Abstract

We demonstrate a simple and robust method for characterizing the temporal coherence of statistically stationary optical sources by using dynamic light scattering. Measurement of the contrast of the fluctuating speckle pattern produced by two counterpropagating beams incident on a scattering medium is used to evaluate their mutual coherence. Important features of this method are high statistical accuracy, the ability to compensate for imperfect spatial coherence, and the possibility of characterizing milliwatt-level optical beams with a wide range of spectral widths. As an example, the squared magnitude of the field autocorrelation function for light emitted by a broadband argon-ion laser is obtained.

© 2005 Optical Society of America

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  1. R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
    [CrossRef]
  2. B. L. Morgan, L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
    [CrossRef]
  3. T. Okoshi, K. Kikuchi, A. Nakayama, “Novel method for high-resolution measurement of laser output spectrum,” Electron. Lett. 16, 630–631 (1980).
    [CrossRef]
  4. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997).
  5. J. Rička, “Dynamic light scattering with single-mode and multimode receivers,” Appl. Opt. 32, 2860–2875 (1993).
    [CrossRef]
  6. B. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  7. I. Flammer, J. Rička, “Dynamic light scattering with single-mode receivers: partial heterodyning regime,” Appl. Opt. 36, 7508–7517 (1997).
    [CrossRef]
  8. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  9. B. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  10. K. Clays, A. Persoons, “Dynamic light scattering with femtosecond laser pulses: potential and limitations toward quasielastic nonlinear light scattering,” J. Chem. Phys. 113, 9706–9713 (2000).
    [CrossRef]
  11. H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
    [CrossRef]
  12. V. Dominic, X. Steve, Y. R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser-pulses in real-time,” Appl. Phys. Lett. 56, 521–523 (1990).
    [CrossRef]

2000 (1)

K. Clays, A. Persoons, “Dynamic light scattering with femtosecond laser pulses: potential and limitations toward quasielastic nonlinear light scattering,” J. Chem. Phys. 113, 9706–9713 (2000).
[CrossRef]

1997 (1)

1993 (1)

1990 (1)

V. Dominic, X. Steve, Y. R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser-pulses in real-time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

1980 (2)

T. Okoshi, K. Kikuchi, A. Nakayama, “Novel method for high-resolution measurement of laser output spectrum,” Electron. Lett. 16, 630–631 (1980).
[CrossRef]

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[CrossRef]

1966 (1)

B. L. Morgan, L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
[CrossRef]

1956 (1)

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

Berne, B.

B. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Brown, R. Hanbury

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

Clays, K.

K. Clays, A. Persoons, “Dynamic light scattering with femtosecond laser pulses: potential and limitations toward quasielastic nonlinear light scattering,” J. Chem. Phys. 113, 9706–9713 (2000).
[CrossRef]

Dominic, V.

V. Dominic, X. Steve, Y. R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser-pulses in real-time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

Eichler, H. J.

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[CrossRef]

Feinberg, J.

V. Dominic, X. Steve, Y. R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser-pulses in real-time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

Flammer, I.

Kikuchi, K.

T. Okoshi, K. Kikuchi, A. Nakayama, “Novel method for high-resolution measurement of laser output spectrum,” Electron. Lett. 16, 630–631 (1980).
[CrossRef]

Klein, U.

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[CrossRef]

Langhans, D.

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[CrossRef]

Mandel, L.

B. L. Morgan, L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
[CrossRef]

Morgan, B. L.

B. L. Morgan, L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
[CrossRef]

Nakayama, A.

T. Okoshi, K. Kikuchi, A. Nakayama, “Novel method for high-resolution measurement of laser output spectrum,” Electron. Lett. 16, 630–631 (1980).
[CrossRef]

Okoshi, T.

T. Okoshi, K. Kikuchi, A. Nakayama, “Novel method for high-resolution measurement of laser output spectrum,” Electron. Lett. 16, 630–631 (1980).
[CrossRef]

Pecora, R.

B. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Persoons, A.

K. Clays, A. Persoons, “Dynamic light scattering with femtosecond laser pulses: potential and limitations toward quasielastic nonlinear light scattering,” J. Chem. Phys. 113, 9706–9713 (2000).
[CrossRef]

Pierce, Y. R. M.

V. Dominic, X. Steve, Y. R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser-pulses in real-time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

Ricka, J.

Saleh, B.

B. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Siegman, A.

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Steve, X.

V. Dominic, X. Steve, Y. R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser-pulses in real-time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

Teich, M.

B. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Twiss, R. Q.

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997).

Appl. Opt. (2)

Appl. Phys. (1)

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[CrossRef]

Appl. Phys. Lett. (1)

V. Dominic, X. Steve, Y. R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser-pulses in real-time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

Electron. Lett. (1)

T. Okoshi, K. Kikuchi, A. Nakayama, “Novel method for high-resolution measurement of laser output spectrum,” Electron. Lett. 16, 630–631 (1980).
[CrossRef]

J. Chem. Phys. (1)

K. Clays, A. Persoons, “Dynamic light scattering with femtosecond laser pulses: potential and limitations toward quasielastic nonlinear light scattering,” J. Chem. Phys. 113, 9706–9713 (2000).
[CrossRef]

Nature (1)

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

Phys. Rev. Lett. (1)

B. L. Morgan, L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
[CrossRef]

Other (4)

B. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997).

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

B. Saleh, M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

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Figures (7)

Fig. 1
Fig. 1

Dual-beam DLS experiment: A polarizer (P) and half-wave plates (HWP) are used to divide the source into right- (R) and left- (L) propagating beams with a relative delay ΔTRL. A single-mode optical fiber (SMF) with a lens defines the observation volume.

Fig. 2
Fig. 2

Detail of the scattering geometry indicating the left- (L) and right- (R) propagating beams, the observation mode (detection), and the respective scattering volumes, VR and VL.

Fig. 3
Fig. 3

Normalized autocorrelation functions obtained for three different values of the relative optical decay ΔTRL. For the data taken when ΔTRL is close to zero (■) the two beams are mutually coherent, giving rise to high-contrast dynamic speckle fluctuations. When ΔTRL is much larger than the source’s coherence time (□), the two beams are mutually incoherent, and the dynamic amplitude of the fluctuating speckle pattern drops to roughly half of its maximum value.

Fig. 4
Fig. 4

Measured dynamic amplitude, β(ΔTRL) of the light scattered from a broadband Ar+ laser when the supply current was 43 A. Also shown are the standard deviations σβ of the dynamic amplitude obtained from three separate experimental runs of 30 s duration at each optical delay position. This procedure was sufficient to obtain values that are accurate to better than 1%. The lower graph displays the individual particle diffusion correlation times τD, which are constant as expected.

Fig. 5
Fig. 5

Magnitude squared temporal autocorrelation function extrapolated form the raw data by use of Eq. (16) for three different supply currents for the argon-ion laser: 20, 30, and 43 A. The solid curves are the fits obtained from the truncated Gaussian model described in the text. Shown in the inset are the corresponding coherence times obtained by use of the power-equivalent width convention, whose values are 174.3 ± 0.7 ps (43 A), 220.1 ± 0.6 ps (30 A), and 248.0 ± 0.8 ps (20 A).

Fig. 6
Fig. 6

Residuals (i.e., difference between the experimental data points and the fit) resulting from the truncated Gaussian fit (○) described in the text and a simple Gaussian fit (■) to the data at a supply current of 43 A. The sums of the squared residuals are 8.5 × 10-4 for the Gaussian fit and 1.9 × 10-4 for the truncated Gaussian fit, respectively.

Fig. 7
Fig. 7

Comparison of the temporal autocorrelation functions obtained with the dual-beam dynamic light-scattering method (■) and the more conventional Michelson interferometer method (□). The residuals in the lower graph were calculated relative to the fit obtained from the truncated Gaussian model described in the text, the respective values being for the dual-beam DLS data (ΔνDop=5.79±0.05 GHz, R=4.65±0.43, A=0.97±0.01, and t0=-0.28±0.58 ps) and the Michelson data (ΔνDop=5.72±0.16 GHz, R=4.02±0.65, A=0.95±0.02, and t0=-0.92±2.23 ps).

Equations (31)

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Fa(r, t)=Ea(t)Sa(r)exp[ika  r]+c.c.,
J(t)=Ka,b=R,LEa(t+Ta)Eb*(t+Tb)×k,lNXa[rk(t)]Xb*[rl(t)]exp{i[qa  rk(t)-qb  rl(t)]},
J(t)=Ka,b=R,L|Ea(t)|2kN|Xa[rk(t)]|2.
k=1N|Xa[rk(t)]|2=ρd3r|Xa[r]|2ρVa,
J(t)=Ka=R,L|Ea(t)|2ρVa.
g(2)(τ)=J(t)J(t+τ)/J(t)2=B+β exp(-2τ/τD).
J(0)J(τ)=K2a,b,c,d=R,LEa(Ta)Eb*(Tb)Ec(Tc+τ)Ed*(Td+τ)×k,l,m,nNXa[rk(t)]Xb*[rl(t)]×Xc[rm(t+τ)]Xd*[rn(t+τ)]×exp{iqa  rk(t)-iqb  rl(t)+iqc  rm(t+τ)-iqd  rn(t+τ)}.
J(0)J(τ)s=K2ρ2a,b=R,LEa(Ta)Ea*(Ta)×Eb(Tb+τ)Eb*(Tb+τ)VaVb,
J(0)J(τ)d=K2ρ2 exp-2ττDa,b=R,LEa(Ta)×Eb*(Tb)Eb(Tb+τ)Ea*(Ta+τ)VaVb.
J(0)J(τ)s=K2ρ2a,b=R,L|Ea(0)|2|Eb(0)|2VaVb=J(0)2,
J(0)J(τ)d=K2ρ2 exp-2ττD{|ER(0)|22VR2+|EL(0)|22VL2+2|ER(ΔTRL)EL*(0)|2VRVL}.
fR=|ER(0)|2VR[|ER(0)|2VR+|EL(0)|2VL],
g(2)(τ)=J(0)J(τ)J(0)2=1+exp-2ττDfR2+fL2+2fRfL |ER(ΔTRL)EL*(0)|2|ER(0)|2|EL(0)|2.
β(ΔTRL)=g(2)(τCohττD)-1=fR2+fL2+2fRfL|γ(ΔTRL)|2,
γ(ΔTRL)=ER(ΔTRL)EL*(0)[|ER(0)|2|EL(0)|2]1/2
β(ΔTRL)=β0(fR2+fL2+2fRfL|γ(ΔTRL)|2).
|γ(ΔTRL)|Gauss2=A exp-π2w2ΔTRL22 ln 2,
Γ(νm)=Γ0 exp{-ln(2)[2(νm-ν0)/ΔνDop]2}×1+I(νm)ISat-1/2.
I(νm)=ISat(R2 exp{-2ln(2)[2(νm-ν0)/ΔνDop]2}-1).
γ(ΔTRL) =m=-NNn=-NNI(νm)I(νn)exp[-i2π(νm-νv)ΔTRL]n=-NNI(νm),
τCoh=dt|γ(t)|2.
Fa(r, t)=xMEa,x(t)Sa,x(r)exp[ika  r]+c.c.
Sa,xSa,y*d3rXa,x(r)Xa,y*(r)=d3rSa,x(r)Sa,y*(r)|Sd(r)|2.
J(0)=Kρa=R,LxM|Ea,x(Ta)|2[Sa,xSa,x*].
J(0)J(τ)s=K2ρ2a,b=R,Lx,yM{Ea,x(Ta)Ea,x*(Ta)×Eb,y(Tb+τ)Eb,y*(Tb+τ)×[Sa,xSa,x*][Sb,ySb,y*]}=J(0)2.
J(0)J(τ)d=K2ρ2 exp-2ττDa,b=R,Lx,yM{Ea,x(Ta)Eb,x*(Tb)Eb,y(Tb+τ)Ea,y*(Ta+τ)×[Sa,xSa,y*][Sb,ySb,x*]},
β(ΔTRL)=g(2)(τCohττD)-1=x,yM|ER,x(0)|2|ER,y(0)|2|SR,xSR,y*|2+|EL,x(0)|2|EL,y(0)|2|SL,xSL,y*|2+2 Re(ER,x(ΔTRL)EL,x*(0)EL,y(0)ER,y*(ΔTRL)[SR,xSR,y*][SL,ySL,x*])xM{|ER,x(0)|2[SR,xSR,x*]+|EL,x(0)|2[SL,xSL,x*]}2.
fR=xM|ER,x(TR)|2[SR,xSR,x*]a=R,LxM|Ea,x(Ta)|2[Sa,xSa,x*],
βR=x,yM|ER,x(0)|2|ER,y(0)|2|SR,xSR,y*|2xM{|ER,x(0)|2[SR,xSR,x*]}2.
β(ΔTRL)=fR2βR+fL2βL+2 Re x,yMER,x(ΔTRL)EL,x*(0)EL,y(0)ER,y*(ΔTRL)[SR,xSR,y*][SL,ySL,x*]xM{|ER,x(0)|2[SR,xSR,x*]+|EL,x(0)|2[SL,xSL,x*]}2.
 β(ΔTRL)=β0(fR2+fL2+2fRfL|γ(ΔTRL)|2),

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