduced by stimulated emission in the presence of gain saturation. M.G.Raymer_TTRL2b_V2_2005 16 of 31
  17. 17. LINEAR OPTICAL SAMPLING IX Photon Number Fluctuationsif the signal is incoherent, no phase sweeping is required 80MHz 1-50kHz Ti:Sapphire Regen. Amplifier λ/2 Electronic Trigger Pulse Sample LO Delay λ/2 Signal Alt. Source PBS1 λ/2 Voltage Charge-Sensitive PBS2 Pulser Pre-Amps Computer Photodiodes n1 Shaper AD/DA Stretcher n2 Shaper M. GPIB controller Balanced Homodyne Detector Munroe M.G.Raymer_TTRL2b_V2_2005 17 of 31
  18. 18. LINEAR OPTICAL SAMPLING X Superluminescent Diode (SLD) Optical Amplifier metal cap o 6 600 µm 3 µm (AR) SiO 2p-clad layer p-contact layerquantum wells ~ ~ undoped, graded ~ ~n-clad layer confining layers n-GaAs substrate Superluminescent(Sarnoff Labs) Emission M. Munroe M.G.Raymer_TTRL2b_V2_2005 18 of 31
  19. 19. LINEAR OPTICAL SAMPLING XI (no cavity) 1.0 (a) (a) 0.8 Intensity (a.u.) 0.6 0.4 0.2 25 0.0Output Power (mW) 810 820 830 840 850 Wavelength (nm) 20 15 10 5 1.0 (b) 0 Intensity (a.u.) 0 100 200 0.5 Drive Current (mA) (b) 0.0 760 800 840 880 Wavelength (nm) M. Munroe M.G.Raymer_TTRL2b_V2_2005 19 of 31
  20. 20. LINEAR OPTICAL SAMPLING XII SLD in the single-pass configuration 3.0 <n(t,t)> 2.4 (2) g (t,t) 2.5 2.2 2.0 2.0 1.8 g(2)(t,t)<n(t)> 1.6 1.5 1.4 Photon Fluctuation is Thermal-like, 1.0 1.2 within a single time 1.0 window (150 fs) 0.5 0 5 10 time (ns) 15 20 M. Munroe M.G.Raymer_TTRL2b_V2_2005 20 of 31
  21. 21. LINEAR OPTICAL SAMPLING XIII SLD in the double-pass with grating configuration 4.0 <n(t)> 14 (2) g (t,t) 3.5 12 3.0 10 2.5 g(2)(t,t)<n(t)> 8 2.0 6 Photon Fluctuation 1.5 4 is Laser-like, within 2 1.0 a single time 0 0.5 window (150 fs) 0 5 10 15 20 time (ns) M. Munroe M.G.Raymer_TTRL2b_V2_2005 21 of 31
  22. 22. Single-Shot Linear Optical Sampling I -- Does not require phase sweeping. Measure both quadratures simultaneously. Dual- DC-Balanced Homodyne Detection LO1 BHD qsignal 50/50 q2 + p2 = n BHD p π/2 phase LO2 shifter M.G.Raymer_TTRL2b_V2_2005 22 of 31
  23. 23. Fiber Implementation of Single-shot Linear Optical Sampling Of Photon NumberMFL: mode-locked Erbium-doped fiber laser. OF: spectral filter.PC: polarization controller. BD: balanced detector. M.G.Raymer_TTRL2b_V2_2005 23 of 31
  24. 24. Measured quadratures(continuous and dashedline) on a 10-Gb/spulse train.Waveform obtained bypostdetection squaringand summing of the twoquadratures. M.G.Raymer_TTRL2b_V2_2005 24 of 31
  25. 25. Two-Mode DC-HOMODYNE DETECTION I LO is in a Superposition of two wave-packet modes, 1 and 2 ˆ (+ ) (r,t) = i c | α L |exp(iθ ) [v1 (r,t)cosα + v 2 (r,t)exp(−iζ )sin α ] ΦLDual temporal modes: 1 2 (temporal, Dual LO spatial, orsignal polarization) BHD Q β = θ −ζ Q = cos(α )[q1 cosθ + p1 sin θ ] + sin(α )[q2 cos β + p2 sin β ] ˆ ˆ ˆ ˆ ˆ ˆ q1θ ˆ q2 β quadrature of mode 1 quadrature of mode 2 M.G.Raymer_TTRL2b_V2_2005 25 of 31
  26. 26. Two-Mode DC-HOMODYNE DETECTION II ultrafast two-time number correlation measurements using dual- LO BHD; super luminescent laser diode (SLD) 1 2 Dual LO signal t1 t2SLD BHD Q two-time second- order coherence : n (t1 ) n (t2 ): ˆ ˆ g (t1,t2 ) = (2) n (t1 ) n (t2 ) ˆ ˆD. McAlister M.G.Raymer_TTRL2b_V2_2005 26 of 31
  27. 27. Two-Mode DC-HOMODYNE DETECTION III Alternative Method using a Single LO. Signal is split and delayed by different times. Polarization rotations can be introduced. signal LOsource BHD Q polarization rotator two-pol., two-time : n i (t1 ) n j (t2 ): ˆ ˆ second-order g (t1,t2 ) = (2) i, j coherence n i (t1 ) n j (t2 ) ˆ ˆA. Funk M.G.Raymer_TTRL2b_V2_2005 27 of 31
  28. 28. Two-Mode DC-HOMODYNE DETECTION IV Single-time, two-polarization correlation measurements on emission from a VCSEL0-2π phasesweepingand timedelay 0-2π relative phase sweeping E. Blansett M.G.Raymer_TTRL2b_V2_2005 28 of 31
  29. 29. Two-Mode DC-HOMODYNE DETECTION V Single-time, two- polarization correlation measurements on emission from a VCSEL at low temp. (10K) : n i (t1 ) n i (t2 ): ˆ ˆ g (t1,t2 ) = (2) i, i n i (t1 ) ni (t2 ) ˆ ˆ : n i (t1 ) n j (t2 ): ˆ ˆ uncorrelatedg (t1,t2 ) = (2) i, j n i (t1 ) n j (t2 ) ˆ ˆE. Blansett M.G.Raymer_TTRL2b_V2_2005 29 of 31
  30. 30. Two-Mode DC-HOMODYNE DETECTION VI Single-time, two- polarization correlation measurements on emission from a VCSEL at room temp. : n i (t1 ) n i (t2 ): ˆ ˆ g (t1,t2 ) = (2) i, i n i (t1 ) ni (t2 ) ˆ ˆ : n i (t1 ) n j (t2 ): ˆ ˆ anticorrelatedg (t1,t2 ) = (2) i, j n i (t1 ) n j (t2 ) ˆ ˆSpin-flip --> gain competition M.G.Raymer_TTRL2b_V2_2005 30 of 31
  31. 31. SUMMARY: DC-Balanced Homodyne Detection1. BHD can take advantage of: high QE and ultrafast timegating.2. BHD can provide measurements of photon meannumbers, as well as fluctuation information (variance,second-order coherence).3. BHD can selectively detect unique spatial-temporalmodes, including polarization states. M.G.Raymer_TTRL2b_V2_2005 31 of 31

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