2 The geometric concept of the DELTA camera

2.1 Synchronous and asynchronous cameras

Like many other photon counting devices, the DELTA camera will use an image intensifier providing a gain of about 1 million, producing detected photo-events as bright spots on a fast decay phosphor. The challenge is to translate these intensified photo-events into numerical coordinates as fast as possible, in order to achieve the highest data flow and temporal resolution.

Most of photon counting cameras, that we will refer to "asynchronous'', like the Ranicon, PAPA, MAMA, or delay-line process only one photon at a time: if two or more intensified photo-events are simultaneously present in the field, the coordinate computation system fails, yielding no data or incorrect coordinates. Except for the MAMA and the delay-line cameras, the data flow is thus limited by: first, either the phosphor decay time (0.5 $\mu$s) or the resistive anode decay (50 $\mu$s), and second, the photon coordinate determination process (1 to 10 $\mu$s).

Other cameras, "synchronous'', like the CP40 use a 2-dimensional ICCD and can process many photons in a single frame. They suffer from a trade-off between spatial resolution and read time of the CCD array: typically 5 to 20 ms. They also suffer from an artifact causing problems in second order moment imaging techniques. Due to the long frame time, there is a non-negligible probability that two detected photons fall close enough in a CCD frame (although not onto the same pixel) to be seen as a single photon, or no photon at all by the coordinate determination electronics.

2.2 The Projection - Back-Projection scheme in the DELTA camera

To solve these problems, the DELTA synchronous camera uses three fast linear CCD chips (each 1024 by 1 pixels and 2.6 $\mu$s frame read time). It may detect and locate several photons simultaneously in each frame, with a 512 by 591 pixels hexagonal field.

The principle is as follows: an intensified frame containing N photons detected between times t and $t+\Delta t$ is described by:

$$F_t(x,y)=\sum_{i=0}^{N-1} \delta (x-x_{i},y-y_{i}),$$ (1)

where $\delta$ represents a Dirac distribution. The photon coordinates to be extracted from the frame are the triplets (x_i,y_i,t_i). t_i is set to t for all the photons in the frame. x_i and y_i are extracted by a Projection - Back-projection scheme.

2.2.1 Projection

Three images of the intensified field are formed. These "images'' are reduced to lines (orthogonal projections of the field) by the optical setup described in Sect. 5.2, and each line is directed to a CCD chip. Let $p_{\theta,{\bf M}}(x,y)$be the projection operator defined by:

$$p_{\theta,{\bf M}}(x,y)=(x-x_{\rm M})\cos \theta+ (y-y_{\rm M})\sin \theta.$$ (2)

It converts the coordinates (x,y) of a photon into its projection on an axis. The axis onto which the projection is made is defined by its direction $\theta$ and the coordinates of its origin ${M}=(x_{\rm M},y_{\rm M})$. The coordinates of a photon could be retrieved from two orthogonal projections. However, if there is more than one photon in the field, there is not a unique solution to the problem of recovering a photon coordinate list from the two projection lists L_x, L_y. The information which associates one element of L_x with the corresponding element of L_y is missing. Therefore, the photon rate would be limited to one per frame. The solution is to reconstruct a "link table'' between the two projection lists by projecting onto a third axis. Let A, B, C be three axes onto which the photons are projected. To each photon coordinate vector ${\vec X}=(x,y)$ will correspond a projection vector ${\vec Y}=(a,b,c)$ such that:

$$\begin{cases} a= p_{\alpha ,{\rm A}}({\vec X})=(x-x_{\rm A})... ...-x_{\rm C})\cos \gamma+(y-y_{\rm C})\sin \gamma .\\ \end{cases}$$ (3)

The projection can be seen as a Radon transform with only three directions. It is done optically with the setup described in Sect. 5.2. It is a virtually null-time operation.

2.2.2 Back-projection

The redundancy in the projection vectors allows the recovery in most cases, the coordinate list from the projection lists. Choosing the projections operators such that $\alpha=0 ; \beta=2\pi/3 ; \gamma=4\pi/3,$and ${\rm A}={\rm B}={\rm C}=(0,0),$yields the relation:

a+b+c=0.

Thus, among all the possible triplets obtained by picking one number in each of the three projection lists, only those having a null-sum will correspond to a photon. This is the basis of the coordinate determination process in the DELTA camera. It can be summarized by: "Project optically the image on the sides of an equilateral triangle, detect the one-dimensional projections, then back-project numerically using the null-sum test''. The equilateral triangle gave its name to the camera.

2.2.3 Field

Let K be the side of the equilateral triangle used for the projections (K also corresponds to the quantization dynamics: each $\Delta$-coordinate is an integer ranging from -K/2 to K/2-1). The set of points (x,y) within the range of the projection $p_{\theta,0}$ onto a segment of length K is:

$$\Omega_\theta=\biggl\{ (x,y) \bigg/ -\frac{K}{2} \leq x\cos\theta+y\sin\theta < +\frac{K}{ 2} \biggr\}.$$ (5)

The field of the detector is: $\Omega_\alpha \cap \Omega_\beta \cap \Omega_\gamma$. As shown in Fig. 1, this field is a hexagon.

  

Figure 1: Intersection of three stripes ($\Omega_\alpha, \Omega_\beta,$ and $\Omega_\gamma$) at $120^\circ$ angles forming the effective field of the DELTA camera (shaded hexagon). Projections (noted a, b, and c) of a point in the field (in white) onto the three corresponding axes