The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Use Numeric Column Types For Numeric Values (query line: 18): Referencing a numeric value (e.g. 0) as a string in a WHERE clause might result in poor performance. Possible impacts of storing numbers as varchars: more space will be used, you won't be able to perform arithmetic operations, the data won't be self-validated, aggregation functions like SUM won't work, the output may sort incorrectly and more. If the column is numeric, remove the quotes from the constant value, to make sure a numeric comparison is done.

Optimal indexes for this query:

CREATE INDEX data_sample_idx_microscope_id_tow_id ON data_entry_sample (microscope_id,tow_id);

The optimized query:

SELECT
        d.tow_id AS text 
    FROM
        data_entry_tow AS d 
    WHERE
        EXISTS (
            SELECT
                1 
            FROM
                data_entry_sample AS d0 
            WHERE
                (
                    d0.tow_id = d.tow_id
                ) 
                AND (
                    (
                        (
                            d0.microscope_id <> '0'
                        ) 
                        OR (
                            d0.microscope_id IS NULL
                        )
                    ) 
                    OR (
                        d0.microscope_id IS NULL
                    )
                )
        )