An alternative optimization scheme for visual performance is given. This scheme provides high performance for the photopicly defined eye. The photopic sensitivity curve given in "Principles of Optics" (auth: Born & Wolf, pub: Pergammon Press) describes the sensitivity of bright light adapted eyesight as a singular curve over the 400 to 750 nm wavelengths of visual light. This curve is utilized to optimize visual performance in refractor telescope design.
Optical design software, such as ZEMAX (ZEMAX Development Corporation, San Diego, CA), allows for a selection of multiple wavelengths for optimization of optical systems. In addition, each wavelength can be provided a "weight" to allow for selective import to be assigned to each wavelength. Ignoring other factors, the RMS pathlength error is provided as the normalized weighted RMS errors for each wavelength.
This feature may be utilized to approximate the integral over a given band of interest. For our purposes, better than 7.5 magnitudes of photopic visual sensitivity are covered in the 33 equally spaced wavelength values covering the range of 400 to 720 nm.
To improve estimation performance, an integration weighting can be applied to the original sensitivity curve. The modification of the normalization factor here can be ignored. The desired value is the location of minimum merit function, which will be independent of that normalization factor. The choice of integration weighting is a 32-degree Richardson's weighting function.
Unfortunately, ZEMAX only allows for up to 12 wavelengths. This is overcome by using multiple configurations, with the only difference between configurations being the value and weight of each wavelength. To equalize the weights between configurations, I selected every third wavelength to be assigned to a given configuration. I arbitrarily assigned the three configurations the names red, green, and blue.
These wavelengths and weights for these configurations are then:
RED GREEN BLUE
num WAVE WLWT WAVE WLWT WAVE WLWT
1 : 0.72 0.0003199077 0.71 0.00304895 0.70 0.001988113
2 : 0.69 0.01190542 0.68 0.01042664 0.67 0.04646019
3 : 0.66 0.02957924 0.65 0.1553513 0.64 0.1066308
4 : 0.63 0.3847484 0.62 0.184749 0.61 0.7302961
5 : 0.60 0.3870125 0.59 1.099074 0.58 0.4218678
6 : 0.57 1.382191 0.56 0.6063014 0.55 1.444622
7 : 0.54 0.4625999 0.53 1.251521 0.52 0.4354657
8 : 0.51 0.7302961 0.50 0.1566245 0.49 0.3019912
9 : 0.48 0.08469529 0.47 0.1321212 0.46 0.02909433
10 : 0.45 0.05517147 0.44 0.01410663 0.43 0.01684182
11 : 0.42 0.001939622 0.41 0.001742257 0.40 0.0001218696
Sometimes, glass refractive indices are not listed to 400 nm. In those cases a shifted version of the weights may be applied. (Note: The wavelength order is also reversed for this weight set, but that is largely irrelevant.)
RED GREEN BLUE
num WAVE WLWT WAVE WLWT WAVE WLWT
1 : 0.43 0.003534219 0.42 0.005807524 0.41 0.0005818866
2 : 0.46 0.08711285 0.45 0.02330661 0.44 0.03339326
3 : 0.49 0.1008604 0.48 0.2018115 0.47 0.055448
4 : 0.52 1.030835 0.51 0.2439075 0.50 0.4689575
5 : 0.55 0.6102653 0.54 1.385094 0.53 0.4179886
6 : 0.58 1.263137 0.57 0.5800994 0.56 1.444626
7 : 0.61 0.2439075 0.60 0.9161368 0.59 0.4642923
8 : 0.64 0.2540792 0.63 0.1285 0.62 0.5531666
9 : 0.67 0.0194982 0.66 0.08856474 0.65 0.05188489
10 : 0.70 0.005952712 0.69 0.005029322 0.68 0.02468198
11 : 0.73 0.0002521509 0.72 0.001524475 0.71 0.0006398155