2018-12-19 Prof. Herbert Gross Danyun Cai, Yi Zhong Friedrich Schiller University Jena Institute of Applied Physics Albert-Einstein-Str 15 07745 Jena

Exercise Solutions: Lens Design II– Part 5

Exercise 5-1: Cooke Triplet In this exercise a Cooke Triplet with a focal length of 52 mm will be optimized as well as the importance of vignetting for photographic lenses will be explained and analyzed. a) Load file ‘Ex15.1 Cooke Triplet-1.zmx’. Optimize the system using the already defined merit function without changing glasses. b) Investigate the impact of the glass choice of the negative lens on the system performance. c) Use the first and the last lens surface to vignette the maximal field angle by 50 percent. Take care that the chief ray still passes the center of the pupil stop. What happens? Solution: a) The Cooke triplet investigated here has a FoV 45.2 deg and an F-number of 3.5 with the object at infinity. Five wavelength equally weigthed are defined: 450 nm, 500 nm, 550 nm, 600 nm and 650 nm. The reference wavelength is 550 nm. The start system looks as follow:

There are exactly eight effective independent variables or degree of freedoms available for the control of optical properties. These major variables are six lens surface curvatures and the two airspaces between the lenses. The six curvatures can als be viewed as three lens powers and three lens bendings. Recall that there are seven primary aberrations (five monochromatic Seidel-aberrations, first-order longitudinal and lateral color). Thus, the Cooke triplet has just enough effective independent variables to correct all mentioned aberrations plus the focal length. But there are no variables available for controlling the higher-order aberrations.

Therefore to balance the higher-order aberrations the low-order aberrations will be not corrrected to zero. Glass selection A Cooke triplet is an achromat. Thus, each of the two positive elements must be made of a crown type glass and the negative element must be made of a flint type glass. For practical reasons and with no loss of performance, both positive crown elements are usually made of the same glass type. The airspaces in a Cooke Triplet are a strong function of the dispersion difference between the crown and the flint glass. A dispersion difference that is too small causes the lens elements to be jammed up. On the other side, a dispersion difference that is too large causes the system to be excessively stretched out. The difference in the refractive index also enters into the optical solution. To help reduce the Petzval sum to flatten the field the positive lenses should be made of a higher-index crown glass and the negative lens should be made of a somewhat lower flint. An excellent high-index crown glass of reasonable cost is N-LAF21. A first guess for a matching flint glass is N-SF15 (See glass map).

In the merit function besides the default merit function for spot size optimization the focal length, axial colour and overall length is defined. After running the optimization the following performance is achieved.

b) The first choice of the negative lens flint type glass was only a guess, to find the best glass choice we will now change the glass type along the glass line in the flint range by hand. (n-sf15 n-sf1 n-sf10 n-sf4 n-sf14). The best performance can be achieved with N-SF4.

c) Like the vast majority of camera lenses the Cooke Triplet is to have mechanical vignetting. Mechanical vignetting is useful for two reasons. First, the smaller lens elements reduce size, weight and cost. Second and more important, vignetting allows better performance. Here a vignetting of about 50 % is achieved setting the aperture of the front surface to 17 mm and of the last surface to 14 mm. With this choice the chief ray of the maximal FoV nearly passes the stop in the center. This is important to avoid a complete vignetted field when stopping down. After the final re-optimization the following system performance is achieved.

Exercise 5-2: Anamorphic Diode Collimator A semiconductor diode with wavelength 650 nm and the divergence / aperture values 0.4 / 0.1 in the fast ans slow axis respectively should be collimated in a circular beam with a diameter of approximarely 8 mm. The collimated beam is now focussed into a fiber with numerical aperture of NA = 0.1.

semiconductor

diode

NAy = 0.4

NAx = 0.1

= 650 nm fiber

NA = 0.1

L1

aspherical

collimator

fast axis

L2

cylindrical lens

L3

L4

focussing lens

circular beamD = 8 mm

cylindrical lens

a) Find a solution for this problem with only available catalog lenses. b) Is the setup diffraction limited ? Explain the shape of the residual spot pattern. What

are the reasons for the residual aberrations in the system ? What can be done to further improved the result ?

c) Discuss possible steps to get a shorter system. What are the consequences of a compact layout ?

Solution If the desired beam diameter after the collimation of the fast axis is 8 mm, the focal length of the first lens is

mmNADf y 10/2/

Since the numerical aperture of the fast axis is high, it is recommended to use an aspherical collimator lens, which is corrected for spherical aberration on axis. If such a lens is found in the lens catalogs, it must be considered: 1. the lens should be used without cover glas plate 2. if a working wavelength near to the 650 nm is found, it is an advantage Possible solution: Catalog Asphericon, lens with the No A12-10HPX

Necessary steps to process this lens: 1. load the lens 2. turn around 3. set NA to 0.4 and vignetting factors in field menu to VCX = 0.75. Alternatively, the front surface of the collimating lens can be established by an elliptical aperture. If the axes of the ellipse are set in a ratio of 1:4, the desired light cone is obtained in approximation. In this case exactly the tan(u) values are related and therefore the numerical apertures as sin(u) values are only roughly obtained.

4. change wavelength to 650 nm 5. optimize first distance to collimate this wavelength (default merit function, with criterion: direction cosines). Alternatively, the option QUICK ADJUST can be used with the first distance as variable and the angle spot as an afocal criterion.

A footprint diagram shows the elliptical beam cross section behind the lens.

In the next step, a Galilean telescope with factor = 4 must be found to enlarge the diameter of the x-section to the same value as in the y-section. First a negative cylindrical lens with a rather short focal length must be found. Possible solution: Lens with 1 inch negative focal length in the catalog of Melles Griot: RCC-25.4-12.7-12.7-C

The lens is inserted behind the collimating asphere and rotated around the x-axis by 90° to work in the x-section.

The distance to the collimator is not very relevant and is fixed to be 5 mm.

For a Galiean telescope with factor 4, the second lens must have a focal length of 4x25.1 mm = 100.4 mm. In the same lens catalog one can found the following lens: RCX-40.0-20.0-50.9-C

The lens is inserted, turned around to get a better performance and also tilted by 90° in the azimuth. A first guess gives a distance of 100-25=75 mm between the telescope lenses to get a collimated x-section. But from the spot diagram with direction cosine option it is seen, that the angle distribution is not equal in both sections. Due to the finite positions of the principal planes of the lenses, the distance must be optimized with an angle criterion default merit function. Again as an alternative, the QUICK ADJUST feature can be used to find the optimal lens distance in the telescope. Spot diagram before and after this focussing operation with the same scale:

The footprint diagram now shows a rather circular cross section. The residual error can be neglected and comes from the fact, that for this wavelengths, the catalog focal lengths are not exact.

The data are now the following:

To focus the beam into a fiber with numerical aperture 0.1, the focal length must be not smaller than f = 4.32 mm / 0.1 = 43.2 mm. A lens of approximately this size can be found in the catalog of Melles Griot as an achromate. This helps in getting a better correction: LAO-44.0-14.0

This lens is inserted to complete the system. Finally the last distance is optimized to get a minimal spot size. It is seen, that the spot is nearly diffraction limited.

Exercise 5-3: Chromatic Confocal Sensor A chromatical confocal sensor is used to detect a work piece with axial resolution by chromatical coding of the signal, that is reflected by the sample. The depth discrimination is created by a system, that has a large axial chromatical aberration. In the detection path, a confocal pinhole or a monomode fiber is used to separate the light from various depths. A system of this type should be designed and evaluated here. Design Specifications:

1. wavelengths 450 / 550 / 650 nm 2. NA in the object side 0.2

3. image sided lateral resolution: 2 m (defined as Airy diameter for simplicity) 4. field is always on axis 5. Measuring range 1 mm 6. Minimal free working distance : 10 mm 7. Only two lenses 8. One aspherical surface is allowed 9. Quality: lateral diffraction limited performance for all depths

Tasks: 1. Optimize the system according to the requirements 2. Check for the linearity of the depth resolution with wavelength, determine the slope

differences of the -z-function in the measuring range. 3. Estimate the accuray of the system by considering the geometrical spot size. Assume that the confocal pinhole is able to detect a 10% increase of the spot size. Remarks: The default merit function with spot criteria should be used for correction for simplicity. Solution: First a two-lens system with only spherical surfaces is used as initial system. To have a large axial color separation, low dispersing glasses are used. To have good conditions for a good spherical correction it makes sense to used high refractive indices. According to the glass map, the glass SF59 is selected for both lenses. To allow for a correction with good spot quality in various distances, a multiconfiguration with the distances 10 to 11 mm is established. The exact value of the intermediate wavelength distance is taken as a parameter/variable. Due to normal dispersion, the shortest distance of 10 mm corresponds to the blue wavelength.

To achieve the resolution of 2 m as the Airy diameter for the longest wavelength 650 nm, we get the numerical aperture: 1.22 0.65 / 2 = 0.4 Initial setup of data:

The aperture in the image space can be requested by: ISNA = 0.4 ( but paraxial, wrong for asphere) PMAG = -0.5 ( but paraxial, wrong for asphere) REAB = -0.4 real, okay

Result of the first optimization run:

It is seen, that the desired numerical aperture is not obtained and the performance is far from beeing diffraction limited. Therefore the weighting is increased here and an asphere of the Q-type is applied for surface number 2 with 4 parameters. The result is

The result is still not satisfactory. The remaining degree of freedom to get a better performance is the selection of the materials. Therefore a Hammer optimization with substitution of the glasses is performed. The result is now fine, the system is diffraction limited.

To visualize all three colors as longitudinal aberrations, the 3 wavelengths must be introduced in one configuration:

To visualize the signal z(), a 2D universal plot with the following settings is calculated. It is seen, that the signal response is nonlinear.

To estimate the nonlinearity, the minimal spot location is calculated with quick focus at 450 and 451 nm as well as at 649 and 650 nm. The results are: 450 nm : z = 10.0000 mm 451 nm: z = 10.0088 mm

z = 8.8 m per = 1 nm at the blue end 649 nm: z = 10.9972 mm 650 nm: z = 11.0000 mm

z = 2.8 m per = 1 nm at the red end This means, for a constant spectral resolution of the sensor, the sensitivity is a factor of approx. 3 larger at the red end for the large distance z = 11.0 mm.

To estimate the z-resolution, a 1D universal plot is calculated for = 550 nm with the spot size

as a function of depth t4 (focus distance). If the numbers are considered, an estimation of z =

0.32 m is obtained. Since the diffraction spreading is neglected here and is approximately a

factor of 3 larger, an accuracy of 1 m seems to be more realistic.

Rmin = 0.2275966 m

1.1 Rmin = 0.25036 m

z = 0.32 m