Диоптрии указаны в рецепте на очки.

Because the sentence is talking about the lens powers as they appear on an eyeglass prescription, and those are normally understood as a set of values.

• диоптрия = diopter / one diopter
• диоптрии = diopters

Russian often uses the plural here because a glasses prescription usually includes more than one number, for example for the left and right eye.

If you were talking about a single value, you could use the singular:

• Диоптрия указана в рецепте. = The diopter value is indicated in the prescription.

Указаны is the short-form passive participle of указать.

It means something like:

• are indicated
• are listed
• are specified

So the structure is:

• Диоптрии = the thing being indicated
• указаны = are indicated / are listed

This form agrees with диоптрии, which is:

• plural
• nominative

That is why it is указаны, not указан or указана.

Compare:

• Номер указан. = The number is indicated.
• Дата указана. = The date is indicated.
• Данные указаны. = The data are indicated.

В рецепте uses the prepositional case after в to mean in or inside something.

Base form:

• рецепт = prescription / recipe

Prepositional singular:

• в рецепте = in the prescription

So:

• в рецепте на очки = in the eyeglass prescription

This is a very common pattern:

• в книге = in the book
• в документе = in the document
• в письме = in the letter

Yes, рецепт can mean both recipe and prescription.

Russian uses the same word in both contexts, and the meaning depends on context.

Examples:

• рецепт супа = a soup recipe
• рецепт от врача = a prescription from a doctor
• рецепт на очки = a prescription for glasses

So in this sentence, because of на очки, it clearly means prescription, not recipe.

Рецепт на что-либо is the normal Russian pattern for a medical prescription for something.

So:

• рецепт на очки = prescription for glasses
• рецепт на лекарство = prescription for medicine

Using для очков would sound less idiomatic here. A learner might think of for glasses, but Russian normally says на очки in this context.

So this is something best learned as a set phrase:

• рецепт на очки

Here очки is in the accusative case after на.

However, очки is one of those nouns where the accusative plural looks the same as the nominative plural.

Base form:

• очки = glasses

In this phrase:

• на очки = for glasses

Because очки is inanimate, the accusative plural is the same as the nominative plural.

So even though the form does not change, the case is still accusative.

Usually yes. Очки is normally used as a plural-only noun when it means glasses.

That is similar to English glasses or scissors, which are also usually plural.

Examples:

• Где мои очки? = Where are my glasses?
• Мне нужны новые очки. = I need new glasses.

There is a singular form очко, but it does not mean a single eyeglass lens in normal modern usage. It has other meanings, so learners should not treat it as the singular of glasses.

Yes. Russian word order is flexible.

You could also say:

• В рецепте на очки указаны диоптрии.

This means the same thing: The diopters are indicated in the eyeglass prescription.

The difference is mostly about emphasis:

• Диоптрии указаны в рецепте на очки.
  Slightly emphasizes диоптрии first.

• В рецепте на очки указаны диоптрии.
  Starts with the location/context: In the eyeglass prescription...

Both are natural.

In a way, there is one built into указаны.

Russian often does not use a separate present-tense verb to be like English does. Instead, forms like указаны can carry the meaning are indicated by themselves.

So Russian says:

• Диоптрии указаны...

English translates this as:

• The diopters are indicated...

There is no separate present-tense быть here.

Both could be translated in some contexts as written, listed, or stated, but they are not exactly the same.

• указаны = indicated, specified, listed
• написаны = written down

In a prescription context, указаны is a good choice because it suggests that the values are officially specified in the document.

So:

• Диоптрии указаны в рецепте sounds natural and slightly formal.
• Диоптрии написаны в рецепте is understandable, but it focuses more literally on the fact that they are written there.

It sounds neutral to slightly formal.

That is because:

• указаны is common in written or official-style language
• the whole sentence sounds like something from a document, clinic, or optical shop

In casual speech, people might say something a bit looser, for example:

• Диоптрии есть в рецепте. = The diopter values are in the prescription.
• В рецепте написаны диоптрии. = The diopter values are written in the prescription.

But your original sentence is perfectly normal and natural.

The stressed syllables are:

• диоптрии
• указаны
• в рецепте
• на очки

A learner-friendly stress version:

• ДиоптрИи указАны в рецЕпте на очкИ.

A rough pronunciation guide:

• диоптрии = dee-ap-TREE-ee
• указаны = oo-ka-ZA-ny
• рецепте = ree-TSEP-tye
• очки = ach-KEE

The most important thing is to stress:

• -трИ- in диоптрии
• -зА- in указаны
• -цЕ- in рецепте
• -кИ in очки

Here are the dictionary forms:

• диоптрия = diopter
• указать = to indicate, to specify
• рецепт = recipe / prescription
• очки = glasses

And here are the forms used in the sentence:

• диоптрии = nominative plural
• указаны = short-form passive participle, plural
• в рецепте = prepositional singular
• на очки = accusative plural

Seeing both the dictionary form and the actual sentence form is very useful for understanding Russian grammar.