Parallax - And how scopes work

Terminology
Parallax
How to set Up a Scope

I've answered questions about scopes and parallax about 900 times, and it's always a long drawn out thing, going several e-mails, and a few phone calls. It doesn't seem to make any difference how long the guy has been shooting, this one always keep screwing guys up.

OK... here goes, and it's gonna be a long one.

There are several things that go on inside a scope, and in the eyes at the same time. Some of them work against each other.

Some terminology first...

And we'll leave out lenses that are there to correct some optical or color errors, but don't have anything to do with image forming. We'll start at the front of it all, and work back.

The "Object"... the "object" (target) that you are looking (shooting) at.
The "Objective". The front lens is called the "Objective"... it forms the first image of the "object" we are looking at (that why they call it the Objective)

It is the lens that "captures" all the light, that is solely responsible for the image quality of the scope... if the objective is poor, you can't fix the poor image later. This lens is usually made of two different types of glasses (called "elements") sandwiched together, and is called an "Achromat". The Achromat is fully color corrected for blue and green. The red wavelengths are partially corrected, but have what is called "residual color errors". These are very minor.

This is the normal type of objective used in shooting and spotting scopes. In quality, they can vary from bad, through sorta OK, to pretty damn good.

If one of the elements is made of an "ED" glass, or a "Fluorite" (CaF) glass, the two element lens can be very good to outstanding.

In some instances, objective lenses are made of three elements, and all three colors (blue, green, and red) are completely corrected. This type of lens is called an "Apochromat", and this is the finest lens that can be bought. The best of these can also have "ED" glass, or Fluorite as one of the elements.
The "First image plane". The Objective focuses the light to make an image of the subject, just like a camera lens. This image is upside down, and right/left reversed. This is the first image plane, but NOT the "First image plane" that is talked about when shooters talk about reticles.
The "Erector lens"... (if it is a group of lenses, it is called the "Erector cell"). Because the first image is upside down/wrong way around, we (as shooters) can't use it... so we flip it around with a simple optical group called the "erector cell". This cell gives us a new image that is right way around, called the second image plane.

But this cell has another very important job. Moving this cell causes this second image plane to move... so micrometer spindles are put against the cell, to get elevation and windage adjustments.

The total amount of elevation/windage available in the scope (MOA from bottom to top) is determined by how much the spindles can move the cell. The amount of movement "per click" is simply determined by the thread of the screw, and the spacing of the detents on the spindle.
The "Second image plane". This is the second real image plane in the scope, and this is the image plane that shooters call the "First image plane" when talking about reticles. In a fixed or variable power scope with a "First image plane reticle", the reticle would be placed in this image plane.
The "Zoom group". In a variable scope with standard (non-magnifying) reticle, the zoom group of optics would follow #5. This group of lenses can change the size of the image plane in #5 and then form a new (third) image plane behind it.
The "Third image plane". In variable power scopes, this is the plane that the reticle is placed in. By being here, it allows the image to change sizes, but the reticle to stay the same size. In the context of reticles, this is the image plane that is referred to as the "second image plane"
The "Eyepiece". This optical group is like a jewelers' loupe. It is (or should be) a super fine magnifier. It's only job in the whole world, is to focus on the reticle.

Let me repeat that for those that live in Rio Linda...

THE ONLY JOB FOR THE EYEPIECE IS TO FOCUS YOUR EYE ON THE RETICLE!!!!

It CANNOT adjust, or compensate for, or do anything else when things look bad in the scope, or when you can't hit the target... and you CANNOT use the eyepiece to try to correct for parallax. That is sheer folly at best, and raw stupidity at worst.

OK... now that you know what the insides are like... let's move on. We'll use the zoom scope for our examples, because if you can understand the zoom scope, then the fixed scope is a walk in the park.

In the scope that is set for infinity range, the object forms an image (upside down, right/left reversed) behind the objective (the first image plane)... the erector cell "sees" that image, and flips it over and makes it right way around in a NEW image plane (the Second image plane). The zoom group adjusts the size of this image plane, and makes a NEW image plane (the Third image plane) that is the desired size. There is a reticle placed in this last image plane, and the eyepiece focuses on the reticle AND the image at the same time. When things are good, that's how the scope works!

But... now the booger falls into the soup... IF the third image plane and the reticle are not exactly, (and I mean EX-ACT-LY) in the same place, then your eye cannot see them LOCKED together as one picture.

It sees them as two separate pictures, and the eye will look at each separately, and the eye can also look AROUND one to see the other.

Lenses are measured in metrics (aka Millimeters). Not because the Europeans wanted the metric system 25 years ago, but because optical strings and chains of lenses (like scopes) are really a string of numbers.

There are constant ratios of "this divided by that's" that give image sizes, "F-ratios", and image locations. It's so damn easy to do the engineering using a 10 based system that the optical guys were using the metric system way back in the 1800's.

The objective has a "Focal length"... this is the distance behind the lens that the first image plane falls when making an image if a subject that is at infinity (or very damn far away).

If the objective has a focal length of 100mm, then the image of that 1000 yd target is 100mm behind the lens.

But the problem with geometric optics (which is what we are dealing with here), is that they follow the laws of geometry... and optics make triangles like rabbits make babies.

AND... in an optical chain, when you change one thing, one angle, one ANYTHING, everything else follows along and the changes are BASED on the ratios involved at THAT stage.

If we take that same target, and move it to 100 yds, the image in the scope moves BACKWARDS, going further into the scope. Not by much, but it doesn't take much, because we are dealing with very small distances inside the scope, and very high magnifications.

How far the image moves back, and what it's new position is, is predictable by the mathematical ratios of the angles formed by the subject and the first image... OR (for us dummies that lost our slip sticks) by the ratio of the distances to the Target and the focal length, multiplied by the focal length, then ADDED to the focal length.

The target is at 100 yds (91440mm), the focal length of the objective is 100, so the displacement is 1/914 x 100, which means that the first image is now at ~100.1mm.

Hmmm only .1mm, that doesn't seem like much.

Read the following paragraph twice...

In a 1x scope, 0.1mm would mean nothing... but this displacement is repeated throughout the chain, AND if any of the optical groups change the image ratio (aka image size), then the displacement (aka ERROR) is changed in direct proportion to the increase in magnification. So in a 3x scope, it would be .3mm, and in a 10x scope, it would be 1mm, and in a 30 power scope, the image would be 3mm behind the reticle. Now, you should have seen a pattern in this last paragraph.

READ THIS TWICE!!

With the same error in the objective (scope focused at 1000, and target at 100), the parallax INCREASES WITH MAGNIFICATION... got it?

If not, READ IT TWO MORE TIMES!!

OK... now, if we do the same math for closer distances, like 50 yds, and 25 yds we will see that the error gets really big, so that with a target at 50 yards, and the scope set at 35 or 65 yds, the parallax makes the combination un-usable.

Parallax is...

When the image of the target, and the reticle, are not in EXACTLY the same plane, and by moving the eye up and down... or side to side, either the target OR the reticle appears to move in relation to the other.

You might see the target move and the reticle stay still, or you might see the target stay still and the reticle move over it... both are exactly the same, and which you see, is only a matter of your OWN perception.

It is NOT possible to have parallax while moving up and down, but not have it when you are moving side to side.

If you think that is what you have, you have other problems... either you are moving the rifle, or you have eye problems.

How to set Up a Scope!

This is the only way to do it...

First...

Screw the eyepiece out (CCW) all the way, until it stops.

If you wear glasses, put them on.

Hold the scope up and look OVER the scope at the sky, and relax your eyes. Then move the scope in front of your eye.

The reticle should look fuzzy

Turn the eyepiece in 1/2 turn, and do the same thing again. You will have to do for a while before the reticle starts to look better. When you start getting close, then turn the eyepiece 1/4 turn each time.

Do this until the reticle is fully sharp and fully BLACK immediately when you look through the scope.

Than back off one turn and do it again to make sure you are in the same place.

Then LOCK the ring on the eyepiece, and leave it alone FOREVER!

Second...

Set the scope down on something solid, where it can see something at a long distance... half a mile or longer is good. It can be on the rifle, and rested in sand bags at the range... but pick something at least 1000 yds away... even further if possible. If the scope has an "AO" Adjustable Objective, then set it for infinity, and look at the distant object, and move your head from one side to the other, or up and down if you prefer. If the reticle seems to move, there is parallax. Change the distance setting and try again... if you are very careful, you can move your eye, and adjust the distance at the same time, seeing which direction gets better.

(Something to know - the graduations/calibrations on the AO of a scope are approximations ONLY... they are to get you close. If shooting is critical, then check for parallax by moving your eye and adjusting until there is NO movement of the crosshair/target images).

With front objective adjustments, you can turn them either way without worry... BUT with side adjustment scopes, like the MK4-M3, the M3-LR, or the other LR family of scopes, the adjustment must ALWAYS be made from the infinity end of the dial. Turn the adjustment all the way until it stops (past infinity), and then start turning it in a little at a time, until there is no parallax. If you "overshoot" the proper setting, you can't just turn back a little, you must go back to stop at the end of the dial, and start over again. While "AO"s dials are locked in place, and if the indicated distance doesn't match the real distance, there's nothing you can do about it... the side focus dials are not locked in place. Once you have found the setting for infinity on the side focus models, then (CAREFULLY) loosen the screws, and set the dial so that little sideways infinity symbol is lined up with the hash mark, so it is calibrated. You can also make little marks or put on a paper tape for other ranges instead of using the round dots that don't match any range.

Now you can set it to infinity, but remember that you MUST turn the dial all the way past infinity to the stop, EVERY TIME before going from a close range to a longer range. If you are set for 500 yds, you can go directly to 100 yds, but if you are set for 100 and want to set it to 500, you MUST go all the way back to the stop, and then go to 500 This is because there is a fair amount of backlash (aka SLOP) in this wheel linkage to the focusing cell, so you can set it only from one direction to make sure the slop is always on one side. The other problem with it is, even if you decided that you wanted to calibrate from the other end... the recoil will push the cell back. SO you must ALWAYS set these dials from the infinity end of their scales.

To make it easy to not have to remember...

I always start from the end stop when I change range, no matter which direction I'm going in... it adds about 0.023 seconds!

That's about it on rifle scopes.

There are thousands of "opinions" on scopes on the web, but this is the science from one that does optics for a living.

Now...
You have a friend that says to set up a scope a different way?
The guy at the next shooting bench at the range said to do it a different way?
You know some guy who's in the Marines says to use your eyepiece to correct parallax?
You got a friend that shoots bench rest and says something different?

Before you take their advice, ask them to explain how a scopes works from the inside out.

This is the way to do it, because this is the way scopes work.

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