I guess the title and the picture explain my question well.
Why is the area in focus in front of the focus distance narrower than behind it?
It isn't. Not always. Just usually for landscape shooters using wider lenses aiming somewhere not that close. :) The 1/3-2/3 proportion rule is basically a rule of thumb that does not apply in all cases.
The larger your aperture, longer your lens, or closer your shooting distance (i.e., thinner the depth of field), the more that proportion will actually range closer to 50/50 (think: hyperfocal distance).
http://www.cambridgeincolour.com/tutorials/depth-of-field.htm (in particular, the table showing how focal length varies the proportion of front/back in the DoF.
Depth of field is all about angles and being able to tell a difference between them. As things get further away, the angles change less and less. If you move forward 1m at a distance of 2m, you may change the angle by 30 degrees. If you move 1m closer at a distance of 2000m, the angle hardly changes at all.
The proportion of near and far DoF is this same principle, but "compacted" around the plane of best focus.
Why is the area in focus in front of the focus distance narrower than behind it?
It isn't aways.
In fact it very rarely is exactly a 1:2 ratio as depicted in your illustration. The rule of thumb you have cited is only approximate. For every focal length and aperture there is only one precise focus distance where the ratio between front and rear Depth of Field is exactly 1:2.
The ratio of the DoF in front of the point of focus to the DoF behind the point of focus will be different for every focus distance when using the same lens and aperture setting (assuming the other conditions are also the same: magnification/display size, viewing distance, assumptions about the viewer's vision, etc.).
At short focus distances the ratio approaches 1:1. A true macro lens that can project a virtual image on the sensor or film that is the same size as the object for which it is projecting the image achieves a 1:1 ratio. Even lenses that can not achieve macro focus will demonstrate a ratio very near to 1:1 at their minimum focus distance.
For example, using a 300mm telephoto lens with a maximum magnification of only .24X and a MFD of 59 inches the DoF calculates to 1:1 within the limits of rounding the distance to one one-hundredth of an inch. With a FF camera and a 300mm lens at f/4 the DoF will be 0.09 inches in front of the focus distance and 0.09 inches behind the focus distance with standard display and viewing conditions. In reality the near DoF will be microscopically smaller than the rear DoF. This difference is not perceptible and utterly meaningless, though. One has to increase focus distance to 133 inches before the near DoF at 0.54 inches is smaller to two significant digits than the rear DoF at 0.55 inches.
With a 30mm lens at f/4 the 1:2 ratio is achieved at a focus distance of 92 inches. At the macro focus distance for a 30mm lens of 2.3622 inches the ratio is 1:1. With a focus distance of 287 inches (just short of the hyperfocal distance) the ratio is 1:61.4 with a near DoF of 141.2 inches and a far DoF of 8674.3 inches.
At longer focus distances the rear of the depth of field reaches all the way to infinity and thus the ratio between front and rear DoF approaches 1:∞. The shortest focus distance at which the rear DoF reaches infinity is called the hyperfocal distance. The near depth of field will very closely approach one half the focus distance. That is, the nearest edge of the DoF will be halfway between the camera and the focus distance.
We must also remember that hyperfocal distance, like the concept of depth of field upon which it is based, is really just an illusion, albeit a rather persistent one. Only a single distance will be at sharpest focus. What we call depth of field are the areas on either side of the sharpest focus that are blurred so insignificantly that we still see them as sharp. Please note that the hyperfocal distance will vary based upon a change to any of the factors that affect DoF: focal length, aperture, magnification/display size, viewing distance, etc. For why this is the case, please see:
Why did manufacturers stop including DOF scales on lenses?
Is there a 'rule of thumb' that I can use to estimate depth of field while shooting?
How do you determine the acceptable Circle of Confusion for a particular photo?
Find hyperfocal distance for HD (1920x1080) resolution?
Why I am getting different values for depth of field from calculators vs in-camera DoF preview?
As well as this answer to Simple quick DoF estimate method for prime lens
You are right, the DOF behind the subject is larger than the DOF before it, but the difference can be very small. That the distance behind is larger can be seen if you look at the formulas for DOF (from mhohner.de: Optical formulas):
Front DOF = cFd^2 / (f^2+cFd)
Rear DOF = cFd^2 / (f^2-cFd)
f is the focal length
d is the focus (or subject) distance
F is the F number of the lens (2.8, 4, 5.6 etc)
c is the circle of confusion (usually around 0.03mm)
You see that the denominator for the rear DOF (f^2-cFd) is always smaller than the denominator for the front DOF (f^2+cFd), which makes the rear DOF larger.
To understand lenses and how they image, you need to know that the lens projects an image of the outside world on the surface of film or digital sensor. Upon close examination you will discover that this image is comprised of countless circles of light. These circles are called circles of confusion because their boundaries are indistinct and they are jumbled end-to-end alongside countless neighboring circles.
We deem an image to be sharp when these image forming circles are so tiny we can’t perceive them as disks. Instead we see a tiny point of light that is not discernable as a disk. This works out to be ½ millimeter viewed from a distance of 500mm. In other words, smaller is better.
Light rays from the subject play on the surface of the lens. As they pass through the lens, the shape of the lens alters their path. They are caused to bend inward (refract). We can trace out this path; it resembles two ice-cream cones set pointy end to pointy end. When we focus, we are adjusting the position of the surface of the digital sensor (or film) as it relates to the lens, so the apex of this cone of light just kisses the light sensitive surface. If achieved, tiny circles result. If the surface is not exactly at the apex of this cone, the circles will be not so small. Sharpness of image is dependent on these circles remaining small.
Now the back-focus (distance lens to image plane) is a variable based on subject distance. If the object is at infinity (as far as the eye can see), the back-focus is at its shortest. We measure the back-focus length when imaging an object at infinity. If the object is closer, the back-focus distance is elongated. The closer the object, the longer the back-focus. At unity (life-size) the back-focus is elongated to twice the focal length. If the object is just shy of infinity, the back-focus is only slightly elongated.
What I am trying to tell you is this: Subject distance dictates the length of the back focus. As you move closer and closer to the subject, the magnitude of the back-focus elongation increases. The bottom line is -- objects far from the camera have nearly the same back focus distance over an extended range. Conversely, objects close to the camera have expanded back-focus, and this distance changes radically with distance changes. It is these back-focus changes that change the size of the circle of confusion. The result is: Depth-of-field extends approximately 2/3 forward and 1/3 back towards the camera from the point focused upon.
The short answer is:
Because it is an exponential relationship.
The long answer... (not that long) is that you do not use linear dimension units like 1 meter, but proportions.
Your image is not telling that the middle is on the center of 2 mts = 1 mt.
It is telling that you have double the distance.