Low-Light Performance Calculator
Compare light gain, perceived brightness, and overall low-light performance
in relation to objective lens diameter, magnification, exit pupil diameter,
light transmission, approximate exit pupil diffraction, and eye pupil diameter.
      For Fixed & Variable Power Scopes

 Your Eye Pupil Diameter at Night: mm 
   How to Measure?     Click for Quick Estimate
 Number of Scopes to Compare:

Enter only two of the three scope properties for each line below:
Use only the first line for fixed-power scopes, or both lines for variable-power scopes

 Scope #1 
 Optional Title: 

 Light Transmission: %    
 Fixed or Minimum Magnification:    
  Magnification: x 
 Objective Lens Diameter: mm 
 Exit Pupil Diameter: mm 
 Optional Maximum Magnification:    
  Magnification: x 
 Objective Lens Diameter: mm 
 Exit Pupil Diameter: mm 

TIP: Resolution, contrast, and realistic light transmission greatly affect the low-light performance.
Because these properties are not available for most scopes, this calculation is best used for comparing scopes among the same brand and scope line.
If you are unsure of the utilized benchmarks for Light Transmission, it may be best to leave Light Transmission fields empty.

 Perceived Brightness Calculation Method: 
Stevens' Power Law 
Weber-Fechner Law 
   How To Choose?



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Environmential Values:
Eye Pupil Diameter: 7 mm
Eye Pupil Area: 38.484510006475 mm˛

 Scope #1 
 Magnification: 
 Objective Lens Diameter: 
 Exit Pupil Diameter: 

 Exit Pupil Area: 
 Eye Pupil Illuminance Area: 
 Exit Pupil Utilized Light Percent: 

 Light Transmission Factor: 
 Light Diffraction Factor: 

 Theoretical Exit Pupil Intensity Gain: 
 Realistic Exit Pupil Intensity Gain: 

 Theoretical Light Gain: 
 Realistic Light Gain: 

 Perceived Brightness: 
 Low Light Performance: 
 3
 56
 18.6667

 273.6676
 38.4845
 14.06

 1
 1

 9
 9

 9
 9

 2.08
 2.5
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 3.5
 56
 16

 201.0619
 38.4845
 19.14

 1
 1

 12.25
 12.25

 12.25
 12.25

 2.31
 2.84
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 4
 56
 14

 153.938
 38.4845
 25

 1
 1

 16
 16

 16
 16

 2.52
 3.17
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 4.5
 56
 12.4444

 121.6301
 38.4845
 31.64

 1
 1

 20.25
 20.25

 20.25
 20.25

 2.73
 3.5
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 5
 56
 11.2

 98.5203
 38.4845
 39.06

 1
 1

 25
 25

 25
 25

 2.92
 3.82
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 5.5
 56
 10.1818

 81.4218
 38.4845
 47.27

 1
 1

 30.25
 30.25

 30.25
 30.25

 3.12
 4.14
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 6
 56
 9.3333

 68.4169
 38.4845
 56.25

 1
 0.9958

 36
 35.85

 36
 35.85

 3.3
 4.45
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 6.5
 56
 8.6154

 58.2961
 38.4845
 66.02

 1
 0.988

 42.25
 41.7441

 42.25
 41.7441

 3.47
 4.75
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 7
 56
 8

 50.2655
 38.4845
 76.56

 1
 0.9768

 49
 47.8632

 49
 47.8632

 3.63
 5.04
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 7.5
 56
 7.4667

 43.7868
 38.4845
 87.89

 1
 0.9634

 56.25
 54.1935

 56.25
 54.1935

 3.78
 5.33
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 8
 56
 7

 38.4845
 38.4845
 100

 1
 0.9488

 64
 60.7231

 64
 60.7231

 3.93
 5.61
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 8.5
 56
 6.5882

 34.0901
 34.0901
 100

 1
 0.9334

 72.25
 67.4415

 64
 59.7406

 3.91
 5.76
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 9
 56
 6.2222

 30.4075
 30.4075
 100

 1
 0.9178

 81
 74.3395

 64
 58.7374

 3.89
 5.91
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 9.5
 56
 5.8947

 27.291
 27.291
 100

 1
 0.902

 90.25
 81.4087

 64
 57.7303

 3.86
 6.06
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 10
 56
 5.6

 24.6301
 24.6301
 100

 1
 0.8864

 100
 88.6418

 64
 56.7307

 3.84
 6.2
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 10.5
 56
 5.3333

 22.3402
 22.3402
 100

 1
 0.871

 110.25
 96.0318

 64
 55.7464

 3.82
 6.33
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 11
 56
 5.0909

 20.3554
 20.3554
 100

 1
 0.856

 121
 103.5728

 64
 54.7823

 3.8
 6.46
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 11.5
 56
 4.8696

 18.6239
 18.6239
 100

 1
 0.8413

 132.25
 111.2591

 64
 53.8418

 3.78
 6.59
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 
 12
 56
 4.6667

 17.1042
 17.1042
 100

 1
 0.827

 144
 119.0855

 64
 52.9269

 3.75
 6.71
x 
 mm 
 mm 

 mm˛ 
 mm˛ 
 % 

x 
x 

x 
x 

x 
x 

x 
x 

Perceived Brightness:
The Perceived Brightness is the perception of the amount of target light through the scope relative to the perception of the amount of target light with the single unscoped eye. Perceived brightness is calculated as the cube root of the light gain, which is the basis for modern computer color space brightness scaling and is also the perceived brightness of a 5-degree target in the dark with a uniformly dark background and surround. If the unscoped eye pupil luminous flux is 1 lumen and the scoped eye pupil luminous flux is 50 lumen, then the Eye Pupil Light Gain is 50x, and 50 times the amount of light from the objects within the field of view of the scope is reaching the eye pupil through the scope than without the scope, although the light from the objects through the scope will appear 3.6 times as bright than with one eye without the scope. If you consider that one eye is closed when looking through the scope, then only 25 times the light is reaching the one eye through the scope than with both eyes open without the scope, and the objects through the scope will appear 2.9 times as bright as with both eyes open without the scope. Both the unscoped eye pupil luminous flux and the scoped eye pupil luminous flux are the amounts of light reaching the eye pupil from only the objects that are within the field of view of the scope.

Perceived Brightness = (Realistic Light Gain)^(1/3)

Low Light Performance:
This calculation derives Low Light Performance as the average of light gain and resolution gain through magnification, as a measure of target image acuity gain in low light similar to Twilight Performance specified by scope manufacturers. Low Light Performance calculated here is much more useful than Twilight Performance, as Twilight performance is the average of the just the objective lens diameter times magnification, while Low Light Performance is the average of the actual Perceived Brightness times magnification, which also includes the exit pupil/eye pupil relation, light transmission, approximated diffraction, as well as the perception of relative light gain. Just as with Twilight Performance, this Low Light Performance calculation does not yet include lens resolution and contrast as factors. Therefore lower quality optics will yield relatively less gains at higher magnifications.

Low Light Performance = (Perceived Brightness x Magnification)^(1/2)

Typical Illuminance Levels:
Although Scotopic vision is more low-light sensitive than Photopic vision, Scotopic vision is not useful for scope use because of the lack of receptors within the 5-degree center of vision which essentially creates a dark blurry view through a scope. Therefore, a primary necessity of a low-light optic is to elevate Scotopic light levels to the Photopic region, or at least to the Mesopic region which is a combination of Photopic and Scotopic vision.

The following table represents typical illuminance levels under various lighting conditions. To estimate a scope's ability to elevate Scotopic light levels to Mesopic or Photopic vision, multiply the scope's Actual Eye Pupil Light Gain (not the Perceived Brightness) times the illuminance in lux. For instance, a scope with an Eye Pupil Light Gain of 50x may be capable of elevating full moon light levels well above twilight light levels, or may be capable of elevating starlight levels just into the range of Mesopic vision that is required for adequate scope use.

P = Photopic Vision
P+S = Mesopic Vision = Photopic Vision + Scotopic Vision
S = Scotopic Vision

 Light Condition  Illuminance 
P Direct sunlight 100,000-130,000 lux 
P Full daylight, indirect sunlight 10,000-20,000 lux 
P Overcast day 1,000 lux 
P Clear sunrise or sunset 500 lux 
P Indoor office 200-400 lux 
P Very dark day 100 lux 
P Hallway 80 lux 
P Twilight 10 lux 
P Pure Photopic Vision 3.4 lux 
P+S Candle at 1 meter 1 lux 
P+S Deep twilight 1 lux 
P+S Full moon overhead at tropical latitudes 1 lux 
P+S Full moon on clear night 0.27 lux 
S Pure Scotopic vision 0.034 lux 
S First or Last Quarter Moon, overhead 0.027 lux 
S Quarter moon 0.01 lux 
S Starlight 0.001 lux 
S Starlight on overcast night 0.0001 lux 
S Threshold of Scotopic vision 0.00001 lux 
NOTES

Development:
Input regarding development of this project and assistance is welcomed. Assistance in performing tests is needed by those with access to many scopes. If there is enough interest, a standard testing method could be devised for testing light transmission, contrast and resolution. This information could then be utilized in a database for making very accurate comparisons among different scope brands and lines with quick selections with pre-entered values for each scope. Please direct comments and questions to Your questions may be added to an FAQ section.

Objective Lens Diameters:
The common use of smaller utilized objective lens diameters at minimum magnifications than full objective lens diameters has no affect on the luminous flux within the eye and the resulting perceived brightness, as the exit pupil diameters near minimum zooms are significantly larger than eye pupil diameters, but with the exact same luminous intensity per magnification. The minimum objective lens diameter is included in this calculation to provide accurate exit pupil diameters at minimum zooms. This calculation also includes automatic estimation of the full objective lens utilization point within the zoom range as the point at 1/3 of the magnification range which typically relates to half of the physical zoom travel, and well under the point at which exit pupil diameters are reduced to the size of eye pupil diameters. For instance, for a 2.5-10x scope the maximum lens utilization point would be calculated at 5x, and for a 3-12x scope the maximum lens utilization point would be at calculated 6x. Actual points that may vary from these predetermined points have no affect on performance calculations, only a slight affect on the exit pupil diameter nearing the point. The actual point can be determined by measuring the exit pupil diameters throughout the zoom range and determining where it first equals the full objective lens diameter divided by the magnification.

USEFUL LINKS

Optics & Vision:
  • The Human Eye
  • Webvision
  • telescope-optics.net
  • A Pupil Primer by Alan M. MacRobert
  • Telescope Function
  • Astronomical Telescope & Optics Formulas
  • Lens Tutorial
  • Optical Generation of the Visual Stimulus
  • The N.A.A. Telescope Calculator

    Brightness Perception:
  • Lightness - Relationship between lightness, value, and luminance
    The evolution of the Steven's Power Law into modern color space lightness scales
  • Considering the Surround in Device-Independent Color Imaging
    The effects of stimulus, background, and surround upon lightness scaling
  • A Probabilistic Explanation of Brightness Scaling
    The essential brightness scaling factors
  • Perceiving the Intensity of Light
    Brightness scaling factors in depth
  • Measuring the Relationship between Perceived Image contrast and Surround Illumination
    Perceived image contrast increases with relative surround luminance
  • A Technical Introduction to Digital Video - Gamma
    The use of power function for gamma correction in 1996
  • The Visual Discrimination of Intensity and The Weber-Fechner Law
    Early published research outlining the inconsistencies of the Weber-Fechner law
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