WINDOW CORRECTIONS: OPTICAL AND IMPEDANCE MISMATCH

A p p e n d i x B

WINDOW CORRECTIONS: OPTICAL AND IMPEDANCE

where

𝑑 𝑋₁ 𝑑 𝑡

=𝑈_𝑠−𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} 𝑑 𝑋₂

𝑑 𝑡

=−𝑈_𝑠 𝑑 𝑋₃

𝑑 𝑡

=0

𝑢_{𝑜 𝑏 𝑠} =−𝜂(𝑈_𝑠 −𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) +𝜂₀𝑈_𝑠 𝑈_𝑠 =𝐶₀+𝑆𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

𝜂 = 𝐴+𝐵 𝜌₁ 𝜌₁= 𝜌₀

1− ^{𝑢𝑎 𝑐 𝑡 𝑢 𝑎𝑙}

𝐶₀+𝑆𝑢𝑎 𝑐 𝑡 𝑢 𝑎𝑙

In the above equations, 𝑋₁, 𝑋₂, 𝑋₃ are the distances shown in Fig. B.1. 𝑈_𝑠 is the shock-velocity in the window in material frame. Since the material ahead of the shock is considered to be under ambient conditions, the material frame and spatial frame shock-velocity are identical for this problem. 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} is the actual target-window interface velocity, and𝑢_{𝑜 𝑏 𝑠} is the measured target-window interface velocity. 𝜂₀is the refractive index of window under ambient conditions, and𝜂is the refractive index of the window material behind the shock-wave. Similarly, 𝜌₀is the density of the window material under ambient conditions, and𝜌₁is the density of the window material behind the shock-wave. 𝐶₀and𝑆 are material parameters used to relate the shock-wave speed in the window to the particle velocity difference across the shock-wave. Using the above equations,𝑢_{𝑜 𝑏 𝑠}can be expressed as a function of 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}[1] as:

𝑢_{𝑜 𝑏 𝑠} =−(𝐴+𝐵 𝜌₁(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙})) (𝑈_𝑠−𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) +𝜂₀𝑈_𝑠(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}). Optical correction for release fan in window

Figure B.2 depicts the release-fan in the LiF window, which is formed when the shock wave is reflected off the free surface of the LiF window. The density, and hence the refractive index of LiF, varies continuously across the release fan. This variation in the refractive index will have to be accounted for in computing the optical path length for a light signal travelling to and from the PDV probe. The procedure adopted to obtain an optical correction for this problem is similar to that used in [2], where optical corrections were determined for a compression-fan in a window. Similar to the case of optical correction due to shock-wave in window, the

PDV Probe η , ρ₁

u_p = u_actual

η0 , ρ0

up ≈ 2uactual

SLG LiF [100]

L_full

X₁ X₂ X₃

Release fan

Figure B.2: Schematic of window-correction for release fan in window.

observed velocity at the interface is related to the actual velocity as:

𝑢_{𝑜 𝑏 𝑠} =𝜂𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} − 2(𝜂₀−1)𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}−

∫ ^𝑥𝑏(𝑡) 𝑥𝑎(𝑡)

𝜕 𝜂

𝜕 𝑡

𝑑𝑥 (B.1)

It remains to evaluate: ∫𝑥𝑏(𝑡) 𝑥𝑎(𝑡)

𝜕 𝜂

𝜕 𝑡 𝑑𝑥.

Consider recasting this integral in material coordinateh, whereh = 0 at LiF-SLG interface, ℎ = ℎ_𝑎(𝑡) at 𝑥 = 𝑥_𝑎(𝑡), and ℎ = ℎ_𝑏(𝑡) at 𝑥 = 𝑥_𝑏(𝑡). For ℎ ≥ ℎ_𝑎(𝑡), the transformation from h to x can be expressed as follows, for 𝑥_𝑖(𝑡) denoting the spatial-coordinate of the SLG-LiF interface:

𝑥(ℎ, 𝑡) =𝑥_𝑖(𝑡) + 𝜌₀ ℎ_𝑎(𝑡) 𝜌₁

+

∫ ^ℎ

ℎ_𝑎(𝑡)

𝜌₀

𝜌(ℎ⁰)𝑑 ℎ⁰ (B.2) For evaluating∫𝑥𝑏(𝑡)

𝑥𝑎(𝑡)

𝜕 𝜂

𝜕 𝑡

𝑑𝑥, we need𝑑𝑥for a constant𝑡. By evaluating this differen- tial using Eq. B.2, we get : 𝑑𝑥 =𝜕 𝑥(ℎ, 𝑡) |𝑡 = _𝜌(ℎ)^𝜌0 𝑑 ℎ. This can be used to evaluate

∫^𝑥𝑏(𝑡) 𝑥_𝑎(𝑡)

𝜕 𝜂

𝜕 𝑡 𝑑𝑥as follows:

∫ ^𝑥𝑏(𝑡) 𝑥𝑎(𝑡)

𝜕 𝜂

𝜕 𝑡

𝜕 𝑥|𝑡 =

∫ ^ℎ𝑏(𝑡) ℎ𝑎(𝑡)

𝜕 𝜂

𝜕 𝑡 𝑥

(ℎ(𝑥 , 𝑡), 𝑡) 𝜌₀ 𝜌(ℎ)𝑑 ℎ

=

∫ ^ℎ𝑏(𝑡) ℎ𝑎(𝑡)

𝜕 𝜂

𝜕 𝜌

𝜕 𝜌

𝜕 𝑡 𝑥

(ℎ(𝑥 , 𝑡), 𝑡) 𝜌₀

𝜌(ℎ)𝑑 ℎ (B.3)

Further,

𝜕 𝜌

𝜕 𝑡 𝑥

(ℎ(𝑥 , 𝑡), 𝑡) = 𝜕 𝜌

𝜕 ℎ 𝑡

𝜕 ℎ

𝜕 𝑡 𝑥

+ 𝜕 𝜌

𝜕 𝑡 ℎ

(B.4) The individual components of Eq. B.4 are evaluated next. To evaluate ^{𝜕 ℎ}_{𝜕 𝑡}

𝑥

, differentiate Eq. B.2 with respect to𝑡keeping𝑥constant.

>

0

𝜕 𝑥(ℎ, 𝑡)

𝜕 𝑡 𝑥

= 𝜕 𝑥_𝑖(𝑡)

𝜕 𝑡

+ 𝜌₀ 𝜌₁

𝜕 ℎ_𝑎(𝑡)

𝜕 𝑡

+ 𝜌₀ 𝜌(ℎ)

𝜕 ℎ

𝜕 𝑡 𝑥

− 𝜌₀ 𝜌(ℎ_𝑎)

𝜕 ℎ_𝑎(𝑡)

𝜕 𝑡

(B.5) Using continuity in the release−fan,

𝜌(ℎ_𝑎) = 𝜌₁ 0=

>

𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

𝜕 𝑥_𝑖(𝑡)

𝜕 𝑡

+ 𝜌₀ 𝜌(ℎ)

𝜕 ℎ

𝜕 𝑡 𝑥

=⇒ 𝜕 ℎ

𝜕 𝑡 𝑥

=−𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} 𝜌(ℎ)

𝜌₀

(B.6)

To evaluate ^{𝜕 𝜌}_{𝜕 ℎ} 𝑡

, consider𝜌= 𝜌(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}, 𝑢), where𝑢 is particle velocity:

𝜕 𝜌

𝜕 ℎ 𝑡

= 𝜕 𝜌

𝜕 𝑢

𝜕 𝑢

𝜕 ℎ 𝑡

(B.7) Consider Fig. B.3 to evaluate Eq. B.7 . Note that although material properties like 𝐶_ℎ and ^𝜕𝐶_{𝜕 𝑢}^ℎ are expressed as functions of particle velocity (𝑢), they are ac- tually functions of both 𝑢 and actual-velocity𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}, i.e, 𝐶_ℎ = 𝐶_ℎ(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}, 𝑢) and

𝜕𝐶ℎ

𝜕 𝑢 = ^{𝜕𝐶ℎ(𝑢}_{𝜕 𝑢}^{𝑎 𝑐 𝑡 𝑢 𝑎𝑙,𝑢)}. The dependence on both variables ensures the material-frame indifference of these material properties. For sake of brevity, henceforth 𝐶_ℎ and its derivative will be expressed only as functions of𝑢, although where required, its dependence on𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} will be utilized.

(𝐿− ℎ) =𝐶_ℎ(𝑢) (𝑡−𝑡₀)

=⇒ 𝐶_ℎ(𝑢) = (𝐿−ℎ) 𝑡−𝑡₀

=⇒ 𝜕𝐶_ℎ(𝑢)

𝜕 𝑢

𝜕 𝑢

𝜕 ℎ 𝑡

= −1 𝑡−𝑡₀

𝜕 𝑢

𝜕 ℎ 𝑡

= −1

𝜕𝐶_ℎ(𝑢)

𝜕 𝑢 (𝑡−𝑡₀)

(B.8)

t

L-h Free Surface of

LiF window L

h

L

Shock-wave Release fan

1 U_s(u_actual) 1

C_h(u_actual) 1

C_h(2u_actual)

t₀

LiF Window

(Length of LiF window) (Material coordinate)

U_s(u) - Shock-wave speed C_h(u) - Release wave speed

u - Particle velocity

Figure B.3: t-X diagram for release fan.

For conditions of uniaxial strain prevalent in plate-impact experiments, the volu- metric strain,𝜀 =1− ^𝜌0

𝜌. Further,

𝑑𝜀= −𝑑 𝑢 𝐶_ℎ(𝑢)

=⇒ 𝜌₀ 𝜌²

𝑑𝜌= −𝑑 𝑢 𝐶_ℎ(𝑢)

=⇒ 𝑑𝜌 𝑑 𝑢

= −𝜌²

𝜌₀𝐶_ℎ(𝑢) (B.9)

Thus, Eq. B.7 becomes:

𝜕 𝜌

𝜕 ℎ 𝑡

= 𝜌²

𝜌₀𝐶_ℎ(𝑢)^{𝜕𝐶ℎ(𝑢)}

𝜕 𝑢 (𝑡−𝑡₀) (B.10)

Finally, to evaluate B.4, it remains to evaluate ^{𝜕 𝜌}_{𝜕 𝑡} ℎ

= ^{𝜕 𝜌}

𝜕 𝑢

𝜕 𝑢

𝜕 𝑡

ℎ

. Next, differentiate the

following equation with respect to𝑡

𝐿−ℎ=𝐶_ℎ(𝑢) (𝑡−𝑡₀)

=⇒ 0= 𝜕𝐶_ℎ

𝜕 𝑢

𝜕 𝑢

𝜕 𝑡 ℎ

(𝑡−𝑡₀) +𝐶_ℎ(𝑢)

=⇒ 𝜕 𝑢

𝜕 𝑡 ℎ

= −𝐶_ℎ(𝑢)

𝜕𝐶_ℎ

𝜕 𝑢 (𝑡−𝑡₀) (B.11)

Using equations B.9 and B.11,

=⇒ 𝜕 𝜌

𝜕 𝑡 ℎ

= 𝜌² 𝜌₀^𝜕𝐶ℎ

𝜕 𝑢 (𝑡−𝑡₀) (B.12) Thus using Eqs. B.10, B.6, and B.12, Eq. B.4 can now be written as:

𝜕 𝜌

𝜕 𝑡 𝑥

(ℎ(𝑥 , 𝑡), 𝑡) = 𝜌² 𝜌₀𝐶_ℎ(𝑢)^{𝜕𝐶ℎ(𝑢)}

𝜕 𝑢 (𝑡−𝑡₀)

−𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} 𝜌(ℎ)

𝜌₀

+ 𝜌²

𝜌₀^𝜕𝐶ℎ

𝜕 𝑢 (𝑡−𝑡₀) The integral (I) given in Eq. B.3 becomes

𝐼𝐼 𝐼 =

∫ ℎ𝑏(𝑡) ℎ𝑎(𝑡)

𝜕 𝜂

𝜕 𝜌

𝜌

𝜕𝐶_ℎ(𝑢)

𝜕 𝑢 (𝑡−𝑡₀)

1− 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} 𝐶_ℎ(𝑢)

𝜌 𝜌₀

𝑑 ℎ

Thus, the apparent particle velocity (𝑢_{𝑜 𝑏 𝑠}) given in Eq. B.1 becomes 𝑢_{𝑜 𝑏 𝑠}=𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}(𝜂−2𝜂₀+2) −I

Further, differentiating both sides with𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} and using Leibniz rule for differenti- ating integrals, we get:

𝑑 𝑢_{𝑜 𝑏 𝑠} 𝑑 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

=(𝜂−2𝜂₀+2) + 𝑑 𝜂 𝑑 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} +

𝜕 𝜂(ℎ_𝑎(𝑡))

𝜕 𝜌

𝜌(ℎ_𝑎(𝑡))

𝜕𝐶_ℎ(𝑢_{𝑎 𝑐 𝑡 𝑢 𝑎𝑙})

𝜕 𝑢 (𝑡−𝑡₀)

1− 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} 𝐶_ℎ(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙})

𝜌(ℎ_𝑎(𝑡)) 𝜌₀

𝑑 ℎ_𝑎(𝑡) 𝑑 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

−

𝜕 𝜂(ℎ_𝑏(𝑡))

𝜕 𝜌

𝜌(ℎ_𝑏(𝑡))

𝜕𝐶ℎ(2𝑢𝑎 𝑐 𝑡 𝑢 𝑎𝑙)

𝜕 𝑢 (𝑡−𝑡₀)

1− 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} 𝐶_ℎ(2𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙})

𝜌(ℎ_𝑏(𝑡)) 𝜌₀

𝑑 ℎ_𝑏(𝑡) 𝑑 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

(B.13) It can be noted that, similarly to what was stated before Eq. B.8, here𝐶_ℎ(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙})= 𝐶_ℎ(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}, 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) and 𝐶_ℎ(2𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) = 𝐶_ℎ(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙},2𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) which, is obtained by setting𝑢 =𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} and𝑢 =2𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}, respectively.

Equation B.13 can be simplified as described next. The refractive index (𝜂) is given by the linear relation 𝜂 = 𝐴+ 𝐵 𝜌 [2, 3]. For LiF[100] windows used with

1550 nm wavelength light, the material properties A = 1.2669 and B = 0.037 are taken from [3]. Further, by continuity at start and end of the release fan, 𝜌(ℎ_𝑎(𝑡)) = 𝜌₁ and 𝜌(ℎ_𝑏(𝑡)) = 𝜌₀. _{𝑑 𝑢}^{𝑑 ℎ𝑎}

𝑎 𝑐 𝑡 𝑢 𝑎𝑙

can be evaluated by differentiating 𝐿−ℎ_𝑎(𝑡)=𝐶_ℎ(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) (𝑡−𝑡₀)on both sides w.r.t𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}.

=⇒ 𝑑 ℎ_𝑎 𝑑 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

= 𝜕𝐶_ℎ(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙})

𝜕 𝑢

(𝑡−𝑡₀) 𝑆𝑖 𝑚𝑖𝑙 𝑎𝑟 𝑙 𝑦,

=⇒ 𝑑 ℎ_𝑏 𝑑 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

=2𝜕𝐶_ℎ(2𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙})

𝜕 𝑢

(𝑡−𝑡₀) The derivative of refractive index can be expressed as:

𝑑 𝜂 𝑑 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

= 𝑑 𝜂 𝑑𝜌₁

𝑑𝜌₁ 𝑑 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

The density 𝜌₁ is evaluated using the shock jump conditions and𝑈_𝑠 −𝑢_𝑝 relation for LiF,𝑈_𝑠 =𝐶₀+𝑆𝑢_𝑝, as:

𝜌₁= 𝜌₀

1− ^{𝑢𝑎 𝑐 𝑡 𝑢 𝑎𝑙}

𝐶₀+𝑆𝑢𝑎 𝑐 𝑡 𝑢 𝑎𝑙

The material wave speeds 𝐶_ℎ(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙})(at peak-strain) and 𝐶_ℎ(2𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙})(at zero- strain) for LiF are evaluated as follows:

𝜎 =𝜌₀ 𝐶₀

1−𝑆𝜀 2

𝜀

𝐶_ℎ(𝑢) = s

1 𝜌₀

𝑑𝜎 𝑑𝜀

=⇒ 𝐶_ℎ(𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) =(𝐶₀+𝑆𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) s

1+ 2𝑆𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} 𝐶₀

=⇒ 𝐶_ℎ(2𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) =𝐶₀

Thus, Eq. B.13 can now be simplified to : 𝑑 𝑢_{𝑜 𝑏 𝑠}

𝑑 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

=(𝐴−2𝜂₀+2) +𝐵 𝜌²

1

𝜌₀ 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}

𝐶₀

(𝐶₀+𝑆𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙})² + 1

(𝐶₀+𝑆𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}) q

1+ ^{2𝑆𝑢𝑎 𝑐 𝑡 𝑢 𝑎𝑙}

𝐶₀

+2

𝜌₀− 𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙}𝜌₀

𝐶₀ (B.14)

Thus Eq. B.14 can be used to plot𝑢_{𝑜 𝑏 𝑠}as a function of𝑢_{𝑎 𝑐𝑡 𝑢 𝑎𝑙} by integration.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 u_actual(km/s)

(km/s)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

uobs

Estimate from release fan correction Jones and Gupta, 2000

Figure B.4: Comparison of z-cut quartz data from [4] and release fan correction (Eq. B.14) for 532 nm wavelength light.

0.2 0.4 0.6 0.8 1

u_actual 0

0.2 0.4 0.6 0.8 1 1.2

u obs

Shock-wave correction Release fan correction

(km/s)

(km/s)

Figure B.5: Observed particle velocity vs. actual particle velocity for shock-waves and release fan in LiF[100] for 1550 nm light.

An experimental validation of Eq. B.14 is performed for z-cut quartz release data provided in [4]. This comparison is provided in Fig. B.4. The parameters for z-cut quartz needed in Eq. B.14 was taken from [3].

The optical corrections due to shock-wave and release fan in LiF[100] window is thus plotted in Fig. B.5.

Corrections due to impedance mismatch between LiF and SLG

As seen in Figure B.6, the particle velocity observed at the SLG-LiF interface, after optical corrections (𝑢𝑖𝑛𝑡 𝑒𝑟 𝑓 𝑎 𝑐 𝑒), is lesser than the particle velocity prevalent in the SLG material (𝑢_𝑖𝑛−𝑚 𝑎𝑡 𝑒𝑟 𝑖 𝑎𝑙) before the shock-wave reaches the interface. In order to

LiF [100]

SLG

Stress

Particle velocity

u_interface uin-material (u)

(σ)

σ_interface σin-material

u'

Figure B.6: Stress-particle velocity of LiF and SLG used to obtain the in-material particle velocity in SLG.

Figure B.7: Impedance mismatch correction for optically corrected data from Expt.

SSL-2 of this work.

construct a stress-strain loading history of the SLG material, the observed velocity profile (𝑢𝑖𝑛𝑡 𝑒𝑟 𝑓 𝑎 𝑐𝑒(𝑡)) is converted to in-material velocities (𝑢𝑖𝑛−𝑚 𝑎𝑡 𝑒𝑟 𝑖 𝑎𝑙(𝑡)) using the following formula:

𝑢_𝑖𝑛−𝑚 𝑎𝑡 𝑒𝑟 𝑖 𝑎𝑙(𝑡) =

𝑢𝑖𝑛𝑡 𝑒𝑟 𝑓 𝑎 𝑐 𝑒(𝑡) +𝑢⁰

2 (B.15)

where u⁰ is shown in Fig. B.6. The 𝑈_𝑠 −𝑢_𝑝 parameters taken from [5] were used to construct LiF [100] stress Hugoniot. The SLG hugoniot was constructed using parameters taken from [6]. An example of impedance mismatch correction is provided in Fig. B.7, for data from Experiment 2 in the main work. The blue curve shown in Fig. B.7, is the in-material particle velocity of SLG.