Appendix - Diffusing wave spectroscopy: a unified treatment on temporal

Chapter VI: Diffusing wave spectroscopy: a unified treatment on temporal

6.6 Appendix

SNR of decorrelation time measurements in spatial ensemble methods

When the light is reflected from the dynamic sample, the light intensity at position 𝑟 and time𝑡, 𝐼_𝑟(𝑡), can be decoupled into two parts,

𝐼_𝑟(𝑡) =𝐼_{𝑟 ,𝑆}(𝑡) +𝑛(𝑡), (6.3) where 𝐼_{𝑟 ,𝑆}(𝑡) is the intensity of one speckle of the signal light that is perturbed by the scattering media, and𝑛(𝑡) is the intensity fluctuation from noise, such as shot

noise and detector noise. By the definition of noise, 𝑛(𝑡) has zero mean. 𝐼_{𝑟 ,𝑆}(𝑡) follows exponential distribution due to speckle statistics [17]. For convenience, we define the AC part of𝐼_{𝑟 ,𝑆}(𝑡)and𝐼_𝑟(𝑡)as ˜𝐼_{𝑟 ,𝑆}(𝑡)and ˜𝐼_𝑟(𝑡), respectively, therefore we have

𝐼_𝑟(𝑡) =𝐼˜_{𝑟 ,𝑆}(𝑡) +𝑛(𝑡). (6.4) Here, both ˜𝐼_{𝑟 ,𝑆}(𝑡)and ˜𝐼_𝑟(𝑡)are zero mean, and

h𝐼˜_{𝑟 ,𝑆}(𝑡)i = q

h𝐼˜_{𝑟 ,𝑆}(𝑡)i = 𝐼₀ (6.5) due to speckle statistics [17]. Here, h·i denotes the expected value and 𝐼₀ is the expected value of 𝐼_{𝑟 ,𝑆}(𝑡).

Define the signal𝑆_𝑟 recorded on the camera pixel located at position𝑟 as 𝑆_𝑟 =

∫ 𝑇 0

𝛼 𝐼_𝑟(𝑡)𝑑 𝑡 , (6.6)

where𝛼is the factor that relates the photon numbers to photoelectrons on camera pixels, including detector quantum efficiency, light collection efficiency, and other experimental imperfections, and𝑇 is the camera exposure time.

The speckle contrast𝛾among the whole camera frame is defined as 𝛾 =

h(𝑆_𝑟− h𝑆_𝑟i)²i

h𝑆_𝑟i² or𝛾²= h(𝑆_𝑟 − h𝑆_𝑟i)²i h𝑆_𝑟i²

. (6.7)

The numerator of the𝛾²is 𝛾²

𝑢 𝑝 =h(𝑆_𝑟 − h𝑆_𝑟i)²i

=h(

∫ ^𝑇

𝛼𝐼˜_𝑟(𝑡)𝑑 𝑡)²i

∫ ^𝑇

𝛼²𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₂)𝑑 𝑡₁𝑑 𝑡₂i.

(6.8)

Substituting equation 6.4 into Eq. 6.8, we have

𝛾²

𝑢 𝑝 =h

∫ ^𝑇

𝛼²𝐼˜_{𝑟 ,𝑆}(𝑡₁)𝐼˜_{𝑟 ,𝑆}(𝑡₂)𝑑 𝑡₁𝑑 𝑡₂i + h

∫ ^𝑇

𝛼²𝑛(𝑡₁)𝑛(𝑡₂)𝑑 𝑡₁𝑑 𝑡₂i

=𝛼²h𝐼˜²

𝑟i

∫ ^𝑇

𝑔_𝑆(𝑡₁−𝑡₂)𝑑 𝑡₁𝑑 𝑡₂+𝛼²h𝑛²i

∫ ^𝑇

𝑔_𝑛(𝑡₁−𝑡₂)𝑑 𝑡₁𝑑 𝑡₂

=𝛼²h𝐼˜²

𝑟i𝑇

∫ ^𝑇

2(1− 𝑡 𝑇

)𝑔_𝑆(𝑡)𝑑 𝑡+𝛼²h𝑛²i𝑇

∫ ^𝑇

2(1− 𝑡 𝑇

)𝑔_𝑛(𝑡)𝑑 𝑡 .

(6.9)

Here, 𝑔_𝑆(𝑡) is the correlation function of the mean-removed signal light intensity, and 𝑔_𝑛(𝑡) is the correlation function of noise. If we assume 𝑔_𝑆(𝑡) = 𝑒⁻²^𝑡/𝜏 and 𝑔_𝑛(𝑡) = 𝑒⁻^𝑡^/^𝜏^𝑛, where𝜏is the decorrelation time of the speckle decorrelation time and 𝜏_𝑛 (related to the detector bandwidth BW, 1/BW) is the noise decorrelation time, the above equation can be simplified as

𝛾²

𝑢 𝑝 =𝛼²h𝐼˜²

𝑟i𝑇 𝜏+2𝛼²h𝑛²i𝑇 𝜏_𝑛. (6.10) If the detector is working under the shot noise dominant scheme, where the mean of the number of photoelectrons is equal to the standard deviation of the number of photoelectrons, we have

𝛼 𝐼₀𝑇 =h(

∫ ^𝑇

𝛼𝑛(𝑡)𝑑 𝑡)²i

=2𝛼²h𝑛²i𝑇 𝜏_𝑛.

(6.11)

Substituting the above equation and Eq. 6.5 into Eq. 6.10, the numerator of the contrast square can be further simplified as

𝛾²

𝑢 𝑝=𝛼²𝐼²

0𝑇 𝜏+𝛼 𝐼₀𝑇 . (6.12)

The denominator of𝛾 is

𝛾_{𝑑𝑜𝑤 𝑛} =h𝑆_𝑟i=𝛼 𝐼₀𝑇 . (6.13)

Hence, the contrast has the expression of 𝛾²=

𝛾²

𝑢 𝑝

𝛾²

𝑑𝑜𝑤 𝑛

= 𝛼²𝐼²

0𝑇 𝜏+𝛼 𝐼₀𝑇 (𝛼 𝐼₀𝑇)²

= 𝜏 𝑇

+ 1

𝛼 𝐼₀𝑇

= 𝜏 𝑇

+ 1 𝑁_𝑇

(6.14)

where𝑁_𝑇 is the number of the photoelectrons in one speckle within the camera exposure time. Conventional speckle statistics without considering shot noise predicts that the speckle contrast scales with respect to 1/p

𝑁𝑝 𝑎𝑡 𝑡 𝑒𝑟 𝑛, where 𝑁𝑝 𝑎𝑡 𝑡 𝑒𝑟 𝑛 is the number of independent decorrelation patterns recorded by the camera sensor within the exposure time. Intuitively,𝑁𝑝 𝑎𝑡 𝑡 𝑒𝑟 𝑛is∼𝑇/𝜏since the ratio provides the number of decorrelation events within the camera exposure time. Here, the extra term 1/𝑁_𝑇

in Eq. 6.14 is due to shot noise, i.e., depending on the photon budget. If the number of photoelectrons is sufficient, i.e., 1/𝑁_𝑇 << 𝜏/𝑇 , we can discard this term and the expression degenerates to the conventional form.

In experiment, we can only collect a finite number of speckles and use the ensemble average to approximate the contrast. Hence, the contrast square calculated from one camera frame ˆ𝛾²is a statistical estimation:

𝛾² = h(𝑆_𝑟 − h𝑆_𝑟i)²i_{𝑓 𝑖𝑛𝑖𝑡 𝑒}

h𝑆_𝑟i_{𝑓 𝑖𝑛𝑖𝑡 𝑒} . (6.15)

Here, h·i_{𝑓 𝑖𝑛𝑖𝑡 𝑒} denotes the ensemble average over the finite speckles in one camera frame. Therefore, both the numerator and denominator of the contrast square ˆ𝛾²are estimated from the finite speckles. To evaluate the accuracy of the estimation, we need to estimate the errors of both numerator and denominator in Eq. 6.15.

Given a random variable𝑋, if we use a sample average 1/𝑁𝑖𝑛𝑑𝑒 𝑝 𝑒𝑛𝑑𝑒𝑛𝑡

Í^𝑁𝑖 𝑛𝑑 𝑒 𝑝 𝑒𝑛𝑑 𝑒𝑛𝑡

𝑖=1 𝑋_𝑖

with𝑁𝑖𝑛𝑑𝑒 𝑝 𝑒𝑛𝑑𝑒𝑛𝑡 independent measurements to estimate its expected valueh𝑋i, the error between the sample average and the expected value is aboutp

𝑉(𝑋)/𝑁𝑖𝑛𝑑𝑒 𝑝 𝑒𝑛𝑑𝑒𝑛𝑡, where 𝑉(·) denotes the variance of the random variable 𝑋. In our calculation, 𝑁𝑖𝑛𝑑𝑒 𝑝 𝑒𝑛𝑑𝑒𝑛𝑡, the number of independent measurements (NIM) in spatial ensemble method, is the number of speckle grains in the camera frame, which is termed 𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}.

Let us first calculate the variance of the numerator (𝛾²

𝑢 𝑝) of the𝛾². The variance of 𝛾²

𝑢 𝑝is 𝑉(𝛾²

𝑢 𝑝)= h(𝑆_𝑟 − h𝑆_𝑟i)⁴i − h(𝑆_𝑟 − h𝑆_𝑟i)²i²

=𝛼⁴

∫ 𝑇 0

h𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₂)𝐼˜_𝑟(𝑡₃)𝐼˜_𝑟(𝑡₄)i𝑑 𝑡₁𝑑 𝑡₂𝑑 𝑡₃𝑑 𝑡₄

−𝛼²h

∫ ^𝑇

𝐼_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₂)𝑑 𝑡₁𝑑 𝑡₂i.

(6.16)

The first term in the above equation takes the expected value of four random variables multiplied together. If ˆ𝐼_𝑟is a Gaussian random variable, the bracket can be expanded as

h𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₂)𝐼˜_𝑟(𝑡₃)𝐼˜_𝑟(𝑡₄)i =h𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₂)i h𝐼˜_𝑟(𝑡₃)𝐼˜_𝑟(𝑡₄)i + h𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₃)i h𝐼˜_𝑟(𝑡₂)𝐼˜_𝑟(𝑡₄)i + h𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₄)i h𝐼˜_𝑟(𝑡₂)𝐼˜_𝑟(𝑡₃)i.

(6.17) Here, even though ˜𝐼_𝑟 is not a Gaussian random variable, we still take the formula as an approximation, and this approximation actually holds with tolerable errors based on our experimental results. Equation 6.16 then becomes

𝑉(𝛾²

𝑢 𝑝) ≈ 𝛼⁴

∫ ^𝑇

(h𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₂)i h𝐼˜_𝑟(𝑡₃)𝐼˜_𝑟(𝑡₄)i

+ h𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₃)i h𝐼˜_𝑟(𝑡₂)𝐼˜_𝑟(𝑡₄)i + h𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₄)i h𝐼˜_𝑟(𝑡₂)𝐼˜_𝑟(𝑡₃)i)𝑑 𝑡₁𝑑 𝑡₂𝑑 𝑡₃𝑑 𝑡₄

−𝛼⁴h

∫ ^𝑇

𝐼_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₂)𝑑 𝑡₁𝑑 𝑡₂i

=2𝛼⁴h

∫ 𝑇 0

𝐼_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₂)𝑑 𝑡₁𝑑 𝑡₂i

=2(𝛾²

𝑢 𝑝)².

(6.18)

Therefore, if there are 𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙} independent speckles in spatial ensemble methods, the numerator of ˆ𝛾²

𝑢 𝑝has a form of ˆ

𝛾²

𝑢 𝑝 =𝛾²

𝑢 𝑝±

√ 2𝛾²

𝑢 𝑝

p𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

=𝛾²

𝑢 𝑝(1± s

2 𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

). (6.19)

Here, the term after the±denotes the standard error of the statistical estimation.

Next, let us calculate the error of the denominator (𝛾_{𝑑𝑜𝑤 𝑛}) of𝛾. It is simply s

𝑉(𝑆_𝑟) 𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

= s

h(𝑆_𝑟 − h𝑆_𝑟i)²i 𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

= s

𝛾_{𝑢 𝑝}² 𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

. (6.20)

Therefore, the denominator ˆ𝛾²

𝑑𝑜𝑤 𝑛has a form of ˆ

𝛾²

𝑑𝑜𝑤 𝑛 = (𝛾_{𝑑𝑜𝑤 𝑛}± s

𝛾²

𝑢 𝑝

𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

)² ≈𝛾²

𝑑𝑜𝑤 𝑛± 2𝛾_{𝑢 𝑝}𝛾_{𝑑𝑜𝑤 𝑛} p𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

. (6.21)

Finally, by combining equations 6.15, 6.19, and 6.21, the expression of the estimation of ˆ𝛾²is

𝛾²= 𝛾ˆ²

𝑢 𝑝

ˆ 𝛾²

𝑑𝑜𝑤 𝑛

≈ 𝛾²

𝑢 𝑝

𝛾²

𝑑𝑜𝑤 𝑛

(1± s

1 𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

2+ 4𝛾²

𝑢 𝑝

𝛾²

𝑑𝑜𝑤 𝑛

)

=(𝜏 𝑇

+ 1 𝑁_𝑇

) (1± s

1 𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

2+4(𝜏 𝑇

+ 1 𝑁_𝑇

)).

(6.22)

Hence, the estimation of the contrast is ˆ

𝛾 = r

𝜏 𝑇

+ 1 𝑁_𝑇

(1± 1 2

s 1

𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙} r

2+4(𝜏 𝑇

+ 1 𝑁_𝑇

)). (6.23)

In SVS, we usually set the camera exposure𝑇 much greater than the decorrelation time𝜏, e.g.,𝑇 >> 𝜏, and the number of photons collected by one camera pixel 𝑁_𝑇

is also much greater than 1, e.g., 𝑁_𝑇 >> 1. In this case, in the above equation, the second term in the second square root in the error part can be dropped and the estimation of the contrast square ˆ𝛾 can be approximated as

ˆ 𝛾 =

r 𝜏 𝑇

+ 1 𝑁_𝑇

(1± s

1 2𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

). (6.24)

Rewriting the above equation, we have

𝜏 =𝑇𝛾ˆ²(1± s

1 2𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

) − 𝑇 𝑁_𝑇

. (6.25)

The SNR of the decorrelation time in spatial ensemble methods is 𝑆 𝑁 𝑅_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙} = 𝜏

𝑒𝑟 𝑟(𝜏) = 𝜏 𝑇𝛾ˆ²

q 1

2𝑀𝑠 𝑝 𝑎𝑡 𝑖 𝑎𝑙

= 1 1+ ^𝑇

𝜏 𝑁𝑇

𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}

2 .

(6.26)

Here, 𝑒𝑟 𝑟(𝜏) is the standard error of 𝜏, which is equal to𝑇𝛾ˆ²

q 1

2𝑀𝑠 𝑝 𝑎𝑡 𝑖 𝑎𝑙. As we define𝑁_𝜏 as the number of photoelectrons on each camera pixel per time interval𝜏, we find𝑁_𝜏 = ^𝑁^𝑇

𝑇

𝜏. The above Eq. 6.26 can be simplified as 𝑆 𝑁 𝑅_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙} = 1

√ 2

1 1+ ¹

𝑁_𝜏

𝑀_{𝑠 𝑝 𝑎𝑡𝑖 𝑎𝑙}. (6.27)

SNR of decorrelation time measurements in temporal sampling methods In temporal sampling methods, a fast photodetector with a sufficient bandwidth, such as a single-photon-counting-module (SPCM), is used to well sample the temporal trace𝐼_𝑟(𝑡), and the decorrelation time𝜏is computed from the intensity correlation function𝐺₂(𝑡):

𝐺₂(𝑡)= 1 𝑇

𝛼²

∫ 𝑇 0

𝐼_𝑟(𝑡₁)𝐼_𝑟(𝑡₁−𝑡)𝑑 𝑡₁. (6.28) In practice, the correlation is performed between the mean-removed intensity fluctuation:

𝐺˜₂(𝑡) = 1 𝑇

𝛼²

∫ ^𝑇

𝐼_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₁−𝑡)𝑑 𝑡₁, (6.29) where ˜𝐺₂(𝑡) denotes the intensity correlation function of the two mean-removed intensity traces, ˜𝐼_𝑟(𝑡)is the AC part of the intensity fluctuation,𝑡₁is the time variable for integral, and𝑡is the time offset between the two intensity traces.

By substituting Eq. 6.4 into Eq. 6.29, we have

𝐺₂(𝑡) = 1 𝑇

𝛼²

∫ ^𝑇

[𝐼˜_{𝑟 ,𝑆}(𝑡₁) +𝑛(𝑡₁)] [𝐼˜_{𝑟 ,𝑆}(𝑡₁−𝑡) +𝑛(𝑡₁−𝑡)]𝑑 𝑡₁. (6.30) The expected value of ˜𝐺₂(𝑡) is

h𝐺˜₂(𝑡)i =𝛼²𝐼²

0𝑔_𝑆(𝑡) +𝛼²h𝑛²i𝑔_𝑛(𝑡). (6.31) Same as the definition before,𝑔_𝑆(𝑡)is the correlation function of the mean-removed signal light intensity, and𝑔_𝑛(𝑡)is the correlation function of noise.

When we use the finite time average to estimate the expected value of ˜𝐺₂(𝑡), we need to calculate the variance of ˜𝐺₂(𝑡):

𝑉[𝐺˜₂(𝑡)] = 1 𝑇

(h𝐺˜₂(𝑡)²i − h𝐺˜₂(𝑡)i²)

= 1 𝑇

∫ ^𝑇

𝛼⁴𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₁−𝑡)𝐼˜_𝑟(𝑡₂)𝐼˜_𝑟(𝑡₂−𝑡)𝑑 𝑡₁𝑑 𝑡₂i − 1 𝑇²

h𝐺˜₂(𝑡)²i²

≈ 2𝛼⁴ 𝑇²

∫ ^𝑇

h𝐼˜_𝑟(𝑡₁)𝐼˜_𝑟(𝑡₂)i²𝑑 𝑡₁𝑑 𝑡₂

= 2𝛼⁴ 𝑇²

∫ ^𝑇

[𝐼²

0𝑔_𝑆(𝑡) + h𝑛²i𝑔_𝑛(𝑡)]²𝑑 𝑡₁𝑑 𝑡₂

≈2(𝛼⁴𝐼⁴

𝜏 2𝑇

+𝛼³𝐼³

1 𝑇

+𝛼²𝐼²

1 𝑇 𝜏_𝑛

(6.32)

Hence, if we calculate the correlation function ˜𝐺₂(𝑡) by using a finite long measurement trace and use it to estimate h𝐺˜₂(𝑡)i, we have the following estimation form

𝐺˜₂(𝑡) =h𝐺˜₂(𝑡)i ± q

𝑉[𝐺˜₂(𝑡)]

= [𝛼²𝐼²

0𝑔_𝑆(𝑡) +𝛼²h𝑛²i𝑔_𝑛(𝑡)] ± r

2(𝛼⁴𝐼⁴

𝜏 2𝑇

+𝛼³𝐼³

1 𝑇

+𝛼²𝐼²

1 𝑇 𝜏_𝑛

(6.33)

Since 𝑔_𝑛(𝑡) usually has much shorter decorrelation time compared to 𝑔_𝑆(𝑡), to estimate the speckle decorrelation time 𝜏, we can use the part of the correlation curve where𝑔_𝑛(𝑡)4 drops close to 0 while 𝑔_𝑆(𝑡) is still close to unity. In this case,

the part of the the correlation curve ˆ𝐺₂(𝑡)is 𝐺ˆ₂(𝑡) = [𝛼²𝐼²

0𝑔_𝑆(𝑡) ± r

2(𝛼⁴𝐼⁴

𝜏 2𝑇

+𝛼³𝐼³

1 𝑇

+𝛼²𝐼²

1 𝑇 𝜏_𝑛

)

=𝛼²𝐼²

0[𝑔_𝑆(𝑡) ± s

2( 𝜏 2𝑇

+ 1

𝛼 𝐼₀𝑇

+ 1

𝛼²𝐼²

0𝑇 𝜏_𝑛 )].

(6.34)

In the experiment,𝜏_𝑛can be approximated as the inverse of the detector bandwidth, or equivalently the time interval Δ𝑇 between two data points. In the following calculation, we will substitute𝜏_𝑛byΔ𝑇.

When we use the decorrelation curve to estimate a parameter associated with the curve, such as decorrelation time, there exist different fitting models to retrieve the parameter. Here, for simplicity, the estimated decorrelation time ˆ𝜏can be chosen by taking the time point where the decorrelation curve drops to 1/𝑒. In this case, the error of the estimated decorrelation time𝑒𝑟 𝑟(𝜏)is

𝑒𝑟 𝑟(𝜏) = 1

|^𝑑𝑔^𝑆

𝑑 𝑡 |𝑔_𝑆(𝑡)=1/𝑒| s

2( 𝜏 2𝑇

+ 1

𝛼 𝐼₀𝑇

+ 1

𝛼²𝐼²

0𝑇Δ𝑇 )

= 𝑒 2𝜏

s 2( 𝜏

2𝑇

+ 1

𝛼 𝐼₀𝑇

+ 1

𝛼²𝐼²

0𝑇Δ𝑇 ).

(6.35)

Hence, the decorrelation time𝜏can be estimated from the calculated decorrelation time𝜏as

𝜏=𝜏ˆ(1± 𝑒

√ 2

s 𝜏 2𝑇

+ 1

𝛼 𝐼₀𝑇

+ 1

𝛼²𝐼²

0𝑇Δ𝑇

). (6.36)

The SNR of the decorrelation time in temporal sampling methods is 𝑆 𝑁 𝑅𝑡 𝑒𝑚 𝑝 𝑜𝑟 𝑎𝑙 = 𝜏

𝑒𝑟 𝑟(𝜏) =

√ 2 𝑒

1 q𝜏

2𝑇 + ¹

𝛼 𝐼₀𝑇 + ¹

𝛼2𝐼²

0𝑇Δ𝑇

. (6.37)

As defined in the main text, the NIM in temporal domain methods 𝑀𝑡 𝑒𝑚 𝑝 𝑜𝑟 𝑎𝑙 = ²^𝑇

𝜏 , and taking the fact that 𝛼 𝐼₀𝑇 = ¹₂𝑀𝑡 𝑒𝑚 𝑝 𝑜𝑟 𝑎𝑙𝑁_𝜏, the SNR equation 6.37 can be rewritten as

𝑆 𝑁 𝑅𝑡 𝑒𝑚 𝑝 𝑜𝑟 𝑎𝑙 =

√ 2 𝑒

1 q1+ ²

𝑁_𝑇 + ²

𝑁²

𝑇

𝜏 Δ𝑇

𝑀𝑡 𝑒𝑚 𝑝 𝑜𝑟 𝑎𝑙. (6.38)

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Dalam dokumen Optical light manipulation and imaging through scattering media (Halaman 145-155)