Parabolical or spherical mirror dish?

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Be carful, contents of this page might be wrong.

Very brief overview:

The mirror diameter (HEGRA) is 0.60m. Assuming an overlap of 2*0,05m per mirror leads to a diameter of HEGRA of 7mirrors*0.5m =3.5m.

For DWARF it's 1/2 row of additional mirrors, so 7.5*0.5m=4m. Of this 4m the "upper side" consists of one additional row compard to the "lower side". This means that the point furthest away from the "centre" of the (former) circle (of the mirror-dish) lies at a distance L=4.5mirrors*0.5m =2.25m.

For a parabola y=ax² the fokal point is known to be at ( 0, 1/(4*a) ), which means that 1/(4*a) is the focal length f.

The length of a curve is defined as

so in case of a parabola

and in case of our parabola L = 2.25.

Get ξ by doing some numerics.

The distance d of this endpoint of the curve (ξ, aξ²) to the focal point can simply be derived by trigonometry.

Doing the same procedure for a circle with a radius R=2*f leads to the distance e of the endpoint of the spheric curve to the focal point.

[(d-e)+(y_parabolic-y_spheric)]/c (where c is the speed of light) is the maximum run time difference of light bounced either by a spherical or a parabolic mirror dish. The term (y_parabolic-y_spheric) is due to the fact, that the light rays are assumed to fall parallel (in -y direction) onto the mirror dishes.

Some figures:

Assumed curve length L=2,25m

focal length f [m]	a [1/m]	ξsph [m]	ysph	d [m]	ξpar [m]	ypar [m]	e [m]	d-e [m]	ypar − ysph [m]	run time diff [ns]
4,000	0,0625	1,883	0,222	4,222	2,220	0,314	4,303	0,081	0,093	0,58
4,250	0,0588	1,885	0,209	4,459	2,224	0,296	4,536	0,077	0,087	0,55
4,500	0,0556	1,886	0,198	4,698	2,227	0,280	4,772	0,074	0,082	0,52
4,750	0,0526	1,888	0,188	4,938	2,229	0,265	5,008	0,071	0,078	0,49
5,000	0,0500	1,889	0,178	5,178	2,231	0,252	5,246	0,068	0,074	0,47

c [m/s] = 299792458

n = 1,00027

c₀ [m/s](=c/n)= 299711535,9

Conclusion: It has been shown above that neither a singel mirror nor the whole mirror dish has to be parabolic in order to preserve the timing information of the light signals, as even for the whole dish the temporal spread is in the orer of 0.5 ns which is quite suitable for a signal readout with 1 GHz.