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University of Illinois at Urbana-Champaign Department of Physics
Physics 402 Laboratory
Experiment A-
Optical Modeling
We are indebted to Breault Research for providing the commercial ASAP software for our teaching labs.
References
- M. V. Klein, T. E. Furtak, Optics , 2nd ed., pp. 129-151, John Wiley, New York (1986).
- E. Hecht, Optics , 2nd ed., Ch. 6.2, Addison-Wesley, Reading, Massachusetts (1987).
- F. A. Jenkins, H. E. White, Fundamentals of Optics , 4th ed., Ch. 8, McGraw-Hill, New York (1976).
- R. Kingslake, Lens Design Fundamentals , Academic Press, New York (1978).
- Levi, Applied Optics , John Wiley, New York (1968).
- W. J. Smith, Modern Optical Engineering , 2nd ed., McGraw-Hill, New York (1990).
- W. J. Smith, Modern Lens Design , McGraw-Hill, New York (1992).
- M. Laiken, Lens Design , Marcel Decker Inc., New York (1991).
- R.E. Fisher & B.T. Tadic-Galeb, McGraw-Hill, New York (2000)
This is an open-ended experiment to learn optical modeling and simulation using the commercial optical modeling ASAP software. You will be given a handout, Introduction to ASAP. It is a step by step procedure to use the software. One source of additional help is in Primer on ASAP which includes fairly extensive documentation about the software. Our goal here is to familiarize with ASAP and use it to create some simple lens systems. After finishing the Introduction to ASAP , with the consent of the instructor, you may proceed to start modeling the telescope and lenses that you had already experimented in A1 and A2. Additional experiments can be attempted after consulting your instructor.
Your report should include the results from the Introduction to ASAP as well as the details of your modeling system.
Introduction To ASAP
Acknowledgement
We are grateful to Breault Research for providing ASAP for this course. ASAP may only be used for educational purposes. You must agree to the terms of the license, which is presented upon startup of ASAP.
Goals:
- Understand what minimal set of conditions must be specified in order to create a computationally consistent optical model.
- Learn how to create and analyze simple optical models in ASAP so you can use ASAP to complete the lab reports for the first two labs.
- Understand the first-order behavior of simple lenses and how the action of lenses can be approximated by their principal planes.
- Understand how simple lenses with identical first-order properties may not act identically.
References:
ASAP Primer : This is a comprehensive, step-by-step introduction to modeling geometric optics in ASAP, starting by assuming no prior experience with ASAP and ending with instructions on how to produce complicated models. The primer does not discuss optics theory, only the mechanics of building and analyzing models in ASAP. Also, it does not cover all of the capabilities of ASAP, and has no information about how to model physical optical effects.
ASAP help : The help menu in the program will open the online documentation while you work. Pressing F1 while highlighting a command will open its specific documentation.
Hecht : No reading from this book is not required, but the topics of first-order optics (aka paraxial optics or Gaussian optics), principal planes, and bending factors are discussed in chapters 5 and 6.
An unresolved bug causes ASAP to crash without warning about every 50 minutes
!^ like clockwork.^ So, constantly save your work at each step!
Important information is placed in a single box for emphasis.
Questions or work to be turned in for grading are in double boxes.
package, but to give you a general understanding – independent of any specific program
- of how optical systems are translated into numerical models.
Ideal Behavior of Lenses:
There are many ways to think about how a simple lens works. Perhaps the easiest definition of imaging, originally from Maxwell, is that a perfect lens would transform light originating from a point on an object to a unique point in an image. Maxwell’s concept of a lens is abstract, without making any assumptions about how the light is actually transformed in real lenses. It’s just a mathematical criterion for consistent imaging, without any connection to the physical laws of light.
Maxwell’s Criteria For Perfect Imaging: A one-to-one map
In a perfect or ideal lens, there is a one-to-one mapping between the image and object points so that rays leaving a particular image point will only converge to a single unique object point. As we’ll see later in the course, for a variety of reasons real lenses never fully achieve this ideal behavior.
Since the mapping between the image and object points is ideally one-to-one, the two points are called conjugate points. Points on the object side of the lens are considered to be in object space. Points on the image side of the lens are in image space. In the ideal case, each point in object space has one and only one conjugate image point.
First, we’ll consider how Maxwell’s abstract criterion for perfect imaging translates into specific geometric requirements for the direction the light rays should take to the image. Next, we’ll consider the direction light rays actually travel when light refracts through a lens according to Snell’s law. What we’ll find is that the Maxwell’s abstract requirement for ideal imaging is not met by the physical mechanism of refraction. This contradiction between the purely geometrical requirements for ideal imaging and the actual behavior of light in a lens is a central point in geometric optics. In lab A2, we’ll quantitatively calculate and measure how real lenses deviate from the ideal.
Ray Directions For Perfect Imaging:
Since light travels in a straight line through media, the abstract condition for perfect imaging also creates a geometric condition for how the rays must converge on the object for ideal imaging. This condition is only a consequence of requiring a one-to-one
mapping between an object and image point and has nothing to do with what physically occurs in a real lens. In an ideal image, for every ray height h, the slopes of the incoming and outgoing rays must be proportional:
Why? Because in the ideal image, all of the incoming rays originate from the same object point and the outgoing rays all converge to the same image point and while at the lens the outgoing ray must originate from the same point on the lens in which the incoming ray arrives. More formally we can say that the slopes of the outgoing rays must be proportional to the incoming rays. If the rays all leave from the object point which is at a distance s and for perfect imaging the rays must all converge to the image point at s’. So on each side of the lens, the distance along the axis is the same for all rays, regardless of the ray height. So, a given set of constant s and s’ are proportional, s = k s’. (Of course, the specific locations of s and s’ are related by the power of the lens in slightly more complicated way, but that’s getting ahead of ourselves.) So,
Image Ray Object Ray '
h h u k k u s s
Well, if the slopes must be proportional for perfect imaging, then the ray angles must also be proportional:
u Image Ray ∝ u Object Ray ⇒ θ '∝θ
Why do we care? Because ray angles are used in Snell’s law, so we can compare the directions rays must travel to meet Maxwell’s requirement for ideal imaging with the directions rays actually travel when they are refracted in a lens.
Ray Directions In Real Lenses:
Of course the physical mechanism by which light is bent in a lens is by refraction. The law of refraction states that the sines of the angles are proportional, not that the angles themselves are proportional:
n 'sin( θ') = n sin( θ )
First-order optics is used to easily calculate the general properties of optical systems, using only the location of the focal points and principal planes of the lenses in the system. Typically, the first-order calculation extended outside the paraxial region and used to describe the entire lens. If a lens is used mainly in the paraxial region, the results are reasonably accurate. Even for more complicated optical systems, first-order calculations can roughly approximate the behavior of the system. Although the first-order calculation lacks accuracy, its advantage is that the results can be calculated analytically by hand. When higher accuracy is needed, numeric methods are required (See lab A8). In lab A1, you’ll use first-order optics to analyze a multi-lens telescope.
A key simplification of first-order optics is that the real refracting surfaces of a lens can be replaced by two imaginary planes where the rays are bent according to first-order theory ( n ' θ '≈ n θ) instead of Snell’s law ( n 'sin( θ') = n sin( θ )). This assumption is even
made outside the paraxial region, where the angle is really too large to approximate sin( θ ).
These fictitious planes are known as principal planes. Notice that, unlike the real rays, the first-order rays are not refracted at the air-glass interfaces of the lens. Instead, the paraxial rays are bent only at the principal planes. From the first principal plane, fictitious rays travel parallel to the optical axis until they intersect the second principal plane. Also notice that in the paraxial region close to the axis, the real rays travel the same path as the first-order rays (at least outside the lens). However, for the rays further
from the optical axis, where the angle of incidence increases such that θ does not
approximate sin( θ ) , the first-order rays focus on the image point while the real rays do
not. So, first-order theory loses some of the details of how the image is formed.
The approximate behavior of a real lens can be entirely captured by the locations of its principal planes and focal points. In lab A1, you’ll see how principal planes make it easy to predict how multiple lenses act together.
Location of the principal planes:
How are the locations of these planes determined? While the locations can be calculated from paraxial theory, the simplest method is to determine the positions graphically. This is the approach we will use in this lab, since paraxial theory has not yet been derived in class.
To determine the location of the principal planes graphically, it’s easiest to consider the case when one of the conjugate points is at infinity.
A conjugate point is considered finite when its position is not infinitely distant from the lens. This is the case of rays either diverging from a point or converging to a point. As the point on the optical axis moves farther from the lens, the slopes of the rays decrease. In the limit where the distance to an on axis point becomes infinite, the slope goes to zero for all rays. When a conjugate point is located infinitely far from the lens, the rays travel in parallel to or from the point (for an off-axis point at infinity, all ray slopes become equal but non-zero). In practice, the conjugate point does not truly need to be an infinite distance away. For many applications, a point located at a distance greater than 20 times the focal length of the lens is far enough to be considered at infinity. Most laser sources are also considered to be located at infinity (regardless of the actual position of the laser), since laser beams usually don’t diverge much and can be approximated by a set of parallel rays. A beam with low divergence is also known as a collimated beam.
For an ideal thin lens, the location of an image is given by the Gaussian Lens Formula:
1 1 1 f s o si
Where f is the focal length of the lens and so is the distance from the object to the first principal plane of the lens and si is the distance from the image to the second principal plane of the lens. You’ve probably seen this before in earlier classes. We won’t derive it here – you’ll do that later in the lectures. It’s important to realize that the focal length of a lens is just a parameter to quantify how much a lens bends light rays. It’s an intrinsic property of the lens which depends on the specific physical mechanism for bending the light. For refractive lenses, the focal length is determined by the Lens Maker’s Equation
- we’ll look at that in the second part of this lab. The main point is that, to first order, the Gaussian Lens Formula will determine the position of an image from the location of the conjugate object point and the focal length of the lens.
There are two important limiting cases: an object at infinity and an object located a focal distance away from the lens. If the object is at infinity, then the image is located exactly 1 focal length from lens. Conversely, if the object is located exactly 1 focal length from
To find the front principal plane, we can use a similar construction with the object at the focal point of the lens and the image at infinity:
Alternatively, we can continue to use an infinite source and just flip the lens or the source to the reverse the ray directions. In this lab, we’ll flip the source from the left side to the right side. This has the advantage of automatically locating the front focal point. Otherwise, we would have to adjust the location of the point source onto the front focal point, the position of which we hadn’t yet determined exactly.
Using the two principal planes and the Gaussian lens equation, the location of the image from an object at any location can be determined:
Part I: Determination of Principal Planes
To perform the analysis in labs A1 and A2, you’ll need to determine the principal planes of lenses you use in the lab. So, for the first part of lab A0, you’ll practice finding the location of the principal planes for four different shapes of lenses using the graphical technique.
You’ll determine the locations of the principal planes for the four different shapes of lenses which are most commonly used lenses in the lab:
For each of the lens shapes, you’ll use ASAP to repeat the graphical construction shown previously in the introduction for the thin biconvex lens.
#1-4. To locate the principal planes, print an ASAP raytrace for each lens shape (described below). You’ll need two different raytraces for each shape: one for the rear principal plane and another for the front plane. Then, on the printout, use a ruler to draw the undeviated rays by hand. Also use the ruler to draw the locations of the front and rear principal planes and the front and rear focal points, like the diagrams in the introduction. Be sure to clearly label the principal planes and focal points. Turn in each of the diagrams for grading (grades are based on whether you’ve located the principal planes and focal points properly).
Before proceeding further, read the handout “ASAP Lab Quickstart.” That handout will tell you how to start, configure, and perform basic operations in ASAP. After you’re done reading the quickstart, set your ASAP working directory and continue with the steps below.
- Now, we must set the units for the model geometry to millimeters. These units are used to specify the size and shape of the elements in the model (lenses, etc…), but not the units of the wavelengths of light to be simulated. To make things more readable, leave a blank row between the Reset command and the Units command.
Set the units to millimeters by double-clicking on the second column in the Units command row.
Type “Watts” in the “Flux Label column.
- Let’s set the wavelength. The default value in the spreadsheet is for 550 nanometers (green), which is OK for our simulation.
Now that we have set up some of the parameters of the model, we can create the geometry of the lens.
- Skip a row and then choose System>>Geometry>>Lenses>>Singlet>>RD. This will create a singlet lens defined by the radius of curvatures of its front and back surfaces (hence the choice, RD for RaDius).
- Check the geometry of your model by clicking the preview all button.
You’ll now get a 3D view of your model:
You can use the mouse in the 3D Viewer to control the display: Left button: Displays name of component under the cursor Right button: Rotates model with mouse movement Control-Right button: Zoom model with mouse movement Shift-Right button: Translate model with mouse movement
You can open the tree on the left. If you right click the BICONVEX_LENS entry, you’ll see all sorts of options for display:
Also, there are useful display control buttons on the toolbar directly above the 3D view:
Be sure the preview of your model looks like the images shown above. If it doesn’t, then check your spreadsheet carefully for errors before continuing.
9) Save your model by clicking the disk icon. From now on be sure to save after every few steps.
In addition to the lens, we’ll need a detector to collect the rays. We’ll create a plane surface after the focus to absorb the photons.
- Choose S ystem>>Geometry>>Surfaces>>Plane>>Axis to create an absorbing plane.
Name the plane “DETECTOR” Leave the plane normal along the Z-axis. Set the location at 70 mm. Leave the shape “ellipse.” Set the semiwidth so the plane is 1
is ok for plotting. For quantitative analysis, the numbers rays must be increased for better statistics. However, for graphically determining the principal planes, we only need a few rays that we can look at in the plot. In fact, we don’t a 3D array, only a 2D set of 5 rays lying in the Y-Z plane. So, set the major axis rays to 1 and the minor axis rays to 5.
- Our source is not yet complete. In the previous step, we specified the location where the photons start, but we must also specify the direction the rays will travel since they are vectors. In specifying the direction of the rays, we also specify their divergence. The divergence of the rays defines the location of the conjugate point of the source. In principle, the calculation of the rays doesn’t have to start at the same location as the conjugate point of the source. As long as the rays are only traveling in free space and have yet to cross an element, we can start the calculation anywhere along the trajectory of the rays.
For locating the rear principal plane, we want a laser like beam of parallel rays with no divergence. This condition is specified by the Rays>>Grids>>Source>>Direction command.
Now we have completed the definition of the model, so we can traces the rays and make the plot needed to locate the principal planes.
- First, we must load the commands into the databases by clicking the run button. This must be done every time before tracing rays. After a trace is run, the results may be analyzed, but to run another simulation the database must be reset by re-running the spreadsheet.
- Now, we’ll trace the rays. Go to the top menu and choose >>Trace>>Trace rays.
A dialog box will appear. We’ll have to change the window values so both plots for the front and rear principal planes will have the same scale:
Uncheck the Auto Scale box in the WINDOW area. Change the horizontal limits to -55 mm and 55 mm. This will give us a large enough region to cover both the front and rear focal points.
Click OK and ASAP will trace the rays. You should see a plot like this with the rays coming from the left and focusing on the right:
If your plot doesn’t look like this, review your spreadsheet carefully so it matches the instructions and keep trying.
