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physics

Optical depth

Optical depth is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Optical depth rather than just read about it. In short: In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material.

Optical depth — main illustration
Optical depth — illustration

Key takeaways

  • Optical depth belongs to physics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Optical depth to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Optical depth from memory before moving on to harder problems.

Reference excerpt

In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material. Spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged. In chemistry, a closely related quantity called "absorbance" or "decadic absorbance" is used instead of optical depth: the common logarithm of the ratio of incident to transmitted radiant power through a material. It is the optical depth divided by loge(10), because of the different logarithm bases used.

Mathematical definitions

Optical depth The optical depth of a material, denoted τ {\textstyle \tau } , is given by: τ = ln ( Φ e i Φ e t ) = − ln ⁡ T {\displaystyle \tau =\ln \!\left({\frac {\Phi _{\mathrm {e} }^{\mathrm {i} }}{\Phi _{\mathrm {e} }^{\mathrm {t} }}}\right)=-\ln T} where

Φ e i {\textstyle \Phi _{\mathrm {e} }^{\mathrm {i} }} is the radiant flux received by that material;

Φ e t {\textstyle \Phi _{\mathrm {e} }^{\mathrm {t} }} is the radiant flux transmitted by that material;

T {\textstyle T} is the transmittance of that material. The absorbance A {\textstyle A} is related to optical depth by: τ = A ln ⁡ 10 {\displaystyle \tau =A\ln {10}}

Spectral optical depth The spectral optical depth in frequency (denoted τ ν {\displaystyle \tau _{\nu }} ) or in wavelength ( τ λ {\displaystyle \tau _{\lambda }} ) of a material is given by:

τ ν = ln ( Φ e , ν i Φ e , ν t ) = − ln ⁡ T ν {\displaystyle \tau _{\nu }=\ln \!\left({\frac {\Phi _{\mathrm {e} ,\nu }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\nu }^{\mathrm {t} }}}\right)=-\ln T_{\nu }}

τ λ = ln ( Φ e , λ i Φ e , λ t ) = − ln ⁡ T λ , {\displaystyle \tau _{\lambda }=\ln \!\left({\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}}\right)=-\ln T_{\lambda },}

where

Φ e , ν t {\displaystyle \Phi _{\mathrm {e} ,\nu }^{\mathrm {t} }} is the spectral radiant flux in frequency transmitted by that material;

Φ e , ν i {\displaystyle \Phi _{\mathrm {e} ,\nu }^{\mathrm {i} }} is the spectral radiant flux in frequency received by that material;

T ν {\displaystyle T_{\nu }} is the spectral transmittance in frequency of that material;

… excerpt ends here. Continue reading the full article.

Illustrations

Optical depth: Aerosol Optical Depth (AOD) at 830 nm measured with the same LED sun photometer from 1990 to 2016 at Geronimo Creek Observatory, Texas. Measurements made at or near solar noon when the Sun is not obstructed by clouds. Peaks indicate smoke, dust and smog. Saharan dust events are measured each summer.
Aerosol Optical Depth (AOD) at 830 nm measured with the same LED sun photometer from 1990 to 2016 at Geronimo Creek Observatory, Texas. Measurements made at or near solar noon when the Sun is not obstructed by clouds. Peaks indicate smoke, dust and smog. Saharan dust events are measured each summer.

Worked examples

Example 1 — a first encounter with Optical depth

Start with the simplest possible case. Write down what Optical depth claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Optical depth before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Optical depth ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Optical depth

In research
Optical depth appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Optical depth in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Optical depth is common in secondary-school and first-year university syllabi. It links to neighbouring topics Optical quantities, Scattering, absorption and radiative transfer (optics), Spectroscopy, so understanding it makes those chapters shorter.
In everyday life
Look for Optical depth outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Optical depth in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Optical depth means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Optical depth out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Optical depth in simple terms?

In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material.

Why does Optical depth matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Optical depth?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Optical depth.

Tags

  • Optical quantities
  • Scattering, absorption and radiative transfer (optics)
  • Spectroscopy
  • Visibility

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