Tank Optics

In the game rules I compute the German gun to-hit accuracy based on ranging error of about 15-25% standard deviation. The UK is 24-26%. The US is 20-29% and the Russian gun to-hit standard deviation ranging error is about 18-26%.  The reason for this German advantage is primarily their optics.  German sights had several advantages over Allied sights.   
All of the articulated, coaxial telescopes on German tanks were designed by Leitz of Wetzlar.  (Zeiss designed the Sturmgeschutz sights which were produced by Leitz under license.)  There are several comprehensive essays on German and Allied WWII optics that are linked from this page.  I will summarize some of the key points.

First there was the clarity of their optical glass.  The Germans recognizing the importance of lenses were world leaders in the research and development of lenses and discovered that the rare earth element Lanthanum in the glass improved the clarity of the lens.  It has been said that the sand/clay used to make the glass was also of a quality not widely found, but the Japanese didn't need the raw material from Europe so that is not confirmed.  The special optical glass was important enough to be included in strategic shipments to Japan[PDF]. Particularly suitable for submarine cargo (Oct. 41 to Sept. 1944 3,069 tons delivered), optical glass was a high priority for transport to Japan. .  Note - The modern standard requirement for US class A optical glass is[PDF] loss of 2% light per cm of glass.

Secondly another advantage to the Germans was that they developed a thin film lens coating that reduced the reflection of light from the lens surface.  History of development of thin film coatings.

The reflection of a polished glass surface is about 4%.  In the 19th century it was observed that weathered glass surfaces have a lower reflection than fresh polished glass. It was found this effect could be produced through an etching process. But, it was several decades before the understanding why it worked. A thin surface layer with a lower refractive index than the glass reduces the reflection through interference at the surface layer. Lower reflection means higher light transmission.   Another 4 percent of the emergent light is lost by reflection at the inner surface.  Thus, both reflection and refraction (the bending of light) occur at any optical interface. In an optical system with many lens to air surfaces a clearer view is had by reducing light reflection and increasing light transmission.  TM 9-258 Elementary Optics[PDF]

The development of producing lenses with these thin layers occurred almost at the same time in Germany and the US.
Milestone discoveries:
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.

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

T-34 model 1940 sight

T-34/85 1944 sight

US 75mm sight on M3 tank

US 90mm sight on combat vehicles

UK Sherman Firefly sight.

Click to enlarge