Online reading assignment: optical instruments

Physics 205B, spring semester 2020
Cuesta College, San Luis Obispo, CA

Students have a bi-weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing presentations on optical instruments and interference.


Selected/edited responses are given below.

Describe what you understand from the assigned textbook reading or presentation preview. Your description (2-3 sentences) should specifically demonstrate your level of understanding.

"Magnifiers make objects appear bigger than they are by increasing the angular size of an object. An object that is farther away appears smaller because the angular size is smaller."

"From this section I learned that telescopes and microscopes have a different set of lenses in order to either magnify an object that is small or an object that is far away. The different lenses in both of the instruments allow each to provide different functions."

"Ray tracings for both microscopes and telescopes create real intermediate images because the rays actually converge. I am beginning to understand that placing the object just outside the focal point of the objective will result in an enlarged image."

"Microscopes use two converging lenses to zoom in on objects that are very small. The first lens, or the objective, creates a real, enlarged image in front of the second lens. This image is then used as the object for the eyepiece, which translates to an upright, virtual view of the original object."

"During microscope magnification, the focal length of the objective lens is decreased and the focal length of the eyepiece is also decreased. Conversely, in telescope magnification, the focal length of the objective lens is increased while the focal length of the eyepiece lens in decreased."

"I understand how a microscope works before reading the text since I had to understand it for biology but now I did learn about the equation for it. I didn't know that it even had an equation before so it was interesting to learn."

"The objective makes an image just inside the eyepiece focal length. The eyepiece projects this first image to an infinite distance so a relaxed eye can see the final image."

"In a compound microscope or telescope, the angular magnification is the angular size of the final image divided by the reference angular size. A difference between them is that the image in a telescope is first diminished before being magnified while a compound microscope enlarges twice."

"I now understand how a compound microscope is designed to work, something I have never thought about before in all of my years of using them in school. They work by having an additional converging lens (objective) 'pre-magnify' the image before it passes through the magnifying glass (eyepiece), ultimately increasing the angular magnification."

"I understand the microscope parts because of taking biology. I understand that the 'tube length' for microscopes is defined as the distance measured between the objective and eyepiece focal points. the 'barrel length' for telescopes is defined as the distance measured between the objective to the eyepiece lenses, which is the same as the sum of their focal points."

"Microscopes and telescopes have some similarities such as the placement of an objective at the front leading to an eyepiece in the back. Moreover, both instruments create the same ray tracing. The main difference between these two instruments is the location of the physical object. For example, the object for a microscope is located just outside of the focal point of the objective while the telescope showcases an object extremely far away."

"There is a negative sign in both the microscope and telescope magnification equations. The negative sign for both denotes that the image produced will be inverted."

Describe what you found confusing from the assigned textbook reading or presentation preview. Your description (2-3 sentences) should specifically identify the concept(s) that you do not understand.

"I don't quite know why but memorizing the ray tracings is still a little difficult."

"I would like a little more information on these ray tracings and how to memorize the differences from the ones we've previously done."

"I do not understand how to draw a telescope ray tracing. I also do not understand why when talking about magnification, that only focal distance is taken into consideration."

"I found confusing the formulas used to calculate the magnifications. Each formula was different depending on the optical instrument."

"I do not fully understand the ray tracings behind the microscope and telescope concepts. I am also unsure how to utilize the equations."

"I would like to go over the angular magnification equations in class. I am a little bit confused about why the magnification factors are negative, primarily."

"I'm not really confused on anything because there wasn't much in the reading, which is what usually confuses me but telescope and microscope are similar. The equations are also distinct enough not to get them confused."

"I believe I understood this topic."

Identify the type for each of these lenses. (Only correct responses shown.)

Microscope objective: converging [94%]
Microscope eyepiece: converging [71%]
Telescope objective: converging [79%]
Telescope eyepiece: converging [59%]

Identify the ray tracing for each of these lenses. (Only correct responses shown.)

Microscope objective: ray tracing 2 [74%]
Microscope eyepiece: ray tracing 3 (or 4) [44%]
Telescope objective: ray tracing 3 (reversed) [0%]
Telescope eyepiece: ray tracing 3 (or 4) [44%]

A (compound) microscope should have a __________ focal length objective lens and a ___________ focal length eyepiece lens in order to maximize its angular magnification.

short; short.	*************** [15]
short; long.	************** [14]
long; short.	* [1]
long; long.	[0]
(Unsure/lost/guessing/help!)	**** [4]

A telescope should have a __________ focal length objective lens and a ___________ focal length eyepiece lens in order to maximize its angular magnification.

short; short.	[0]
short; long.	** [2]
long; short.	********************** [22]
long; long.	****** [6]
(Unsure/lost/guessing/help!)	**** [4]

Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.

"A little extra help with these two-part ray tracings would be awesome!"

"Please explain the initial and final images produced by a microscope."

"May we review how to compare real-life examples to ray tracings?"

"Can we review how to actually use the equations for the magnification of the microscope and telescope please?"

"Cool concepts, but a tad difficult."

"Why are we learning about microscopes and telescopes?" (We're just finishing up the optics section of this course with some cool stuff.)

"Why can't we use the same angular magnification formula for both the microscope and telescope?" (Even though both the microscope and telescope use an eyepiece the same way (to magnify the images made but the objective lenses), the objective lens for a microscope looks at an object very close to it, just outside its focal point, while the objective lens for a telescope looks at an object very far away, essentially out at infinity. So the differences between the microscope and telescope equations object distances primarily result from the different uses of the objective lenses.)

"When looking at the ray tracing, the ray tracing to the eyepiece looks like converging lens tracing #3. However, the rays never intersect on this ray tracing. How is this possible?" (Your relaxed eye is set to look at things infinitely far away. The rays from an object at infinity are parallel as they hit your eye. So an eyepiece will take an object at its focal point and its rays will be parallel (which would normally not produce an image), but your eye will take those parallel rays as something it thinks is infinitely far away, and easily focus those rays to interpret it as being from a distant location (but apparently with a very large angular size.)

Online reading assignment: corrective optics, magnifiers

Physics 205B, spring semester 2020
Cuesta College, San Luis Obispo, CA

Students have a bi-weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing presentations on corrective optics and magnifiers.


Selected/edited responses are given below.

Describe what you understand from the assigned textbook reading or presentation preview. Your description (2-3 sentences) should specifically demonstrate your level of understanding.

"I understood the presentation on myopia. I think it was super cool to see how we are entry-level optometrists after this."

"Refractive power is used by optometrists to measure the degree at which light is converged or diverged denoted by a (+) or (–) sign. It is also the inverse of the focal length."

"The combined total refractive power for two lenses is addition of each refractive power."

"Negative values for the focal length and refractive power mean that corrective lenses should be diverging, to correct for the person's myopia. Corrective lenses with a positive focal length and refractive power will be converging lenses, to correct for hyperopia."

"Magnifiers 'magnify' by increasing the angular size of the object thus making the object appear bigger to the eye. Increasing the distance between the object and the eye provides a smaller angular size and makes the object appear smaller."

"Angular size is the size of an object at a certain distance. The larger the object, the larger the angular size. Magnification isn't really magnifying but rather the lens allows the eye to be able to focus on the object at a distance where it other wise wouldn't have been able to."

Describe what you found confusing from the assigned textbook reading or presentation preview. Your description (2-3 sentences) should specifically identify the concept(s) that you do not understand.

"Corrective optics. It all seems overwhelming, but simplification will help. Especially refractive power and power addition equations."

"Still figuring out how to work with the thin-lens equations. The steps for correcting myopia and hyperopia were a little confusing."

"I am a little confused on the important difference(s) between linear and angular magnification. I think that I get the basic idea of both but want to make sure I clearly understand how they differ."

"Maybe a little more clarification on the mathematics on angular magnification."

"I did not understand how the eye can perceive the penny to be bigger than the moon. I also did not understand the idea of doing multiple thin lens equations for two lens."

"A review of the equations for angular size and angular magnification would be very helpful. Also I am not really sure about what it exactly means when 'the image is at infinity.'"

"The math seems a little confusing to me right now per usual. But it's late and I'm sleepy."

In general, a converging lens will produce virtual, upright images located __________ the original object.

closer than.	******* [7]
at the same distance as.	** [2]
farther than.	********************** [22]
(Unsure/lost/guessing/help!)	* [1]

In general, a diverging lens will produce virtual, upright images located __________ the original object.

closer than.	************************** [26]
at the same distance as.	[0]
farther than.	***** [5]
(Unsure/lost/guessing/help!)	* [1]

Identify the type of lens used for these optics. (Only correct responses shown.)

Glasses/contacts to correct for myopia: diverging [75%]
Glasses/contacts to correct for hyperopia: converging [78%]
Glasses/contacts to correct for presbyopia: converging [29%]
Magnifying lenses: converging [74%]

State the units of refractive power for lenses, and briefly describe the relationship between refractive power P and focal length f.

"Diopters. Refractive power is 1/f."

"The unit of refractive power for lenses is diopters, which is the inverse of the focal length."

"Units for P are meters^–1."

"I don't understand what this is asking. "

Explain the difference between the two types of magnification, m and M.

"Linear magnification is 'm' and angular magnification is 'M.'"

"m is the ratio of the object's distance or height to the images distance or height at convergence of the light rays h_i/h_o or –d_i/d_o. M is how much the apparent size of something is change by a magnifying lens. The apparent size is an illusion and relative to the viewer."

"m is the ratio of image height to object height while M is a distinct ratio of angular size of some image produced by an optical instrument divided by the reference angular size of an object viewed without an optical instrument."

"I'm not sure what the answer is."

"Again, I am pretty lost."

"Don't remember."

A nominal, relaxed eye is set to focus on objects located at:

+∞.	************ [12]
+25 cm (at your near point).	*********** [11]
+f (at the focal point of the lens).	**** [4]
(Unsure/lost/guessing/help!)	***** [5]

If an object is brought closer to your eye, its angular size will:

increase.	**************************** [28]
decrease.	* [1]
remain unchanged.	[0]
(Unsure/lost/guessing/help!)	*** [3]

When a converging lens is used as a simple magnifier, the object is placed at a distance d_o = __________ in front of (to the left of) the lens.

+∞.	**** [4]
+25 cm (at your near point).	************* [13]
+f (at the focal point of the lens).	********** [10]
(Unsure/lost/guessing/help!)	***** [5]

The ray tracing that best matches when a converging lens is used as a simple magnifier is:

#3.	* [1]
#4.	*************************** [27]
(Unsure/lost/guessing/help!)	**** [4]

Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.

"Can you show a drawing demonstrating the difference between microscope and telescope lens placement and rays? I was confused with presbyopia and how that age-onset disorder works with myopia. (1) In that case someone would be unable to see far--myopia--while simultaneously being unable to see close--presbyopia? Also (2) can someone experience hyperopia and presbyopia or would the condition default to the latter? Finally, while partially off-topic, (3) is there a physics-based description of blindness?" ((1) Yes, someone with myopia (can see near, can't see far) who develops presbyopia (can't see near) would then require bifocals to correct both problems. (2) If you can't see near as a child, you have congenital hyperopia. If you could see near when you were young, but gradually lost that ability as you got older, then you have presbyopia, which is basically age-onset hyperopia, and they are both corrected with a (stronger) converging lens prescription. (3) Not really, I would think that's more of a biology/anatomy/physiology thing.)

"Can we go over problems or see these problems done? When I see these equations being used it makes it easier to understand."

"I have some questions about what a relaxed eye is meant to focus on."

"Please help to understand the angular size and tie all the equations to the ray tracing, this is getting complex."

"I'm not too confident on my answer for how close a simple converging lens used as a magnifier should be."

"I would love more examples of how to guess what kind of ray tracing would apply to real-life situations like a magnifier."

"I think angular magnification is the hardest part of this section, especially when comparing them to ray tracings." (It really is. More on angular magnification next week.)

"Can the human eye control the amount of light that comes in?" (It make less light come in (by making the pupil smaller, or by squinting your eyelids), but it can't make more light coming in than is already there.)

"Do you meditate? Moreover, if you do meditate, do you call it 'total internal reflection?' (Sorry I'm a couple chapters late on this joke.)" (I think my meditation chill is more like something called 'frustrated total internal reflection.' It's when total internal reflection is supposed to happen, but introducing an external object with a new index of refraction that barely touches that surface ruins the total internal reflection, allowing a transmitted ray to go through.)

"How long have you taught physics?" (My first physics class as an instructor was at UC-Davis 26 years ago, when I was still a graduate student.)

Online reading assignment: optical instruments

Physics 205B, spring semester 2019
Cuesta College, San Luis Obispo, CA

Students have a bi-weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing presentations on optical instruments and interference.


Selected/edited responses are given below.

Describe what you understand from the assigned textbook reading or presentation preview. Your description (2-3 sentences) should specifically demonstrate your level of understanding.

"The objective lens is nearest to the object and the eyepiece (or magnifying glass) is nearest to the eye for both the microscope and telescope."

"For a two-lens model of a microscope and a telescope, the object uses light from the object to create a real image that will become the object for the eye piece. For a microscope, since the object is close to the focal point, the image produced is highly enlarged. On a telescope, however, the object is going to be very distant and the ray tracing becomes parallel before it is focused by the objective."

"The similarities between microscopes and telescopes: they both have tubes that hold two lenses apart from each other and have an objective lens in the front and an eyepiece in the back. The ray tracings are different, however."

"The first image/second object produced by the objective for both a microscope and telescope is near the focal point of the eyepiece. Because it is near the focal point of the eyepiece, the light transmitted through the eyepiece is nearly parallel. The final image produced is enlarged, virtual and near infinity."

"Both a microscope and telescope consists of an objective and an eyepiece. Angular magnification of a compound microscope: the M is greater when f_o and f_e are as short as possible and when the distance between lenses, L, is large. angular magnification for telescope: the objective should have a long focal length and the eyepiece a short focal length."

Describe what you found confusing from the assigned textbook reading or presentation preview. Your description (2-3 sentences) should specifically identify the concept(s) that you do not understand.

"I reviewed the lecture, but I still do not feel like I understand the difference between a microscope and telescope. I know the first object is a different distance from the objective and the image it creates is around the focal point of the eyepiece. But what are the lenses (or focal points?) doing differently?"

"I am not sure what type of focal lengths you would want for an objective lens for a microscope or telescope. I am not sure type of focal lengths I would want for the microscope and the telescope."

"The differences between the telescope and microscope magnification equations. Possibly it is due to the objects being placed at different focal lengths but I am not sure. To me it seems like both instruments essentially do the same thing so their magnification equations shouldn't be different."

"The ray tracings for the telescope and microscope. Also how the ray tracings for both are so similar."

"Why the formulas for the magnification equations of microscopes and telescopes differ."

"I'm a little confused because now we have two objects and two lenses involved in our ray tracings."

"When I read the readings I will know what I know, but I will also know what I do not know and that will go here."

"I really struggle with these sections and am not 100% sure what I'm confused on. Talking it out with examples in class always helps, but working on this section at home is a struggle for me."

"Nothing really too confusing."

Identify the type for each of these lenses. (Only correct responses shown.)

Microscope objective: converging [85%]
Microscope eyepiece: converging [64%]
Telescope objective: converging [85%]
Telescope eyepiece: converging [67%]

Identify the ray tracing for each of these lenses. (Only correct responses shown.)

Microscope objective: ray tracing 2 [46%]
Microscope eyepiece: ray tracing 3 (or 4) [41%]
Telescope objective: ray tracing 3 (reversed) [13%]
Telescope eyepiece: ray tracing 3 (or 4) [38%]

A (compound) microscope should have a __________ focal length objective lens and a ___________ focal length eyepiece lens in order to maximize its angular magnification.

short; short.	********************** [22]
short; long.	****** [6]
long; short.	*** [3]
long; long.	[0]
(Unsure/lost/guessing/help!)	******** [8]

A telescope should have a __________ focal length objective lens and a ___________ focal length eyepiece lens in order to maximize its angular magnification.

short; short.	* [1]
short; long.	*** [3]
long; short.	********************** [22]
long; long.	****** [6]
(Unsure/lost/guessing/help!)	******* [7]

Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.

"Could you review a little more about the angular magnification in class? I am still struggling with that a little bit."

"No questions today but I am still a little confused on the magnification equations for both instruments."

"What does it mean when an image 'goes out into infinity?'"

Online reading assignment: corrective optics, magnifiers

Physics 205B, spring semester 2019
Cuesta College, San Luis Obispo, CA

Students have a bi-weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing presentations on corrective optics and magnifiers.


Selected/edited responses are given below.

Describe what you understand from the assigned textbook reading or presentation preview. Your description (2-3 sentences) should specifically demonstrate your level of understanding.

"Ray tracings are accurate models of how light passes through a lens to create an altered image of the object. The way that the three rays interact with the lens and the focal points determine if the image of the object is: upright or inverted and magnified or diminished."

"The refractive power of contact lens is expressed as the inverse of the focal length. To correct shortsightedness, a diverging lens is prescribed, and for hyperopia, a converging lens is used. The angular magnification measures the angular enlargement or reduction of an object through a lens."

"So basically the two-lens method consists of just getting an image from one lens and then using that image as the next object and thus get another image through the second lens. Which happens with any corrections to your eyesight such as contacts or glasses. I just learned why prescriptions have either the ± sign in front, I can describe how happy I am that I know this now."

"That glasses or contacts over eyes means the light is going through two lenses, the eyes and the contacts. I understood that angular size is not the actual size but the size it relates to the eye from your viewpoint."

"Magnifying glasses do not actually magnify, instead they 'bring' the object closer to the eye. Magnifying glasses allow the eyes to focus on an object when it is closer than 25 cm, the normal point of vision."

"I feel like angular magnification is more intuitive than linear magnification. The angular size of an object is not it's actual size, but it is how large the object appears to be on the retina. This size is given by the angle at which light from the object enters the lens. This is why huge object appear small if they are far away. The far distance creates a small angle."

"The best way to approach optics is to take them one lens at a time. An object goes through one lens to produce an image and then that image is the new object for the second lens. Angular magnification is the way magnifiers work, which allow you to see the object at a greater angular size. This image appears closer and is able to keep the clarity."

"I promise I will read about it this weekend."

Describe what you found confusing from the assigned textbook reading or presentation preview. Your description (2-3 sentences) should specifically identify the concept(s) that you do not understand.

"I still have trouble applying ray tracings to examples. I think I just have to keep practicing and applying it to things to understand as time goes on."

"I found how optometrists make prescriptions confusing as well as how contacts/glasses actually work when you look at the numbers."

"How the two-lens setup corrects for myopia and hyperopia. I know that the contacts take the object and create an image that the eye is able to see, but I do not understand how the parameters change (focal points and image distance)."

"I don't understand how diopters work, and how the concept of refractive power factors in. I am also a little confused on converging and diverging lenses."

"I am still confused by magnifiers and how they actually work with a relaxed human eye closer or further away from the near focal point of +25 cm. how does this concept compare or relate to the ray tracings?"

"I found everything from the two-lens systems confusing. I don't even know enough to explain what I do or don't know."

"I'm sure you'll go over it in lecture but I could use a bit more explanation on the angular magnification equation."

"Utilizing the equations in problems like homework has also been confusing."

"It would be really helpful if you go over how magnifiers work and what it means when it says focused at infinity."

In general, a converging lens will produce virtual, upright images located __________ the original object.

closer than.	**** [4]
at the same distance as.	** [2]
farther than.	********************** [22]
(Unsure/lost/guessing/help!)	***** [5]

In general, a diverging lens will produce virtual, upright images located __________ the original object.

closer than.	************************** [26]
at the same distance as.	* [1]
farther than.	* [1]
(Unsure/lost/guessing/help!)	***** [5]

Identify the type of lens used for these optics. (Only correct responses shown.)

Glasses/contacts to correct for myopia: diverging [58%]
Glasses/contacts to correct for hyperopia: converging [64%]
Glasses/contacts to correct for presbyopia: converging [33%]
Magnifying lenses: converging [61%]

State the units of refractive power for lenses, and briefly describe the relationship between refractive power P and focal length f.

"m^–1. Refractive power is used to tell the focal length of contact lenses and is the inverse of focal length."

"Diopters. P = 1/f."

"I really don't know...I need help."

Explain the difference between the two types of magnification, m and M.

"Magnification (m) is the ratio of image height to object height and magnification (M) denotes how much larger the angular size of something appears as seen through a magnifier."

"Lower case m is the linear magnification which is the ratio of image length to object length. Capital M magnification is the angular magnification and denotes how much larger the angular size of something appears through a magnifier vs the naked eye."

A nominal, relaxed eye is set to focus on objects located at:

+∞.	****** [6]
+25 cm (at your near point).	************** [14]
+f (at the focal point of the lens).	******* [7]
(Unsure/lost/guessing/help!)	****** [6]

If an object is brought closer to your eye, its angular size will:

increase.	******************* [24]
decrease.	* [1]
remain unchanged.	*** [3]
(Unsure/lost/guessing/help!)	***** [5]

When a converging lens is used as a simple magnifier, the object is placed at a distance d_o = __________ in front of (to the left of) the lens.

+∞.	***** [5]
+25 cm (at your near point).	***** [5]
+f (at the focal point of the lens).	************** [14]
(Unsure/lost/guessing/help!)	********* [9]

The ray tracing that best matches when a converging lens is used as a simple magnifier is:

#3.	******** [8]
#4.	************** [14]
(Unsure/lost/guessing/help!)	*********** [11]

Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.

"I think more ray tracing in lecture would be helpful. I'm starting to get it, but there's still some doubt."

"This material is harder to understand so hopefully class will clarify my confusion."

"I'm sorry! I have a calculus test and a chemistry test and I need to study really bad :/ "

"Favorite anime?" (Recent favorites? Kill la Kill, Arpeggio of Blue Steel, FLCL (the original series). If you want to get old school, then The Super Dimension Fortress Macross, Space Battleship Yamato, Space Pirate Captain Harlock.) "

Presentation: corrective optics

In the previous presentation we considered vision problems caused by defects in the curvature (and focal length) of the eye. By augmenting the eye with a second lens, here either using a contact lens...

...or glasses, we can compensate for these common vision defects.

In general, we could use a two-lens approach, where light passes through the first lens 1, and then subsequently through the second lens 2.

So for any two-lens system, the main idea is to just take it one lens at a time. The object 1 in front of lens 1 will produce an (intermediate) image 1, and the thin lens equation is applied to to just lens 1.

This image 1 will then become the object 2 for lens 2, which will produce the (final) image 2, and the thin lens equation is again applied to just lens 2.

Keep in mind that the intermediate image 1 from the first lens is subsequently "fed" to the second lens as its object 2. We will be using this general two-step approach to two-lens systems for a magnifying glass (lens 1) held in front of an eye (lens 2); or a microscope (objective lens 1 and eyepiece lens 2); or a telescope (objective lens 1 and eyepiece lens 2).

However, for the specific case of a two-lens system of a contact lens and an eye, we can simplify the two-step approach and instead use a "combinatorial" equation, as a contact lens is placed directly on the eye, and as a result these two lenses can behave together as a single "stacked" lens. As a result, we will compress four years of post-graduate optometry school into this presentation, and be able to prescribe corrective optics for common vision defects.

First, applying the combinatorial model to contacts and eyes.

If you have a prescription from your optometrist, or the box that your contact lenses came in, check out the refractive power P ("diopters") listed there, but note that the focal length is nowhere to be found.

The focal length of contact lenses or glasses are typically not specified, instead they are rated in terms of refractive power P in "diopters," which are merely the inverse of the focal length f (measured in meters). Wait, why do optometrists do that?

The sign convention for refractive power is the same as for focal lengths: positive values for converging contact lenses, and negative values for diverging contact lenses.

For two lenses that are held together in a "stack," the inverse of their combined total focal length will be the addition of the inverse of their individual focal lengths. Note that this is only strictly true for contact lenses and eyes, as they are "in contact" with each other. For glasses and eyes, the space between them means that we can't really consider them a "stack," and must resort to the general two-step two-lens approach mentioned earlier. So for the purposes of prescribing corrective optics, we'll be only looking at contact lenses using the "stack" equation.

Since the refractive power is the reciprocal of the focal length, note that for two lenses held together in a "stack," their combined total refractive power is just the simple addition of their individual refractive powers. This is why optometrists use refractive powers instead of focal lengths, as the math is easier (after you have done the conversions from focal lengths into refractive powers.)

Second, applying this "stacking" equation of combining two lenses to prescribe contacts to correct common vision defects.

We only need to measure your actual far point and your actual near point. If the measured far point is less than the nominal value of infinity (i.e., some measurably finite value), and/or the measured near point is greater than the nominal value of 25 cm...

...then you will need corrective optics, which we can solve for using thin lens equations.

Let's set things up for a Physics 205B student with myopia (nearsightedness). This student has an uncorrected far point (say, 5.0 m), which is the farthest distance that this student's unaided eye can focus on (instead of the nominal farthest distance of ∞). So light from an object at the far point will go through this student's eye, and form an image on the retina.

When the student is wearing contacts to correct for this defect, the student can now focus on normal distant objects at ∞. So now light from an object at ∞ will go through the "stacked" contact lens (lens 1) and student's eye (lens 2), and form an image on the retina. This is all conceptual set-up, the actual math to find the prescription for these contact lenses using thin lens equations is discussed here.

When you do solve for the focal length of contact lenses that correct for myopia, you should find a negative value for the focal length (and refractive power), which means that contact lenses (and glasses) to correct for myopia are diverging lenses. This should make sense because diverging lenses will take light from a distant object to make a closer, upright virtual image that the eye can look at. (Which ray tracing(s) best match this?)

Now let's set things up for a Physics 205B student with hyperopia (farsightedness). This student has an uncorrected near point (say, 0.45 m), which is the nearest distance that this student's unaided eye can focus on (instead of the nominal nearest distance of 0.25 m). So light from an object at the near point will go through this student's eye, and form an image on the retina.

When the student is wearing contacts to correct for this defect, the student can now focus on normal nearby objects at 0.25 m. So now light from an object at 0.25 m will go through the "stacked" contact lens (lens 1) and student's eye (lens 2), and form an image on the retina. This is all conceptual set-up, the actual math to find the prescription for these contact lenses using thin lens equations is discussed here.

When you do solve for the focal length of contact lenses that correct for hyperopia, you should find a positive value for the focal length (and refractive power), which means that contact lenses (and glasses) to correct for myopia are converging lenses. This should make sense because converging lenses will take light from a nearby object to make a farther away, upright virtual image that the eye can look at. (Which ray tracing(s) best match this?)

Now what happens if a Physics 205B student is myopic (nearsighted, can see near, but can't see far) due to defects in curvature of the eye, who as a result of aging develops presbyopia, losing the ability to accommodate and see nearby objects? You would need to prescribe separate diverging and converging lenses to correct for myopia and presbyopia...

...or these two lenses could be combined into bifocal glasses, with the compromise that looking down (at reading distances) would have a diverging lens which only corrects for presbyopia, and looking straight (for far distances) would have a converging lens which only corrects for myopia.