balanced homodyne detector

8/18/2019 Balanced Homodyne Detector

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PART ! :

BALANCED HOMODYNEDETECTION

Michael G" Raymer

Oregon Center for Optics# University of Oregon

[email protected]

M.G.Raymer_TTRL2b_V2_20051 of 31


2/31

OUTLINE

PART $

$" Noise Properties of Photodetectors

!" Quantization of Light

%" Direct Photodetection and Photon Counting

PART !

&" Balanced Homodyne Detection

'" Ultrafast Photon Number Sampling

PART %

(" Quantum State Tomography



3/31

DC)BALANCED HOMODYNE DETECTION I

Goal )) measure quadrature amplitudes with high

Q "E " and temporal)mode selectivity

E S * signal field +!O,# $ ) $--- photons

E L * laser reference field +local oscillator, +!O,# $- photons

N D ! E

1

(" )(t " #

d ) E

1

(+)(t ) dt

" E 2

(" )(t " #

d )$ E 2(+)(t ) dt

E S (t )

E L(t )

n1

n2

!

BS

dt

dt

N

E $ *

E S . E L

P

P

E ! *

E S ) E L

delay

" d



4/31

DC)BALANCED HOMODYNE DETECTION IIintegrator circuit

n1

n2

#

dt

dt

N

P

P



5/31

DC)BALANCED HOMODYNE DETECTION III

$ S * signal amplitude; $ L * laser reference amplitude

E S (t )

E L(t )

n1

n2

!

BS

dt

dt

N

delay

" d

ˆ N D

= dt 0

T

d 2 x

Det "̂ L

(# )( x,0,t # $

d ) % "̂ S

(+)

( x,0, t ) + h.c.

!̂ S (+ )

(r, t ) = i ck

" âk v k (r,t )

v k (r,t ) = j

! C k j u j (r) exp("i# j t )

$ S

$ L

overlap

integral

c dt

0

T

! d 2 x v *k ( x,0, t ) " v m ( x,0, t ) Det

! =# k mwave)packet

modesM.G.Raymer_TTRL2b_V2_20055 of 31


6/31

DC)BALANCED HOMODYNE DETECTION IV

ˆ N D ! dt 0T

" d 2 x

Det " #̂ L

($ )

( x,0, t $ % d ) &k ' â

k v k ( x,0, t ) + h.c.

wave)packet modes

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.

Assume that the LO pulse is a strong coherent state of a particular

localized wave packet mode:

ˆ N D(! ) = |"

L| (â e

# i! + â

†e

i! )

!̂ L(+ )

(r,t ) " |# L | exp(i$ ) v L (r,t ) + vacuum

LO phase

â =

k

! âk c dt 0T

" d 2 x Det " v * L ( x,0,t # $ d ) % v k ( x,0,t ) = âk = L



7/31

DC)BALANCED HOMODYNE DETECTION V

wave)packet

modessignal :

quadrature operators:

q̂! "

ˆ N D(! )

|# L | 2=

â e$ i!

+ â†e

i!

2

q̂!

ˆ p!

# $

& ' =

cos! sin!

(sin! cos!

# $

& ' q̂

ˆ p# $

& '

q̂ = (â+ â†) / 2

1/2

ˆ p = (â ! â†) / i2

1/2

!̂ S

(+ )

(r,t )"

ˆa v

L (r,t )

+

k # ˆa

k v

k (r,t )

q̂! "

ˆ N D(! )|# L | 2

= q̂ cos! + ˆ p sin!

detected

quantity:LO phase



8/31

ULTRAFAST OPTICAL SAMPLING

Conventional Approach:

Ultrafast Time Gating of Light Intensity by

NON)LINEAR OPTICAL SAMPLING

strong short

pump +!p

,

weak signal+!s

,

sum)frequency +!p

. !s

,

second)order NL crystal

delay



9/31

q̂ = (â+ â†) / 2

1/2 ˆ p = (â ! â†) / i2

1/2

LINEAR OPTICAL SAMPLING I

BHD for Ultrafast Time Gating of Quadrature Amplitudes

q̂! "

ˆ N D(! )

|# L

| 2= q̂ cos! + ˆ p sin!

detected

quantity: LO phase

â =

k ! âk c dt 0

T

" d 2 x

Det " v * L ( x,0,t # $ d ) % v k ( x,0,t ) = âk = L

t

signalLO

#



10/31

LINEAR OPTICAL SAMPLING II

Ultrafast Time Gating of Quadrature Amplitudes

v L( x,0, t ) ! " L v L ( x) f L (t # $ d )

ˆ N D(! d ) = "i c# L*

dt 0

T

f L*(t " ! d ) % S (t ) + h.c.

! S (t ) = d

2 x

Det

v L *( x) # $̂ S (+)

( x,0, t )

ˆ N D(! d ) " # L* ˜ f L

*($ )

d %

2& $ ' B /2

$ + B /2

( exp('i%! d ) )̃ S (% ) + h.c.

" # L* ˜ f L

*($ ) ) S (! d ) + h.c.

LO mode:

f L (t )! (1 / t )sin( B t / 2)

if signal is band)limited and

LO covers the band# e"g"

%&'/2 %+'/2 !

signal

LO

exact samplingM.G.Raymer_TTRL2b_V2_200510 of 31


11/31

LINEAR OPTICAL SAMPLING III

M. E. Anderson, M. Munroe, U. Leonhardt, D. Boggavarapu, D. F. McAlister and M. G. Raymer, Proceedings of

Generation, Amplification, and Measurment of Ultrafast Laser Pulses III, pg 142-151 (OE/LASE, San Jose, Jan.

1996) (SPIE, Vol. 2701, 1996).

q̂! (t ) "

Ultrafast

Laser

Spectral

Fi lter

Time

Delay

Signal

Source

Balanced

Homodyne

Detector

Computer

LO

Signal

Phase

Adjustment

(optical or

elect. synch.)

n1 n2

mean quadrature

amplitude in sampling

window at time t

# " d

Reference +LO,

Signal



12/31

LINEAR OPTICAL SAMPLING IV

Sample: Microcavity

exciton polariton

scan LOdelay " d

/&- nm# $0- fs

q̂! (t ) "

#

LO

Balanced

Homodynedetector

coherent

signal



13/31

LINEAR OPTICAL SAMPLING V

Mean Quadrature Measurement ) sub ps Time Resolution

0.01

0.1

1

10

100

1000

10000

<

n ( t ) >

121086420

Time (ps)

5

4

3

2

1

0

- 1

g ( 2 )

( t , t )

mean

quadratureamplitude

1q2 at

time t

LO delay " d + ps ,

Sample: Microcavity

exciton polaritonq̂! (t ) "

q̂! +" /2

(t )# = ˆ p

! (t )

# $ 0

coherent field ))2 M.G.Raymer_TTRL2b_V2_200513 of 31


14/31

LINEAR OPTICAL SAMPLING VI

Phase Sweeping for Indirect Sampling of Mean

Photon Number and Photon Number Fluctuations

q̂! "

ˆ N D(! )

|# L

| 2= q̂ cos! + ˆ p sin!

detected

quantity: +# * LO phase,

Relation with photon)number operator:

n̂ = â†

â =1

2 q̂ ! i ˆ p( ) q̂ + i ˆ p( ) = q̂2

+ ˆ p2

+

1

2

Phase)averaged quadrature)squared:

q̂!

2

! =1

" q̂!

2

d ! 0

"

# =1

" q̂ cos! + ˆ p sin! ( )

2

d ! 0

"

# =1

2q̂

2+ ˆ p

2( )

n̂ = q̂!

2

! "

1

2n̂(t )

! = q̂"

2(t )

" ! #

1

2ensemble

average

works also for incoherent field +no fixed phase,M.G.Raymer_TTRL2b_V2_200514 of 31


15/31

LINEAR OPTICAL SAMPLING VII

Phase Sweeping ))2 Photon Number Fluctuations

q̂! " ˆ N D(! )

|# L | 2= q̂ cos! + ˆ p sin! detectedquantity:

n(r )

!

= [n(n "1)...(n " r +1)]n= 0

#

$ p(n) = (â†)r (â)r!

=(r!)

2

2r(2r)!

d %

2& 0

2&

' H 2r (q̂% ) !

Richter’s formula for Factorial Moments:

n̂(t )! = q̂"

2(t )

" ! #

1

2

H 0( x) =1, H 1( x) = 2 x, H 3( x) = 4 x

2! 2Hermite Polynomials:

n(1)

= â†â =

1

4

d !

2" 0

2"

# 4 q̂! 2$ 2

%

n(2)

= â†2

â2

=d !

2" 0

2"

# 2

3q̂!

4$ 2 q̂!

2+1

2 % M.G.Raymer_TTRL2b_V2_2005

15 of 31


16/31

Variance of Photon Number in Sampling TimeWindow: var+n,*1 n 2 ) 1 n 2

LINEAR OPTICAL SAMPLING VIII

Phase Sweeping ))2 Photon Number Fluctuations

var(n) =d !

2" 0

2"

# 2

3q̂!

4$ q̂

!

2$ q̂

!

2 2

+1

4

& ' ) *

Second)Order Coherence of Photon Number in

Sampling Time Window:

g+ ,+t #t ,*31 n ! 2 ) 1 n 2451 n 2

corresponds to thermal light, i.e. light produced

primarily by spontaneous emission.

corresponds to light with Poisson statistics, i.e., light

produced by stimulated emission in the presence of gain saturation.

g(2)

(t , t ) = 2

g(2)

(t , t ) =1



17/31

LINEAR OPTICAL SAMPLING IX

Photon Number Fluctuations

PBS1

LO

Signal

PBS2

PhotodiodesComputer

n1

n2 Shaper

Charge-SensitivePre-Amps

Stretcher

Balanced Homodyne Detector

!/2

!/2

80MHz 1-50kHz

Ti:Sapphire

Shaper

AD/DA

!/2

Alt. Source

ElectronicDelay

Voltage

Pulser

Trigger Pulse

GPIB controller

Regen.Amplifier

Sample

M"

Munroe

if the signal is incoherent# no phase sweeping is required



18/31

LINEAR OPTICAL SAMPLING XSuperluminescent Diode +SLD, Optical Amplifier

M" Munroe

~~

~~

6o

SiO 2

p-contact layer

metal cap

n-GaAs substrate

p-clad layer

undoped, gradedconfining layers

quantum

wells

n-clad layer

3 µm

600 µm

(AR)

Superluminescent

Emission+Sarnoff Labs

,M.G.Raymer_TTRL2b_V2_2005

18 of 31


19/31

LINEAR OPTICAL SAMPLING XI

M" Munroe

25

20

15

10

5

0 O u t p u t P o w e r ( m W )

2001000

Drive Current (mA)

1.0

0.5

0.0

I n t e n s i t y

( a . u . )

880840800760

Wavelength (nm)

(b)

1.0

0.8

0.6

0.4

0.2

0.0

I n t e n s i t y

( a . u . )

850840830820810

Wavelength (nm)

(a)+a,

+b,

+no cavity,



20/31

LINEAR OPTICAL SAMPLING XIISLD in the single-pass configuration

Photon Fluctuationis Thermal)like#

within a single time

window +$'- fs,

M" Munroe

3.0

2.5

2.0

1.5

1.0

0.5

< n ( t ) >

20151050

time (ns)

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

g ( 2 ) (

t , t )

g(2)(t,t)



21/31

LINEAR OPTICAL SAMPLING XIIISLD in the double-pass with grating configuration

14

12

10

8

6

4

2

0

< n ( t ) >

20151050

time (ns)

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

g ( 2 ) (

t , t )

g(2)(t,t)

Photon Fluctuationis Laser)like# within

a single time

window +$'- fs,

M" MunroeM.G.Raymer_TTRL2b_V2_2005

21 of 31


22/31

Single)Shot Linear Optical Sampling I

)) Does not require phase sweeping"Measure both quadratures simultaneously"

Dual) DC)Balanced Homodyne Detection

(/2 phase

shifter

BHD

BHD

signal

LO$

LO!

q

p

'-5'- q . p * n



23/31

Fiber Implementation of Single)shot Linear Optical

Sampling Of Photon Number

MFL: mode)locked Erbium)doped fiber laser" OF: spectral filter"

PC: polarization controller" BD: balanced detector"



24/31

Measured quadratures

+continuous and dashed

line, on a $-)Gb 5 s

pulse train"

Waveform obtained by

postdetection squaring

and summing of the twoquadratures"



25/31

Two)Mode DC)HOMODYNE DETECTION I

BHD

signal

Q

LO is in a Superposition of two wave)packet modes# $ and !

$ !

!̂ L(+ )

(r, t ) = i c |" L |exp(i# ) v1(r, t )cos" + v 2(r, t )exp($i% )sin" [ ]

Q̂ = cos(! ) q̂1cos" + ˆ p1 sin" [ ]+ sin(! ) q̂2 cos# + ˆ p2 sin# [ ]

q̂1!

=" #

Dual temporal modes:

q̂2!

quadrature of mode $ quadrature of mode !

+temporal#

spatial# or

polarization,

Dual LO



26/31

Two)Mode DC)HOMODYNE DETECTION II

ultrafast two-time number correlation measurements using dual-

LO BHD; super luminescent laser diode (SLD)

two-time second-

order coherence

g(2)

(t 1, t 2) =:n̂(t 1) n̂(t 2):

n̂(t 1) n̂(t

2)

BHD

signal

Dual LO

Q

$ !

SLD

t

t

D" McAlisterM.G.Raymer_TTRL2b_V2_2005

26 of 31


27/31

Two)Mode DC)HOMODYNE DETECTION III

two-pol., two-time

second-order

coherence

BHD

signalLO

Q

g i, j (2)

(t 1, t 2) =:n̂i(t 1) n̂ j (t 2):

n̂i(t 1) n̂ j (t 2)

source

polarization rotator

Alternative Method using a Single LO.

Signal is split and delayed by different times.Polarization rotations can be introduced.

A" Funk M.G.Raymer_TTRL2b_V2_200527 of 31


28/31

Two)Mode DC)HOMODYNE DETECTION IV

E" Blansett

Single)time# two)polarization correlation measurements on

emission from a VCSEL

-)!( phase

sweeping

and time

delay

-)!(

relative phase sweepingM.G.Raymer_TTRL2b_V2_200528 of 31


29/31

Two)Mode DC)HOMODYNE DETECTION V

Single)time# two)

polarization correlationmeasurements on


at low temp" +$-K,

E" Blansett

g i, j (2)

(t 1, t 2) =:n̂i(t 1) n̂ j (t 2):

n̂i(t 1) n̂ j (t 2)

g i, i(2)

(t 1, t

2) =

:n̂i(t 1) n̂i(t 2):

n̂i(t 1) n̂i(t 2)

uncorrelated



30/31

Two)Mode DC)HOMODYNE DETECTION VI

Single)time# two)

polarization correlationmeasurements on


at room temp"

g i, j (2)

(t 1, t 2) =:n̂i(t 1) n̂ j (t 2):

n̂i(t 1) n̂ j (t 2)

g i, i(2)

(t 1, t

2) =

:n̂i(t 1) n̂i(t 2):

n̂i(t 1) n̂i(t 2)

anticorrelated

Spin)flip ))2 gain competition M.G.Raymer_TTRL2b_V2_200530 of 31


31/31

SUMMARY: DC)Balanced Homodyne Detection

$" BHD can take advantage of: high QE and ultrafast time

gating"

!" BHD can provide measurements of photon mean

numbers# as well as fluctuation information +variance#second)order coherence,"

%" BHD can selectively detect unique spatial)temporal

modes# including polarization states"


balanced homodyne detector

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