Is real image formed by a single optical element always inverted and virtual image always erect?

optics reflection geometric-optics lenses

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Solution 1

A gradient index rod lens will form an erect or inverted image, depending on its length. A SELFOC lens is usually designed to transfer an image from the flat surface on one end of a rod, to the flat surface at the other end of the rod. A SELFOC lens is usually designed to produce an inverted image, but when made twice as long (that is, with pitch 1.0) it forms an erect image. The ray paths for a single pixel on one surface go through the rod as shown below. They come to an inverted focus in the middle of the rod, then back to an erect image at the other end of the rod.

This image is one frame from a video on the Stemmer Imaging website, though there are several companies who provide gradient index rod lenses. This sort of device is sometimes called a "Contact Imaging Sensor", and would be one element of a "line scan bar".

Edit #1 Some other approaches might meet the uniform refractive index, single-element condition:

Retroreflection: A ball lens with a refractive index of 2.0 and a highly reflective coating on its back half will return light to its source. That means that an erect real image is formed from an erect source image.
Fly's Eye lens: A slab of glass or plastic with small lenses on both sides can form an erect image. Each lens on one face of the device forms a small field-of-view erect real image via the corresponding lens on the opposite side of the slab. The lenses and the slab thickness are designed to ensure that the imaging is precisely 1:1, and the images all add up. This is the principle of the contact image sensors (line scan bars), and is a variation on the principle illustrated by @BarsMonster: just build an array of the rod lenses he described, and fuse them together to eliminate boundaries between the rods, leaving only lens bumps on the surfaces. Of course this element can be cast in plastic.

I'm pretty sure two optical surfaces are required, when the refractive index is positive, because we can show that an single optical surface (reflective or positive-refractive) can only form an inverted real image. HOWEVER, negative index materials, in principle, can do it with only a flat, parallel-faced slab. See "Negative refraction makes a perfect lens" by Pendry.

Solution 2

I can only think of multiple optical devices combined in a single entity in a way to not violate your restrictions of being single device. Ether double lenses merged into one, or prism/mirror merged with a lens, or first surface curved mirrors with multiple reflections, while all this still being made from 1 piece of glass.

Solution 3

Correct me if I am wrong, by "single optical device" you meant the device should either converge or diverge or do nothing to the parallel rays. If we want to create a real image of a point light source, all the divergent light rays from the source must converge again to a point after passing through the stated device. In order to get an erect image, rays diverging from the point source, if present above central line, must also converge the above the central line. Consider a ray coming from the source and passing through the center. This ray must follow a V-Shaped path to get an image above the center line, which is impossible (because it won't be convergent anymore). I tried to depict my argument by a figure, it's not a very good drawing though.

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Vishnu

Updated on January 24, 2020

Comments

Vishnu over 3 years

Question:

Is real image always inverted and virtual image always erect with respect to the object? Is there any single optical device (a mirror, or a lens, or anything else) which could produce a real and erect image or a virtual and inverted image? Till now, I haven't encountered such a device. Does it mean this is not possible? Is there any fundamental law which prevents the existence of an optical device which could produce a real and erect image or a virtual and inverted image?

I know that we could produce a real and erect image by using two convex lenses by placing them one after the other, where the inverted image formed by the first lens acts as an object for the second lens which again inverts it to give a resultant erect image with respect to the original object. Here we get an erect image due to two inversions. This question is about a single device which could produce a real and erect image without any intermediate inversion steps.

Edit:

It was asked in the comments what is meant by a "single optical device". By this term, I meant that the device is made of a single piece of material of same refractive index throughout. So a combination of lenses or mirrors which may produce real and erect images do not count.
- tpg2114 over 3 years
  
  As a reminder, please do not post answers or attempted answers in the comments. If you can answer the question, please post it as an answer. Thanks!
- Ruslan over 3 years
  
  Your definition of a "single optical device" still allows e.g. a piece of glass with an air bubble (or other hollow shape) inside, which can in effect function as a pair of lenses.
- Vishnu over 3 years
  
  @Ruslan: Thanks for pointing that out. I thought "...single piece of material of same refractive index throughout" would eliminate such a case. I was unable to frame a better definition of a "single optical device". However, I hope you understood the real intention of my question. If there's anything else to improve, kindly let me know.
Vishnu over 3 years

Thank you for your answer. If I understood properly, "gradient index" means the device is made of a material of non-uniform refractive index. I understood that we could produce a real and erect image using a SELFOC lens through an inversion process as depicted in the ray diagram. Is it possible to produce a real and erect image without any intermediate inversions? Further, is there any optical device of uniform refractive index which could satisfy the conditions mentioned in the question?
Vishnu over 3 years

Thank you for your answer. I'm glad you understood my definition of a "single optical device". I think the ray diagram in your answer produces real and erect image through an intermediate inversion process (at the middle of the device where blue is below green which is below red). Even though the diagram clearly answers the question, I'm curious to know is it possible to produce a real and erect image as a direct process without any intermediate inversions?
S. McGrew over 3 years

I have added edits that I think provide a complete answer - and a solution to the problem.
Ruslan over 3 years

"See Science Direct article" — which one?
S. McGrew over 3 years

I changed the link to an original paper by Pendry, and the image to an image from Pendry's paper.
Vishnu over 3 years

Thank you for your answer. Can you explain why the real erect image in your diagram is impossible?
Vishnu over 3 years

@S.McGrew: Thanks for making the edits :) So, the answer to the question "Is real image always inverted and virtual image always erect with respect to the object?" is no. And you provide examples of SELFOC lens, negative index materials etc., which produce real and erect image (well through intermediate inversion steps). However, the main intention of my first comment remains the same - Is it possible to produce real, erect images or virtual, inverted images by an optical element of uniform index without any intermediate inversions?
Vishnu over 3 years

To be honest, I really learnt many new interesting things from your answer. It's really informative and it belongs to a level much higher than what I expected when I placed the bounty. Thank you for all your efforts.
Vikash Kumar over 3 years

@GuruVishnu I explained the same with my text. By definition, single optical device can't converge and diverge both for a same light source but different rays.
Vishnu over 3 years

Thanks. I understood the general optical elements we find in textbooks like convex, concave lenses could not produce real, erect or virtual, inverted images. Your answer explains that really well. However, as stated in the question, I believe if we aren't aware of such devices that doesn't imply real, erect or otherwise couldn't be produced. Are you saying real, erect image couldn't be formed by a single optical element without any intermediate inversion processes? If yes, it would be helpful if you could specify why it's not possible to have real, erect images?
Vikash Kumar over 3 years

All I am saying a real erect image is possible only when multiple divergent rays from a same light source behave differently after passing through a single optical device: at least one should converge and all other should diverge which is highly illogical. The simple device would follow the same logic in terms of bending the light.
Vishnu over 3 years

I'll give it some more thought. However, I've upvoted your answer. Thanks.