# Why is fused silica superior to N-BK7 in terms of thermal lensing?

In optics lab, the two commonly used types of glasses are N-BK7 and fused silica. The lab wisdom has been that fused silica is superior to N-BK7 in terms of thermal lensing.

And indeed it seems so. Our collimator lens for a 10W fiber amplifier looks like it's made of N-BK7, and when a beam profiler is situated 0.5 meter away from the collimator, the initially collimated beam with 1.7 mm diameter focuses to 0.6 diameter.

But from what I learned from the RP Photonics entry on thermal lensing, the figure-of-merit for thermal lensing is the ratio of thermo-optic coefficient to thermal conductivity. That is, you want the ratio $\frac{dn}{dT} / \kappa$ tobe as small as possible.

However, if you look at the thermal properties of different glasses (at 1064 nm), fused silica has $\frac{dn}{dT} \sim 11\times 10^{-6} / K$ and N-BK7 has $\frac{dn}{dT} \sim 2 \times 10^{-6} / K$. On the other hand, they have similar thermal conductivity ($1.4\, W/(m\cdot K)$ for FS and $1.1\,W/(m\cdot K)$ for N-BK7).

Based on what I've said so far, it seems N-BK7 should have better resistance to thermal lensing! But it does not in practice. Why?

EDIT: Based on the answer below, I looked out for extinction coefficient $k$ (or absorption coefficient $\alpha = \frac{4\pi k}{\lambda}$, whichever is available).

According to this data, N-BK7 has absorption coefficient of roughly $10^{-3}/cm$ at 1064 nm.

According to this website, fused silica has absorption coefficient between $10^{-4}$ and $10^{-5}$, per cm, at 1064 nm.

EXTRA QUESTION: For those familiar with optics industry, can you tell me why N-BK7 is popular, and in what uses it is superior to fused silica?

• A study on thermal lensing effect in fused silica: arxiv.org/pdf/1904.01965.pdf. It is reported that up to several hundred watts (for ~mm waist beam?) fused silica exhibits little thermal lensing. – wcc Apr 4 at 2:11

"But from what I learned from the RP Photonics entry on thermal lensing, the figure-of-merit for thermal lensing is the ratio of thermo-optic coefficient to thermal conductivity."

You should also take into account the extinction coefficient: if there were no losses, the medium would not heat up at all. However, I did not compare extinction of the two media.

Note that your source suggests the "figure-of-merit" "for high-power gain media", whereas your collimator lens is probably a passive device. I don't know if this is important though.

• The principle behind the thermal lensing equation is, you calculate the thermal gradient across the lens, and from it refractive index gradient caused by heat. It doesn't matter whether my medium is passive or gain. – wcc Aug 2 '18 at 7:08
• I didn't think about extinction coefficient, thanks. At 1064 nm, N-BK7 has absorption coefficient of 0.0013/cm. For FS, it has $10^{-5}$/cm at 1000 nm, so this is roughly two orders of magnitude lower. Wow... – wcc Aug 2 '18 at 7:20
• @IamAStudent : "It doesn't matter whether my medium is passive or gain." The sources of power input are different for passive and gain: extinction for the former and pumping power for the latter. And remember that pumping efficiency is typically (very) low. Sorry. don't have time for details now. – akhmeteli Aug 2 '18 at 14:31
• do you mean, if I take a realistic gain medium with four levels, the dominant source heat comes from the non-radiative relaxations between (2)->(3) and (4)->(1)? (where pumping happens on (1)->(2) and lasing happens on (3)->(4))? If so I agree with you. – wcc Aug 2 '18 at 17:45

BK7 is an OK material to use for most applications. I would use it for standard applications in the visible spectrum.

For off-the-shelf optics, the cost is often 20-30% less than for UV grade fused silica. This adds up when considering buying a large set of optics.