1. IV Practical Aspects of Lens Design October 2008
Rudi Rottenfusser – Carl Zeiss MicroImaging

2. The Objective The Most Important Microscope Component
The objective is the microscope’s most important optical element. The final image quality is always limited by the quality of the objective. No other elements which follow will be able to improve a mediocre image from the objective!
To provide for an image which comes as close as possible to the way the object looks like, the objective doesn’t look even close to the single bi-convex lenses we have been looking at for the purpose of understanding optics. A fair amount of individual lenses may be needed to produce the best image possible.
The example on the next slide is a cross-section through a Plan-Apochromat >

4. Lenses

5. Glass Parameters (excerpt)
Refractive Index
Dispersion
Thermal Expansion Coeff.
Spectral Transmission
No Autofluorescence
No Schlieren, bubbles, inclusions
Reflectivity
Film Adhesion (AR coatings)
Chemical Resistance
Resistance to Humidity
Availability

6. Topics Airy Disk / Point Spread Function
Resolution Criteria – Rayleigh, Sparrow, etc.
Definition: Depth of field / focus
Aberrations
The Objective
What do the markings mean
What to consider when selecting an objective
Website - References

7. 1) No Lens Aberrations (“perfect lens”)
What happens to the image of the object when it travels through the various microscope components?
1) No Lens Aberrations (“perfect lens”)

8. On Axis image Wave fronts
Before we get to the aberrations of optical systems, we need to determine what the limits of a perfectly corrected optical system are. For this purpose, one starts out with a bright source point (A) which is in size below the resolution limit of the objective. If we consider the wave nature of light as it travels from this point to the objective and then towards the intermediate image plane, we can picture that all waves coming from this spot will be in phase at the drawn radial lines So and S1. As the rays continue towards the intermediate image, all of these in-phase rays come together at the center and form the image of the source point.
Wave fronts

9. In phase
½λ Out of phase
If we add actual waves to our ray diagram, we see in the left drawing how the waves from every point of S1 interfere constructively at the center of the intermediate image; they generate a bright spot.
If however, we observe an area in the intermediate image which is off axis, one can see that the distance traveled by the waves, will be different, and at some angle the waves will be out of phase or shifted by 180 degrees. Destructive interference now causes these waves to cancel each other out. As we increase the angle even more, we will get again into a condition for constructive interference, and further out again destructive interference.
Since our drawing is only showing this offset in one dimension, the dark spot at one point in space becomes a circle in 2-D. The bright area in the center, before we get to that dark ring, is called the “Airy” disk, named after Sir George Airy, an English astronomer (1801–92). Airy calculated the distribution of light from the center to the outer areas mathematically.
In other words, the Airy disk, in monochromatic light, consists of a central point of maximum intensity surrounded by alternate circles of light and darkness caused by the constructive and destructive interference of diffracted rays. The light areas are called maxima and the dark areas minima.
The “Airy” disk forms a basis for determining the resolving power of an ideal lens system. The diameter of the disk depends largely on the aperture of the lens. The diffraction of light causing the Airy disk is a factor limiting the resolution of a well corrected optical system.
If the optical system’s only limitation is the size of the Airy disk, we speak of “diffraction-limited optics”.

10. D Relative sizes of Airy disk (D) as a function of Numerical Aperture
NA:
The size of the Airy disk is directly related to the Numerical Aperture of the objective. In other words, the point source becomes the image of a disk surrounded by a series of concentric rings. The center maximum will be smaller as the objective aperture increases.

11. Airy Disk D
According to Airy’s calculations, the diameter of the Disk is given by the displayed formula 1.22 times lambda divided by the numerical aperture of the objective.
When we generate an intensity profile of the Airy disk as shown, we obtain a figure which is commonly referred to as “Point Spread Function”. In this case, we see, how we would like the PSF of an objective to look like!
D = Diameter of Airy disk in image plane

12. Resolution in z as defined by the “Airy Body” is
Airy Disk D
Rayleigh Limit of Lateral Resolution
d = ½ D
Resolution in z as defined by the “Airy Body” is

13. Airy Disks of 2 clearly imaged separate points:
Rayleigh Criterion for resolution
Intensity X
dmin
Minimum distance dmin is reached, when the principal maximum of object 1 (center of Airy Disk) coincides with first minimum of object 2
Intensity of maxima = 20% higher than intensity of “dip” between maxima
Up to now we had observed the transformation of one diffraction-limited spot by an objective into a point spread function
If we wish to determine the resolution of two adjacent diffraction limited spots, we are observing two overlapping point spread functions. At a stage where the maximum of one PSF overlaps with the first minimum of the other PSF, we have a condition which is called the Rayleigh Limit. Sir Lord Rayleigh ( ), an English professor of Physics, stated that the ultimate resolution limit of a system is reached when the center of one Airy Disk is just one radius away from the center of another Airy Disk.
Two points at minimum distance to be “resolved”
Rayleigh Limit of lateral resolution
d = ½ D (radius)

14. There are other criteria which are sometimes used for comparison of systems, for instance the the Sparrow criterion and “half width at half maximum”.
The Raleigh limit is the most widely used one.
Now that we understand how an ideal point spread function will look like, we can explore
Common aberrations, which are part of all optical systems
How these aberrations affect the ideal PSF and, therefore, the performance of an optical system

15. Objectives - Definitions: Depth of Field / Focus
Different formulas (e.g. Berek 1927, Shillaber 1944, Françon 1961, Martin 1966, Michel 1981, Piller 1977)
T = Depth of field (µm)
λ = Wavelength (µm)
n = Refractive Index
M = Magnification (Image Ratio)
e = diffraction-limited resolution
d in image plane (µm)
From Shinya Inoué / Kenneth R. Spring book:
“Video Microscopy Fundamentals - 2nd edition”
Chapter 2.4.6
Example: C-Apochromat 40x/1,2W
1 Rayleigh unit = 0,42 µm in object plane
= 0,668 mm in image plane
Depth of field / depth of focus is not an exact science. It is a bit arbitrary at which point one decides that the image which is in perfect focus at one point is still to be considered in focus as one changes the focus of the microscope. I tried to come up with a universal definition but realized that even Dr. Inoue acknowledged different formulas for this by Berek, Francon, Martin, Piller and Michel. The formula I am listing here is from Dr. Inoue’s book chapter
Important to know is that at high NA’s the depth of field it is primarily determined by wave optics, while at lower NA’s it is determined by the so-called “circle of confusion”.
We refer to depth of field to the phenomena on the side of the specimen and depth of focus to this phenomena on the image side. Practically speaking, at high NA the depth of field is very small and the depth of focus very large. This reverses at low magnifications.
In general: At high NA the depth of field is small and the depth of focus at the image side is large. This reverses at low magnifications!

16. 2) Considering Aberrations
What happens to the image of the object when it travels through the various microscope components?
2) Considering Aberrations

17. Aberrations Spherical Aberration Chromatic Aberration (axial)
Chromatic Aberration (lateral or radial)
Curvature of Field
Astigmatism
Distortion
Internal Reflexes
So now that we know how important NA is, and we have that special high NA objective in our hands – what can go wrong with the image quality of an objective:
Aberrations!

18. Plan-Apochromat 40x/0.95 corr.
Spherical Aberration
Plan-Apochromat 40x/0.95 corr.
Correction Collar set at 0.21mm

19. Plan-Apochromat 40x/0.95 corr.
Spherical Aberration
Plan-Apochromat 40x/0.95 corr.
Correction Collar set at 0.17mm

20. Spherical Aberration
Infinite number of prisms with different angles and, therefore, different refractive powers

21. Spherical Aberration
Due to the spherical character of the lens, rays do not cross over at the same Focal Point

22. Spherical Aberration is reduced by smaller aperture
Less confused “Zone of Confusion”

23. Fixing Spherical Aberration
Combination between a positive and negative lens (doublet)

24. Reducing Spherical Aberration
Multiple Elements
Another approach requires multiple lens elements, or to utilize an “aspheric” lens element which increases the throughput by eliminating glass/air interfaces. Many spherical lenses can be processed in one batch, but only 1 aspheric lens can be ground and polished at a time because it requires a computer-controlled process which changes the angle of the polishing wheel as a function of the radius. For large numbers of aspheres unique methods have been developed such as applying pressure to still soft spherical lens elements, but the tooling is very expensive and only cost-efficient for volumes of hundred thousands or more – this is not the case for microscope optics!
Aspheric Lens
Exaggerated

26. How to generate Spherical Aberration:
Incorrect Cover Glass
Maximum Intensity in an image of a point object

27. How to generate Spherical Aberration:
Incorrect Cover Glass
(Full Width, Half Max)
Resolution [µm]

28. Use 0.170 mm thick cover slips !
Choose the right cover glass!
Types and Thickness Ranges
No mm
No mm
No mm
No mm
No mm
No mm
No mm
No mm
Most objectives for biological applications are corrected for cover slips which are 0.17mm thick. This relates to a cover glass type 1.5. Some specialty lenses are made for different thicknesses. It is very important to match these requirements, otherwise color fringes will occur, reducing the image quality.
The exact match of cover slip thickness to a specific objective becomes more critical as the numerical aperture of the objective increases. At very high NA values, such as a 0.95 dry objective, it even becomes necessary to move internal elements of the objective via a correction with collar, to compensate for the thickness tolerances of cover slips.
Use mm thick cover slips !

30. How to generate Spherical Aberration:
Focusing deeper into the sample
40x/1.3 Oil immersion objective – Energy at different depths of penetration z in water

31. Benefit of Water Immersion Objectives (with cover slip correction)
Cover Slip n=1.52
Aquaeus Medium
n~1.3

32. Chromatic Aberration (Axial)
Remember “Dispersion” of Light!

33. Fixing Chromatic Aberration
The classic “Achromat” (Doublet)

34. Corrected Wavelength (nm):
Objectives -
Definitions: l
Corrected Wavelength (nm):
UV VIS IR
Plan Neofluar (435)
Plan Apochromat
C-Apochromat
IR C-Apochromat

35. Objectives -
Best Focus
1 RU
RU = Rayleigh Unit
480 nm
546 nm
644 nm

36. Lateral Chromatic Aberration (LCA) (Chromatic Magnification Difference)
Image
The main advantage of the Zeiss approach lies in “simplicity”. If the objective does not have to correct for another aberration, additional glass may not be needed or other corrections can be made in the available space. This is particularly important in all fluorescence and reflected light applications where the rays go through the objective twice, first for illumination, and then for imaging.
This animation shows the effect of lateral chromatic aberration. Images are projected into the same plane but different colors appear at different magnifications, hence the term “chromatic magnification difference”.

37. Lateral Chromatic Aberration (LCA)

38. Lateral Chromatic Aberration (LCA)
Different manufacturers correct for LCA in different ways:
Leica:
The tube lens corrects for a fixed amount of LCA
Nikon:
The objectives themselves are fully corrected
Olympus:
Zeiss:
The tube lens corrects for objectives with different LCA’s
While all four microscope manufacturers have adopted the “Infinity” optics concept, different optical and mechanical properties in the design of the microscopes exist. Obviously, best results are to be expected when one matches the microscope brand with the same brand objective. But if this is not done, it may be useful to know what to expect...
The major optical differences boil down to
Different ways to correct for LCA (Lateral Chromatic Aberration). In a mismatch, color fringes may appear towards the periphery of the observed field
Different focal lengths of the tube lens. This will affect the magnification and brightness of the microscope
The patented Zeiss ICS (Infinity Color Corrected System) Principle is unique because it doesn’t require objectives to be corrected for LCA which eliminates additional elements (throughput!!) in the objectives. The placement of the Zeiss tube lens at a certain distance from the nosepiece (~100mm) will automatically correct for the different amounts of LCA which are present in objectives of different magnification.

39. Astigmatism Tangential Sagittal
Cylindrical error = Difference in position of the focus between horizontal and vertical bundles of light rays when the object is off axis. One of the oblique bundles is focused at a point while the other bundle forms a line in the same plane and vice versa.
Effect: blurred image, double lines
Solution: anastigmatic lens system

40. Intensity Distribution in Airy Disk
Spherical Aberration
Astigmatism
These three images of point spread functions show quite well how different aberrations affect the point spread function of a system. If you observe a diffraction limited spot in the microscope, you move it around the field of view, and you see its shape change as shown in the graphs, you will be able to figure out the type of aberration it is experiencing.
Coma

41. Curvature of field: Flat object does not project a flat image
(Problem: Camera Sensors are flat) f1 f2
image
Another aberration is the lack of flatness of field in the image (curvatore of field). Peripheral rays will be imaged in sharp focus but not in a flat plane. In optics which is not “flat-field”, a curved field of view results. This becomes especially apparent when specimens are very thin and flat. For microscopy in thick tissues, this may not even be noticeable, and the benefit of a “simpler” lens may be more important than having a flat field.
object

42. Objectives - Flatness at 435nm: SF 18 SF 25 Plan Neofluar 1 R
Definitions: Flatness
Flatness at 435nm:
SF 18 SF 25
Plan Neofluar R
Plan Apochromat < 0,5 R
C-Apochromat 0,6 - 1 R R
Objectives such as Achromats or Fluars have a pronounced curvature of field, which can be minimized by choosing eyepieces with a lower field of view number. Plan-Apochromats are corrected to be flat all the way to the periphery of a 25mm field-of-view eyepiece, within better than one Rayleigh unit. This is one of the reasons why they are very popular with pathologists and especially in the field of hematology.

43. Distortion Pincushion Barrel
Another aberration is “distortion”. It shows up as converting a perfectly square sample into a square, which is “distorted” to look like a “pincushion” or a “barrel”. In biomedical applications while studying tissues small amounts of distortion are no problem. This becomes more an issue if straight lines are imaged such as in semiconductors. These lines will be straight in the center, and towards the periphery of the field of view they become more and more curved.

44. Internal Straylight

45. Reflexes (unwanted reflections)4
uncoated
Anti Reflection (AR) Coating 2
Single layer
~ 1%
Internal reflexes are not aberrations per se, but they may be very detrimental to the contrast of an image.
Most lenses, therefore, will be coated with a very thin film to eliminate reflexes from the surface of the glass. Uncoated glass transmits about 96% of light which means it reflects 4%. By using a single layer coating, the reflex can be reduced to about 1%. Multi-layer coatings may reduce reflexes as to as little as 0.1%.
Anti-reflex coatings were invented by Zeiss and first applied in Current optics are coated wherever the likelihood of reflexes exists. This can make a very big difference in systems which contain multiple air glass interfaces, such as highly corrected objectives.
Multi layer
~ 0.1%
400
700
l [nm]

46. Internal Straylight

47. Anti-Reflex (AR) Coating
How does it work?
l/4
How does an Anti Reflex (AR) coating work?
If the goal is to eliminate the reflex in the part of the spectrum which the eyes are most sensitive in, green, about 540nm, an appropriate coating material is chosen will be evaporated on the glass to a thickness of lambda /4 for green light, which is 135nm.
Now, the reflex coming from the interface of the glass and the coating, will be out of phase with the direct reflex from the coating, because it went twice through the coating, a distance of half lambda.
We will get total extinction of the green light if the amplitudes of the two reflexes are the same. This means the material for the coating has to be carefully chosen. Single reflex coatings can be recognized easily by tilting a lens and observing the reflection of light on its surface. If it is “purplish”, it means that green light has been eliminated, leaving the 2 extremes of the visible spectrum 400nm, 700nm, a mixture of deep purple and deep red, unaffected.
Modern multi-coatings may consist of dozens of individual layers, each one affecting another wavelength range.

48. Questions?

49. Mounting Distance (Specimen to Flange):
Objective Markings
Thread Diameter
0.8”x1/36” (RMS) 27mm, 25mm
Mounting Distance (Specimen to Flange):
22, 45, 60, 75mm, others?

50. “Standard” Sequence Magnification Why these strange numbers?
1.251
1.252
1.253
1.254
1.255
1.256
1.257
1.258
1.259
1.2510
1.2511 =
1.25x
1.6x 2x
2.5x
3.2x 4x 5x
6.3x 8x
10x
12.5x…
Why these
strange numbers?
There is a reason behind the sequence for magnifications. It is actually an industrial standard which applies to many devices which have to be available in different sizes. Electrical fuses, motors, come to mind right away.
The basic number in this sequence called “R10” is the tenth root from the number 10. This number is, when it is rounded off, 1.25.
As you multiply 1.25 ten times by itself, you arrive at “10”. When you continue multiplying, you will get to the same numbers, only multiplied by 10 now, and so on. Every step represents the same relative increase which is 25% from number to number. What’s convenient is that when you multiply or divide any number into any other, you will still arrive at a number within this sequence…
So that’s why you can buy a 1.6A fuse or a motor with a power of 25kW, and an objective 63x. Zeiss uses this sequence throughout its products, because it makes sense.

51. What to consider when selecting an objective:
Magnification
Working Distance
Numerical Aperture (NA) – Resolution / Depth of Field
Image Quality – minimized Aberrations (spherical, chromatic, flatness of field, astigmatism, coma, distortion)
Adaptation to specific Applications (Contrasting Techniques, Cover Slips, Chambers, Shape of front lens for Access)
Spectral Transmission (Visible,IR,UV?)
No Autofluorescence
No Strain (Pol)
Temperature Tolerance
Temperature Isolation (heating!)
Chemical Resistance
Electrical Shielding
Minimal Path Gradient (2-photon)
Perfect Parfocality and Parcentricity
Compact yet durable
Inexpensive
What to consider when selecting an objective:
It should be clear that it is not possible for any one lens to perform all of the above requirements.
Any of these topics may be paramount to success in an imaging situation, and so this listing may serve as a basis for discussing a customer’s needs. The following website is very useful for these discussions. >

52. For a comprehensive lookup of objectives, consult Websites!
(Example for Zeiss:

53. Description of Classes of Objectives

54. Example (Screenprint)
Please refer to appropriate web sites from
Leica
Nikon
Olympus

55. Thank you, and do enjoy your microscopes !
Rudi Rottenfusser
Office: 508/
Cell: 508/
Microscopy Support: 800/
Imaging Support: 800/
Website:
Educational Site: