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Riflescope Background


The major optical components of a riflescope are shown in Figure 1, which is a modified scanned image from reference (1), page 195. A riflescope consists of a front objective that collects the light and focuses the image in the first focal plane. A few riflescopes will put the reticle in the first focal plane, but most do not. Placing the reticle in the first focal place allows the reticle image magnification to be changed when the user adjusts the zoom ring on a variable scope. Most users DO NOT like to see their crosshairs change in size, when they change magnification, so most scopes have the reticle placed in the second focal plane. For the rest of this paper, we will assume the reticle and field stop are in the second focal plane, although the derivation of parallax error equation does not matter if the reticle is in the first focal plane. The typical Galileo type refracting astronomical telescope produces an inverted image, which isn’t usually a problem when viewing astronomical objects, but is a problem for terrestrial viewing. A riflescope addresses this problem by using erector optics (two doublets) to invert the image. The erect image is then focused at the second focal plane. The reticle crosshairs are attached to the field stop ring. The field stop diameter is carefully chosen by the optical designer, and ALWAYS determines a riflescope’s field of view (FOV). The last part of the riflescope is the ocular (aka eyepiece). The ocular acts as a magnifying glass, and is adjusted to bring in focus the reticle and field stop edge located at the second focal plane. Any image located in the reticle/field stop plane will also be in focus. Figure 1 has left out some things that all modern riflescopes have. For example, it does not show that the erector optics are held inside a tube, called the erector tube. This erector tube is allowed to pivot , and is held in place at four points: the elevation and windage screws in the front, a pivot point in the back, and a strong spring that applies a force to hold the erector tube against the elevation and windage screws. It also does not show that in a variable riflescope, the two erector optics (doublets) are allowed to move relative to each at different rates. The movement of the erector optics relative to each other creates the change in magnification for a variable riflescope. The fact that they move at different rates keeps the target image at the same focus point (second focal plane) over the magnification range. Optical zooms that keep the image at the same focus point throughout the zoom range are called parafocal optics. Figure 1 is also missing additional optical elements that might be present to correct optical aberrations. Many variable scopes also include a negative field lens, located between the objective lens and erector optics, that increases the elevation and windage adjustment ranges by optically magnifying the offsets (see Weaver US Patent 3423146 for more information).

Fig 1
Figure 1: Riflescope optics

What is PE (PARALLAX ERROR) in a riflescope?


Optical systems can focus only at a single distance. To allow the user to change focus, some riflescopes have a means to adjust focus. The most popular focusing mechanism is called a front AO (adjustable objective), and consists of a rotating front collar, that moves the front objective along the optical axis when turned. Another method is moving a focusing lens, located between the front objective and erector optics using a SF (side focus) knob that is usually located opposite of the windage control knob. In some rare cases, the focusing lens element is located in the rear, between the erector optics and the ocular, and is controlled by a rotating collar that is normally located where the magnification zoom collar is located in a variable riflescope.

The focused target image is located at the second focal plane, in the same plane as the reticle. For scopes that do not have adjustable focus, the focus is factory preset and is called the riflescope’s parallax range (aka focus range). Typical preset parallax ranges for a riflescope are: 50, 60, 75, 100, and 150 yards. If the target is located at a different range than the parallax range, the target image WILL NOT be in the reticle plane, but instead will be in front or behind the reticle plane. Because the target image and reticle are not in the same plane, one can see a parallax shift (aka PARALLAX ERROR) between reticle image and target image, if ones eyes are not exactly on the optical axis.

It is the purpose of this white paper to calculate the parallax error. To calculate the maximum parallax error, we must know how far the eyeball can move off the optical axis. The MAXIMUM distance that the eyeball can move off the optical axis, before losing the image, is determined by the optical system’s EXIT PUPIL.

What is the EXIT PUPIL?


The EXIT PUPIL for a riflescope is the IMAGE of the front objective lens, or any front or back aperture stop that might define a limit to the objective lens. Unlike camera lens, riflescope don’t normally have apertures, so the EXIT PUPIL is the image of the front OBJECTIVE. Some readers might be wondering how you can FOCUS on something as close as the objective lens and create an image if the objective lens defines the exit pupil? Looking at Figure 1, and for the moment lets forget that the erector optics are there, if we let the ocular act as an objective lens, it can focus an image of the front objective, so long the front objective is further away than the ocular focal length (Don’t worry it has to be this way in any magnifying telescope system). Most riflescope oculars have a focal length of about 2 inches, meaning that a close object, like the front objective, will be in focus further away than the eyepiece focal length, usually about 3 inches. The front objective image distance from the ocular is defined as the EYE RELIEF. One can measure the eye relief of a riflescope, by pointing the scope outside a bright window, and by placing a piece of paper about 3 inches from the back of the eyepiece. You will see a circle of light, and by moving the paper back and forth, can bring the circle of light into sharp focus (smallest size). The distance of sharpest focus of this circle of light is called the EYE RELIEF distance, and that circle of light is the IMAGE of the front objective lens. If a fly were to land on the front objective, you would see a tiny in focus silhouette image of the fly, inside the circle of light on the paper.

This circle (or cone) of light, called the EXIT PUPIL, defines the boundary of ALL the light collected by the front objective. The EXIT PUPIL does not define FOV (remember I said that the field stop defines that!). If the eyeball’s pupil where to leave this circle of light, the image would black out (some people call it winking out). So the EXIT PUPIL puts a limit on the eyeballs off axis range for defining PARALLAX ERROR (PE). One interesting property of the exit pupil, given without proof, is that the riflescope’s MAGNIFICATION is OBJECTIVE DIAMETER divided by EXIT PUPIL. For example, if we assume Figure 1 was drawn to scale, it looks like the EXIT PUPIL is about half the diameter of the OBJECTIVE, giving a MAGNIFICATION of 2x for this system.

Derivation of PARALLAX ERROR Equation


Figure 2 shows the parallax free distance, p, indicated by an object point P0. The target object, T0, is located distance t from objective lens, and is located beyond the parallax free distance p. The difference between the target and parallax focus distance is given by (t-p). The objective lens focuses the target object T0, as an image located at ti. The effective focal length of the objective lens is fi. The objective lens will focus an image of an object, P0, located at the parallax distance p, at the point indicated by pi. The reticle, by definition is located in the same plane as the parallax focus distance pi. The angle that suspends half the diameter of the objective lens, as seen from the parallax distance p, is called alpha. Notice that the erector optics and ocular are missing from Figure 2.
Fig2
Figure 2: Parallax Equation

To solve the parallax error of a riflescopes requires properly identifying the EXIT PUPIL. The front objective lens defines the EXIT PUPIL. The EXIT PUPIL in Figure 1, is shown right of the ocular and is an image of the objective lens created by the ocular. The eye’s range is BOUNDED by the limits of this EXIT PUPIL as shown in Figure 1. The EXIT PUPIL is the objective lens as shown in Figure 2, so one can easily handle the parallax conditions and calculations at the objective lens rather than exit pupil at the ocular. Using Figure 2, it is apparent that the Parallax Error as measured from the optical axis to one of the extreme edges of the objective EXIT PUPIL is given as:

Equation 1   Maximum PE = TAN (alpha) (t-p)

From Figure 2, notice that TAN(alpha) = 0.5 D/p, where D is the objective diameter. Substituting this value for TAN(alpha) into Equation 1 gives:

Equation 2   Maximum PE = 0.5 (D) (t-p)/p

If t and p have the same units (for example yards), then D can be expressed in any other units such as millimeters. I leave it as an exercise for the reader to confirm that similar results occur if the target is located inside the parallax distance p, so the parallax error can be given in general form as:

Equation 3   Maximum PE = 0.5 D (ABS(t-p))/p

Where ABS is the absolute value of the argument. We can approximate the angular parallax error in radians, rather than linear error, by just dividing maximum PE by t, where t >> D. For the typical objective diameters of 2 inches or less, Equation 4 is accurate to 3 significant digit if t > 5 yards.

Equation 4   Maximum Angular PE = Maximum PE/t

Equation 4 is given in units of radians. To convert radians to MOA (Minutes of Angle) just multiply by conversion factor of 180*60/PI, which is equal to 3438.

EXAMPLE 1:
Assume we have a 3-9X32mm scope, where objective diameter, D = 32mm, parallax range of scope (p) = 100 yards, and the user desires to calculate maximum parallax error for target ranges (t) at 25, 50, 150,175, 200 and 300 yards. Using Equation 3, we get:

Maximum PE @ 25 yards = 0.5 * 32* ABS(25-100)/100 = 12mm or 0.47 inches,
Maximum PE @ 50 yards = 0.5 * 32* ABS(50-100)/100 = 8mm or 0.32 inches,
Maximum PE @ 150 yards = 0.5 * 32* ABS(150-100)/100 = 8mm or 0.32 inches,
Maximum PE @ 175 yards = 0.5 * 32* ABS(175-100)/100 = 12mm or 0.47 inches,
Maximum PE @ 200 yards = 0.5 * 32* ABS(200-100)/100 = 16mm or 0.63 inches,
Maximum PE @ 300 yards = 0.5 * 32* ABS(300-100)/100 = 32mm or 1.26 inches.

To convert this to an angular error, in MOA units, divide the maximum PE by t, where t MUST be in same units as linear Maximum PE, and multiply by 3438 to convert radians to MOA. The value t at 25yards = 22860mm, 50yards = 45720mm, 150yards = 137160mm, 175yards = 160020, 200yards = 182880mm, and 300yards = 274320mm.

Maximum PE @25 yards in MOA = 12mm *3438/ 22860mm = 1.80 MOA,
Maximum PE @50 yards in MOA = 8mm *3438/ 45720mm = 0.90 MOA,
Maximum PE @150 yards in MOA = 8mm *3438/ 137160mm = 0.20 MOA,
Maximum PE @175 yards in MOA = 12mm *3438/ 160020mm = 0.26 MOA,
Maximum PE @200 yards in MOA = 16mm *3438/ 182880mm = 0.30 MOA,
Maximum PE @300 yards in MOA = 32mm *3438/ 274320mm = 0.40 MOA.

Some people will think that parallax error is greater as target gets an equivalent distance closer to the riflescope from parallax free distance and less for similar distance further away from the parallax free distance. This is not true, when measuring linear parallax error: 12mm at 25 yards VERSUS same 12mm at 175 yards in this example, but is true if we compare angular parallax error: 1.80 MOA at 25 yards VERSUS 0.26 MOA at 175 yards in this example.

EXAMPLE 2:
Assume we have a 2-7X28mm scope, where objective diameter, D = 28mm, parallax range of scope (p) = 60 yards, and the user desires to calculate the maximum parallax error for target ranges (t) of 20 and 100 yards. Using Equation 3, we get:

Maximum PE @20 yards = 0.5 * 28* ABS(20-60)/60 = 9.3mm
Maximum PE @100 yards = 0.5 * 28* ABS(100-60)/60 = 9.3mm.
To convert this to an angular error, in MOA units, divide the MAXIMUM PE by t, where t MUST be in same units as linear Maximum PE, and multiply by 3438 to convert radians to MOA. The value t at 20yards = 18288mm and 100yards = 914400mm

Maximum PE @20 yards in MOA = 9.3mm * 3438/ 18288mm = 1.75 MOA.
Maximum PE @100 yards in MOA = 9.3mm * 3438/ 91440mm = 0.35 MOA.

Example 2, like Example 1, demonstrates that if a common offset from the parallax free distance is used (40 yards in this case), the maximum linear PE is the same, however the angular maximum PE are different. The closer in target will have a larger angular PE than the target located further away.

Things to know about the Maximum PARALLAX EQUATION


Some interesting things to notice about Equation 3 is that for distances closer than the parallax free distance (p) of the scope, the maximum linear parallax error will never exceed half the objective diameter (0.5 D). For distances more than twice the parallax setting of the scope, the error becomes larger than half the objective diameter. For the case of t approaching infinity, the maximum angular PE is given as 0.5 (D)/(p) radians, where p and D must have same units. Likewise, if t approaches zero (gets closer to the objective), then maximum angular PE approaches ARCTAN(0.5 (D)/p). The maximum PE calculated in Equation 3 is measured from the optical axis to one of the EXIT PUPIL extreme limits. If you want to measure the total maximum PE as measured from one EXIT PUPIL (EP) extreme to the other, then the value in Equation 3 must be doubled. MAG (MAGNIFICATION) is a factor when calculating parallax error, but NOT when calculating the Maximum PE.

PARALLAX EQUATION with Magnification term


EXIT PUPIL and MAGNIFICATION are inversely proportional to each other, i.e. as MAGNIFICATION goes up, the EXIT PUPIL size decreases. If we are interested in PARALLAX ERROR due to a specific offset, then we must consider the MAGNIFICATION. We can easily add the magnification factor by remembering that the EXIT PUPIL size at the ocular is defined as follows (see reference (1) for derivation):

Equation  5 EP = D/MAG

The shooter’s eye is located at the riflescope’s EYE RELIEF, and is offset from the optical axis by distance X, as shown in Figure 1. The parallax error is just the ratio of X, as measured at the ocular EXIT PUPIL, to half the value of EP, multiplied by the maximum PARALLAX ERROR as given in Equation 3.

PE @ offset X = X/(0.5 EP) ( 0.5 D (ABS(t-p))/p)

Substituting Equation 5 into above equation gives us the PE equation that is dependent on magnification and offset distance X:

Equation 6  PE @ offset X = (X) (MAG) ABS(t-p)/(p); for X < EP/2

EXAMPLE 3:
We have a 3-9X40mm scope, where objective diameter, D = 40mm, parallax range of scope (p) = 100 yards, and target range (t) is 25 yards. What is the parallax error at 3X and 9X if eyeball is 1 millimeter off optical axis at the eye relief of the scope? Again, p and t units must be the same, and X units can be anything you desire, although most people will probably use millimeters. The first thing to do is verify that the offset is less than or equal to EP/2. If it is great than EP/2, then your eyeball offset is outside the EXIT PUPIL and you can’t see any image. Using Equation 5, the EXIT PUPIL at 3X is equal to 40/3 = 13.3mm and the EXIT PUPIL at 9X is 40/9 = 4.4mm, so X is indeed less than EP/2 in both cases. Using Equation 6, we calculate the PARALLAX ERROR as:

PE @ 25 yards with a 1mm offset from optical axis @ 3X = (1.0) (3) ABS(25-100)/100 = 2.25mm
PE @ 25 yards with a 1mm offset from optical axis @ 9X = (1.0) (9) ABS(25-100)/100 = 6.75 mm

To convert these PARALLAX ERRORS from linear to angular errors, in MOA units, divide by t, where t MUST be in same units as PE (100 yards = 91440mm in this case), and multiply by 3438 to convert radians to MOA.

PE in MOA @ 25 yards with a 1mm offset from optical axis @ 3X = (2.25)(3438)/ 91440 = 0.085 MOA
PE in MOA @ 25 yards with a 1mm offset from optical axis @ 9X = (6.75)(3438)/ 91440 = 0.254 MOA
Example 3 shows that the parallax error at 9X power is three times larger than at 3X power for a common linear offset from the optical axis. The two things to remember is that Equation 6 is only valid if the eye is located at the eye relief of the system, and that the offset is measured from the optical axis. If the offset is measure both ways from optical axis, (i.e. X offset value is range above and below the optical axis), then the value of Equation 6 must be doubled.

Acronym List:

AKA Also Known As
ABS Absolute value function
AO Adjustable Objective
ARCTAN Arc (Inverse) Tangent function
D Diameter of Objective lens
EP Exit Pupil
FOV Field of View
MAG Magnification
MOA Minutes of Angle (60 MOA = 1 degree)
p Parallax free distance
PE Parallax Error
SF Side Focus
t Target Distance
TAN Tangent function

References:
1. Optics, by Eugene Hecht, 2nd Ed., 1987.
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