Achromatic Doublet Design Using a Custom Sequential Ray Tracer
From first-order chromatic correction to nonparaxial ray-tracing analysis
The Problem:
The refractive index of materials is a function of wavelenght. Thus if incoming light is made of many wavelenghts, a single lens would not be able to focus it at the same spot. Thus the deisgn problem becomes - to maintain the focal length while reducing the chromatic aberrration.
Design Criteria:
Effective focal length |
75 mm |
Entrance pupil diameter |
7.5 mm |
F-number |
f/10 |
Field |
On-axis, 0° |
Object distance |
Infinity |
Crown glass |
N-BK7 |
Flint glass |
F2 |
Design wavelengths |
486.1, 587.6, 656.3 nm |
Problems with a single lens:
We start with a single lens of N-BK7 glass. This lens can match the focal lenght to our deisgn criteria but would give a chromatic focal shift as every wavelenght would be bent differently due to different refractive index. This is visible in the figure below. Red and blue wavelengths come to focus at ~1.1 mm away from each other.
Solution:
Lets first tackle this in a first order way.
Achromat condition is :
\[ \frac{\Phi_1}{V_1} + \frac{\Phi_2}{V_2} = 0 \]
And thus the optical power of the first lens is :
\[ \Phi_1 = \frac{V_1}{V_1-V_2}\Phi \]
and the optical power of the second lens is :
\[ \Phi_2 = -\frac{V_2}{V_1-V_2}\Phi \]
We use the net power and the abbe number of the two lens materials to get their respective powers. Once we know the respenctive powers, we can start with some assumption of radii of surface 1 and predict every other radii using thin lens formula.
We now set up our doublet system, whose parameters have been calculated to fulfill achromat criteria at least wrt thin paraxial lens optics. We now run the ray tracer to see what happens to the axial spread of focii in this doublet case.
The longitudinal axis spread changes from 1159 µm to 42 µm. Note that for the former singlet case, the two extreme cases are the two extreme wavelenghts (red and blue). However, for the latter case, the two extremes in axial spread are not red and blue as is also visible in the figure below. The spread between red and blue is only 18 µm.
The rms polychromatic spot size of the system changed from 18.14 µm to 1.3 µm when we moved to achromatic doublet.. These spots are at the best focus plane of the primary wavelenght. This shows that adding more elements while ensuring abbe condition is satisfied reduced the weighted polychromatic RMS spot radius.
We started with an assumed R1. So we need to optimize over R1 and check which R1 (not that R2, R3 are decided through abbe condition and net power criteria) gives the lower axial color (span between the the point at the maximum focus and the point at minimum focus - may not necessarily align with wavelength change - especially in the case of an achromatic doublet like ours). The figure below shows how the axial color changes as as function of bending of R1.
We choose the R1 at which the axial color is minimized and use that to form our doublet. Using that R, we retrace the system. Very small improvements in the axial color and the RMS spot radius is observed.
The following figure shows the comparison of axial color between this optimized doublet and the initial singlet we started with.
The followign figure shows a comparison of the polychromatic spot diagram for the singlet and doublet case.
Thus, by adding more elements whose refractive index and power are such that abbe condition is satisfied, the axial spread in color can be reduced (Figure below)
The final lens layout was shown at the starting of the page and prescription of the lens is shown below.
Limitations
This design is made for on-axis field only. This also mean it only has spherical aberrations in addition to chromatic and none of the off-axis aberrations like coma, astigmatism, field curvature or distortion. The sphericalm aberrations were very low on account for slow lens (high f/#)
A multi-variable numerical optimization has not been performed. In the optimization, only the radius of surface 1 was changed and other radii were calculated from that using first order paraxial optics.
No tolerance and sensitivity analysis has been done.
Conclusion:
In conclusion, here using a in-house custom sequential ray tracer, a cemented N-BK7/F2 achromatic doublet with a 75 mm effective focal length and an f/10 aperture was designed.
The first-order design was derived from the achromatic condition relating power and Abbe numbers of the two glasses. Nonparaxial exact ray-tracing analysis demonstrated a substantial reduction in longitudinal chromatic aberration relative to an N-BK7 singlet with the same nominal focal length and aperture.
This doublet decreased the axial color from 1159 µm to 37 µm, a reduction of 96.8%.
The doublet also decresead the polychromatic RMS spot radius from 18.09 µm to 1.16 µm, a reduction of approximately 93.6%.
This study demonstrates the application of analytical optical design principles, wavelength-dependent material properties, and numerical ray tracing to evaluate chromatic aberration correction in a multi-element lens.
#Adding Singlet lens system (here ot is opt ray tracer library)
L = ot.System([486.1, 587.6, 656.3], # F, d, C [nm]
object_distance=float('inf'),
field_type='angle', weights=None)
L.set_primary(1) # 587.6 nm (d-line) is the reference
L.set_fields([(0.0, 0.0)]) # on-axis field only
EFL = 75.0 # target focal length [mm]
n_d = abbe('N-BK7')[0] # N-BK7 index at the d-line (587.6 nm)
R = 2 * (n_d - 1) * EFL # equiconvex radius
t = 0.05 * R # center-thickness : 5% of the radius
R1, R2, t1, D = R, -R, t, 7.5 #D is the Aperture stop diameter calc using f/# and F
L.add(radius=R1, thickness=t1, glass='N-BK7', semi_dia=D/2, stop=True) # surface 1 (stop)
L.add(radius=R2, thickness=75.0, glass='air', semi_dia=D/2) # surface 2 -> image space
#Running a trace of rays onto the above system
res = L.trace(ot.Pupil(n_rays=1500))[0] # polychromatic SpectralResult, on-axis