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Derivation Of Lens Maker Formula

Derivation Of Lens Maker Formula

Edited By Vishal kumar | Updated on Jul 02, 2025 04:27 PM IST

The refractive index and curvatures of lens surfaces relate the focal length with Lens Maker's formula. Its significance in designing lenses that achieve certain focal lengths is important to the optics industry. It is based on the ideas behind refraction at spherical surfaces and the geometry of thin lenses. The assumption is that they are all made from one type of material which has a different refractive index than air (the most common) or water (less frequently). This equation applies both convex and concave lenses helping understand how the curvature of these two shapes with several materials will affect image making.

This Story also Contains
  1. Lens Maker Formula
  2. What is a Thin Lens?
  3. What is Focal length?
  4. Derivation of Lens Maker’s Formula
  5. Image Formation With a Thin Lens: Characteristics
  6. Limitations of the Lens Maker’s Formula
Derivation Of Lens Maker Formula
Derivation Of Lens Maker Formula

Lens Maker Formula

For various optical equipment, lenses of varying focal lengths are utilized. The focal length of a lens is determined by the refractive index of the lens's material and the curvature radii of the two surfaces. The lens maker formula is derived here to help applicants better comprehend the subject. The lens maker formula is often used by lens manufacturers to create lenses with the appropriate focal length.

For spherical lenses, the lens equation or lens formula is an equation that links the focal length, image distance, and object distance.
Lens Formula 1f=1v1u is how it's written. where. v is the image's distance from the lens, u is the object distance and f is the focal length.

If the relationship between a lens' focal length, the refractive index of its material, and the radii of curvature of its two surfaces is known as the lens maker's formula. Lens manufacturers utilize it to build lenses with a specific power from glass with a specific refractive index.

1f=(n1)(1R11R2)

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What is a Thin Lens?

A thin lens is defined as one whose thickness is insignificant in comparison to its curvature radii. The thickness (t) is significantly lower than the two curvature radiii R1 and R2
The focal length, image distance, and object distance are all connected in the lens formula for concave and convex lenses. The formula 1f=1v+1u can be used to establish this link.
The focal length of the lens is f, and the distance of the generated image from the lens' optical centre is v in this equation. Finally, u is the distance between an item and the optical centre of this lens. For convex lenses, this is the lens equation.

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There are two types of thin lenses

To create a thin lens formula, you must first understand the difference between converging and diverging lenses.

  • Converging lenses allow light rays that are parallel to the optic axis to pass through and converge at a common point behind them. The focal point (f) or focus is the name given to this point.
  • Diverging - These lenses have the opposite purpose as converging lenses. Light rays parallel to the optic axis travel through and diverge here. It creates an optical illusion by giving the impression that the lights are coming from the same source (f) in front of the lens.

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What is Focal length?

The focal length of an optical system is the inverse of the system's optical power; it measures how strongly the system converges or diverges light. A system with a positive focal length converges light, while one with a negative focal length diverges light.

What is the Formula of the Focal Lens?

The formula 1f=1v+1u gives the focal length of a double convex lens, where u is the distance between the object and the lens and v is the distance between the image and the lens.

OR,

F=R/2
Where,
F is the focal length, and
R is the radius of curvature of the lens

Derivation of Lens Maker’s Formula

Lens Maker Formula

The lens maker formula is derived using the assumptions listed below

Consider the thin lens in the picture above, which has two refracting surfaces with curvature radii R1 and R2, respectively. Assume that the surrounding medium and the lens material have refractive indices of n1 and n2, respectively. The whole derivation of the lens maker formula is provided further below. We can say that, using the formula for refraction at a single spherical surface,

For the first surface,
n2v1r1u=n2r1R1
For the second surface,

n1vn2v1=n1n2R2
Now adding equation (1) and (2),

n1/vn1/u=(n2n1)[1/R11/R2]

on simplifying we get,

1/v1/u=(n2/n11)[1/R11/R2]
When u= and v=f

1f=(n2n11)[1R11R2]

But also,
1v1u=1f
Therefore, we can say that,

1f=(μ1)(1R11R2)
Where μ is the material's refractive index.

Where μ is the material's refractive index.

This is the derivation of the lens maker formula. Examine the constraints of the lens maker's formula to have a better understanding of the lens maker's formula derivation.

Image Formation With a Thin Lens: Characteristics

It's not enough to know the thin lens formula for convex lenses. The characteristics of a ray of light going through converging and diverging lenses must be understood.

  1. On the opposite side, parallel rays going through converging lenses will intersect at point f.
  1. Parallel rays seem to emerge from point f in front of diverging lenses.
  1. The direction of light rays travelling through the centre of converging or diverging lenses does not change.
  1. Light rays that enter a converging lens through its focal point always exit parallel to the lens's axis.
  1. On the other side of a diverging lens, a light ray going towards the focal point will also emerge parallel to its axis.

A concave or divergent lens has a negative focal length. When the picture is generated on the side where the object is positioned, the image distance is also negative. The image is virtual in this case. A converging or convex lens, on the other hand, has a positive focal length.

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Limitations of the Lens Maker’s Formula

  • The lens should not be thick so that the space between the 2 refracting surfaces can be small.
  • The medium used on both sides of the lens should always be the same.

Frequently Asked Questions (FAQs)

1. How is the Lens Maker's Formula derived?

The formula is derived from the refraction of light at the two spherical surfaces of the lens. By applying the refraction equations at each surface and combining them, the formula 1f=(n1)(1R11R2) is obtained, which relates the lens's focal length to its curvature and refractive index.

2. When is a Concave Lens' Image Virtual?

Only when object along with image are on same side of lens is the picture generated by a concave lens virtual.

3. When Do Convex Lenses Act Like Combined Lenses?

The combined lens works as a convex lens if focal length of second lens is greater than focal length of first lens.

4. What is the answer to the lens formula ?

The lens formula is relationship between object's distance u, image's distance v, as well as lens's focal length f. With the right sign conventions, this law can be applied to both concave and convex lenses.

5. Concave lenses come in a variety of shapes and sizes.

Concave lenses can be found in a variety of real-world applications.

Telescopes and binoculars

Nearsightedness can be corrected with eyeglasses.

Cameras.

Flashlights.

6. What is the Lens Maker's Formula?

The Lens Maker's Formula calculates the focal length of a lens based on its curvature and the refractive index of its material. It is expressed as 1f=(n 1) (1R11R2), where f is the focal length, n is the refractive index of the lens material, and R1 and R2 are the radii of curvature of the lens surfaces.

7. Why do we need to consider two surfaces in the lens maker's formula?
We consider both surfaces because refraction occurs twice: when light enters the lens and when it exits. Each surface contributes to the overall bending of light, and thus to the focal length of the lens. Even if one surface is flat, it's still accounted for in the formula.
8. How does the lens maker's formula relate to the concept of optical path difference?
While the lens maker's formula doesn't explicitly use optical path difference, it's based on the same principles. The formula ensures that light rays from a point source converge to a single point by equalizing their optical path lengths, which is achieved through the specific curvatures and refractive index of the lens.
9. Can the lens maker's formula be used for contact lenses?
Yes, the formula can be applied to contact lenses, but with some modifications. Since contact lenses conform to the eye's surface, one side typically has a curvature matching the cornea. The other side is then designed to achieve the desired optical correction.
10. Why doesn't the lens maker's formula include the diameter of the lens?
The lens diameter doesn't affect the focal length, which is what the lens maker's formula calculates. However, the diameter is important for other optical properties like light-gathering power and resolution, which are not addressed by this formula.
11. Can the lens maker's formula be used to design achromatic lenses?
The basic formula doesn't directly address chromatic aberration. However, by applying the formula to different wavelengths and combining lenses with different dispersions, we can design achromatic doublets that minimize chromatic aberration.
12. What is the lens maker's formula and why is it important?
The lens maker's formula is an equation that relates the focal length of a lens to its refractive index and the radii of curvature of its surfaces. It's important because it allows us to design lenses with specific focal lengths, which is crucial in optical systems like cameras, microscopes, and eyeglasses.
13. How does changing the refractive index affect the focal length in the lens maker's formula?
Increasing the refractive index of the lens material (while keeping other factors constant) decreases the focal length. This is because a higher refractive index causes light to bend more sharply, bringing it to a focus closer to the lens.
14. How does the lens maker's formula account for the shape of a lens?
The lens maker's formula incorporates the radii of curvature of both surfaces of the lens. These radii determine the lens shape, which affects how light is refracted. For a convex surface, the radius is positive, while for a concave surface, it's negative.
15. Why does the lens maker's formula use the difference in refractive indices?
The formula uses the difference between the refractive index of the lens material and the surrounding medium (usually air) because refraction occurs at the boundary between these two materials. This difference determines how much the light bends when entering and leaving the lens.
16. What happens to the focal length if one of the lens surfaces is flat?
If one surface is flat, its radius of curvature is infinite. In the lens maker's formula, the term 1/R for that surface becomes zero. This simplifies the formula and typically results in a longer focal length compared to a lens with two curved surfaces of similar curvature.
17. How does the lens maker's formula relate to the power of a lens?
The power of a lens is the reciprocal of its focal length. Since the lens maker's formula gives us the focal length, we can easily calculate the power by taking its reciprocal. This relationship is crucial in optometry for prescribing corrective lenses.
18. How does the lens maker's formula relate to the concept of vergence?
Vergence is the reciprocal of distance and is measured in diopters, just like lens power. The lens maker's formula gives us the focal length, whose reciprocal is the lens power or vergence. This allows us to easily calculate how a lens will change the vergence of incoming light.
19. How does the lens maker's formula account for chromatic aberration?
The basic lens maker's formula doesn't directly account for chromatic aberration. However, it shows that focal length depends on refractive index, which varies with wavelength. By using different refractive indices for different colors, we can see how focal length varies, explaining chromatic aberration.
20. Why does the lens maker's formula use the reciprocal of focal length instead of focal length directly?
Using the reciprocal of focal length (1/f) allows the formula to be expressed as a simple sum of terms involving the radii of curvature and refractive index. This makes it easier to calculate the focal length and relates directly to the concept of optical power, which is measured in diopters (1/meters).
21. Can the lens maker's formula be applied to mirrors?
While the lens maker's formula is specifically for lenses, a similar principle applies to mirrors. For mirrors, the focal length is simply half the radius of curvature, and there's no need to consider refractive indices since reflection, not refraction, is involved.
22. Can the lens maker's formula be applied to all types of lenses?
The basic lens maker's formula applies to thin lenses, where the thickness is negligible compared to the focal length. For thick lenses, a modified version of the formula is used that takes into account the lens thickness and the refractive indices of the lens material and surrounding medium.
23. How does the lens maker's formula relate to the thin lens approximation?
The lens maker's formula is derived using the thin lens approximation, which assumes the thickness of the lens is negligible compared to its focal length and the radii of curvature of its surfaces. This simplifies the calculations but may introduce errors for very thick lenses.
24. How does the lens maker's formula relate to the concept of principal planes?
The lens maker's formula assumes thin lenses where the principal planes coincide at the center of the lens. For thick lenses, where principal planes are separated, a modified version of the formula is needed that takes into account the lens thickness and the position of the principal planes.
25. How does the lens maker's formula relate to the concept of conjugate points?
While the lens maker's formula doesn't directly deal with conjugate points, it provides the focal length, which is crucial in determining conjugate points using other equations like the thin lens equation. The focal length is the key parameter linking object and image distances.
26. Can the lens maker's formula be used for Fresnel lenses?
The standard lens maker's formula isn't directly applicable to Fresnel lenses, which use a series of concentric grooves instead of a continuous curved surface. However, each groove in a Fresnel lens can be approximated as a small section of a conventional lens, so modified versions of the formula can be applied.
27. Why do we use the approximation that sin θ ≈ θ in deriving the lens maker's formula?
This approximation, known as the small-angle approximation, is used because lenses typically deal with small angles of refraction. It simplifies the mathematics while maintaining accuracy for most practical applications. However, for very curved lenses or wide-angle optics, this approximation may introduce significant errors.
28. How does the lens maker's formula account for astigmatism?
The basic lens maker's formula doesn't account for astigmatism, which occurs when a lens has different curvatures in different planes. For astigmatic lenses, separate calculations are needed for each principal meridian, resulting in two different focal lengths.
29. Can the lens maker's formula be used for non-spherical lenses?
The basic lens maker's formula assumes spherical surfaces. For non-spherical lenses (like parabolic or aspherical lenses), more complex equations are needed. However, the basic formula can sometimes be used as an approximation for slightly non-spherical lenses.
30. Can the lens maker's formula be used for gradient-index lenses?
The standard lens maker's formula assumes a uniform refractive index throughout the lens. For gradient-index lenses, where the refractive index varies within the lens, more complex equations are needed. However, the basic formula can sometimes be used as an approximation for slightly graded lenses.
31. Why do we use the paraxial approximation in deriving the lens maker's formula?
The paraxial approximation, which assumes small angles and heights relative to the optical axis, simplifies the mathematics significantly. It allows us to use simple trigonometry and make linear approximations, making the derivation tractable while still providing accurate results for most practical lenses.
32. How does the lens maker's formula account for aspherical surfaces?
The standard lens maker's formula assumes spherical surfaces. For aspherical surfaces, which are often used to reduce aberrations, more complex equations are needed. However, for slightly aspherical surfaces, the formula can sometimes be used as an approximation, with the understanding that it may introduce some error.
33. Can the lens maker's formula be used for diffractive lenses?
The lens maker's formula is based on refraction and doesn't apply directly to diffractive lenses, which use diffraction to focus light. Diffractive lenses require different design equations based on the principles of wave optics rather than geometric optics.
34. Why do we use the small-angle approximation in deriving the lens maker's formula?
The small-angle approximation (sin θ ≈ θ for small θ) simplifies the trigonometry involved in the derivation. This approximation is valid for most practical lenses where the angles of refraction are small. It allows us to derive a simple, algebraic formula that's accurate for a wide range of lenses.
35. Why doesn't the lens maker's formula include terms for lens material dispersion?
The basic formula uses a single refractive index and doesn't account for dispersion (variation of refractive index with wavelength). To consider dispersion, we would need to use different refractive indices for different wavelengths. This is important when dealing with chromatic aberration.
36. How does the lens maker's formula account for coma aberration?
The standard lens maker's formula doesn't account for coma, which is an off-axis aberration. The formula assumes all light rays pass close to the optical axis (paraxial approximation). More advanced optical design techniques are needed to address coma and other off-axis aberrations.
37. How does the lens maker's formula account for birefringence?
The standard lens maker's formula doesn't account for birefringence, where a material has different refractive indices depending on the polarization and direction of light. For birefringent materials, separate calculations would be needed for the different refractive indices, potentially resulting in different focal lengths for different polarizations.
38. How does the lens maker's formula relate to the lensmaker's equation?
The terms "lens maker's formula" and "lensmaker's equation" are often used interchangeably. Both refer to the same equation that relates a lens's focal length to its refractive index and radii of curvature. The equation form is what's typically called the "formula."
39. How does the lens maker's formula account for the medium surrounding the lens?
The formula includes the refractive index of the surrounding medium (usually air, with n ≈ 1) in the term (n - 1), where n is the refractive index of the lens material. If the lens is in a medium other than air, this term would become (n_lens - n_medium).
40. How does the sign convention work in the lens maker's formula?
In the lens maker's formula, convex surfaces have positive radii of curvature, while concave surfaces have negative radii. The focal length is positive for converging lenses and negative for diverging lenses. Consistency in applying these conventions is crucial for correct results.
41. Why do we assume parallel light rays when deriving the lens maker's formula?
Parallel light rays are assumed because they represent light from an infinitely distant object. This simplifies the derivation and gives us the focal length, which is defined as the distance at which parallel rays converge after passing through the lens.
42. How does the lens maker's formula account for spherical aberration?
The basic lens maker's formula doesn't directly account for spherical aberration. It assumes all light rays focus to a single point, which is only true for paraxial rays. In reality, spherical lenses suffer from spherical aberration, where rays far from the optical axis focus at different points.
43. How does the lens maker's formula account for the index of refraction changing with wavelength?
The basic formula doesn't explicitly account for this variation. However, by using different refractive indices for different wavelengths, we can see how the focal length changes with color. This is crucial for understanding and correcting chromatic aberration in optical systems.
44. Why doesn't the lens maker's formula include terms for lens thickness?
The standard lens maker's formula assumes thin lenses where the thickness is negligible compared to the radii of curvature and focal length. For thick lenses, a modified version of the formula is used that includes terms for the lens thickness and the positions of the principal planes.
45. How does the lens maker's formula relate to the concept of back focal length?
The lens maker's formula calculates the effective focal length, which is measured from the principal plane of the lens. The back focal length, measured from the rear vertex of the lens, can differ from this. For thin lenses, these are approximately the same, but for thick lenses, they can differ significantly.
46. Can the lens maker's formula be applied to compound lenses?
The basic formula applies to single lenses. For compound lenses (multiple lenses in combination), we typically apply the formula to each lens individually and then use other methods (like the thick lens formula or matrix methods) to combine their effects.
47. How does the lens maker's formula relate to the concept of numerical aperture?
While the lens maker's formula doesn't directly involve numerical aperture, it provides the focal length, which is a key parameter in calculating numerical aperture. The numerical aperture depends on both the focal length and the diameter of the lens, the latter of which isn't part of the lens maker's formula.
48. Why do we use the refractive index relative to air in the lens maker's formula?
We use the relative refractive index (n) because refraction occurs at the boundary between the lens material and the surrounding medium (usually air). If the lens is in a medium other than air, we would use the ratio of the refractive indices of the lens material and the surrounding medium.
49. How does the lens maker's formula relate to the concept of cardinal points?
The lens maker's formula provides the focal length, which is one of the cardinal points (specifically, the focal points). Other cardinal points like principal points and nodal points are assumed to coincide at the lens center in the thin lens approximation used by the formula. For thick lenses, additional calculations are needed to find these points.
50. How does the lens maker's formula account for field curvature?
The basic lens maker's formula doesn't account for field curvature, which is an aberration where the image forms on a curved surface rather than a flat plane. The formula assumes all image points lie on a plane perpendicular to the optical axis. More advanced optical design techniques are needed to address field curvature.
51. Can the lens maker's formula be used for liquid lenses?
Yes, the lens maker's formula can be applied to liquid lenses. The key is to use the appropriate refractive index for the liquid and the radii of curvature of the surfaces containing the liquid. For variable focus liquid lenses, the formula can be used to understand how changing the surface curvature affects the focal length.
52. How does the lens maker's formula relate to the concept of depth of field?
While the lens maker's formula doesn't directly calculate depth of field, it provides the focal length, which is a crucial parameter in determining depth of field. Shorter focal lengths (as calculated by the formula) generally result in greater depth of field, all else being equal.
53. Can the lens maker's formula be used for meniscus lenses?
Yes, the lens maker's formula can be applied to meniscus lenses. For a meniscus lens, one radius of curvature will be positive and the other negative. The formula will correctly calculate the focal length, which could be positive or negative depending on whether the meniscus lens is converging or diverging.
54. How does the lens maker's formula relate to the concept of telecentricity?
The lens maker's formula doesn't directly address telecentricity, which is a property of certain optical systems where the chief rays are parallel to the optical axis. However, the focal length calculated by the formula is crucial in designing telecentric systems, as it affects the position and size of apertures needed to achieve telecentricity.
55. Why do we assume the lens material is homogeneous in the lens maker's formula?
Assuming a homogeneous material simplifies the mathematics significantly. It allows us to use a single refractive index throughout the lens. For inhomogeneous materials or gradient-index lenses, more complex equations are needed to account for the varying refractive index within the lens.

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