Thursday, April 05, 2012

A Layman’s Understanding on Obtaining the Optimum Pinhole Diameter for Sharpness




Making a pinhole camera is very simple.  Cameras can be made from just about any light tight container.  Folks have made them using oatmeal boxes, paint cans, match boxes and even digital SLR body caps.  Just about any pinhole will create an image.  However, to get  the sharpest  image, you have to know what you are doing.
I was disappointed with the images created by my first pinhole camera.  As a result I decided to learn how to select the optimum pinhole diameter for the next camera I would build.  I knew there would be math involved and I had serious misgivings. but my need to understand, pushed me forward.

With very little understanding, I began with the most common formula for obtaining the optimum pinhole diameter.  This first formula was refined by John Strutt, Third Barron of Reyleigh.
d = 1.9 x sqrt(f x w)

Where “d” is the pinhole diameter, 1.9 is a constant which is multiplied by focal length (f) times the wavelength (w) of light squared.  This simple little formula puts your image close to obtaining a sharper pinhole image.  Soon after discovering this formula, I found a pinhole enthusiast (online name of Ren-on) who was successfully using it, although varying the constant number.  This he changed depending on his desired results.  For sharpness he favored the constant number of 1.562.  His images were strikingly sharp.  

My next camera was also made from a cigar box, but this one was covered in leather and trimmed out with polished brass to resemble a camera of old.  This time I used the Reyleigh calculation with the 1.562 constant.  The first image was very sharp and a significant improvement over my first camera.
As I continued working with this calculation, I came across another formula.  This new formula worked on the concept that a pinhole should be proportionate to the diffraction disk it produces in order to obtain the sharpest image.  The formula proposed was as follows:

diameter = sqrt (2.44 x focal length x wavelength)
  
With nothing to lose, I gave this formula a try.  Lo and behold I got the exact same pinhole diameter that I had obtained using Reyleigh’s formula with the 1.562 constant.  I knew I was onto something, but I did not know what yet.  So I compared both formulas side by side.

Reyleigh         d = 1.562 x sqrt(f x w)
New                d = sqrt(2.44 x f x w)

The square root of 2.44 is 1.562!  The calculations were essentially the same.  Both equations calculate the optimum pinhole diameter for sharpness based on the size of the Airy Disk.  More correctly, the square root of the Airy Disk.

Airy Pattern and Disk

So let’s talk about the Airy Pattern and Disk so as to understand where these calculations are coming from.  Light travels in waves, just like the ocean.  For example, the light reflected from your photo subject-matter, let’s say a building, is transmitted to your pinhole camera in waves, light-waves.  Just like the ocean, light-waves behave in a fairly predictable manner.  If an object blocks these light-waves, let’s say you forgot to remove the lens cap from your camera, the light-waves bounce off.  The lens cap effectively blocks the transmission of the light waves from reaching the film.

Getting back to your photo of the building.  This time you remembered to remove the lens cap. and the light reflected from your image makes it into your pinhole camera.  However, like the lens cap, the edges around the pinhole interfere with the light-waves.  As the light-waves pass through the camera opening, they are disrupted (this is called diffraction) and their travel changes radically from nice straight light waves to bowed waves that interfere with each other as they travel forward.  The reason for this is that light-waves tend to bend back onto themselves as they come into contact with a fixed object.  In this case the edges around the opening of your pinhole.  The resulting bowed light waves and how they interfere with each other creates an Airy Pattern.  In 1835, an Englishman by the name of Sir George B. Airy first described (in his "On the Diffraction of an Object-glass with Circular Aperture") how light behaves when it passed through a small opening.  Not surprisingly, this pattern was named after him.



               DIFFRACTION                      AIRY PATTERN (disk is center)

In the image above left, see how the light waves become bowed and wavy.  One can see how these waves interfere with each other as they bend back.  Notice the pattern of lines radiating from the entry point.  These are created by the bending light waves blocking preceding light waves.  They show up as dark rings on the Airy Pattern.  The light that is not blocked or is less disturbed shows up as white rings and at the center, the white Airy Disk are the least disturbed light waves.  This is, in essence, the “sweet spot” of the image, again, where the light is least diffracted.  I think it makes sense that this is the best place to retrieve our image, if we want it to be sharp.

In order to use this “sweet spot” (Airy Disk) to produce sharper pinhole images, one must know how to define it mathematically.  That is to say where is this “sweet spot?”  The following is the simplest formula for defining this “sweet spot.”

spot = 2.44 x focal length x wave length

You might notice that this formula looks familiar.  Let me put it side by side with the three formulas I’ve found.

spot = 2.44 x focal length x wave length                    ←-----------Airy Disk
d = 1.562 x sqrt(focal length x wave length)               ←-----------Reyleigh
d = sqrt (2.44 x focal_length x wavelength)                ←-----------New

Based on these formulas, it is clear that the optimum pinhole diameter is the square root of the Airy Disk size.  To be fair, Lord Reyleigh did not call upon the 1.562 constant specifically, but instead said the constant could be between 1 and 2.  I found the use of 1.562 initially from an online user, but ultimately discovered others who used it in their calculations.

So now that we understand, let us pretend we are making a big pinhole camera from a small room.  The room is 12 x 9.78 feet.  We will be projecting the image onto the 12 foot wall, so our focal length will be 9.78 feet.  We plug this into our Airy Disk formula (converting feet to millimeters) and it looks like this:

2.44 x . 2980.626 x 00055 = 4 mm  ←------This is the size of the Airy Disk produced.

Now we get the square root of 4 mm:

sqrt(2.44 x 2980.626 x .00055) = 2   mm  ←-------This is the optimal pinhole diameter.

Using this size hole will produce the sharpest image one can achieve without using a lens.  This final equation is the one to use.  The Airy Disk formula previous to this was simply shown to illustrate the Airy Disk size on its own.

d = sqrt(2.44 x f x w)        ←--------Use this formula.

That is how I understand it.  Please feel free to correct me, remember I am a layman.

Humbly submitted,
Henry Chavez






References:
  1. John Strutt, Lord Reyleigh http://www.westminster-abbey.org/our-history/people/john-strutt,-lord-rayleigh
  2. Lord Reyleigh on Pinhole Photography http://www.westminster-abbey.org/our-history/people/john-strutt,-lord-rayleigh
  3. Interesting Thread on Optimal Pinhole Diameter - Renon taking the lead.  http://www.f295.org/Pinholeforum/forum/Blah.pl?v-print/m-1289932551/

Paterfamilias

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