Deep-Sea Exploration: Absolute Refractive Index and Dispersion Sensor for an Unknown Liquid

Problem Statement

To investigate the optical properties of a special liquid (liquid F) filling the deep ocean of an unknown ocean planet, a sensor made of a special optical crystal was lowered into the ocean.

The key element of the sensor is a prism with an equilateral triangular cross section $\text{PQR}$PQR (apex angle $\angle\text{P} = 60^\circ$∠P=60∘). The prism is fully submerged in liquid F.

A laser beam is fired from the exploration vessel into liquid F and incident on face $\text{PQ}$PQ (face 1). The light refracts and travels inside the prism, then reaches face $\text{PR}$PR (face 2).

The absolute refractive index of this crystal varies with the wavelength of the incident light (dispersion), but the absolute refractive index $n_f$nf​ of liquid F is assumed to be constant across the visible wavelength range.

Two experiments were performed:

[Experiment 1: Blue Light] For blue light, the crystal's absolute refractive index is $n_{cB}$ncB​. As the angle of incidence $\theta_0$θ0​ on face $\text{PQ}$PQ was gradually increased from $0^\circ$0∘, the angle of incidence on face $\text{PR}$PR decreased (due to geometry). When $\theta_0$θ0​ reached $\theta_{0B}$θ0B​, the angle of incidence on face $\text{PR}$PR decreased to the critical angle for total internal reflection, and transmitted light just began to emerge from face $\text{PR}$PR into liquid F.

[Experiment 2: Red Light] For red light, the crystal's absolute refractive index is $n_{cR}$ncR​. Similarly increasing $\theta_0$θ0​ from $0^\circ$0∘, the angle of incidence on face $\text{PQ}$PQ at the moment transmitted light just begins to emerge from face $\text{PR}$PR (critical condition) is $\theta_{0R}$θ0R​.

All light propagation occurs in the plane containing triangle $\text{PQR}$PQR. The absolute refractive index $n_f$nf​ of liquid F is greater than $1$1 but smaller than both $n_{cB}$ncB​ and $n_{cR}$ncR​.

Find $\sin\theta_{0R}$sinθ0R​, the sine of the critical angle of incidence on face $\text{PQ}$PQ for red light.

Constraints

• Prism apex angle: $\angle\text{P} = 60^\circ$∠P=60∘
• Crystal refractive index for blue light: $n_{cB} = 2.8$ncB​=2.8
• Crystal refractive index for red light: $n_{cR} = 2.5$ncR​=2.5
• Critical condition for blue light: $\sin\theta_{0B} = 0.6$sinθ0B​=0.6
• Angle of incidence $\theta_0$θ0​ is measured from the normal ($0^\circ \le \theta_0 < 90^\circ$0∘≤θ0​<90∘)
• $n_f$nf​ is constant (wavelength-independent)

Input Format

The value of $\sin\theta_{0R}$sinθ0R​ is expressed in the rationalized form:

$$\sin\theta_{0R} = \frac{\sqrt{A} - B}{C}$$sinθ0R​=CA​−B​

where $A, B, C$A,B,C are positive integers with $\gcd(A, B, C) = 1$gcd(A,B,C)=1, and $A$A has no perfect-square factor greater than $1$1.

Give the value of $A \times B \times C$A×B×C as a positive integer.