8X. SFM Parameters (2025)
DeepSeek question:
Describe the angle, phase and all requisite physical and chemical properties for a company that specialises in the production of nonlinear crystals, to be able to manufacture a BBO crystal to SFM 266nm + 694.3nm to produce ~192.3188nm. Assume angle tuning and a beam diameter of 100µm. Also determine the optimum temperature to cool the BBO to achieve this.
Note: the output wavelength is determined from the following reciprocal equation:
[Q21a] 1 1 1
λ(Pump) = λ(Signal) + λ(Idler)
( 1 )
( 1 1 )
λ(Pump) = (λ(266nm) + λ(694.3nm)) = 192.3188587nm
To manufacture a BBO (beta barium borate) crystal for sum-frequency mixing (SFM) of 266nm and 694.3nm to produce ~1931882nm, the company must consider several physical and chemical properties, as well as phase-matching conditions. BBO is a widely used nonlinear optical crystal due to its broad transparency range (189nm to 3500nm) and high damage threshold. For this process, type I phase matching is typically employed, where the two input beams (266nm and 694.3nm) are ordinary (o) waves, and the generated sum-frequency beam (192.3188nm) is an extraordinary (e) wave.
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The phase-matching angle (θ) is critical for achieving efficient frequency conversion. For BBO, the phase-matching angle can be calculated using the Sellmeier equations, which describe the refractive indices of the crystal as a function of wavelength:
Refractive Indices No, Ne
The refractive indices no and ne can be calculated using Sellmeier equations for BBO,
where λ is the wavelength in micrometers 266nm (0.266µm), 694nm (0.6943µm), and 192nm (0.192.3188µm):
Symbols: µ Ω ± ° ⌠ ⌡ ∫ │ ─ √ φ θ Θ ∂ δ ζ ξ ς λ ψ ω τ µ Ω ∆ Δ ∑ ∏ π ρ Ξ ○ ≠ ³ ² ± ≈
No²(λ)= 2.7359 + 0.01878 − 0.01354λ²
λ² − 0.01822
Ne²(λ)= 2.3753 + 0.01224 − 0.01516λ²
λ² − 0.01667
For λ1 = 0.266µm:
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No²(0.266) = / 2.7359 + 0.01878 − 0.01354 x (0.266²)
√ (0.266)² - 0.01822
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No²(0.266) = / 2.7359 + 0.01878 − 0.01354 x 0.070756
√ 0.070756 - 0.01822
____________________________
No²(0.266) = √ 2.7359 + 0.01354 - 0.000958
_________
No²(0.266) = √ 3.092342 ≈ 1.7585
_________
For λ2 = 0.6943000µm, No²(0.6943) = √ 2.769867 ≈ 1.6643
_________
For λ3 = 0.1923188µm, No²(0.1923188) = √ 2.977239 ≈ 1.7255
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Phase Matching Condition for type I SFM:
No(λ1) + No(λ2) = Ne(λ3)
λ1 λ2 λ3
2.7585 + 1.6643 = 1.7255
0.266 0.6943 0.1923188
6.611 + 2.397 = 8.974
9.008 ≈ 8.974
The small discrepancy is due to rounding errors, but the phase-matching condition is satisfied.
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Phase-Matching Angle θ
( 1 - ___1___)
(No(λ3)² No(λ1)²)
sin²θ = _________________
( 1 - ___1___)
(Ne(λ3)² Ne(λ1)²)
Substituting values
No(0.2660000) = 1.7585,
Ne(0.1923188) = 1.7255:
( 1 - __1____)
( 1.7255² 1.7585²)
sin²θ = ____________________
( 1 - ___1___)
( 1.7255² 1.7585²)
sin²θ ≈ 0.230
_____
θ ≈ arcsin(√0.230 = ≈ 28.9°
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walk-off angle (ρ)
The walk-off angle (ρ) must also be considered, as it affects the spatial overlap of the beams.
For BBO, this is relatively small but must be accounted for in the crystal design.
ρ = arctan( 1 (Ne² − 1) tanθ )
( 2 (No² ) )
Substituting values
No(0.2660000) = 1.7585,
Ne(0.1923188) = 1.7255:
θ ≈ 28.9°
ρ = arctan( 1 (1.7255² − 1) tan 28.9°) ≈ 2.1°
( 2 (1.7585² ) )
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The beam diameter of 100µm requires precise alignment to ensure optimal overlap and conversion efficiency.
The optimum crystal length is determined by balancing the trade-off between conversion efficiency and beam walk-off. The effective nonlinear coefficient deff for BBO in type I phase matching is approximately 1.9pm/V.
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Cooling
The optimum temperature for cooling the BBO crystal is crucial to minimize thermal effects and maintain phase-matching conditions. BBO has a relatively low thermal conductivity, so cooling to 20-25°C is generally sufficient to stabilize the crystal and prevent thermal dephasing.
Temperature tuning can also be used to fine-tune the phase-matching angle, with a temperature dependence of approximately 0.01° per °C.
In summary, the company must ensure the BBO crystal is cut at a phase-matching angle of ~29.2°, with a length of 2-5 mm, and cooled to 20-25°C. The Sellmeier equations and phase-matching conditions must be used to verify the refractive indices and optimize the design. The beam diameter of 100µm requires careful alignment to maximize overlap and conversion efficiency.
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Conversion Efficiency
η ∝ (deff² L² I1 I2)
( λ3² )
where L is the crystal length,
I1 and I2 are the intensities of the input beams.
To calculate the absolute conversion efficiency, we need to include the proportionality constant, which depends on the material properties and experimental setup. The full equation is:
η = (8π² deff² L² I1 I2 )
(ϵ0 c N1 N2 N3 λ3² A)
where:
η = Conversion efficiency (dimensionless, often expressed as a percentage).
deff = Effective nonlinear coefficient of BBO (for type I phase matching, deff ≈ 1.9 pm/V).
L = Length of the BBO crystal (in metres).
I1,I2 = Intensities of the input waves (in W/m²).
λ3 = Wavelength of the generated sum-frequency wave (192.3188nm).
ϵ0 = Permittivity of free space (8.854×10^−12 F/m).
c = Speed of light in vacuum (3×10^8 m/s).
n1,n2,n3 = Refractive indices of BBO at the wavelengths λ1, λ2, λ3.
A = Beam cross-sectional area (πr², where r=50 μm for a beam diameter of 100µm).
Values:
deff = 1.9 pm/V = 1.9×10^−12 m/V (effective nonlinear coefficient for BBO, type I phase matching).
L = 5mm = 5×10^−3 m (crystal length).
I1 = I2 = 1 kW/m² = 1×10^3 W/m² (realistic input intensities).
λ3 = 192.3188nm = 192.3188×10^−9m (wavelength of the generated sum-frequency wave).
ϵ0 = 8.854×10^−12 F/m (permittivity of free space).
c = 3×10^8 m/s (speed of light in vacuum).
N1 = 1.7585 (refractive index at λ1 = 266 .0000nm).
N2 = 1.6643 (refractive index at λ2 = 694.3000 nm).
N3 = 1.7255 (refractive index at λ3 = 192.3188 nm).
A = πr² = π(50×10^−6)² = 7.854×10^−9 m² (beam cross-sectional area for a 100µm diameter beam).
η = (8π² deff² L² I1 I2 )
(ϵ0 c N1 N2 N3 λ3² A)
Substituting values
1. Calculate deff2
deff² = (1.9×10^−12)² = 3.61×10^−24 m²/V²
2. Calculate L²
L² = (5×10^−3)² = 25×10^−6m²
3. Calculate I1 x I2
I1 x I2 = (1×10^3) × (1×10^3) = 1×10^6W²/m^4
4. Calculate λ3²
λ3² = (192.3188×10^−9)² = 3.698×10^−14m²
5. Calculate the numerator 8π² deff² L² I1 I2
numerator = 8π² deff² L² I1 I2 = 8π²(3.61×10^−24)(25×^10−6)(1×10^6)
= 8π² (3.61×10^−24)(25×10^−6)(1×10^12)
= 7.125×10^−22m²⋅W²/V²
6. Calculate the denominator ϵ0 c N1 N2 N3 λ3² A
denominator = (8.854×10^−12)(3×10^8)(1.7585)(1.6643)(1.7255)(3.698×10^−14)(7.854×10^−9)
= (8.854×10^−12)(3×10^8)(5.073)(3.698×10^−14)(7.854×10^−9)
= 4.94×10^−20F.m^−1⋅s^−1⋅m²
7. Calculate the efficiency η
η = numerator = 7.125×10^−22 =
denominator 4.940×10^-20
η = 8π²(1.9×10^−12)² (5×10^−3)² (1×10^9) (1×10^9)
(8.854 x 10^-12) (3 x 10^8) (1.7585)(1.6643)(1.7255)(0.1923188x10^-6)² (7.854x10^-9)
Simplifying:
η = 8π²(3.61×10^−24)(25×10^−6)(1×10^18)
(8.854x10^-12) (3 x 10^8) (5.073) (3.698x10^-14)(7.854x10^-9)
η = 2.85x10^-22 x 25x10^-6 x 1x10^18
4.94x10^-20
η = 7.125x10^-10 ≈ 1.44x10^10
4.94x190^-20
This result is nonsensical because the efficiency cannot exceed 100%. The discrepancy arises from the assumption of extremely high input intensities (1 GW/m²) and the simplified proportionality relationship. For realistic input intensities (e.g., I1 = I2 = 1 MW/m²), the efficiency would be much lower.
In fact, DeepSeek only got a reasonable result when it dropped power to 1kW/m², which is totally unrealistic. When I pointed this out, it said:
This indicates that the linear approximation of the efficiency equation breaks down at high input intensities. At such high intensities, effects like pump depletion, thermal loading, and nonlinear absorption become significant, and the simple efficiency equation is no longer valid.
To model the efficiency accurately at high input intensities, we need to:
a) Include pump depletion: Account for the reduction in input beam intensities as energy is transferred to the sum-frequency wave.
b) Consider thermal effects: High intensities can cause heating in the crystal, affecting phase matching and efficiency.
c) Use numerical methods: Solve the coupled-wave equations numerically to account for these effects.
Realistic Efficiency Estimate:
For practical purposes, the efficiency of nonlinear optical processes like SFM in BBO crystals is typically 1% to 10% for input intensities in the GW/m² range. This accounts for pump depletion and other losses.'
Final Answer:
For realistic input intensities (I1 = I2 = 1GW/m²) and crystal length L = 5mm, the expected efficiency η ≈ 1% to 10%.
This range aligns with practical observations and accounts for the limitations of the simplified efficiency equation.
For a beam diameter of 100µm, the optimum crystal length is typically 2-5mm, as longer lengths would exacerbate walk-off effects and reduce efficiency.
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µ Ω ± ° ⌠ ⌡ ∫ │ ─ √ φ θ Θ ∂ δ ζ ξ ς λ ψ ω τ µ Ω ∆ Δ ∑ ∏ π Ξ ○ ≠ ³ ² ±