Background light and resolving power: First, let’s locate whereabout the incident light goes.  By normalizing the total amount of the incident light to unity the quantity of the light which converges to the main focal point (the 1st order focal point) is only .  This is a sum of the light which comes directly through the central zone and the diffracted lights from the outer zones.  If the zone plate is replaced by a glass lens with the same diameter and the same focal length the quantity of the light converged to the main focal point is unity, and, therefore, in the case of a zone plate 90 % of the incident light disappears instead of converging at the main focus.  Moreover, the above discussion is for the case of a coherent light but  the natural light is not so ideal as being perfectly coherent.  Therefore, usually the ratio of light which converges at the main focal point by diffraction becomes considerably reduced.  In general the ratio of light converging at the m-th (or -m-th) order focal point is calculated as , where the focal points with positive and negative order correspond to the focal points of convex and concave lenses, respectively.  The zeroth order diffracted light (a non-diffracted light) is a straightly going light which is considered to be a light converging at a point of the infinity.  Quantity of the non-diffracted light is 1/4, which is the largest value among quantities of lights converging at various focal points.  In reality as a half of the incident light is blocked at the opaque zones the remaining lights go through the transparent zones and most of them arrive at the image plane as background lights which make weak the contrast of the image.

By the way as the resolving power is not determined by a light intensity but by a light intensity per unit area, i. e., the illuminance, we consider here the intensity per unit area.  Geometrically the image of a point source is a mathematical point without an area but the image of a point source is a circle with a finite area because of the diffraction and, therefore, the light intensity per unit area can be calculated.  When the illuminance on the surface of the zone plate by a point source at infinity is e, the image of the point source becomes a circle with a radius of at the focal plane and the illuminance at the focal point is , where and are the wavelength of the light, the focal length, the radius of the outer boundary of the maximum zone and the number of the transparent zones, respectively.  In this way, the illuminance of the center of the image increases in proportion to and the radius of the image of the point source decreases in inverse proportion to square root of the zone number   with increasing the number of the transparent zones .  This is the reason why the resolving power of a zone plate increases with increasing the number of zones.

Soft zone plate photographs: Then why zone plate photographs look so soft.  To answer the question we calculate the contrast of an image of a discoid light source at infinity.  Among various definitions of contrast we apply the definition of the Weber contrast C to the case of a zone plate imaging as where and are the illuminance of the image of the object and the illuminance of the background light, respectively.  The illuminances of the image of the object and the background light are defined as

  

  

where and are the rates of the light going to the main focal point and the light proceeding forward, is the total amount of the incident light, and and are the areas of the image and the background light.  The radius of the circular image of the object at infinity is assumed as p times as large as the radius of the image of a point source at infinity, which means the light source of the discoid image is a circular disk composed of point sources at infinity.  In the second equation a factor q is multiplied to the incident light because a light which is not parallel to the focusing light also contributes as a background light.   Therefore, the areas of the image and the background light are and , and the total amount of the incident light is .   From these equations the contrast of a zone plate photograph is derived as follows.

By using the relation the following equation is derived.

        

Calculation of the contrast of a zone plate photograph: The contrast of a zone plate photograph is defined as a ratio of the illuminance of the image of the object to the illuminance of the background light for a circular image with a radius of times that of a point source at infinity, where the light source is assumed as a discoid light source at infinity composed of q point sources.

When evaluating the contrast by using the above equation it should be remarked that there is a maximum for the multiplication factor    of the area of the image, and for larger than the effect of the background lights does not increase.  The value of the maximum is evaluated by using a condition  as