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The development of producing lenses with these thin layers occurred almost at the same time in Germany and the US.
Milestone discoveries:
1935 Zeiss, Olexander Smakula inventor - German Patent claimed on vapor deposition of CaF2.
1936 Strong, CaF2 layer formula specified. The reflection reduced from 4.25% to 0.6% using this type film coating.
1938 Cartwright and Turner published a whole class of usable materials including MgF2 and Cryolite (Na3AlF6, sodium hexafluoroaluminate).
1940,1941 Zeiss, Schott and Steinkeil, experimented with double and triple layers; double layers never got into production in Germany or the U.S until after the war.
In WWII Germany the chemically inert Argon gas was used (as it still is) as a "carrier" in electrospray ionization which allows the deposition of ultra-thin coatings. Through bombardment with Argon ions, the material (Cryolite) gets sputtered*. The great advantage of this method is, that it is a cold procedure, where the film material doesn't have to be heated. It was kept as a military secret until after the war.
The result is that light transmission which normally lost 4.25% light per lens surface could only lose as little as 0.6%. Telescopic sights had around 10-16 lenses so this decrease in light transmission could really add up. (In reality this is progressive since calculations of light loss per surface would go 0.992 x 0.992 x 0.992 x 0.992 ….)
*Sputter- A momentum-transfer process where ionized Argon gas is accelerated towards a negatively biased target and the ions dislodge (sputter) particles from the target to deposit on the substrate.
Source for following summary:
Maybe some were thinking the Germans had higher magnifications on their sights than the Allies. Allied optics for the most part were not inferior in magnification to German sights. Originally the British 2 pounder gun and associated tanks and anti-tank guns had a 1.9x magnified sight, which was somewhat inferior to the 2.4x magnified German sights on tanks. American and British tank sights (6 pdr and 75mm.) had a 3x magnified sight which was slightly superior in zoom to the 2.4 x magnified German sight on more of their common tanks such as the Mark III and early Mark IV. German Mark IVH, IVJ, late Tiger I, Tiger II and Panthers had selectable powers of 2.5x or 5x.. The Sherman Firefly was equipped with a selectable 3x or 6x magnified sight.
The 2.4x magnified German sight on Mark IIIs, Mark IVs, and early Tigers had a wide 25° field of view. In comparison allied 3x magnified optics had only a 13° field of view (FOV). The adjustable German 2.5x and 5x optics also had an wide FOV compared to allied optics. The 2.5x sight had a 25-28° field of view, while the 5x optics were 14° FOV. In short German 5x optics had slightly better FOV than allied 3x, and German 2.4x and 2.5x optics had roughly double the FOV. The late war high powered 6x sight for the Firefly had a 9° field limited view. The only allied optic that compared to German sights in magnification and FOV was the 5x optic put in the Sherman 76 series, which had a 13° FOV, similar to the 5x magnification 14° field of view of the German adjustable optics. . One equation I found that could be used as a comparison was magnification = apparent FOV/true FOV. This leads to apparent FOV = magnification x true FOV. The best apparent FOV is what you want.
Another advantage of the German optics was their design which created a 'Mili-radian' sight[PDF]. That is a sight window with a number of triangles of two sizes. Comparing sizes to that of known tank size allowed the gunner to calculate a rough range of the tank target without even having to take a ranging shot. This allowed German gunners to have a high chance of getting first shot hits. Especially, when coupled with their high velocity flat trajectory guns.
Something is wrong with the British evaluation of German sights.
American optics were simple if not primitive in comparison, consisting of a line down the middle with crossing lines representing 400 yard intervals. They were nonadjustable and had to be lined up on the enemy tank.
British optics on their post 2-pounder armed tanks were similar to the American optics in that they had all the ranges listed in the sight, but it was adjustable so at least you didn't have the problem with the US optics.
Early Soviet sights were better designed than early US and UK sights. A Russian report on an unsubstantiated US report (see definition of 'hearsay') says the sights on the T-34 sent were the best constructed sights some engineers at Aberdeen had ever seen (at least up until when the T-34 was tested during the summer of 1942 through spring of 1943). Prior to the war the Soviets purchased an obsolete Zeiss plant and technology though the resulting sights produced were only of adequate quality. (Rifle sights made with this machinery appeared by 1932.) The Germans and Russians had a falling out and by the end of 1933 they ended their military co-operation. It might be reasonable to assume this technology transfer didn't include the lens coating inventions of 1935 and 1936. An example in poor light transmission was the Soviet sights on the model T-34s. Preliminary Report No. 2/0 Feb, 1944 on the T-34 the British made concerning the sample T-34 model 1942 (produced spring or summer 1942 at factory #183 in Nizhniy Tagil.) tank the Russians sent them. The 2.5 x telescope sight has a light transmission of 39.2% and the 2.5 x periscope dial sight only of 26.3%. As the war progressed it looks like the Soviets improved their design with the addition of the mili-radian feature of German sights. Finnish evaluation of T-34-85 sight. "Observation devices: the refractive telescopic sight is greatly superior to the sight of the model 1942-1943 T-34 tank. The clarity of the sight is on the level of the German 75 mm model 1940 gun. The field of view was increased by 15%. The markings are more convenient when firing on tanks up to 1000 meters." Reference link
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